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• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. Ask a question

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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Chapter 3: Developing a Research Question

3.4 Hypotheses

When researchers do not have predictions about what they will find, they conduct research to answer a question or questions with an open-minded desire to know about a topic, or to help develop hypotheses for later testing. In other situations, the purpose of research is to test a specific hypothesis or hypotheses. A hypothesis is a statement, sometimes but not always causal, describing a researcher’s expectations regarding anticipated finding. Often hypotheses are written to describe the expected relationship between two variables (though this is not a requirement). To develop a hypothesis, one needs to understand the differences between independent and dependent variables and between units of observation and units of analysis. Hypotheses are typically drawn from theories and usually describe how an independent variable is expected to affect some dependent variable or variables. Researchers following a deductive approach to their research will hypothesize about what they expect to find based on the theory or theories that frame their study. If the theory accurately reflects the phenomenon it is designed to explain, then the researcher’s hypotheses about what would be observed in the real world should bear out.

Sometimes researchers will hypothesize that a relationship will take a specific direction. As a result, an increase or decrease in one area might be said to cause an increase or decrease in another. For example, you might choose to study the relationship between age and legalization of marijuana. Perhaps you have done some reading in your spare time, or in another course you have taken. Based on the theories you have read, you hypothesize that “age is negatively related to support for marijuana legalization.” What have you just hypothesized? You have hypothesized that as people get older, the likelihood of their support for marijuana legalization decreases. Thus, as age moves in one direction (up), support for marijuana legalization moves in another direction (down). If writing hypotheses feels tricky, it is sometimes helpful to draw them out and depict each of the two hypotheses we have just discussed.

Note that you will almost never hear researchers say that they have proven their hypotheses. A statement that bold implies that a relationship has been shown to exist with absolute certainty and there is no chance that there are conditions under which the hypothesis would not bear out. Instead, researchers tend to say that their hypotheses have been supported (or not). This more cautious way of discussing findings allows for the possibility that new evidence or new ways of examining a relationship will be discovered. Researchers may also discuss a null hypothesis, one that predicts no relationship between the variables being studied. If a researcher rejects the null hypothesis, he or she is saying that the variables in question are somehow related to one another.

Quantitative and qualitative researchers tend to take different approaches when it comes to hypotheses. In quantitative research, the goal often is to empirically test hypotheses generated from theory. With a qualitative approach, on the other hand, a researcher may begin with some vague expectations about what he or she will find, but the aim is not to test one’s expectations against some empirical observations. Instead, theory development or construction is the goal. Qualitative researchers may develop theories from which hypotheses can be drawn and quantitative researchers may then test those hypotheses. Both types of research are crucial to understanding our social world, and both play an important role in the matter of hypothesis development and testing.  In the following section, we will look at qualitative and quantitative approaches to research, as well as mixed methods.

Text attributions

This chapter has been adapted from Chapter 5.2 in Principles of Sociological Inquiry , which was adapted by the Saylor Academy without attribution to the original authors or publisher, as requested by the licensor, and is licensed under a CC BY-NC-SA 3.0 License .

Research Methods for the Social Sciences: An Introduction Copyright © 2020 by Valerie Sheppard is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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What Is Formulation of Hypothesis in Research? Key Concepts and Steps

Formulating a hypothesis is a crucial part of any research project. It acts like a roadmap, guiding the direction of the study. By making a prediction based on existing knowledge, researchers can design experiments and collect data to test their ideas. This article will explore the key concepts and steps involved in creating a solid hypothesis.

Key Takeaways

• A hypothesis is a prediction that guides the research process.
• Formulating a hypothesis helps focus data collection and analysis.
• Background research is essential for developing a good hypothesis.
• There are different types of hypotheses, like null and alternative.
• Ethical considerations are important when making a hypothesis.

Understanding the Concept of Hypothesis in Research

A hypothesis is a statement that predicts what you expect to find in your research. It is a testable statement that explains what is happening or observed. The hypothesis proposes the relationship between the various participating variables. In scientific research, a hypothesis must meet certain criteria to be considered acceptable. If a hypothesis is disregarded, the research may be rejected by the scientific community.

Importance of Hypothesis Formulation in Research

Guiding the research process.

Formulating a hypothesis is crucial as it guides the entire research process . It provides a clear direction and helps you stay focused on your research objectives. By having a hypothesis, you can systematically plan your study and ensure that every step is aligned with your research goals.

Providing a Focus for Data Collection

A well-defined hypothesis helps in determining what data needs to be collected. It acts as a blueprint, ensuring that you gather relevant information that directly addresses your research question. This focused approach not only saves time but also enhances the efficiency of your research.

Facilitating Data Analysis

When you have a hypothesis, it simplifies the data analysis process. You can use statistical methods to test your hypothesis and draw meaningful conclusions. This is particularly important in hypothesis testing , where you assess the validity of your assumptions based on the collected data.

Investigating Background Research

Reviewing existing literature.

Before you start your research, it's crucial to review existing literature . This step helps you understand what has already been studied and where there might be gaps. You can use various sources like books, academic journals, and online databases. Knowing how to find literature efficiently will save you time and effort.

Identifying Research Gaps

Once you've reviewed the literature, the next step is identifying research gaps . These are areas that haven't been explored yet or need further investigation. Recognizing these gaps can inspire focused and relevant research questions. Discussing your ideas with peers or mentors can also help refine your questions.

Formulating Research Questions

After identifying the gaps, you can start formulating your research questions . These questions should be specific and feasible. They will guide your entire research process, from data collection to analysis. A well-defined research question is the foundation of a strong research proposal .

Developing a Theoretical Framework

A [ theoretical framework provides the theoretical assumptions](https://resources.nu.edu/c.php?g=1109615&p=10328334) for the larger context of a study, and is the foundation or 'lens' by which a study is developed. It helps you understand the theories related to your research topic and integrate them into your hypothesis formulation. This framework must demonstrate an understanding of theories and concepts that are relevant to the topic of your research paper and that relate to your study's objectives. Creating an effective theoretical framework involves establishing a research design aligned with objectives , ensuring quality and rigor in data collection.

Steps in Formulating a Hypothesis

Formulating a hypothesis is a crucial step in the research process. It involves several key steps that help in shaping a clear and testable statement. Each step is essential for ensuring that your hypothesis is well-founded and researchable.

Identifying Variables

The first step in formulating a hypothesis is identifying the variables involved in your study. Variables are the elements that you will measure, manipulate, or control in your research. These can be classified into independent variables (which you manipulate) and dependent variables (which you measure). Understanding the difference between these variables is fundamental to demystifying research .

Establishing Relationships Between Variables

Once you have identified your variables, the next step is to establish the relationships between them. This involves determining how the independent variable might affect the dependent variable. This step is crucial for creating clear statements and focusing on specific research questions. It is important to distinguish and formulate clear objectives in research to ensure that your hypothesis is testable.

Predicting Outcomes

The final step in formulating a hypothesis is predicting the outcomes of your research. This involves making an educated guess about what you expect to happen during your experiment. This step is often referred to as stating your hypothesis . Your prediction should be based on existing literature and theoretical frameworks related to your research topic. This is crucial for informed decision-making in research and helps in designing experiments to test hypotheses effectively.

Types of Hypotheses in Research

When conducting research, you will encounter various types of hypotheses, each serving a unique purpose in guiding your investigation . Understanding these types will help you formulate your own hypotheses more effectively.

Testing the Hypothesis

Testing hypotheses is a crucial part of research. It’s where you see if your ideas hold up in the real world. Good clinical research starts from a plausible hypothesis supported by contemporary scientific knowledge that makes a testable prediction. Let's explore the main steps in hypothesis testing:

Common Challenges in Hypothesis Formulation

When formulating a hypothesis, it's crucial to remain objective. Bias can skew your results and lead to incorrect conclusions. To avoid this, challenge your assumptions and evaluate how likely they are to affect your decisions and actions .

Creating untestable hypotheses is a common pitfall. Hypotheses that can't be empirically tested, either due to abstract constructs or lack of measurement methods, pose significant challenges. Ensure all variables can be measured or manipulated with existing research methods.

Research often involves complex variables that can be difficult to define and measure. Clearly operationalize abstract concepts and consider the feasibility of empirical testing during the hypothesis formulation stage .

Examples of Hypotheses in Various Research Fields

Hypotheses in social sciences.

In social sciences, hypotheses often explore relationships between social behaviors and societal factors. For instance, a hypothesis might state that increased social media use leads to higher levels of anxiety among teenagers. This type of hypothesis helps in understanding complex social dynamics and can guide interventions.

Hypotheses in Natural Sciences

Natural sciences frequently use hypotheses to explain natural phenomena. For example, a hypothesis in biology might propose that [a specific gene affects flower color ](https://www.examples.com/english/hypothesis.html), predicting that altering this gene will change the flower's hue. Such hypotheses are crucial for advancing scientific knowledge and can lead to significant discoveries.

Hypotheses in Applied Sciences

In applied sciences, hypotheses are often practical and solution-oriented. An example could be hypothesizing that a new type of renewable energy source will reduce carbon emissions more effectively than current methods. These hypotheses drive innovation and can result in real-world applications that address pressing issues.

Ethical Considerations in Hypothesis Formulation

Ensuring integrity and honesty.

When formulating a hypothesis, it is crucial to maintain integrity and honesty . This means you should honestly report data, results, methods, and procedures . Avoid manipulating data to fit your hypothesis, as this compromises the validity of your research. Remember, both the question and the hypothesis should be formulated before the study is planned and should not be generated retrospectively based on data .

Avoiding Plagiarism

Plagiarism is a serious ethical violation in research. Always give proper credit to the original authors of the ideas and findings you use. This not only respects the intellectual property of others but also upholds the academic standards of your work. Ethical considerations in Ph.D. thesis research are essential for protecting participants' rights, maintaining integrity, and upholding academic standards .

Respecting Confidentiality

Respecting the confidentiality of your research participants is paramount. Ensure that personal information is kept secure and used only for the purposes of your study. This is especially important when dealing with sensitive data. Ethical considerations and unforeseen variables in experimental research emphasize integrity, transparency, and adaptability .

When forming a hypothesis, it's crucial to think about the ethical side of things. This means making sure your research is fair and honest. If you're a student struggling with this, don't worry! Our Thesis Action Plan can guide you through every step. Visit our website to learn more and get started today.

In summary, formulating a hypothesis is a crucial step in the research process. It involves investigating background research, developing a theory, and determining how to test it. This process helps researchers make predictions and guide their studies. By following these steps, researchers can create testable hypotheses that provide a clear direction for their work. Understanding how to formulate a hypothesis is essential for conducting effective and meaningful research.

Frequently Asked Questions

What are the steps in formulating a hypothesis.

To form a hypothesis, researchers usually follow these steps: 1. Investigate background research in the area of interest. 2. Develop or examine a theory. 3. Decide how the theory will be tested and predict what the researcher expects to find based on previous studies.

Why is formulating a hypothesis important in research?

Formulating a hypothesis is crucial because it guides the research process, provides a focus for data collection, and makes it easier to analyze data.

What is a hypothesis in research?

A hypothesis is a predictive statement about what the researcher expects to find when testing the research question. It is based on background research and theories.

What are the characteristics of a good hypothesis?

A good hypothesis should be clear, testable, and based on existing theories or knowledge. It should also be specific and focused on a particular relationship between variables.

What are the different types of hypotheses in research?

There are several types of hypotheses, including null hypotheses, alternative hypotheses, and directional vs. non-directional hypotheses.

How do researchers test a hypothesis?

Researchers test a hypothesis by designing experiments, collecting data, and analyzing the results to see if they support the hypothesis.

What challenges do researchers face when formulating a hypothesis?

Common challenges include avoiding bias, ensuring the hypothesis is testable, and dealing with complex variables.

What ethical considerations are involved in formulating a hypothesis?

Researchers must ensure integrity and honesty, avoid plagiarism, and respect confidentiality when formulating a hypothesis.

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What is a Research Hypothesis: How to Write it, Types, and Examples

Any research begins with a research question and a research hypothesis . A research question alone may not suffice to design the experiment(s) needed to answer it. A hypothesis is central to the scientific method. But what is a hypothesis ? A hypothesis is a testable statement that proposes a possible explanation to a phenomenon, and it may include a prediction. Next, you may ask what is a research hypothesis ? Simply put, a research hypothesis is a prediction or educated guess about the relationship between the variables that you want to investigate.  

It is important to be thorough when developing your research hypothesis. Shortcomings in the framing of a hypothesis can affect the study design and the results. A better understanding of the research hypothesis definition and characteristics of a good hypothesis will make it easier for you to develop your own hypothesis for your research. Let’s dive in to know more about the types of research hypothesis , how to write a research hypothesis , and some research hypothesis examples .  

Table of Contents

What is a hypothesis ?  

A hypothesis is based on the existing body of knowledge in a study area. Framed before the data are collected, a hypothesis states the tentative relationship between independent and dependent variables, along with a prediction of the outcome.  

What is a research hypothesis ?  

Young researchers starting out their journey are usually brimming with questions like “ What is a hypothesis ?” “ What is a research hypothesis ?” “How can I write a good research hypothesis ?”   

A research hypothesis is a statement that proposes a possible explanation for an observable phenomenon or pattern. It guides the direction of a study and predicts the outcome of the investigation. A research hypothesis is testable, i.e., it can be supported or disproven through experimentation or observation.     

Characteristics of a good hypothesis  

Here are the characteristics of a good hypothesis :  

• Clearly formulated and free of language errors and ambiguity  
• Concise and not unnecessarily verbose  
• Has clearly defined variables  
• Testable and stated in a way that allows for it to be disproven  
• Can be tested using a research design that is feasible, ethical, and practical   
• Specific and relevant to the research problem  
• Rooted in a thorough literature search  
• Can generate new knowledge or understanding.  

How to create an effective research hypothesis  

A study begins with the formulation of a research question. A researcher then performs background research. This background information forms the basis for building a good research hypothesis . The researcher then performs experiments, collects, and analyzes the data, interprets the findings, and ultimately, determines if the findings support or negate the original hypothesis.  

Let’s look at each step for creating an effective, testable, and good research hypothesis :  

• Identify a research problem or question: Start by identifying a specific research problem.   
• Review the literature: Conduct an in-depth review of the existing literature related to the research problem to grasp the current knowledge and gaps in the field.   
• Formulate a clear and testable hypothesis : Based on the research question, use existing knowledge to form a clear and testable hypothesis . The hypothesis should state a predicted relationship between two or more variables that can be measured and manipulated. Improve the original draft till it is clear and meaningful.  
• State the null hypothesis: The null hypothesis is a statement that there is no relationship between the variables you are studying.   
• Define the population and sample: Clearly define the population you are studying and the sample you will be using for your research.  
• Select appropriate methods for testing the hypothesis: Select appropriate research methods, such as experiments, surveys, or observational studies, which will allow you to test your research hypothesis .  

Remember that creating a research hypothesis is an iterative process, i.e., you might have to revise it based on the data you collect. You may need to test and reject several hypotheses before answering the research problem.  

How to write a research hypothesis  

When you start writing a research hypothesis , you use an “if–then” statement format, which states the predicted relationship between two or more variables. Clearly identify the independent variables (the variables being changed) and the dependent variables (the variables being measured), as well as the population you are studying. Review and revise your hypothesis as needed.  

An example of a research hypothesis in this format is as follows:  

“ If [athletes] follow [cold water showers daily], then their [endurance] increases.”  

Population: athletes  

Independent variable: daily cold water showers  

Dependent variable: endurance  

You may have understood the characteristics of a good hypothesis . But note that a research hypothesis is not always confirmed; a researcher should be prepared to accept or reject the hypothesis based on the study findings.  

Research hypothesis checklist  

Following from above, here is a 10-point checklist for a good research hypothesis :  

• Testable: A research hypothesis should be able to be tested via experimentation or observation.  
• Specific: A research hypothesis should clearly state the relationship between the variables being studied.  
• Based on prior research: A research hypothesis should be based on existing knowledge and previous research in the field.  
• Falsifiable: A research hypothesis should be able to be disproven through testing.  
• Clear and concise: A research hypothesis should be stated in a clear and concise manner.  
• Logical: A research hypothesis should be logical and consistent with current understanding of the subject.  
• Relevant: A research hypothesis should be relevant to the research question and objectives.  
• Feasible: A research hypothesis should be feasible to test within the scope of the study.  
• Reflects the population: A research hypothesis should consider the population or sample being studied.  
• Uncomplicated: A good research hypothesis is written in a way that is easy for the target audience to understand.  

By following this research hypothesis checklist , you will be able to create a research hypothesis that is strong, well-constructed, and more likely to yield meaningful results.  

Types of research hypothesis  

Different types of research hypothesis are used in scientific research:  

1. Null hypothesis:

A null hypothesis states that there is no change in the dependent variable due to changes to the independent variable. This means that the results are due to chance and are not significant. A null hypothesis is denoted as H0 and is stated as the opposite of what the alternative hypothesis states.   

Example: “ The newly identified virus is not zoonotic .”  

2. Alternative hypothesis:

This states that there is a significant difference or relationship between the variables being studied. It is denoted as H1 or Ha and is usually accepted or rejected in favor of the null hypothesis.  

Example: “ The newly identified virus is zoonotic .”  

3. Directional hypothesis :

This specifies the direction of the relationship or difference between variables; therefore, it tends to use terms like increase, decrease, positive, negative, more, or less.   

Example: “ The inclusion of intervention X decreases infant mortality compared to the original treatment .”   

4. Non-directional hypothesis:

While it does not predict the exact direction or nature of the relationship between the two variables, a non-directional hypothesis states the existence of a relationship or difference between variables but not the direction, nature, or magnitude of the relationship. A non-directional hypothesis may be used when there is no underlying theory or when findings contradict previous research.  

Example, “ Cats and dogs differ in the amount of affection they express .”  

5. Simple hypothesis :

A simple hypothesis only predicts the relationship between one independent and another independent variable.  

Example: “ Applying sunscreen every day slows skin aging .”  

6 . Complex hypothesis :

A complex hypothesis states the relationship or difference between two or more independent and dependent variables.   

Example: “ Applying sunscreen every day slows skin aging, reduces sun burn, and reduces the chances of skin cancer .” (Here, the three dependent variables are slowing skin aging, reducing sun burn, and reducing the chances of skin cancer.)  

7. Associative hypothesis:  

An associative hypothesis states that a change in one variable results in the change of the other variable. The associative hypothesis defines interdependency between variables.  

Example: “ There is a positive association between physical activity levels and overall health .”  

8 . Causal hypothesis:

A causal hypothesis proposes a cause-and-effect interaction between variables.  

Example: “ Long-term alcohol use causes liver damage .”  

Note that some of the types of research hypothesis mentioned above might overlap. The types of hypothesis chosen will depend on the research question and the objective of the study.  

Research hypothesis examples  

Here are some good research hypothesis examples :  

“The use of a specific type of therapy will lead to a reduction in symptoms of depression in individuals with a history of major depressive disorder.”  

“Providing educational interventions on healthy eating habits will result in weight loss in overweight individuals.”  

“Plants that are exposed to certain types of music will grow taller than those that are not exposed to music.”  

“The use of the plant growth regulator X will lead to an increase in the number of flowers produced by plants.”  

Characteristics that make a research hypothesis weak are unclear variables, unoriginality, being too general or too vague, and being untestable. A weak hypothesis leads to weak research and improper methods.   

Some bad research hypothesis examples (and the reasons why they are “bad”) are as follows:  

“This study will show that treatment X is better than any other treatment . ” (This statement is not testable, too broad, and does not consider other treatments that may be effective.)  

“This study will prove that this type of therapy is effective for all mental disorders . ” (This statement is too broad and not testable as mental disorders are complex and different disorders may respond differently to different types of therapy.)  

“Plants can communicate with each other through telepathy . ” (This statement is not testable and lacks a scientific basis.)  

Importance of testable hypothesis  

If a research hypothesis is not testable, the results will not prove or disprove anything meaningful. The conclusions will be vague at best. A testable hypothesis helps a researcher focus on the study outcome and understand the implication of the question and the different variables involved. A testable hypothesis helps a researcher make precise predictions based on prior research.  

To be considered testable, there must be a way to prove that the hypothesis is true or false; further, the results of the hypothesis must be reproducible.  

Frequently Asked Questions (FAQs) on research hypothesis  

1. What is the difference between research question and research hypothesis ?  

A research question defines the problem and helps outline the study objective(s). It is an open-ended statement that is exploratory or probing in nature. Therefore, it does not make predictions or assumptions. It helps a researcher identify what information to collect. A research hypothesis , however, is a specific, testable prediction about the relationship between variables. Accordingly, it guides the study design and data analysis approach.

2. When to reject null hypothesis ?

A null hypothesis should be rejected when the evidence from a statistical test shows that it is unlikely to be true. This happens when the test statistic (e.g., p -value) is less than the defined significance level (e.g., 0.05). Rejecting the null hypothesis does not necessarily mean that the alternative hypothesis is true; it simply means that the evidence found is not compatible with the null hypothesis.  

3. How can I be sure my hypothesis is testable?  

A testable hypothesis should be specific and measurable, and it should state a clear relationship between variables that can be tested with data. To ensure that your hypothesis is testable, consider the following:  

• Clearly define the key variables in your hypothesis. You should be able to measure and manipulate these variables in a way that allows you to test the hypothesis.  
• The hypothesis should predict a specific outcome or relationship between variables that can be measured or quantified.   
• You should be able to collect the necessary data within the constraints of your study.  
• It should be possible for other researchers to replicate your study, using the same methods and variables.   
• Your hypothesis should be testable by using appropriate statistical analysis techniques, so you can draw conclusions, and make inferences about the population from the sample data.  
• The hypothesis should be able to be disproven or rejected through the collection of data.  

4. How do I revise my research hypothesis if my data does not support it?  

If your data does not support your research hypothesis , you will need to revise it or develop a new one. You should examine your data carefully and identify any patterns or anomalies, re-examine your research question, and/or revisit your theory to look for any alternative explanations for your results. Based on your review of the data, literature, and theories, modify your research hypothesis to better align it with the results you obtained. Use your revised hypothesis to guide your research design and data collection. It is important to remain objective throughout the process.  

5. I am performing exploratory research. Do I need to formulate a research hypothesis?  

As opposed to “confirmatory” research, where a researcher has some idea about the relationship between the variables under investigation, exploratory research (or hypothesis-generating research) looks into a completely new topic about which limited information is available. Therefore, the researcher will not have any prior hypotheses. In such cases, a researcher will need to develop a post-hoc hypothesis. A post-hoc research hypothesis is generated after these results are known.  

6. How is a research hypothesis different from a research question?

A research question is an inquiry about a specific topic or phenomenon, typically expressed as a question. It seeks to explore and understand a particular aspect of the research subject. In contrast, a research hypothesis is a specific statement or prediction that suggests an expected relationship between variables. It is formulated based on existing knowledge or theories and guides the research design and data analysis.

7. Can a research hypothesis change during the research process?

Yes, research hypotheses can change during the research process. As researchers collect and analyze data, new insights and information may emerge that require modification or refinement of the initial hypotheses. This can be due to unexpected findings, limitations in the original hypotheses, or the need to explore additional dimensions of the research topic. Flexibility is crucial in research, allowing for adaptation and adjustment of hypotheses to align with the evolving understanding of the subject matter.

8. How many hypotheses should be included in a research study?

The number of research hypotheses in a research study varies depending on the nature and scope of the research. It is not necessary to have multiple hypotheses in every study. Some studies may have only one primary hypothesis, while others may have several related hypotheses. The number of hypotheses should be determined based on the research objectives, research questions, and the complexity of the research topic. It is important to ensure that the hypotheses are focused, testable, and directly related to the research aims.

9. Can research hypotheses be used in qualitative research?

Yes, research hypotheses can be used in qualitative research, although they are more commonly associated with quantitative research. In qualitative research, hypotheses may be formulated as tentative or exploratory statements that guide the investigation. Instead of testing hypotheses through statistical analysis, qualitative researchers may use the hypotheses to guide data collection and analysis, seeking to uncover patterns, themes, or relationships within the qualitative data. The emphasis in qualitative research is often on generating insights and understanding rather than confirming or rejecting specific research hypotheses through statistical testing.

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• Knowledge Base
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• How to Write a Strong Hypothesis | Guide & Examples

How to Write a Strong Hypothesis | Guide & Examples

Published on 6 May 2022 by Shona McCombes .

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more variables . An independent variable is something the researcher changes or controls. A dependent variable is something the researcher observes and measures.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1: ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2: Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalise more complex constructs.

Step 3: Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

Step 4: Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

Step 5: Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in if … then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

Step 6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis. The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis is not just a guess. It should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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Definition of a Hypothesis

What it is and how it's used in sociology

• Key Concepts
• Major Sociologists
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• Research, Samples, and Statistics
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A hypothesis is a prediction of what will be found at the outcome of a research project and is typically focused on the relationship between two different variables studied in the research. It is usually based on both theoretical expectations about how things work and already existing scientific evidence.

Within social science, a hypothesis can take two forms. It can predict that there is no relationship between two variables, in which case it is a null hypothesis . Or, it can predict the existence of a relationship between variables, which is known as an alternative hypothesis.

In either case, the variable that is thought to either affect or not affect the outcome is known as the independent variable, and the variable that is thought to either be affected or not is the dependent variable.

Researchers seek to determine whether or not their hypothesis, or hypotheses if they have more than one, will prove true. Sometimes they do, and sometimes they do not. Either way, the research is considered successful if one can conclude whether or not a hypothesis is true. 

Null Hypothesis

A researcher has a null hypothesis when she or he believes, based on theory and existing scientific evidence, that there will not be a relationship between two variables. For example, when examining what factors influence a person's highest level of education within the U.S., a researcher might expect that place of birth, number of siblings, and religion would not have an impact on the level of education. This would mean the researcher has stated three null hypotheses.

Alternative Hypothesis

Taking the same example, a researcher might expect that the economic class and educational attainment of one's parents, and the race of the person in question are likely to have an effect on one's educational attainment. Existing evidence and social theories that recognize the connections between wealth and cultural resources , and how race affects access to rights and resources in the U.S. , would suggest that both economic class and educational attainment of the one's parents would have a positive effect on educational attainment. In this case, economic class and educational attainment of one's parents are independent variables, and one's educational attainment is the dependent variable—it is hypothesized to be dependent on the other two.

Conversely, an informed researcher would expect that being a race other than white in the U.S. is likely to have a negative impact on a person's educational attainment. This would be characterized as a negative relationship, wherein being a person of color has a negative effect on one's educational attainment. In reality, this hypothesis proves true, with the exception of Asian Americans , who go to college at a higher rate than whites do. However, Blacks and Hispanics and Latinos are far less likely than whites and Asian Americans to go to college.

Formulating a Hypothesis

Formulating a hypothesis can take place at the very beginning of a research project , or after a bit of research has already been done. Sometimes a researcher knows right from the start which variables she is interested in studying, and she may already have a hunch about their relationships. Other times, a researcher may have an interest in ​a particular topic, trend, or phenomenon, but he may not know enough about it to identify variables or formulate a hypothesis.

Whenever a hypothesis is formulated, the most important thing is to be precise about what one's variables are, what the nature of the relationship between them might be, and how one can go about conducting a study of them.

Updated by Nicki Lisa Cole, Ph.D

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2.1 Approaches to Sociological Research

Learning objectives.

By the end of this section, you should be able to:

• Define and describe the scientific method.
• Explain how the scientific method is used in sociological research.
• Describe the function and importance of an interpretive framework.
• Describe the differences in accuracy, reliability and validity in a research study.

When sociologists apply the sociological perspective and begin to ask questions, no topic is off limits. Every aspect of human behavior is a source of possible investigation. Sociologists question the world that humans have created and live in. They notice patterns of behavior as people move through that world. Using sociological methods and systematic research within the framework of the scientific method and a scholarly interpretive perspective, sociologists have discovered social patterns in the workplace that have transformed industries, in families that have enlightened family members, and in education that have aided structural changes in classrooms.

Sociologists often begin the research process by asking a question about how or why things happen in this world. It might be a unique question about a new trend or an old question about a common aspect of life. Once the question is formed, the sociologist proceeds through an in-depth process to answer it. In deciding how to design that process, the researcher may adopt a scientific approach or an interpretive framework. The following sections describe these approaches to knowledge.

The Scientific Method

Sociologists make use of tried and true methods of research, such as experiments, surveys, and field research. But humans and their social interactions are so diverse that these interactions can seem impossible to chart or explain. It might seem that science is about discoveries and chemical reactions or about proving ideas right or wrong rather than about exploring the nuances of human behavior.

However, this is exactly why scientific models work for studying human behavior. A scientific process of research establishes parameters that help make sure results are objective and accurate. Scientific methods provide limitations and boundaries that focus a study and organize its results.

The scientific method involves developing and testing theories about the social world based on empirical evidence. It is defined by its commitment to systematic observation of the empirical world and strives to be objective, critical, skeptical, and logical. It involves a series of six prescribed steps that have been established over centuries of scientific scholarship.

Sociological research does not reduce knowledge to right or wrong facts. Results of studies tend to provide people with insights they did not have before—explanations of human behaviors and social practices and access to knowledge of other cultures, rituals and beliefs, or trends and attitudes.

In general, sociologists tackle questions about the role of social characteristics in outcomes or results. For example, how do different communities fare in terms of psychological well-being, community cohesiveness, range of vocation, wealth, crime rates, and so on? Are communities functioning smoothly? Sociologists often look between the cracks to discover obstacles to meeting basic human needs. They might also study environmental influences and patterns of behavior that lead to crime, substance abuse, divorce, poverty, unplanned pregnancies, or illness. And, because sociological studies are not all focused on negative behaviors or challenging situations, social researchers might study vacation trends, healthy eating habits, neighborhood organizations, higher education patterns, games, parks, and exercise habits.

Sociologists can use the scientific method not only to collect but also to interpret and analyze data. They deliberately apply scientific logic and objectivity. They are interested in—but not attached to—the results. They work outside of their own political or social agendas. This does not mean researchers do not have their own personalities, complete with preferences and opinions. But sociologists deliberately use the scientific method to maintain as much objectivity, focus, and consistency as possible in collecting and analyzing data in research studies.

With its systematic approach, the scientific method has proven useful in shaping sociological studies. The scientific method provides a systematic, organized series of steps that help ensure objectivity and consistency in exploring a social problem. They provide the means for accuracy, reliability, and validity. In the end, the scientific method provides a shared basis for discussion and analysis (Merton 1963). Typically, the scientific method has 6 steps which are described below.

Step 1: Ask a Question or Find a Research Topic

The first step of the scientific method is to ask a question, select a problem, and identify the specific area of interest. The topic should be narrow enough to study within a geographic location and time frame. “Are societies capable of sustained happiness?” would be too vague. The question should also be broad enough to have universal merit. “What do personal hygiene habits reveal about the values of students at XYZ High School?” would be too narrow. Sociologists strive to frame questions that examine well-defined patterns and relationships.

In a hygiene study, for instance, hygiene could be defined as “personal habits to maintain physical appearance (as opposed to health),” and a researcher might ask, “How do differing personal hygiene habits reflect the cultural value placed on appearance?”

Step 2: Review the Literature/Research Existing Sources

The next step researchers undertake is to conduct background research through a literature review , which is a review of any existing similar or related studies. A visit to the library, a thorough online search, and a survey of academic journals will uncover existing research about the topic of study. This step helps researchers gain a broad understanding of work previously conducted, identify gaps in understanding of the topic, and position their own research to build on prior knowledge. Researchers—including student researchers—are responsible for correctly citing existing sources they use in a study or that inform their work. While it is fine to borrow previously published material (as long as it enhances a unique viewpoint), it must be referenced properly and never plagiarized.

To study crime, a researcher might also sort through existing data from the court system, police database, prison information, interviews with criminals, guards, wardens, etc. It’s important to examine this information in addition to existing research to determine how these resources might be used to fill holes in existing knowledge. Reviewing existing sources educates researchers and helps refine and improve a research study design.

Step 3: Formulate a Hypothesis

A hypothesis is an explanation for a phenomenon based on a conjecture about the relationship between the phenomenon and one or more causal factors. In sociology, the hypothesis will often predict how one form of human behavior influences another. For example, a hypothesis might be in the form of an “if, then statement.” Let’s relate this to our topic of crime: If unemployment increases, then the crime rate will increase.

In scientific research, we formulate hypotheses to include an independent variables (IV) , which are the cause of the change, and a dependent variable (DV) , which is the effect , or thing that is changed. In the example above, unemployment is the independent variable and the crime rate is the dependent variable.

In a sociological study, the researcher would establish one form of human behavior as the independent variable and observe the influence it has on a dependent variable. How does gender (the independent variable) affect rate of income (the dependent variable)? How does one’s religion (the independent variable) affect family size (the dependent variable)? How is social class (the dependent variable) affected by level of education (the independent variable)?

Taking an example from Table 12.1, a researcher might hypothesize that teaching children proper hygiene (the independent variable) will boost their sense of self-esteem (the dependent variable). Note, however, this hypothesis can also work the other way around. A sociologist might predict that increasing a child’s sense of self-esteem (the independent variable) will increase or improve habits of hygiene (now the dependent variable). Identifying the independent and dependent variables is very important. As the hygiene example shows, simply identifying related two topics or variables is not enough. Their prospective relationship must be part of the hypothesis.

Step 4: Design and Conduct a Study

Researchers design studies to maximize reliability , which refers to how likely research results are to be replicated if the study is reproduced. Reliability increases the likelihood that what happens to one person will happen to all people in a group or what will happen in one situation will happen in another. Cooking is a science. When you follow a recipe and measure ingredients with a cooking tool, such as a measuring cup, the same results is obtained as long as the cook follows the same recipe and uses the same type of tool. The measuring cup introduces accuracy into the process. If a person uses a less accurate tool, such as their hand, to measure ingredients rather than a cup, the same result may not be replicated. Accurate tools and methods increase reliability.

Researchers also strive for validity , which refers to how well the study measures what it was designed to measure. To produce reliable and valid results, sociologists develop an operational definition , that is, they define each concept, or variable, in terms of the physical or concrete steps it takes to objectively measure it. The operational definition identifies an observable condition of the concept. By operationalizing the concept, all researchers can collect data in a systematic or replicable manner. Moreover, researchers can determine whether the experiment or method validly represent the phenomenon they intended to study.

A study asking how tutoring improves grades, for instance, might define “tutoring” as “one-on-one assistance by an expert in the field, hired by an educational institution.” However, one researcher might define a “good” grade as a C or better, while another uses a B+ as a starting point for “good.” For the results to be replicated and gain acceptance within the broader scientific community, researchers would have to use a standard operational definition. These definitions set limits and establish cut-off points that ensure consistency and replicability in a study.

We will explore research methods in greater detail in the next section of this chapter.

Step 5: Draw Conclusions

After constructing the research design, sociologists collect, tabulate or categorize, and analyze data to formulate conclusions. If the analysis supports the hypothesis, researchers can discuss the implications of the results for the theory or policy solution that they were addressing. If the analysis does not support the hypothesis, researchers may consider repeating the experiment or think of ways to improve their procedure.

However, even when results contradict a sociologist’s prediction of a study’s outcome, these results still contribute to sociological understanding. Sociologists analyze general patterns in response to a study, but they are equally interested in exceptions to patterns. In a study of education, a researcher might predict that high school dropouts have a hard time finding rewarding careers. While many assume that the higher the education, the higher the salary and degree of career happiness, there are certainly exceptions. People with little education have had stunning careers, and people with advanced degrees have had trouble finding work. A sociologist prepares a hypothesis knowing that results may substantiate or contradict it.

Sociologists carefully keep in mind how operational definitions and research designs impact the results as they draw conclusions. Consider the concept of “increase of crime,” which might be defined as the percent increase in crime from last week to this week, as in the study of Swedish crime discussed above. Yet the data used to evaluate “increase of crime” might be limited by many factors: who commits the crime, where the crimes are committed, or what type of crime is committed. If the data is gathered for “crimes committed in Houston, Texas in zip code 77021,” then it may not be generalizable to crimes committed in rural areas outside of major cities like Houston. If data is collected about vandalism, it may not be generalizable to assault.

Step 6: Report Results

Researchers report their results at conferences and in academic journals. These results are then subjected to the scrutiny of other sociologists in the field. Before the conclusions of a study become widely accepted, the studies are often repeated in the same or different environments. In this way, sociological theories and knowledge develops as the relationships between social phenomenon are established in broader contexts and different circumstances.

Interpretive Framework

While many sociologists rely on empirical data and the scientific method as a research approach, others operate from an interpretive framework . While systematic, this approach doesn’t follow the hypothesis-testing model that seeks to find generalizable results. Instead, an interpretive framework, sometimes referred to as an interpretive perspective , seeks to understand social worlds from the point of view of participants, which leads to in-depth knowledge or understanding about the human experience.

Interpretive research is generally more descriptive or narrative in its findings. Rather than formulating a hypothesis and method for testing it, an interpretive researcher will develop approaches to explore the topic at hand that may involve a significant amount of direct observation or interaction with subjects including storytelling. This type of researcher learns through the process and sometimes adjusts the research methods or processes midway to optimize findings as they evolve.

Critical Sociology

Critical sociology focuses on deconstruction of existing sociological research and theory. Informed by the work of Karl Marx, scholars known collectively as the Frankfurt School proposed that social science, as much as any academic pursuit, is embedded in the system of power constituted by the set of class, caste, race, gender, and other relationships that exist in the society. Consequently, it cannot be treated as purely objective. Critical sociologists view theories, methods, and the conclusions as serving one of two purposes: they can either legitimate and rationalize systems of social power and oppression or liberate humans from inequality and restriction on human freedom. Deconstruction can involve data collection, but the analysis of this data is not empirical or positivist.

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How to Write a Great Hypothesis

Hypothesis Definition, Format, Examples, and Tips

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis.

• Operationalization

Hypothesis Types

Hypotheses examples.

• Collecting Data

A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process.

Consider a study designed to examine the relationship between sleep deprivation and test performance. The hypothesis might be: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

At a Glance

A hypothesis is crucial to scientific research because it offers a clear direction for what the researchers are looking to find. This allows them to design experiments to test their predictions and add to our scientific knowledge about the world. This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. At this point, researchers then begin to develop a testable hypothesis.

Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore numerous factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk adage that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

How to Formulate a Good Hypothesis

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

The Importance of Operational Definitions

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

Operational definitions are specific definitions for all relevant factors in a study. This process helps make vague or ambiguous concepts detailed and measurable.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in various ways. Clearly defining these variables and how they are measured helps ensure that other researchers can replicate your results.

Replicability

One of the basic principles of any type of scientific research is that the results must be replicable.

Replication means repeating an experiment in the same way to produce the same results. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. For example, how would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

To measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming others. The researcher might utilize a simulated task to measure aggressiveness in this situation.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type suggests a relationship between three or more variables, such as two independent and dependent variables.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative population sample and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."
• "Children who receive a new reading intervention will have higher reading scores than students who do not receive the intervention."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "There is no difference in anxiety levels between people who take St. John's wort supplements and those who do not."
• "There is no difference in scores on a memory recall task between children and adults."
• "There is no difference in aggression levels between children who play first-person shooter games and those who do not."

Examples of an alternative hypothesis:

• "People who take St. John's wort supplements will have less anxiety than those who do not."
• "Adults will perform better on a memory task than children."
• "Children who play first-person shooter games will show higher levels of aggression than children who do not." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when  conducting an experiment is difficult or impossible. These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a  correlational study  can examine how the variables are related. This research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Thompson WH, Skau S. On the scope of scientific hypotheses .  R Soc Open Sci . 2023;10(8):230607. doi:10.1098/rsos.230607

Taran S, Adhikari NKJ, Fan E. Falsifiability in medicine: what clinicians can learn from Karl Popper [published correction appears in Intensive Care Med. 2021 Jun 17;:].  Intensive Care Med . 2021;47(9):1054-1056. doi:10.1007/s00134-021-06432-z

Eyler AA. Research Methods for Public Health . 1st ed. Springer Publishing Company; 2020. doi:10.1891/9780826182067.0004

Nosek BA, Errington TM. What is replication ?  PLoS Biol . 2020;18(3):e3000691. doi:10.1371/journal.pbio.3000691

Aggarwal R, Ranganathan P. Study designs: Part 2 - Descriptive studies .  Perspect Clin Res . 2019;10(1):34-36. doi:10.4103/picr.PICR_154_18

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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• v.24(1); Jan-Mar 2019

Formulation of Research Question – Stepwise Approach

Simmi k. ratan.

Department of Pediatric Surgery, Maulana Azad Medical College, New Delhi, India

1 Department of Community Medicine, North Delhi Municipal Corporation Medical College, New Delhi, India

2 Department of Pediatric Surgery, Batra Hospital and Research Centre, New Delhi, India

Formulation of research question (RQ) is an essentiality before starting any research. It aims to explore an existing uncertainty in an area of concern and points to a need for deliberate investigation. It is, therefore, pertinent to formulate a good RQ. The present paper aims to discuss the process of formulation of RQ with stepwise approach. The characteristics of good RQ are expressed by acronym “FINERMAPS” expanded as feasible, interesting, novel, ethical, relevant, manageable, appropriate, potential value, publishability, and systematic. A RQ can address different formats depending on the aspect to be evaluated. Based on this, there can be different types of RQ such as based on the existence of the phenomenon, description and classification, composition, relationship, comparative, and causality. To develop a RQ, one needs to begin by identifying the subject of interest and then do preliminary research on that subject. The researcher then defines what still needs to be known in that particular subject and assesses the implied questions. After narrowing the focus and scope of the research subject, researcher frames a RQ and then evaluates it. Thus, conception to formulation of RQ is very systematic process and has to be performed meticulously as research guided by such question can have wider impact in the field of social and health research by leading to formulation of policies for the benefit of larger population.

I NTRODUCTION

A good research question (RQ) forms backbone of a good research, which in turn is vital in unraveling mysteries of nature and giving insight into a problem.[ 1 , 2 , 3 , 4 ] RQ identifies the problem to be studied and guides to the methodology. It leads to building up of an appropriate hypothesis (Hs). Hence, RQ aims to explore an existing uncertainty in an area of concern and points to a need for deliberate investigation. A good RQ helps support a focused arguable thesis and construction of a logical argument. Hence, formulation of a good RQ is undoubtedly one of the first critical steps in the research process, especially in the field of social and health research, where the systematic generation of knowledge that can be used to promote, restore, maintain, and/or protect health of individuals and populations.[ 1 , 3 , 4 ] Basically, the research can be classified as action, applied, basic, clinical, empirical, administrative, theoretical, or qualitative or quantitative research, depending on its purpose.[ 2 ]

Research plays an important role in developing clinical practices and instituting new health policies. Hence, there is a need for a logical scientific approach as research has an important goal of generating new claims.[ 1 ]

C HARACTERISTICS OF G OOD R ESEARCH Q UESTION

“The most successful research topics are narrowly focused and carefully defined but are important parts of a broad-ranging, complex problem.”

A good RQ is an asset as it:

• Details the problem statement
• Further describes and refines the issue under study
• Adds focus to the problem statement
• Guides data collection and analysis
• Sets context of research.

Hence, while writing RQ, it is important to see if it is relevant to the existing time frame and conditions. For example, the impact of “odd-even” vehicle formula in decreasing the level of air particulate pollution in various districts of Delhi.

A good research is represented by acronym FINERMAPS[ 5 ]

Interesting.

• Appropriate
• Potential value and publishability
• Systematic.

Feasibility means that it is within the ability of the investigator to carry out. It should be backed by an appropriate number of subjects and methodology as well as time and funds to reach the conclusions. One needs to be realistic about the scope and scale of the project. One has to have access to the people, gadgets, documents, statistics, etc. One should be able to relate the concepts of the RQ to the observations, phenomena, indicators, or variables that one can access. One should be clear that the collection of data and the proceedings of project can be completed within the limited time and resources available to the investigator. Sometimes, a RQ appears feasible, but when fieldwork or study gets started, it proves otherwise. In this situation, it is important to write up the problems honestly and to reflect on what has been learned. One should try to discuss with more experienced colleagues or the supervisor so as to develop a contingency plan to anticipate possible problems while working on a RQ and find possible solutions in such situations.

This is essential that one has a real grounded interest in one's RQ and one can explore this and back it up with academic and intellectual debate. This interest will motivate one to keep going with RQ.

The question should not simply copy questions investigated by other workers but should have scope to be investigated. It may aim at confirming or refuting the already established findings, establish new facts, or find new aspects of the established facts. It should show imagination of the researcher. Above all, the question has to be simple and clear. The complexity of a question can frequently hide unclear thoughts and lead to a confused research process. A very elaborate RQ, or a question which is not differentiated into different parts, may hide concepts that are contradictory or not relevant. This needs to be clear and thought-through. Having one key question with several subcomponents will guide your research.

This is the foremost requirement of any RQ and is mandatory to get clearance from appropriate authorities before stating research on the question. Further, the RQ should be such that it minimizes the risk of harm to the participants in the research, protect the privacy and maintain their confidentiality, and provide the participants right to withdraw from research. It should also guide in avoiding deceptive practices in research.

The question should of academic and intellectual interest to people in the field you have chosen to study. The question preferably should arise from issues raised in the current situation, literature, or in practice. It should establish a clear purpose for the research in relation to the chosen field. For example, filling a gap in knowledge, analyzing academic assumptions or professional practice, monitoring a development in practice, comparing different approaches, or testing theories within a specific population are some of the relevant RQs.

Manageable (M): It has the similar essence as of feasibility but mainly means that the following research can be managed by the researcher.

Appropriate (A): RQ should be appropriate logically and scientifically for the community and institution.

Potential value and publishability (P): The study can make significant health impact in clinical and community practices. Therefore, research should aim for significant economic impact to reduce unnecessary or excessive costs. Furthermore, the proposed study should exist within a clinical, consumer, or policy-making context that is amenable to evidence-based change. Above all, a good RQ must address a topic that has clear implications for resolving important dilemmas in health and health-care decisions made by one or more stakeholder groups.

Systematic (S): Research is structured with specified steps to be taken in a specified sequence in accordance with the well-defined set of rules though it does not rule out creative thinking.

Example of RQ: Would the topical skin application of oil as a skin barrier reduces hypothermia in preterm infants? This question fulfills the criteria of a good RQ, that is, feasible, interesting, novel, ethical, and relevant.

Types of research question

A RQ can address different formats depending on the aspect to be evaluated.[ 6 ] For example:

• Existence: This is designed to uphold the existence of a particular phenomenon or to rule out rival explanation, for example, can neonates perceive pain?
• Description and classification: This type of question encompasses statement of uniqueness, for example, what are characteristics and types of neuropathic bladders?
• Composition: It calls for breakdown of whole into components, for example, what are stages of reflux nephropathy?
• Relationship: Evaluate relation between variables, for example, association between tumor rupture and recurrence rates in Wilm's tumor
• Descriptive—comparative: Expected that researcher will ensure that all is same between groups except issue in question, for example, Are germ cell tumors occurring in gonads more aggressive than those occurring in extragonadal sites?
• Causality: Does deletion of p53 leads to worse outcome in patients with neuroblastoma?
• Causality—comparative: Such questions frequently aim to see effect of two rival treatments, for example, does adding surgical resection improves survival rate outcome in children with neuroblastoma than with chemotherapy alone?
• Causality–Comparative interactions: Does immunotherapy leads to better survival outcome in neuroblastoma Stage IV S than with chemotherapy in the setting of adverse genetic profile than without it? (Does X cause more changes in Y than those caused by Z under certain condition and not under other conditions).

How to develop a research question

• Begin by identifying a broader subject of interest that lends itself to investigate, for example, hormone levels among hypospadias
• Do preliminary research on the general topic to find out what research has already been done and what literature already exists.[ 7 ] Therefore, one should begin with “information gaps” (What do you already know about the problem? For example, studies with results on testosterone levels among hypospadias
• What do you still need to know? (e.g., levels of other reproductive hormones among hypospadias)
• What are the implied questions: The need to know about a problem will lead to few implied questions. Each general question should lead to more specific questions (e.g., how hormone levels differ among isolated hypospadias with respect to that in normal population)
• Narrow the scope and focus of research (e.g., assessment of reproductive hormone levels among isolated hypospadias and hypospadias those with associated anomalies)
• Is RQ clear? With so much research available on any given topic, RQs must be as clear as possible in order to be effective in helping the writer direct his or her research
• Is the RQ focused? RQs must be specific enough to be well covered in the space available
• Is the RQ complex? RQs should not be answerable with a simple “yes” or “no” or by easily found facts. They should, instead, require both research and analysis on the part of the writer
• Is the RQ one that is of interest to the researcher and potentially useful to others? Is it a new issue or problem that needs to be solved or is it attempting to shed light on previously researched topic
• Is the RQ researchable? Consider the available time frame and the required resources. Is the methodology to conduct the research feasible?
• Is the RQ measurable and will the process produce data that can be supported or contradicted?
• Is the RQ too broad or too narrow?
• Create Hs: After formulating RQ, think where research is likely to be progressing? What kind of argument is likely to be made/supported? What would it mean if the research disputed the planned argument? At this step, one can well be on the way to have a focus for the research and construction of a thesis. Hs consists of more specific predictions about the nature and direction of the relationship between two variables. It is a predictive statement about the outcome of the research, dictate the method, and design of the research[ 1 ]
• Understand implications of your research: This is important for application: whether one achieves to fill gap in knowledge and how the results of the research have practical implications, for example, to develop health policies or improve educational policies.[ 1 , 8 ]

Brainstorm/Concept map for formulating research question

• First, identify what types of studies have been done in the past?
• Is there a unique area that is yet to be investigated or is there a particular question that may be worth replicating?
• Begin to narrow the topic by asking open-ended “how” and “why” questions
• Evaluate the question
• Develop a Hypothesis (Hs)
• Write down the RQ.

Writing down the research question

• State the question in your own words
• Write down the RQ as completely as possible.

For example, Evaluation of reproductive hormonal profile in children presenting with isolated hypospadias)

• Divide your question into concepts. Narrow to two or three concepts (reproductive hormonal profile, isolated hypospadias, compare with normal/not isolated hypospadias–implied)
• Specify the population to be studied (children with isolated hypospadias)
• Refer to the exposure or intervention to be investigated, if any
• Reflect the outcome of interest (hormonal profile).

Another example of a research question

Would the topical skin application of oil as a skin barrier reduces hypothermia in preterm infants? Apart from fulfilling the criteria of a good RQ, that is, feasible, interesting, novel, ethical, and relevant, it also details about the intervention done (topical skin application of oil), rationale of intervention (as a skin barrier), population to be studied (preterm infants), and outcome (reduces hypothermia).

Other important points to be heeded to while framing research question

• Make reference to a population when a relationship is expected among a certain type of subjects
• RQs and Hs should be made as specific as possible
• Avoid words or terms that do not add to the meaning of RQs and Hs
• Stick to what will be studied, not implications
• Name the variables in the order in which they occur/will be measured
• Avoid the words significant/”prove”
• Avoid using two different terms to refer to the same variable.

Some of the other problems and their possible solutions have been discussed in Table 1 .

Potential problems and solutions while making research question

G OING B EYOND F ORMULATION OF R ESEARCH Q UESTION–THE P ATH A HEAD

Once RQ is formulated, a Hs can be developed. Hs means transformation of a RQ into an operational analog.[ 1 ] It means a statement as to what prediction one makes about the phenomenon to be examined.[ 4 ] More often, for case–control trial, null Hs is generated which is later accepted or refuted.

A strong Hs should have following characteristics:

• Give insight into a RQ
• Are testable and measurable by the proposed experiments
• Have logical basis
• Follows the most likely outcome, not the exceptional outcome.

E XAMPLES OF R ESEARCH Q UESTION AND H YPOTHESIS

Research question-1.

• Does reduced gap between the two segments of the esophagus in patients of esophageal atresia reduces the mortality and morbidity of such patients?

Hypothesis-1

• Reduced gap between the two segments of the esophagus in patients of esophageal atresia reduces the mortality and morbidity of such patients
• In pediatric patients with esophageal atresia, gap of <2 cm between two segments of the esophagus and proper mobilization of proximal pouch reduces the morbidity and mortality among such patients.

Research question-2

• Does application of mitomycin C improves the outcome in patient of corrosive esophageal strictures?

Hypothesis-2

In patients aged 2–9 years with corrosive esophageal strictures, 34 applications of mitomycin C in dosage of 0.4 mg/ml for 5 min over a period of 6 months improve the outcome in terms of symptomatic and radiological relief. Some other examples of good and bad RQs have been shown in Table 2 .

Examples of few bad (left-hand side column) and few good (right-hand side) research questions

R ESEARCH Q UESTION AND S TUDY D ESIGN

RQ determines study design, for example, the question aimed to find the incidence of a disease in population will lead to conducting a survey; to find risk factors for a disease will need case–control study or a cohort study. RQ may also culminate into clinical trial.[ 9 , 10 ] For example, effect of administration of folic acid tablet in the perinatal period in decreasing incidence of neural tube defect. Accordingly, Hs is framed.

Appropriate statistical calculations are instituted to generate sample size. The subject inclusion, exclusion criteria and time frame of research are carefully defined. The detailed subject information sheet and pro forma are carefully defined. Moreover, research is set off few examples of research methodology guided by RQ:

• Incidence of anorectal malformations among adolescent females (hospital-based survey)
• Risk factors for the development of spontaneous pneumoperitoneum in pediatric patients (case–control design and cohort study)
• Effect of technique of extramucosal ureteric reimplantation without the creation of submucosal tunnel for the preservation of upper tract in bladder exstrophy (clinical trial).

The results of the research are then be available for wider applications for health and social life

C ONCLUSION

A good RQ needs thorough literature search and deep insight into the specific area/problem to be investigated. A RQ has to be focused yet simple. Research guided by such question can have wider impact in the field of social and health research by leading to formulation of policies for the benefit of larger population.

Financial support and sponsorship

Conflicts of interest.

There are no conflicts of interest.

R EFERENCES

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10 Research questions and testing hypotheses

Soumyajit Patra

This module will teach you about the importance of research questions and hypothesis in social science research. At the end of this module, you will find some digital resources and a bibliography for your further study.

• Introduction

Knowledge is contextual and much of it depends on agreement. It is contextual as it has a time-space dimension – knowledge varies from time to time, region to region and from society to society. Take the example of the imperishable plastic bags. These were thought to be very useful even a few years back. But today its harmful effects on environment are known to all and we have started thinking the other way round. The old knowledge has been replaced by the new one.

Knowledge is a matter of agreement as well. A significant portion of what we know is a matter of agreement and belief. Little of it is based on our personal experience and discovery. As we grow up through the process of socialization, we learn to accept (to take it for granted) what everybody around us already knows. If we start questing everything we are taught and try to test instead of accepting what is given, life would be impossible to bear with (Babbie 2004: 5). We have to learn where and how to raise ‘questions’ in our everyday life. This is equally true in case of scientific knowledge. We need to clarify, in this context, the distinction between scientific knowledge and common sense as well as the purpose of scientific research.

2.1 Scientific Knowledge and Common Sense

The distinction between scientific knowledge and common sense would be relevant here. The former is based on logic and is verifiable. The foundation of scientific knowledge is systematic and critical questioning, observation and reasoning. But as Majumdar (2005: 10) defines it, common sense does not ‘take us beyond what are observable. It limits us to events and conclusions that are widely believed as true.’ So, for obvious reasons, knowledge gathered through scientific inquiries may oppose the common sense. Social scientists often emphasize upon the explanatory nature of science that, to a large extent, involves refined and fundamental questioning of the existing knowledge.

2.2 Scientific Research

The purpose of scientific research is to modify or contribute to the existing stock of knowledge through proper inquiry directed by properly framed research questions and reasoning. It is widely believed that there is no ‘absolute’ in science. Scientific knowledge is inclusive and is always open to further investigation and revision. In the words of Das (2004: 21), ‘research may be described as systematic and critical investigation of phenomena towards increasing the stream of knowledge.’ In a similar way Majumdar (2005: 25) writes: ‘The obvious function of research is to add new knowledge to our existing store. Therefore, scientific research is a cumulative process. Since new insights are obtained into the problem investigated, we need to review or modify our earlier beliefs and postulates’.

• Learning Outcome

This Module will help you understand different types of research questions and hypotheses that give rise to reliable scientific knowledge. You will also learn how to formulate them.

• What are Research Questions?

Researchers have many queries and curiosities in their mind and they try to reach at some satisfactory and valid answers and solutions of these after a careful and meticulous analysis of the relevant data collected through appropriate methods. Research questions are specific questions framed during the initial phase of the research, the answers of which a researcher tries to find out. The research questions set the direction of the entire research process. We can argue following Bryman (2012: 9) that

“A research question is a question that provides an explicit statement of what it is the researcher wants to know about. A research purpose can be presented as a statement … but a question forces the researcher to be more explicit about what is to be investigated”.

4.1 How can Research Questions be framed?

Mere selection of research topic does not direct a researcher to the actual methods to be followed and the specific areas to look at for collecting data. As Patrick White (2009: 33) has argued,

“It is usually much easier to decide upon a topic or area of interest than it is to produce a set of well-structured questions”.

It may be worth noting here that research involves certain definite stages and a researcher starts framing research questions and hypothesis after selection of topic and review of existing literature. The following diagram shows the stages of research before and after the research questions:

Selection of research topic, which is the elementary task of any research, is, however, not an easy task. One has to go through the existing literatures to find out gaps in research. Indian Council of Social Science Research (ICSSR) however publishes trend reports of research done on important themes in sociology and makes us aware of what has so far been done and what needs to be done (Singh 2013). Despite such literature, a researcher has to be very precise in focusing his or her attention while framing research questions. When, after going through the existing literatures on the concerned area, the researcher finally specifies the objectives of the study, he or she is better able to frame his or her research questions. Research questions can also be framed on the basis common sense.

• De Vaus (2002) has provided us with some examples that can guide us in developing research questions particularly for descriptive researches. These are:
• What is the time frame of our interest?
• What is the geographical location of our interest?
• Is our interest in broad description or in comparing and specifying patterns for subgroups?
• What aspect of the topic are we interested in?
• How abstract is our interest?

The research questions of explanatory studies mostly focus on delving the causal relationships between different variables. Naturally the ‘why-questions’ are more important in explanatory researches than the ‘what-questions’, which forms the basis of descriptive studies. According to Babbie (2008: 99), descriptive studies answer questions of what, where, when and how, exploratory studies questions of why? However, the suggestions of Ramkrishna Mukherjee (1993) can be helpful for formulating research questions for any type research in social sciences.

Calling his approach as ‘inductive-inferential method’ Mukherjee ( Ibid .) argues that a social scientist should try to find out the answer of the following questions:

What is it?

What will it be?

What should it be?

For obvious reasons when a researcher deals with the first two questions, i.e. ‘what is it?’ and ‘how is it?’ the orientation of her research is descriptive and classificatory (see also Bose 1997). As soon as she incorporates the question ‘why is it’, her work becomes more explanatory in its spirit. When a social scientist’s research questions include the fourth and fifth questions as well, it becomes a diagnostic study. There can be a reasonable debate among the positivists regarding the inclusion of the fifth question as they believe that the questions like ‘what should it be?’ involve value judgements. However, you canunderstand that a comprehensive research should be based on all the questions mentioned above. In many cases, the researchers deal with a number of research questions, but do not clearly state which questions are more important, or how the questions are related. Such a multiplicity of questions can lead to the problem of lack of focus (Andrews 2003). The researchers should select the number of research questions for her or his study considering the time-cost-labour components of the work. Time, labour and cost of the study would proportionately be increased with the increase in the number of research questions. Too many research questions are difficult to manage as well.

• 2 Features of Research Questions?

Good research questions that lead to proper research findings have some important features (White 2009; De Vaus 2002; Andrews 2003). The following are the most important among them:

 Research questions should be interrogative – Research questions should be interrogative in nature, it should not be declarative. For example it should be like this: ‘What is the relationship between educational level and attitude towards the freedom of media?’ A statement like: ‘There may be some relationship between educational level and attitude towards the freedom of media’ is not a research question.

 Research questions should be based on the objectives of the study – Research questions should not divert the attention of the researcher from the basic objectives of the study. It should rather try to delve deep into the problem.

 Research questions should be specific – There should not be any ambiguity in the research questions. It should be easily understandable and precise as much as possible.

 It should be simple but well-structured – Much of the success of a research depends on the research questions. It should be focused and precisely framed. The ‘fallacy of many questions’ (i.e. aiming at ‘more than one inquiry in a single question’) should be avoided (White 2009). The questions should be structured in such a manner that they help the researcher unveil a specific dimension of the problem.

Self-check Exercise -1:

• What do you mean by research?

Research is a scientific process of inquiry by which the existing stock of knowledge is either enriched or modified.

• Distinguish between scientific knowledge and common sense.

Scientific knowledge is based on logic and is verifiable. The foundation of scientific knowledge is systematic and critical questioning, observation and reasoning. But common sense is gathered from direct day to day experience. Although common sense is not gathered through scientific inquiry it can be helpful in many research works.

• What are research questions?

Research questions are specific questions emerged out of the broad problems of research, the answers of which a researcher tries to find out. The research questions set the direction of the entire research process.

• What are the features of good research questions?

Research questions should be interrogative. It should not be a statement. A research question should also be specific, understandable and well-structured. Good research questions are based on the objectives of the study.

• Distinguish between descriptive and explanatory research.

Descriptive research tries to describe a phenomenon or a situation or a problem. It generally deals with ‘what’ and ‘how’ questions. Explanatory research, on the other hand, tries to explain the ‘cause-effect’ relationships between different variables. This type of research also involves ‘why’ questions along with the ‘what’ and ‘how’ questions.

• Research questions and hypothesis

Both research questions and hypotheses are useful in social science research. According to White (2009), the difference between them is that while research questions are interrogative in its form, hypotheses are declarative statements which are intended to be tested during the course of research. Hypotheses can be restructured in the form of questions. But then one should not call it hypothesis.

5.1 What is hypothesis?

When a researcher conceptualizes her research problems, she thinks about it in general terms. Research questions or hypotheses help look at the specific aspects of the problem. So hypotheses or research questions enable us to carry out meaningful analysis. Hypotheses are specific statements about the problem made at the initial stage of the research, which may be proved right or wrong also include things such as households, cities, organizations, schools, and nations. If an attribute does not vary, it is a constant” (Bryman 2012: 48).

Once a hypothesis assumes a relationship between two or more variables, the validity of such assumption, made on the basis of the personal experiences, knowledge and insights of the researcher, is tested through suitable statistical techniques. Hence, hypothesis is ‘[a]n informed speculation, which is set up to be tested, about the possible relationship between two or more variables’ Bryman (2012: 712). If the primary assumptions are proved correct after the analysis of data, they become part of the theory. So it is said that ‘hypothesis provides the link between the empirical world and the theory’ (Majumdar 2005: 78). Hypothesis formulation and testing are closely associated with the quantitative approach to study social phenomena (Jupp 2006).

5.2 Features of a good hypothesis

‘A hypothesis is a specified testable expectation about empirical reality that follows from a more general proposition’ (Babbie 2004: 44). It is the assumption made about the relationship between different variables on the basis of existing knowledge or common sense. But all declarative statements or assumptions are not hypotheses. Let us discuss some examples:

• ‘The rate of dropout is higher among the girl students’.
• ‘The rate of dropout varies with gender with the girl students having a higher dropout rate’.

The first assumption is not an example of a good hypothesis as it does not clearly state the two variables. But, the second one is a better one because it clearly mentions gender and rate of dropout as two variables and a relationship between them is assumed.

The features of good hypotheses are as follows:

• Hypothesis generally states (predicts) the relationship between two variables.
• It is expressed as a statement and not as a question (Payne and Payne 2005: 112)
• Hypothesis should be clearly stated, specific and conceptually clear.
• It should be consistent with the known laws of nature (Majumdar 2005)
• Hypothesis is testable (after the final analysis it may prove to be correct or incorrect).

5.3 Soures of hypotheses

Hypotheses are not ordinary or casual statements about the empirical reality. They emerge through a systematic and logical process. According to Goode and Hatt (1981), there are four possible sources from which hypotheses can emerge. These are:

 Culture can furnish hypotheses – Every human society has some distinctive cultural traits. Many social science researches focus on human behaviour or on meaningful social actions. Folkways, mores, values, customs, belief patterns can help formulate hypotheses in these studies. at the end of the analysis (Henn et al 2006).

Hypotheses are formulated at the third stage of the research process (see Diagram 1). According to Goode and Hatt (1981: 56), ‘[a] hypothesis states what we are looking for.’ They write ‘[i]t is a proposition which can be put to a test to determine its validity.’ Hypotheses are primary assumptions about the interrelations of different variables which set the direction of the entire research process. It may be noted that “a variable is simply an attribute on which cases vary. ‘Cases’ can obviously be people, but they can

Hypotheses can emerge from the science itself – In the backdrop of any research there should be one or more theories. Hypotheses are often deducted from a theory to verify or modify some of its basic conclusions. Goode and Hatt ( ibid .) opine that the socialization process, that a student of a particular discipline undergoes, teaches her/him about the promising areas, paradigms, laws, analytical categories, concepts and methods of that particular discipline. This knowledge can help the student to assume some possible causal relationships between some variables that she or he can put to a test for verification.

 Hypotheses can be formulated from analogies – Analogies between human society and nature, between two different types of communities are often a fertile source of hypotheses. For obvious reasons, the researcher should take care in making such analogies. Analogies should not be illogical, it should, on the other hand, be consistent with the known laws of nature.

 Hypotheses can come out from idiosyncratic, personal experiences of the researcher – The scientist lives in a particular culture or she can encounter some cultural traits of some other cultures. Her personal experiences can help her formulate effective hypotheses.

5.4 Types of hypothesis

Hypothesis can be classified in many ways. Goode and Hatt (1981) categorize them into three types on the basis of the level of abstraction.

• pothesis that state the existence of empirical uniformities – Generally these hypotheses are framed when the researchers want to test the ‘common-sense propositions’. In other words, sometimes the researchers are interested to establish the parallels between what people think about a phenomenon and what actually exists. These often lead to the observations of simple differences. In these hypotheses, sometimes, common sense ideas are put into well-defined concepts and then the hypotheses are statistically verified.
• Hypothesis that is concerned with complex ideal types – These hypotheses try to focus on the logically assumed relationships existing among empirical uniformities. In particular, these hypotheses ‘lead to specific coincidences of observations’ ( ibid .: 62). For obvious reasons, these types of hypotheses deal with a higher level of abstraction than the hypotheses that are concerned with the existence of empirical uniformities.
• thesis that is concerned with the relation of analytical variables – According to Goode and Hatt ( ibid .) these hypotheses deal with the highest level of abstraction. In this case, the researcher analytically formulates a hypothesis that shows a relationship between changes in one aspect of the phenomenon with the actual or assumed changes in another aspect.

Majumdar (2005) has categorized hypothesis into two types – eliminative (or analytic) induction and enumerative induction. In the former case hypotheses are formulated as ‘universal generalization’ and the presence of any contrary evidence leads to its rejection. In case of enumerative induction, a complete enumeration is required to accept or reject the hypothesis. Look at the following examples:

Hypothesis I: Female students score better in Research Methodology course than the male students.

This is an example of eliminative or analytic induction. If any male student is found, who has scored more than the female students, there would be no reason to accept the hypothesis.

Hypothesis II: Ten Percent female students score better in Research Methodology course than the male students.

This is an example of enumerative induction. To accept or reject the hypothesis a complete enumeration is necessary.

5.5 Hypothesis Testing

A researcher formulates a number of hypotheses (sometimes called experimental hypotheses) and all these hypotheses are tested on the basis of data collected for the study. When a researcher wants to test the hypothesis with the help of some statistical techniques, he or she frames what is called null hypothesis. According to Babbie (2004: 49), in connection with hypothesis testing and tests of statistical significance, the hypothesis that suggests that there is no relationship among the variables under study is null hypothesis . Sometimes null hypothesis states that there is no difference between two variables.

Null Hypothesis (denoted by H 0 ): There is no difference between the percentage of male students and the percentage of female students who have got 60 per cent marks in Research Methodology course.

If the data collected for the study show, for example, that in reality there are differences between the percentage of male students and percentage of female students who have scored 60 per cent in Research Methodology course, there are statistical techniques to determine whether the difference found is statistically significant, or we can ignore the difference attributing it simply to chance factors and accept the null hypothesis (H0). If the difference obtained from the collected data is statistically significant the researcher rejects the null hypothesis and accepts the alternative hypothesis. For obvious reasons there may be more than one alternative hypotheses (denoted by: H1, H2, H3 etc) the researcher has to select any one from among the alternatives if the null hypothesis (H0) is rejected. The following are the examples:

Alternative Hypothesis (H 1 ): There is significant difference between the percentage of male students and the percentage of female students who have got 60 per cent marks in Research Methodology course.

Or, Alternative Hypothesis (H 2 ): The percentage of male students is higher than the percentage of female students who have got 60 per cent marks in Research Methodology course.   Or,

Alternative Hypothesis (H 3 ): The percentage of female students is higher than the percentage of male students who have got 60 per cent marks in Research Methodology course.

It is not always easy to accept a hypothesis from among the alternatives. The researchers often has to find out what is called crucial instance to take a final decision regarding the acceptance of a hypothesis from among a number of options (alternative hypotheses). Sometimes they have to go through an experiment to decide what actually would be the alternative hypothesis (in the above example whether H2 is correct or H3 is correct. It should be noted that both H2 and H3 cannot be correct at the same time.) The experiment which finally helps to come to a final decision regarding which one should be accepted reasonably from among the hypotheses is called experimentum crucis (Babbie 2004; Majumdar 2005) . There are a number of statistical techniques like Z-test, t-test, χ2-test etc to test the null hypothesis.

Self-check Exercise – 2:

• Distinguish between research question and hypothesis.

Both research questions and hypothesis are framed at the initial stage of the research and both help to look at the research problem in a very specific manner. But while the research question is a specific question the answer of which is sought, hypothesis is a declarative statement framed on the basis of the initial assumptions the validity of which is tested with the help of some statistical techniques. Hypotheses are formulated mainly in case of quantitative research.

• State any two features of a good hypothesis.

A good hypothesis is specific and it indicates the relationship between two variables.

• What is null hypothesis?

In case of hypothesis testing, the hypothesis that states that there is no difference or relationship between two variables under study is called null hypothesis. It is denoted by Ho. The statistical testing of hypothesis starts with null hypothesis. If the test-result tells the researcher to reject the null hypothesis, the researcher accepts alternative hypothesis.

• What are alternative hypotheses?

Alternative hypotheses are formulated against the null hypothesis that states that there is some relationship or difference between the variables. These are denoted by H1, H2 etc.

• Type I error and Type II error

Although in case of quantitative research, the researcher specifies the variables and puts the null hypothesis to test using some statistical techniques, there are dangers of reaching at wrong decisions even if the researcher resort to scientific techniques in the testing of hypotheses. He or she can commit two types of errors – Type I error and Type II error. When the researcher accepts a hypothesis when it is actually incorrect it is called Type I error. In case of Type II error the test-result tells the researcher to reject a hypothesis when it is actually correct. Let us look into the following examples:

Suppose you are interested to know whether there is any relationship between the education of the mother and that of their daughters. Sociologists generally use χ2 test to determine such relationships statistically. The null hypothesis of such a test would be like this:

H 0: There is no relationship between mothers’ education and daughters’ education.

The alternative hypothesis would be:

H 1: There is a definite relationship between mothers’ education and daughters’ education.

Suppose the calculated value of χ2 forces the researcher to accept the null hypothesis and come to the conclusion that there is no relationship between the education of the mother and that of their daughters. But in reality these two are highly related. This is the example of Type I error.

Now, suppose you are interested to know about the relationship between distance of home from the nearest high road and the number of children of the married women.

The null hypothesis of such a test would be like this:

H 0: There is no relationship between distance of home from the high road and the number of children.

H 1: There is significant relationship between distance of home from the high road and the number of children.

Suppose the calculated value of χ2 forces the researcher to reject the null hypothesis and come to the conclusion that there is significant relationship between distance of home from the nearest high road and the number of children married women have. It is not difficult to understand that any such relationship between these two is absurd. This is the example of Type II error.

• Hypothesis and qualitative research

It has been said that hypothesis is generally associated with quantitative research. But it would be wrong to assume that in qualitative studies, they are irrelevant. According to Jupp (2006), some qualitative researches aim at describing the nature, contexts and consequences of social interactions, social relationships and the process of creations of meanings. These studies also start with some assumptions about the social realities which can be treated as hypotheses. Obviously, these hypotheses do not indicate the relationships between variables. Hypothesis testing in ‘qualitative research is a continuous process, involving the search for cases or contexts that do not square with the assertions being made, rather than a once-and-for-all event’ (Jupp 2006: 138). This is the process of analytical induction and when contrary evidences or what is called crucial instances challenge the conclusions of previous study they are modified or rejected. New hypotheses are then framed in the light of new information or experiences and again their validity is checked.

The topic of any research, which the title of the dissertation signifies, indicates the broad area of research. It is often easy to decide about a topic of research. But, a researcher has to be precise in focusing his or her attention while framing his/her research questions. Research questions and hypotheses are framed to specify the areas in which the researcher concentrates. Research questions are interrogative whereas hypotheses are declarative statements. When the researcher finalizes the specific objectives of the study, he or she is better able to frame his research questions or hypotheses. The researcher tries to find out the answers of the research questions framed at the beginning of the study. Hypotheses or the assumptions.

• Andrews, R. Research Questions. London: Continuum, 2003.
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• Bose, P. K. “Problems and Paradoxes of Inductive Social Science: A Critique of Ramkrishna Mukherjee”.
• Sociological Bulletin 46, no. 2 (1997): 153-171.
• Bryman, A, Quantity and Quality in Social Research. London: Routledge, 1988.
• …….. Social Research Methods. Oxford: Oxford University Press, 2012. Das, D. K. L. Practice of Social Research. Jaipur: Rawat Publications, 2004.
• Goode, W. J. and Hatt, P. K. Methods in Social Research. Auckland: McGraw – Hill International Book Company, 1981.
• Henn, M.  et. al. A Short Introduction to Social Research, London: Sage Publications, 2006.
• Jupp, V. The Sage Dictionary of Social Research Methods. London: Sage Publications, 2006.
• Majumdar, P. K. Research Methods in Social Science. New Delhi: Viva Books Pvt. Ltd., 2005.
• Mukherjee, R. Systematic Sociology. New Delhi: Sage Publications, 1993.
• Payne, G. and Payne, J. Key Concepts in Social Research. London: Sage Publications, 2005.
• Singh, Yogendra. ICSSR Research Surveys and Explorations: Indian Sociology. Box Set, Vols 1-3. New Delhi: Oxford, 2013.
• Vaus, De D, Surveys in Social Research. London: Routledge, 2002.
• White, P. Developing Research Questions: A Guide for Social Scientists. New York: Palgrave Macmillan, 2009.

Hypothesis: Functions, Problems, Types, Characteristics, Examples

Basic Elements of the Scientific Method: Hypotheses

The Function of the Hypotheses

A hypothesis states what one is looking for in an experiment. When facts are assembled, ordered, and seen in a relationship, they build up to become a theory. This theory needs to be deduced for further confirmation of the facts, this formulation of the deductions constitutes of a hypothesis. As a theory states a logical relationship between facts and from this, the propositions which are deduced should be true. Hence, these deduced prepositions are called hypotheses.

Problems in Formulating the Hypothesis

There are three major difficulties in the formulation of a hypothesis, they are as follows:

Sometimes the deduction of a hypothesis may be difficult as there would be many variables and the necessity to take them all into consideration becomes a challenge. For instance, observing two cases:

Deduction: This situation holds much more sense to the people who are in professions such as psychotherapy, psychiatry and law to some extent. They possess a very intimate relationship with their clients, thus are more susceptible to issues regarding emotional strains in the client-practitioner relationship and more implicit and explicit controls over both participants in comparison to other professions.

2. Principle: Extensive but relatively systematized data show the correlation between members of the upper occupational class and less unhappiness and worry. Also, they are subjected to more formal controls than members of the lower strata.

Deduction: There can numerous ways to approach this principle, one could go with the comparison applying to martial relationships of the members and further argue that such differential pressures could be observed through divorce rates. This hypothesis would show inverse correlations between class position and divorce rates. There would be a very strong need to define the terms carefully to show the deduction from the principle problem.

Types of Hypothesis

Science and hypothesis.

“The general culture in which a science develops furnishes many of its basic hypotheses” holds true as science has developed more in the West and is no accident that it is a function of culture itself. This is quite evident with the culture of the West as they read for morals, science and happiness. After the examination of a bunch of variables, it is quite easy to say that the cultural emphasis upon happiness has been productive of an almost limitless range.

The hypotheses originate from science; a key example in the form of “socialization” may be taken. The socialization process in learning science involves a feedback mechanism between the scientist and the student. The student learns from the scientist and then tests for results with his own experience, and the scientist in turn has to do the same with his colleagues.

Analogies are a source of useful hypotheses but not without its dangers as all variables may not be accounted for it as no civilization has a perfect system.

Hypotheses are also the consequence of personal, idiosyncratic experience as the manner in which the individual reacts to the hypotheses is also important and should be accounted for in the experiment.

The Characteristics for Usable Hypotheses

The formulation of a hypothesis is probably the most necessary step in good research practice and it is very essential to get the thought process started. It helps the researcher to have a specific goal in mind and deduce the end result of an experiment with ease and efficiency. History is evident that asking the right questions always works out fine.

Also Read: Research Methods – Basics

Kartik is studying BA in International Relations at Amity and Dropped out of engineering from NIT Hamirpur and he lived in over 5 different countries.

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Practice  Questions  – Write short note on Importance and Sources of Hypothesis in Sociological Research. [ UPSC 2008]

Approach –  Introduction, What makes Hypothesis relevant in a sociological research?, What are the sources which aids us to derive hypothesis?, Conclusion

INTRODUCTION

A hypothesis is a prediction of what will be found at the outcome of a research project and is typically focused on the relationship between two different variables studied in the research. It is usually based on both theoretical expectations about how things work and already existing scientific evidence.

We know that research begins with a problem or a felt need or difficulty. The purpose of research is to find a solution to the difficulty. It is desirable that the researcher should propose a set of suggested solutions or explanations of the  difficulty which the research proposes to solve. Such tentative solutions formulated as a proposition are called hypotheses. The suggested solutions formulated as hypotheses may or may not be the real solutions to the problem. Whether they are or not is the task of research to test and establish.

DEFINTITIONS

• Lundberg- A Hypothesis is a tentative generalisation, the validity of which remains to be tested. In its most elementary stages, the hypothesis may be any hunch, guess imaginative idea or Intuition whatsoever which becomes the basis of action or Investigation.
• Bogardus- A Hypothesis is a proposition to be tested.
• Goode and Hatt- It is a proposition which can be put to test to determinants validity.
• P. V. Yaung- The idea of ​a temporary but central importance that becomes the basis of useful research is called a working hypothesis.

TYPES OF HYPOTHESIS

i)  Explanatory Hypothesis : The purpose of this hypothesis is to explain a certain fact. All hypotheses are in a way explanatory for a hypothesis is advanced only when we try to explain the observed fact. A large number of hypotheses are advanced to explain the individual facts in life. A theft, a murder, an accident are examples.

ii) Descriptive Hypothesis:  Some times a researcher comes across a complex phenomenon. He/ she does not understand the relations among the observed facts. But how to account for these facts? The answer is a descriptive hypothesis. A hypothesis is descriptive when it is based upon the points of resemblance of some thing. It describes the cause and effect relationship of a phenomenon e.g., the current unemployment rate of a state exceeds 25% of the work force. Similarly, the consumers of local made products constitute asignificant market segment.

iii) Analogical Hypothesis : When we formulate a hypothesis on the basis of similarities (analogy), it is called an analogical hypothesis e.g., families with higher earnings invest more surplus income on long term investments.

iv) Working hypothesis : Some times certain facts cannot be explained adequately by existing hypotheses, and no new hypothesis comes up. Thus, the investigation is held up. In this situation, a researcher formulates a hypothesis which enables to continue investigation. Such a hypothesis, though inadequate and formulated for the purpose of further investigation only, is called a working hypothesis. It is simply accepted as a starting point in the process of investigation.

v) Null Hypothesis:  It is an important concept that is used widely in the sampling theory. It forms the basis of many tests of significance. Under this type, the hypothesis is stated negatively. It is null because it may be nullified, if the evidence of a random sample is unfavourable to the hypothesis. It is a hypothesis being tested (H0). If the calculated value of the test is less than the permissible value, Null hypothesis is accepted, otherwise it is rejected. The rejection of a null hypothesis implies that the difference could not have arisen due to chance or sampling fluctuations.

USES OF HYPOTHESIS

i) It is a starting point for many a research work. ii) It helps in deciding the direction in which to proceed. iii) It helps in selecting and collecting pertinent facts. iv) It is an aid to explanation. v) It helps in drawing specific conclusions. vi) It helps in testing theories. vii) It works as a basis for future knowledge.

ROLE  OF HYPOTHESIS

In any scientific investigation, the role of hypothesis is indispensable as it always guides and gives direction to scientific research. Research remains unfocused without a hypothesis. Without it, the scientist is not in position to decide as to what to observe and how to observe. He may at best beat around the bush. In the words of Northrop, “The function of hypothesis is to direct our search for order among facts, the suggestions formulated in any hypothesis may be solution to the problem, whether they are, is the task of the enquiry”.

First ,  it is an operating tool of theory. It can be deduced from other hypotheses and theories. If it is correctly drawn and scientifically formulated, it enables the researcher to proceed on correct line of study. Due to this progress, the investigator becomes capable of drawing proper conclusions. In the words of Goode and Hatt, “without hypothesis the research is unfocussed, a random empirical wandering. The results cannot be studied as facts with clear meaning. Hypothesis is a necessary link between theory and investigation which leads to discovery and addition to knowledge.

Secondly,  the hypothesis acts as a pointer to enquiry. Scientific research has to proceed in certain definite lines and through hypothesis the researcher becomes capable of knowing specifically what he has to find out by determining the direction provided by the hypothesis. Hypotheses acts like a pole star or a compass to a sailor with the help of which he is able to head in the proper direction.

Thirdly , the hypothesis enables us to select relevant and pertinent facts and makes our task easier. Once, the direction and points are identified, the researcher is in a position to eliminate the irrelevant facts and concentrate only on the relevant facts. Highlighting the role of hypothesis in providing pertinent facts, P.V. Young has stated, “The use of hypothesis prevents a blind research and indiscriminate gathering of masses of data which may later prove irrelevant to the problem under study”. For example, if the researcher is interested in examining the relationship between broken home and juvenile delinquency, he can easily proceed in the proper direction and collect pertinent information succeeded only when he has succeed in formulating a useful hypothesis.

Fourthly , the hypothesis provides guidance by way of providing the direction, pointing to enquiry, enabling to select pertinent facts and helping to draw specific conclusions. It saves the researcher from the botheration of ‘trial and error’ which causes loss of money, energy and time.

Finally,  the hypothesis plays a significant role in facilitating advancement of knowledge beyond one’s value and opinions. In real terms, the science is incomplete without hypotheses.

STAGES OF HYPOTHESIS TESTING

• EXPERIMENTATION   : Research study focuses its study which is manageable and approachable to it and where it can test its hypothesis. The study gradually becomes more focused on its variables and influences on variables so that hypothesis may be tested. In this process, hypothesis can be disproved.
• REHEARSAL TESTING :   The researcher should conduct a pre testing or rehearsal before going for field work or data collection.
• FIELD RESEARCH :  To test and investigate hypothesis, field work with predetermined research methodology tools is conducted in which interviews, observations with stakeholders, questionnaires, surveys etc are used to follow. The documentation study may also happens at this stage.
• PRIMARY & SECONDARY DATA/INFORMATION ANALYSIS :  The primary or secondary data and information’s available prior to hypothesis testing may be used to ascertain validity of hypothesis itself.

Formulating a hypothesis can take place at the very beginning of a research project, or after a bit of research has already been done. Sometimes a researcher knows right from the start which variables she is interested in studying, and she may already have a hunch about their relationships. Other times, a researcher may have an interest in ​a particular topic, trend, or phenomenon, but he may not know enough about it to identify variables or formulate a hypothesis. Whenever a hypothesis is formulated, the most important thing is to be precise about what one’s variables are, what the nature of the relationship between them might be, and how one can go about conducting a study of them.

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• Technical Report
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• Published: 06 September 2024

Dissociative and prioritized modeling of behaviorally relevant neural dynamics using recurrent neural networks

• Omid G. Sani   ORCID: orcid.org/0000-0003-3032-5669 1 ,
• Bijan Pesaran   ORCID: orcid.org/0000-0003-4116-0038 2 &
• Maryam M. Shanechi   ORCID: orcid.org/0000-0002-0544-7720 1 , 3 , 4 , 5  

Nature Neuroscience ( 2024 ) Cite this article

Metrics details

• Brain–machine interface
• Dynamical systems
• Machine learning
• Neural decoding
• Neural encoding

Understanding the dynamical transformation of neural activity to behavior requires new capabilities to nonlinearly model, dissociate and prioritize behaviorally relevant neural dynamics and test hypotheses about the origin of nonlinearity. We present dissociative prioritized analysis of dynamics (DPAD), a nonlinear dynamical modeling approach that enables these capabilities with a multisection neural network architecture and training approach. Analyzing cortical spiking and local field potential activity across four movement tasks, we demonstrate five use-cases. DPAD enabled more accurate neural–behavioral prediction. It identified nonlinear dynamical transformations of local field potentials that were more behavior predictive than traditional power features. Further, DPAD achieved behavior-predictive nonlinear neural dimensionality reduction. It enabled hypothesis testing regarding nonlinearities in neural–behavioral transformation, revealing that, in our datasets, nonlinearities could largely be isolated to the mapping from latent cortical dynamics to behavior. Finally, DPAD extended across continuous, intermittently sampled and categorical behaviors. DPAD provides a powerful tool for nonlinear dynamical modeling and investigation of neural–behavioral data.

Understanding how neural population dynamics give rise to behavior is a major goal in neuroscience. Many methods that relate neural activity to behavior use static mappings or embeddings, which do not describe the temporal structure in how neural population activity evolves over time 1 . In comparison, dynamical models can describe these temporal structures in terms of low-dimensional latent states embedded in the high-dimensional space of neural recordings. Prior dynamical models have often been linear or generalized linear 1 , 2 , 3 , 4 , 5 , 6 , 7 , thus motivating recent work to develop support for piece-wise linear 8 , locally linear 9 , switching linear 10 , 11 , 12 , 13 or nonlinear 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 models of neural dynamics, especially in applications such as single-trial smoothing of neural population activity 9 , 14 , 15 , 16 , 17 , 18 , 19 and decoding behavior 20 , 21 , 22 , 23 , 24 , 26 . Once trained, the latent states of these models can subsequently be mapped to behavior 1 , 25 to learn an overall dynamical transformation from neural activity to behavior. However, multiple challenges hinder the dynamical modeling and interpretation of neural–behavioral transformations.

First, the neural–behavioral transformation can exhibit nonlinearities, which the dynamical model should capture. Moreover, these nonlinearities can be in one or more different elements within the dynamical model, for example, in the dynamics of the latent state or in its embedding. Enabling hypothesis testing regarding the origin of nonlinearity (that is, where the nonlinearity can be isolated to within the model) is important for interpreting neural computations and developing neurotechnology but remains largely unaddressed in current nonlinear models. Second, neural dynamics related to a given behavior often constitute a minority of the total neural variance 28 , 29 , 30 , 31 , 32 , 33 . To avoid missing or confounding these dynamics, nonlinear dynamical models need to dissociate behaviorally relevant neural dynamics from other neural dynamics and prioritize the learning of the former, which is currently not possible. Indeed, existing nonlinear methods for modeling neural activity either do not explicitly model temporal dynamics 34 , 35 , 36 or do not prioritize behaviorally relevant dynamics 16 , 37 , 38 , or have a mixed objective 18 that may mix behaviorally relevant and other neural dynamics in the same latent states ( Discussion and Extended Data Table 1 ). Our prior method, termed PSID 6 , has enabled prioritized dissociation of behaviorally relevant neural dynamics but for linear dynamical models. Third, for broad applicability, in addition to continuous behaviors, dynamical models should admit categorical (for example, choices) or intermittently sampled behaviors (for example, mood reports), which are not supported by existing dynamical methods with a mixed objective 18 or by PSID. To date, learning nonlinear dynamical models of neural population activity that can address the above challenges has not been achieved.

Here, we develop dissociative prioritized analysis of dynamics (DPAD), a nonlinear dynamical modeling framework using recurrent neural networks (RNNs) that addresses all the above challenges. DPAD models both behaviorally relevant and other neural dynamics but dissociates them into separate latent states and prioritizes the learning of the former. To do so, we formulate a two-section RNN as the DPAD nonlinear dynamical model and develop a four-step optimization algorithm to train it. The first RNN section learns the behaviorally relevant latent states with priority, and the second section learns any remaining neural dynamics (Fig. 1a and Supplementary Fig. 1 ). Moreover, DPAD adjusts these optimization steps as needed to admit continuous-valued, categorical or intermittently sampled data ( Methods ). Furthermore, to capture nonlinearity in the neural–behavioral transformation and enable hypothesis testing regarding its origins, DPAD decomposes this transformation into the following four interpretable elements and allows each element to become linear or nonlinear (Fig. 1a,b ): the mapping from neural activity to the latent space (neural input), the latent state dynamics within this space (recursion) and the mappings of the state to neural activity and behavior (neural and behavior readouts). Finally, we formulate the DPAD model in predictor form such that the learned model can be directly used for inference, enabling causal and computationally efficient decoding for data, whether with or without a fixed-length trial structure ( Methods ).

a , DPAD decomposes the neural–behavioral transformation into four interpretable mapping elements. It learns the mapping of neural activity ( y k ) to latent states ( x k ), termed neural input in the model; learns the dynamics or temporal structure of the latent states, termed recursion in the model; dissociates the behaviorally relevant latent states ( \({x}_{k}^{\left(1\right)}\) ) that are relevant to a measured behavior ( z k ) from other states ( \({x}_{k}^{\left(2\right)}\) ); learns the mapping of the latent states to behavior and to neural activity, termed behavior and neural readouts in the model; and allows flexible linear or nonlinear mappings in any of its elements. DPAD additionally prioritizes the learning of behaviorally relevant neural dynamics to learn them accurately. b , Computation graph of the DPAD model consists of a two-section RNN whose input is neural activity at the current time step and whose outputs are the predicted behavior and neural activity in the next time step ( Methods ). This graph assumes that computations are Markovian, that is, with a high enough dimension, latent states can summarize the information from past neural data that is useful for predicting future neural–behavioral data. Each of the four mapping elements from a has a corresponding parameter in each section of the RNN model, indicated by the same colors and termed as introduced in a . c , We developed a four-step optimization method to learn all the model parameters from training neural–behavioral data (Supplementary Fig. 1a ). Further, each model parameter can be specified via the ‘nonlinearity setting’ to be linear or nonlinear with various options to implement the nonlinearity (Supplementary Fig. 1b,c ). After a model is learned, only past neural activity is used to decode behavior and predict neural activity using the computation graph in b . d , DPAD also has the option of automatically selecting the ‘nonlinearity setting’ for the data by fitting candidate models and comparing them in terms of both behavior decoding and neural self-prediction accuracy ( Methods ). In this work, we chose among 90 candidate models with various nonlinearity settings ( Methods ). We refer to this automatic selection of nonlinearity as ‘DPAD with flexible nonlinearity’.

To show its broad utility, we demonstrate five distinct use-cases for DPAD across four diverse nonhuman primate (NHP) datasets consisting of both population spiking activity and local field potentials (LFPs). First, DPAD more accurately models the overall neural–behavioral data than alternative nonlinear and linear methods. This is due both to DPAD’s prioritized and dynamical modeling of behaviorally relevant neural dynamics and to its nonlinearity. Second, DPAD can automatically uncover nonlinear dynamical transformations of raw LFP that are more predictive of behavior than traditional LFP power band features and in some datasets can even outperform population spiking activity in terms of behavior prediction. Further, DPAD reveals that among the neural modalities, the degree of nonlinearity is greatest for the raw LFP. Third, DPAD enables nonlinear and dynamical neural dimensionality reduction while preserving behavior information, thus extracting lower-dimensional yet more behavior-predictive latent states from past neural activity. Fourth, DPAD enables hypothesis testing regarding the origin of nonlinearity in the neural–behavioral transformation. Consistently across our movement-related datasets, doing so revealed that summarizing the nonlinearities just in the behavior readout from the latent state is largely sufficient for predicting the neural–behavioral data (see Discussion ). Fifth, DPAD extends to categorical and intermittently observed behaviors, which is important for cognitive neuroscience 11 , 39 and neuropsychiatry 40 , 41 , 42 . Together, these results highlight DPAD’s broad utility as a dynamical modeling tool to investigate the nonlinear and dynamical transformation of neural activity to specific behaviors across various domains of neuroscience.

Overview of DPAD

Formulation.

We model neural activity and behavior jointly and nonlinearly ( Methods ) as

where k is the time index, \({y}_{k}\in {{\mathbb{R}}}^{{n}_{y}}\) and \({z}_{k}\in {{\mathbb{R}}}^{{n}_{z}}\) denote the neural activity and behavior time series, respectively, \({x}_{k}\in {{\mathbb{R}}}^{{n}_{x}}\) is the latent state, and e k and \({{\epsilon }}_{k}\) denote neural and behavior dynamics that are unpredictable from past neural activity. Multi-input–multi-output functions A ′ (recursion), K (neural input), C y (neural readout) and C z (behavior readout) are parameters that fully specify the model and have interpretable descriptions ( Methods , Supplementary Note 1 and Fig. 1a,b ). The adjusted formulation for intermittently sampled and noncontinuous-valued (for example, categorical) data is provided in Methods . DPAD supports both linear and nonlinear modeling, which will be termed linear DPAD and nonlinear DPAD (or just DPAD), respectively.

Dissociative and prioritized learning

We further expand the model in Eq. ( 1 ) in two sections, as depicted in Fig. 1b (Eq. ( 2 ) in Methods and Supplementary Note 2 ). The first and second sections describe the behaviorally relevant neural dynamics and the other neural dynamics with latent states \({x}_{k}^{(1)}\in {{\mathbb{R}}}^{{n}_{1}}\) and \({x}_{k}^{(2)}\in {{\mathbb{R}}}^{{n}_{x}-{n}_{1}}\) , respectively. We specify the parameters of the two RNN sections with superscripts (for example, K (1) and K (2) ) and learn them all sequentially via a four-step optimization ( Methods , Supplementary Fig. 1a and Fig. 1b ). The first two steps exclusively learn neural dynamics that are behaviorally relevant with the objective of behavior prediction, whereas the optional last two steps learn any remaining neural dynamics with the objective of residual neural prediction ( Methods and Supplementary Fig. 1 ). We implement DPAD in Tensorflow and use an ADAM 43 optimizer ( Methods ).

Comparison baselines

As a baseline, we compare DPAD with standard nonlinear RNNs fitted to maximize neural prediction, unsupervised with respect to behavior. We refer to this baseline as nonlinear neural dynamical modeling (NDM) 6 or as linear NDM if all RNN parameters are linear. NDM is nondissociative and nonprioritized, so comparisons with NDM show the benefit of DPAD’s prioritized dissociation of behaviorally relevant neural dynamics. We also compare DPAD with latent factor analysis via dynamical systems (LFADS) 16 and with two concurrently 44 developed methods with DPAD named targeted neural dynamical modeling (TNDM) 18 and consistent embeddings of high-dimensional recordings using auxiliary variables (CEBRA) 36 in terms of neural–behavioral prediction; however, as summarized in Extended Data Table 1 , these and other existing methods differ from DPAD in key goals and capabilities and do not enable some of DPAD’s use-cases (see Discussion ).

Decoding using past neural data

Given DPAD’s learned parameters, the latent states can be causally extracted from neural activity by iterating through the RNN in Eq. ( 1 ) ( Methods and Supplementary Note 1 ). Note that this decoding always only uses neural activity without seeing the behavior data.

Flexible control of nonlinearities

We allow each model parameter (for example, C z ) to be an arbitrary multilayer neural network (Supplementary Fig. 1c ), which can universally approximate any smooth nonlinear function or implement linear matrix multiplications ( Methods and Supplementary Fig. 1b ). Users can manually specify which parameters will be learned as nonlinear and with what architecture (Fig. 1c ; see application in use-case 4). Alternatively, DPAD can automatically determine the best nonlinearity setting for the data by conducting a search over nonlinearity options (Fig. 1d and Methods ), a process that we refer to as flexible nonlinearity. For a fair comparison, we also implement this flexible nonlinearity for NDM. To show the benefits of nonlinearity, we also compare with linear DPAD, where all parameters are set to be linear, in which case Eq. ( 1 ) formulates a standard linear state-space model in predictor form ( Methods ).

Evaluation metrics

We evaluate how well the models can use the past neural activity to predict the next sample of behavior (termed ‘decoding’) or the next sample of neural activity itself (termed ‘neural self-prediction’ or simply ‘self-prediction’). Thus, decoding and self-prediction assess the one-step-ahead prediction accuracies and reflect the learning of behaviorally relevant and overall neural dynamics, respectively. Both performance measures are always computed with cross-validation ( Methods ).

Our primary interest is to find models that simultaneously reach both accurate behavior decoding and accurate neural self-prediction. But in some applications, only one of these metrics may be of interest. Thus, we use the term ‘performance frontier’ to refer to the range of performances achievable by those models that compared to every other model are better in neural self-prediction and/or behavior decoding or are similar in terms of both metrics ( Methods ).

Diverse neural–behavioral datasets

We used DPAD to study the behaviorally relevant neural dynamics in four NHPs performing four different tasks (Fig. 2 and Methods ). In the first task, the animal made naturalistic three-dimensional (3D) reach, grasp and return movements to diverse locations while the joint angles in the arm, elbow, wrist and fingers were tracked as the behavior (Fig. 2a ) 6 , 45 . In the second task, the animal made saccadic eye movements to one of eight possible targets on a screen, with the two-dimensional (2D) eye position tracked as the behavior (Fig. 2d ) 6 , 46 . In the third task, the animal made sequential 2D reaches on a screen using a cursor controlled with a manipulandum while the 2D cursor position and velocity were tracked as the behavior (Fig. 2g ) 47 , 48 . In the fourth task, the animal made 2D reaches to random targets in a virtual-reality-presented grid via a cursor that mirrored the animal’s fingertip movements, for which the 2D position and velocity were tracked as the behavior (Fig. 2i ) 49 . In tasks 1 and 4, primary motor cortical activity was modeled. For tasks 2 and 3, prefrontal cortex and dorsal premotor cortical activities were modeled, respectively.

a , The 3D reach task, along with example true and decoded behavior dimensions, decoded from spiking activity using DPAD, with more example trajectories for all modalities shown in Supplementary Fig. 3 . b , Cross-validated decoding accuracy correlation coefficient (CC) achieved by linear and nonlinear DPAD. Results are shown for spiking activity, raw LFP activity and LFP band power activity ( Methods ). For nonlinear DPAD, the nonlinearities are selected automatically based on the training data to maximize behavior decoding accuracy (that is, flexible nonlinearity). The latent state dimension in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak decoding in the training data among all state dimensions ( Methods ). Bars show the mean, whiskers show the s.e.m., and dots show all data points ( N  = 35 session-folds). Asterisks (*) show significance level for a one-sided Wilcoxon signed-rank test (* P  < 0.05, ** P  < 0.005 and *** P  < 0.0005); NS, not significant. c , The difference between the nonlinear and linear results from b shown with the same notations. d – f , Same as a – c for the second dataset with saccadic eye movements ( N  = 35 session-folds). g , h , Same as a and b for the third dataset, which did not include LFP data, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). Behavior consists of the 2D position and velocity of the cursor, denoted as ‘hand kinematics’ in the figure. i – k , Same as a – c for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip movement ( N  = 35 session-folds). For all DPAD variations, only the first two optimization steps were used in this figure (that is, n 1  =  n x ) to only focus on learning behaviorally relevant neural dynamics.

Source data

In all datasets, we modeled the Gaussian smoothed spike counts as the main neural modality ( Methods ). In three datasets that had LFP, we also modeled the following two additional modalities: (1) raw LFP, downsampled to the sampling rate of behavior (that is, 50-ms time steps), which in the motor cortex is known as the local motor potential 50 , 51 , 52 and has been used to decode behavior 6 , 50 , 51 , 52 , 53 ; and (2) LFP power in standard frequency bands from delta (0.1–4 Hz) to high gamma (130–170 Hz (refs. 5 , 6 , 40 ); Methods ). Similar results held for all three modalities.

Numerical simulations validate DPAD

We first validate DPAD with linear simulations here (Extended Data Fig. 1 ) and then present nonlinear simulations under use-case 4 below (Extended Data Fig. 2 and Supplementary Fig. 2 ). We simulated general random linear models (not emulating any real data) in which only a subset of state dimensions contributed to generating behavior and thus were behaviorally relevant ( Methods ). We found that with a state dimension equal to that of the true model, DPAD achieved ideal cross-validated prediction (that is, similar to the true model) for both behavior and neural signals (Extended Data Fig. 1b,d ). Moreover, even given a minimal state dimension equal to the true behaviorally relevant state dimension, DPAD still achieved ideal prediction for behavior (Extended Data Fig. 1c ). Finally, across various regimens of training samples, linear DPAD performed similarly to the linear-algebraic-based PSID 6 from our prior work (Extended Data Fig. 1 ). Thus, hereafter, we use linear DPAD as our linear modeling benchmark.

Use-case 1: DPAD enables nonlinear neural–behavioral modeling across modalities

Dpad captures nonlinearity in behaviorally relevant dynamics.

We modeled each neural modality (spiking, raw LFP or LFP power) along with behavior using linear and nonlinear DPAD and compared their cross-validated behavior decoding (Fig. 2b,e,h,j and Supplementary Fig. 3 ). Across all neural modalities in all datasets, nonlinear DPAD achieved significantly higher decoding accuracy than linear DPAD. This result suggests that there is nonlinearity in the dynamical neural–behavioral transformation, which DPAD successfully captures (Fig. 2b,e,h,j ).

DPAD better predicts the neural–behavioral data

Across all datasets and modalities, compared to nonlinear NDM or linear DPAD, nonlinear DPAD reached higher behavior decoding accuracy while also being as accurate or better in terms of neural self-prediction (Fig. 3 , Extended Data Fig. 3 and Supplementary Fig. 4 ). Indeed, compared to these, DPAD was always on the best performance frontier for predicting the neural–behavioral data (Fig. 3 and Extended Data Fig. 3 ). Additionally, DPAD was always on the best performance frontier for predicting the neural–behavioral data compared to long short-term memory (LSTM) networks as well as a concurrently 44 developed method with DPAD termed CEBRA 36 on our four datasets (Fig. 4a–h ) in addition to a fifth movement dataset 54 analyzed in the CEBRA paper (Fig. 4i,j ). These results suggest that DPAD provides a more accurate description for neural–behavioral data.

a , The 3D reach task. b , Cross-validated neural self-prediction accuracy (CC) achieved by each method shown on the horizontal axis versus the corresponding behavior decoding accuracy on the vertical axis for modeling spiking activity. Latent state dimension for each method in each session, and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). The plus on the plot shows the mean self-prediction and decoding accuracy across sessions and folds ( N  = 35 session-folds), and the horizontal and vertical whiskers show the s.e.m. for these two measures, respectively. Capital letter annotations denote the methods according to the legend to make the plots more accessible. Models whose self-prediction and decoding accuracy measures lead to values toward the top-rightmost corner of the plot lie on the best performance frontier (indicated by red arrows) as they have better performance in both measures and thus better explain the neural–behavioral data ( Methods ). c , d , Same as a and b for the second dataset with saccadic eye movements ( N  = 35 session-folds). e , f , Same as a and b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). g , h , Same as a and b for the fourth dataset with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, the first 16 latent state dimensions are learned using the first two optimization steps, and the remaining dimensions are learned using the last two optimization steps (that is, n 1  = 16). For nonlinear DPAD/NDM, we fit models with different combinations of nonlinearities and then select a final model among these fitted models based on either decoding or self-prediction accuracy in the training data and report both sets of results (Supplementary Fig. 1 and Methods ). DPAD with nonlinearity selected based on neural self-prediction was better than all other methods overall ( b , d , f and h ).

a – h , Figure content is parallel to Fig. 3 (with pluses and whiskers defined in the same way) but instead of NDM shows CEBRA and LSTM networks as baselines ( Methods ). i , j , Here, we also add a fifth dataset 54 ( Methods ), where in each trial an NHP moves a cursor from a center point to one of eight peripheral targets ( i ). In this fifth dataset ( N  = 5 folds), we use the exact CEBRA hyperparameters that were used for this dataset from the paper introducing CEBRA 36 . In the other four datasets ( N  = 35 session-folds in b , d and h and N  = 15 session-folds in f ), we also show CEBRA results for when hyperparameters are picked based on an extensive search ( Methods ). Two types of LSTM networks are shown, one fitted to decode behavior from neural activity and another fitted to predict the next time step of neural activity (self-prediction). We also show the results for DPAD when only using the first two optimization steps. Note that CEBRA-Behavior (denoted by D and F), LSTM for behavior decoding (denoted by H) and DPAD when only using the first two optimization steps (denoted by G) dedicate all their latent states to behavior-related objectives (for example, prediction or contrastive loss), whereas other methods dedicate some or all latent states to neural self-prediction. As in Fig. 3 , the final latent dimension for each method in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). Across all datasets, DPAD outperforms baseline methods in terms of cross-validated neural–behavioral prediction and lies on the best performance frontier. For a summary of the fundamental differences in goals and capabilities of these methods, see Extended Data Table 1 .

Beyond one-step-ahead predictions, we next evaluated DPAD in terms of multistep-ahead prediction of neural–behavioral data, also known as forecasting. To do this, starting with one-step-ahead predictions (that is, m  = 1), we pass m -step-ahead predictions of neural data using the learned models as the neural observation in the next time step to obtain ( m  + 1)-step-ahead predictions ( Methods ). Nonlinear DPAD was consistently better than nonlinear NDM and linear dynamical systems (LDS) modeling in multistep-ahead forecasting of behavior (Extended Data Fig. 4 ). For neural self-prediction, we used a naive predictor as a conservative forecasting baseline, which reflects how easy it is to predict the future in a model-free way purely based on the smoothness of neural data. DPAD significantly outperformed this baseline in terms of one-step-ahead and multistep-ahead neural self-predictions (Supplementary Fig. 5 ).

Use-case 2: DPAD extracts behavior-predictive nonlinear transformations from raw LFP

We next used DPAD to compare the amount of nonlinearity in the neural–behavioral transformation across different neural modalities (Fig. 2 and Supplementary Fig. 3 ). To do so, we compared the gain in behavior decoding accuracy when going from linear to nonlinear DPAD modeling in each modality. In all datasets, raw LFP activity had the highest gain from nonlinearity in behavior decoding accuracy (Fig. 2c,f,k ). Notably, using nonlinear DPAD, raw LFP reached more accurate behavior decoding than traditional LFP band powers in all tasks (Fig. 2b,e,j ). In one dataset, raw LFP even significantly surpassed spiking activity in terms of behavior decoding accuracy (Fig. 2e ). Note that computing LFP powers involves a prespecified nonreversible nonlinear transformation of raw LFP, which may be discarding important behaviorally relevant information that DPAD can uncover directly from raw LFP. Interestingly, linear dynamical modeling did worse for raw LFP than LFP powers in most tasks (compare linear DPAD for raw LFP versus LFP powers), suggesting that nonlinearity, captured by DPAD, was required for uncovering the extra behaviorally relevant information in raw LFP.

We next examined the spatial pattern of behaviorally relevant information across recording channels. For different channels, we compared the neural self-prediction of DPAD’s low-dimensional behaviorally relevant latent states (Extended Data Fig. 5 ). We computed the coefficient of variation (defined as standard deviation divided by mean) of the self-prediction over recording channels and found that the spatial distribution of behaviorally relevant information was less variable in raw LFP than spiking activity ( P  ≤ 0.00071, one-sided signed-rank test, N  = 35 for all three datasets with LFP). This could suggest that raw LFPs reflect large-scale network-level behaviorally relevant computations, which are thus less variable within the same spatial brain area than spiking, which represents local, smaller-scale computations 55 .

Use-case 3: DPAD enables behavior-predictive nonlinear dynamical dimensionality reduction

We next found that DPAD extracted latent states that were lower dimensional yet more behavior predictive than both nonlinear NDM and linear DPAD (Fig. 5 ). Specifically, we inspected the dimension required for nonlinear DPAD to reach almost (within 5% of) peak behavior decoding accuracy in each dataset (Fig. 5b,g,l,o ). At this low latent state dimension, linear DPAD and nonlinear and linear NDM all achieved much lower behavior decoding accuracy than nonlinear DPAD across all neural modalities (Fig. 5c–e,h–j,m,p–r ). The lower decoding accuracy of nonlinear NDM suggests that the dominant dynamics in spiking and LFP modalities can be unrelated to the modeled behavior. Thus, behaviorally relevant dynamics can be missed or confounded unless they are prioritized during nonlinear learning, as is done by DPAD. Moreover, we visualized the 2D latent state trajectories learned by each method (Extended Data Fig. 6 ). Consistent with the above results, DPAD extracted latent states from neural activity that were clearly different for different behavior/movement conditions (Extended Data Fig. 6b,e,h,k ). In comparison, NDM extracted latent states that did not as clearly dissociate different conditions (Extended Data Fig. 6c,f,i,l ). These results highlight the capability of DPAD for nonlinear dynamical dimensionality reduction in neural data while preserving behaviorally relevant neural dynamics.

a , The 3D reach task. b , Cross-validated decoding accuracy (CC) achieved by variations of linear/nonlinear DPAD/NDM for different latent state dimensions. For nonlinear DPAD/NDM, the nonlinearities are selected automatically based on the training data to maximize behavior decoding accuracy (flexible nonlinearity). Solid lines show the average across sessions and folds ( N  = 35 session-folds), and the shaded areas show the s.e.m.; Low-dim., low-dimensional. c , Decoding accuracy of nonlinear DPAD versus linear DPAD and nonlinear/linear NDM at the latent state dimension for which DPAD reaches within 5% of its peak decoding accuracy in the training data across all latent state dimensions. Bars, whiskers, dots and asterisks are defined as in Fig. 2b ( N  = 35 session-folds). d , Same as c for modeling of raw LFP ( N  = 35 session-folds). e , Same as c for modeling of LFP band power activity ( N  = 35 session-folds). f – j , Same as a – e for the second dataset with saccadic eye movements ( N  = 35 session-folds). k – m , Same as a – c for the third dataset, which did not include LFP data, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). n – r , Same as a – e for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, only the first two optimization steps were used in this figure (that is, n 1  =  n x ) to only focus on learning behaviorally relevant neural dynamics in the dimensionality reduction regimen.

Next, we found that at low dimensions, nonlinearity could improve the accuracy of both behavior decoding (Fig. 5b,g,l,o ) and neural self-prediction (Extended Data Fig. 7 ). However, as the state dimension was increased, linear methods reached similar neural self-prediction performance as nonlinear methods across modalities (Fig. 3 and Extended Data Fig. 3 ). This was in contrast to behavior decoding, which benefited from nonlinearity regardless of how high the dimension was (Figs. 2 and 3 ).

Use-case 4: DPAD localizes the nonlinearity in the neural–behavioral transformation

Numerical simulations validate dpad’s localization.

To demonstrate that DPAD can correctly find the origin of nonlinearity in the neural–behavioral transformation (Extended Data Fig. 2 and Supplementary Fig. 2 ), we simulated random models where only one of the parameters was set to a random nonlinear function ( Methods ). DPAD identifies a parameter as the origin if models with nonlinearity only in that parameter are on the best performance frontier when compared to alternative models, that is, models with nonlinearity in other parameters, models with flexible/full nonlinearity and fully linear models (Fig. 6a ). DPAD enables this assessment due to (1) its flexible control over nonlinearities to train alternative models and (2) its simultaneous neural–behavioral modeling and evaluation ( Methods ). In all simulations, DPAD identified that the model with the correct nonlinearity origin was on the best performance frontier compared to alternative nonlinear models (Extended Data Fig. 2 and Supplementary Fig. 2 ), thus correctly revealing the origin of nonlinearity.

a , The process of determining the origin of nonlinearity via hypothesis testing shown with an example simulation. Simulation results are taken from Extended Data Fig. 2b , and the origin is correctly identified as K . Pluses and whiskers are defined as in Fig. 3 ( N  = 20 random models). b , The 3D reach task. c , DPAD’s hypothesis testing. Cross-validated neural self-prediction accuracy (CC) for each nonlinearity and the corresponding decoding accuracy. DPAD variations that have only one nonlinear parameter (for example, C z ) use a nonlinear neural network for that parameter and keep all other parameters linear. Linear and flexible nonlinear results are as in Fig. 3 . Latent state dimension in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). Pluses and whiskers are defined as in Fig. 3 ( N  = 35 session-folds). Annotated arrows indicate any individual nonlinearities that are on the best performance frontier compared to all other models. Results are shown for spiking activity here and for raw LFP and LFP power activity in Supplementary Fig. 6 . d , e , Same as b and c for the second dataset with saccadic eye movements ( N  = 35 session-folds). f , g , Same as b and c for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). h , i , Same as b and c for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, the first 16 latent state dimensions are learned using the first two optimization steps, and the remaining dimensions are learned using the last two optimization steps (that is, n 1  = 16).

DPAD consistently localized nonlinearities in the behavior readout

Having validated the localization of nonlinearity in simulations, we used DPAD to find where in the model nonlinearities could be isolated to in our real datasets. We found that having the nonlinearity only in the behavior readout parameter C z was largely sufficient for achieving high behavior decoding and neural self-prediction accuracies across all our datasets and modalities (Fig. 6b–i and Supplementary Fig. 6 ). First, for spiking activity, models with nonlinearity only in the behavior readout parameter C z reached the best behavior decoding accuracy compared to models with other individual nonlinearities (Fig. 6c,e,i ) while reaching almost the same decoding accuracy as fully nonlinear models (Fig. 6c,e,g,i ). Second, these models with nonlinearity only in the behavior readout also reached a self-prediction accuracy that was unmatched by other types of individual nonlinearity (Fig. 6c,e,g,i ). Overall, this meant that models with nonlinearity only in the behavior readout parameter C z were always on the best performance frontier when compared to all other linear or nonlinear models (Fig. 6c,e,g,i ). This result interestingly also held for both LFP modalities (Supplementary Fig. 6 ).

Consistent with the above localization results, DPAD with flexible nonlinearity also, very frequently, automatically selected models with nonlinearity in the behavior readout parameter (Supplementary Fig. 7 ). However, critically, this observation on its own cannot conclude that nonlinearities can be isolated in the behavior readout parameter. This is because in the flexible nonlinearity approach, parameters may be selected as nonlinear as long as this nonlinearity does not hurt the prediction accuracies, which does not imply that such nonlinearities are necessary ( Methods ); this is why we need the hypothesis testing procedure above (Fig. 6a ). Of note, using an LSTM for the recursion parameter A ′ is one of the nonlinearity options that is automatically considered in DPAD (Extended Data Fig. 3 ), but we found that LSTM was rarely selected in our datasets as the recursion dynamics in the flexible search over nonlinearities (Supplementary Fig. 7 ). Finally, note that fitting models with a nonlinear behavior readout via a post hoc nonlinear refitting of linear DPAD models (1) cannot identify the origin of nonlinearity in general (for example, other brain regions or tasks) and (2) even in our datasets resulted in significantly worse decoding than the same models being fitted end-to-end as done by nonlinear DPAD ( P  ≤ 0.0027, one-sided signed-rank test, N  ≥ 15).

Together, these results highlight the application of DPAD in enabling investigations of nonlinear processing in neural computations underlying specific behaviors. DPAD’s machinery can not only fit fully nonlinear models but also provide evidence for the location in the model where the nonlinearity can be isolated ( Discussion ).

Use-case 5: DPAD extends to noncontinuous and intermittent data

Dpad extends to intermittently sampled behavior observations.

DPAD also supports intermittently sampled behaviors ( Methods ) 56 , that is, when behavior is measured only during a subset of time steps. We first confirmed in numerical simulations with random models that DPAD correctly learns the model with intermittently sampled behavioral data (Supplementary Fig. 8 ). Next, in each of our neural datasets, we emulated intermittent sampling by randomly discarding up to 90% of behavior samples during learning. DPAD learned accurate nonlinear models even in this case (Extended Data Fig. 8 ). This capability is important, for example, in affective neuroscience or neuropsychiatry applications where the behavior consists of sparsely sampled momentary ecological assessments of mental states such as mood 40 . We next simulated a mood decoding application and found that with as low as one behavioral (for example, mood survey) sample per day, DPAD still outperformed NDM even when NDM had access to continuous behavior samples (Extended Data Fig. 9 ). These results suggest the potential utility of DPAD in such applications, although substantial future validation in data is needed 7 , 40 , 41 , 42 .

DPAD extends to noncontinuous-valued observations

DPAD also extends to modeling of noncontinuous-valued (for example, categorical) behaviors ( Methods ). To demonstrate this, we modeled the transformation from neural activity to the momentary phase of the task in the 3D reach task: reach, hold, return or rest (Fig. 7 ). Compared to nonlinear NDM (which is dynamic) or nonlinear nondynamic methods such as support vector machines, DPAD more accurately predicted the task phase at each point in time (Fig. 7 ). This capability can extend the utility of DPAD to categorical behaviors such as decision choices in cognitive neuroscience 39 .

a , In the 3D reach dataset, we model spiking activity along with the epoch of the task as discrete behavioral data ( Methods and Fig. 2a ). The epochs/classes are (1) reaching toward the target, (2) holding the target, (3) returning to resting position and (4) resting until the next reach. b , DPAD’s predicted probability for each class is shown in a continuous segment of the test data. Most of the time, DPAD predicts the highest probability for the correct class. c , The cross-validated behavior classification performance, quantified as the area under curve (AUC) for the four-class classification, is shown for different methods at different latent state dimensions. Solid lines and shaded areas are defined as in Fig. 5b ( N  = 35 session-folds). AUC of 1 and 0.5 indicate perfect and chance-level classification, respectively. We include three nondynamic/static classification methods that map neural activity for a given time step to class label at the same time step (Extended Data Table 1 ): (1) multilayer neural network, (2) nonlinear support vector machine (SVM) and (3) linear discriminant analysis (LDA). d , Cross-validated behavior classification performance (AUC) achieved by each method when choosing the state dimension in each session and fold as the smallest that reaches peak classification performance in the training data among all state dimensions with that method ( Methods ). Bars, whiskers, dots and asterisks are defined as in Fig. 2b ( N  = 35 session-folds). e , Same as d when all methods use the same latent state dimension as DPAD (best nonlinearity for decoding) does in d ( N  = 35 session-folds). c and e show DPAD’s benefit for dimensionality reduction. f , Cross-validated neural self-prediction accuracy achieved by each method versus the corresponding behavior classification performance. Here, the latent state dimension for each method in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). Pluses and whiskers are defined as in Fig. 3 ( N  = 35 session-folds).

Finally, we applied DPAD to nonsmoothed spike counts, where we compared the results with two noncausal sequential autoencoder methods, termed LFADS 16 and TNDM 18 (Supplementary Fig. 9 ), both of which have Poisson observations that model nonsmoothed spike counts 16 , 18 . TNDM 18 , which was developed after LFADS 16 and concurrently with our work 44 , 56 , adds behavioral terms to the objective function for a subset of latents but unlike DPAD does so with a mixed objective and thus does not completely dissociate or prioritize behaviorally relevant dynamics (Extended Data Table 1 and Supplementary Note 3 ). Compared to both LFADS and TNDM, DPAD remained on the best performance frontier for predicting the neural–behavioral data (Supplementary Fig. 9a ) and more accurately predicted behavior using low-dimensional latent states (Supplementary Fig. 9b ). Beyond this, TNDM and LFADS also have fundamental differences with DPAD and do not address some of DPAD’s use-cases ( Discussion and Extended Data Table 1 ).

We developed DPAD for nonlinear dynamical modeling and investigation of neural dynamics underlying behavior. DPAD can dissociate the behaviorally relevant neural dynamics and prioritize their learning over other neural dynamics, enable hypothesis testing regarding the origin of nonlinearity in the neural–behavioral transformation and achieve causal decoding. DPAD enables prioritized dynamical dimensionality reduction by extracting lower-dimensional yet more behavior-predictive latent states from neural population activity and supports modeling noncontinuous-valued (for example, categorical) and intermittently sampled behavioral data. These attributes make DPAD suitable for diverse use-cases across neuroscience and neurotechnology, some of which we demonstrated here.

We found similar results for three neural modalities: spiking activity, LFP band powers and raw LFP. For all modalities, nonlinear DPAD more accurately learned the behaviorally relevant neural dynamics than linear DPAD and linear/nonlinear NDM as reflected in its better decoding while also reaching the best performance frontier when considering both behavior decoding and neural self-prediction. Notably, the raw LFP activity benefited the most from nonlinear modeling using DPAD and outperformed LFP powers in all tasks in terms of decoding. This suggests that automatic learning of nonlinear models from raw LFP using DPAD reveals behaviorally relevant information that may be discarded when extracting traditionally used features such as LFP band powers. Also, nonlinearity was necessary to recover the extra information in raw LFP, as, unlike DPAD modeling, linear dynamical modeling of raw LFP did not outperform that of LFP powers in most datasets. These results highlight another use-case of DPAD for automatic dynamic feature extraction from LFP data.

As another use-case, DPAD enabled an investigation of which element in the neural–behavioral transformation was nonlinear. Interestingly, consistently across our four movement-related datasets, DPAD models with nonlinearity only in the behavior readout performed similarly to fully nonlinear models, reaching the best performance frontier for predicting future behavior and neural data using past neural data. The consistency of this result across our datasets is interesting because, as demonstrated in simulations (Extended Data Fig. 2 , Supplementary Fig. 2 and Fig. 6a ), the detected origin of nonlinearity could have technically been in any one (or more) of the following four elements (Fig. 1a,b ): neural input, recurrent dynamics and neural or behavior readouts, all of which were correctly localized in simulations (Extended Data Fig. 2 and Supplementary Fig. 2 ). Thus, the consistent localization results on our neural datasets provide evidence that across these four tasks, neural dynamics in these recorded cortical areas may be largely describable with linear dynamics of sufficiently high dimension, with additional nonlinearities introduced somewhere between the neural state and behavior. This finding may be consistent with (1) introduction of nonlinear processing along the downstream neuromuscular pathway that goes from the recorded cortical area to the measured behavior or any of the convergent inputs along this pathway 57 , 58 , 59 or (2) cognition intervening nonlinearly between these latent neural states and behavior, for example, by implementing context-dependent computations 60 . This result illustrates how DPAD can provide new hypotheses and the machinery to test them in future experiments that would record from multiple additional brain regions (for example, both motor and cognitive regions) and use DPAD to model them together. Such analyses may narrow down or revise the origin of nonlinearity for the wider neural–behavioral measurement set; for example, the state dynamics may be found to be nonlinear once additional brain regions are added. Localization of nonlinearity could also guide the design of competitive deep learning architectures that are more flexible or easier to implement in neurotechnologies such as brain–computer interfaces 61 .

Interestingly, the behavior decoding aspect of the localization finding here is consistent with a prior study 22 that explored the mapping of the motor cortex to an electromyogram (EMG) during a one-dimensional movement task with varying forces and found that a fully linear model was worse than a nonlinear EMG readout in decoding the EMG 22 . However, as our simulations show (Extended Data Fig. 2b and Fig. 6a ), comparing a linear model to a model that has nonlinear behavior readout is not sufficient to conclude the origin of nonlinearity, and a stronger test is needed (see Fig. 6a for a counter example and details in Methods ). Further, this previous study 22 used a specific condition-dependent nonlinearity for behavior readout rather than a universal nonlinear function approximator that DPAD enables. Finally, to conclude localization, the model with that specific nonlinearity should perform similarly to fully nonlinear models; however, unlike our results, a fully nonlinear LSTM model in some cases appears to outperform models with nonlinear readout in this prior study (see Fig. 7a,b in ref. 22 versus Fig. 9c in ref. 22 ); it is unclear if this result is due to this prior study’s specific readout nonlinearity being suboptimal or to the nonlinear origin being different in its dataset 22 . DPAD can address such questions by (1) allowing for training and comparison of alternative models with different nonlinear origins and (2) enabling a general (versus specific) nonlinearity in model parameters.

When hypothesis testing about where in the model nonlinearity can be isolated to, it may be possible to equivalently explain the same data with multiple types of nonlinearities (for example, with either a nonlinear neural input or a nonlinear readout). Such nonidentifiability is a common limitation for latent models. However, when such equivalence exists, we expect all equivalent nonlinear models to have similar performance and thus lie on the best performance frontier. But this was not the case in our datasets. Instead, we found that the nonlinear behavior readout was in most cases the only individual nonlinear parameter on the best performance frontier, providing evidence that no other individual nonlinear parameter was as suitable in our datasets. Alternatively, the best model describing the data may require two or more of the four parameters to be nonlinear. But in our datasets, models with nonlinearity only in the behavior readout were always on the best performance frontier and could not be considerably outperformed by models with more than one nonlinearity (Fig. 6 ). Nevertheless, we note that ultimately our analysis simply provides evidence for one location of nonlinearity resulting in a better fit to data with a parsimonious model, but it does not rule out other possibilities for explaining the data. For example, one could reformulate a nonlinear readout model by adding latent states and representing the readout nonlinearity as a recursion nonlinearity for the additional states, although such an equivalent but less parsimonious model may need more data to be learned as accurately. Finally, we also note that our conclusions were based on the datasets and family of nonlinear models (recursive RNNs) considered here, and thus we cannot rule out different conclusions in other scenarios and/or brain regions. Nevertheless, by providing evidence for a nonlinearity configuration, DPAD can provide testable hypotheses for future experiments that record from more brain regions.

Sequential autoencoders, spearheaded by LFADS 16 , have been used to smooth single-trial neural activity 16 without considering relevance to behavior, which is a distinct goal as we showed in comparison to PSID in our prior work 6 . Notably, another sequential autoencoder, termed TNDM, has been developed concurrently with our work 44 , 56 that adds a behavior term to the optimization objective 18 . However, these approaches do not enable several of the use-cases of DPAD here. First, unlike DPAD’s four-step learning approach, TNDM and LFADS use a single learning step with a neural-only objective (LFADS) 16 or a mixed neural–behavioral objective (TNDM) 18 that does not fully prioritize the behaviorally relevant neural dynamics (Extended Data Table 1 and Supplementary Note 3 ). DPAD’s prioritization is important for accurate learning of behaviorally relevant neural dynamics and for preserving them in dimensionality reduction, as our results comparing DPAD to TNDM/LFADS suggest (Supplementary Fig. 9 ). Second, TNDM and LFADS 16 , 18 , like other prior works 16 , 18 , 20 , 23 , 24 , 26 , 61 , do not provide flexible nonlinearity or explore hypotheses regarding the origin of nonlinearities because they use fixed nonlinear network structures (use-case 4). Third, TNDM considers spiking activity and continuous behaviors 18 , whereas DPAD extends across diverse neural and behavioral modalities: spiking, raw LFP and LFP powers and continuous, categorical or intermittent behavioral modalities. Fourth, in contrast to these noncausal sequential autoencoders 16 , 18 and some other nonlinear methods 8 , 14 , DPAD can process the test data causally and without expensive computations such as iterative expectation maximization 8 , 14 or sampling and averaging 16 , 18 . This causal efficient processing is also important for real-time closed-loop brain–computer interfaces 62 , 63 . Of note, noncausal processing is also implemented in the DPAD code library as an option ( Methods ), although it is not shown in this work. Finally, unlike these prior methods 14 , 16 , 18 , DPAD does not require fixed-length trials or trial structure, making it suitable for modeling naturalistic behaviors 5 and neural dynamics with trial-to-trial variability in the alignment to task events 64 .

Several methods can in some ways prioritize behaviorally relevant information while extracting latent embeddings from neural data but are distinct from DPAD in terms of goals and capabilities. One group includes nondynamic/static methods that do not explicitly model temporal dynamics 1 . These methods build linear maps (for example, as in demixed principal component analysis (dPCA) 34 ) or nonlinear maps, such as convolutional maps in a concurrently 44 developed method with DPAD named CEBRA 36 , to extract latent embeddings that can be guided by behavior either as a trial condition 34 or indirectly as a contrastive loss 36 . These nondynamic mappings only use a single sample or a small fixed window around each sample of neural data to extract latent embeddings (Extended Data Table 1 ). By contrast, DPAD can recursively aggregate information from all past neural data by explicitly learning a model of temporal dynamics (recursion), which also enables forecasting unlike in static/nondynamic methods. These differences may be one reason why DPAD outperformed CEBRA in terms of neural–behavioral prediction (Fig. 4 ). Another approach is used by task aligned manifold estimation (TAME-GP) 9 , which uses a Gaussian process prior (as in Gaussian process factor analysis (GPFA) 14 ) to expand the window of neural activity used for extracting the embedding into a complete trial. Unlike DPAD, methods with a Gaussian process prior have limited support for nonlinearity, often do not have closed-forms for inference and thus necessitate numerical optimization even for inference 9 and often operate noncausally 9 . Finally, the above methods do not provide flexible nonlinearity or hypothesis testing to localize the nonlinearity.

Other prior works have used RNNs either causally 20 , 22 , 23 , 24 , 26 or noncausally 16 , 18 , for example, for causal decoding of behavior from neural activity 20 , 22 , 23 , 24 , 26 . These works 20 , 22 , 23 , 24 , 26 have similarities to the first step of DPAD’s four-step optimization (Supplementary Fig. 1a ) in that the RNNs in these works learn dynamical models by solely optimizing behavior prediction. However, these works do not learn the mapping from the RNN latent states to neural activity, which is done in DPAD’s second optimization step to enable neural self-prediction (Supplementary Fig. 1a ). In addition, unlike what the last two optimization steps in DPAD enable, these prior works do not model additional neural dynamics beyond those that decode behavior and thus do not dissociate the two types of neural dynamics (Extended Data Table 1 ). Finally, as noted earlier, these prior works 9 , 20 , 23 , 24 , 26 , 36 , 61 , similar to prior sequential autoencoders 16 , 18 , have fixed nonlinear network structures and thus cannot explore hypotheses regarding the origin of nonlinearities or flexibly learn the best nonlinear structure for the training data (Fig. 1c,d and Extended Data Table 1 ).

DPAD’s optimization objective functions are not convex, similar to most nonlinear deep learning methods. Thus, as usual with nonconvex optimizations, convergence to a global optimum is not guaranteed. Moreover, as with any method, quality and neural–behavioral prediction of the learned models depend on dataset properties such as signal-to-noise ratio. Thus, we compare alternative methods within each dataset, suggesting that (for example, Fig. 4 ) across the multiple datasets here, DPAD learns more accurate models of neural–behavioral data. However, models in other datasets/scenarios may not be as accurate.

Here, we focused on using DPAD to model the transformation of neural activity to behavior. DPAD can also be used to study the transformation between other signals. For example, when modeling data from multiple brain regions, one region can be taken as the primary signal ( y k ) and another as the secondary signal ( z k ) to dissociate their shared versus distinct dynamics. Alternatively, when modeling the brain response to electrical 7 , 41 , 42 or sensory 41 , 65 , 66 stimulation, one could take the primary signal ( y k ) to be the stimulation and the secondary signal ( z k ) to be neural activity to dissociate and predict neural dynamics that are driven by stimulation. Finally, one may apply DPAD to simultaneously recorded brain activity from two subjects as primary and secondary signals to find shared intersubject dynamics during social interactions.

Model formulation

Equation ( 1 ) simplifies the DPAD model by showing both of its RNN sections as one, but the general two-section form of the model is as follows:

This equation separates the latent states of Eq. ( 1 ) into the following two parts: \({x}_{k}^{\left(1\right)}\in {{\mathbb{R}}}^{{n}_{1}}\) denotes the latent states of the first RNN section that summarize the behaviorally relevant dynamics, and \({x}_{k}^{\left(2\right)}\in {{\mathbb{R}}}^{{n}_{2}}\) , with \({n}_{2}={n}_{x}-{n}_{1}\) , denotes those of the second RNN section that represent the other neural dynamics (Supplementary Fig. 1a ). Here, A ′(1) , A ′(2) , K (1) , K (2) , \({C}_{y}^{\,\left(1\right)}\) , \({C}_{y}^{\,\left(2\right)}\) , \({C}_{z}^{\,\left(1\right)}\) and \({C}_{z}^{\,\left(2\right)}\) are multi-input–multi-output functions that parameterize the model, which we learn using a four-step numerical optimization formulation expanded on in the next section (Supplementary Fig. 1a ). DPAD also supports learning the initial value of the latent states at time 0 (that is, \({x}_{0}^{\left(1\right)}\) and \({x}_{0}^{\left(2\right)}\) ) as a parameter, but in all analyses in this paper, the initial states are simply set to 0 given their minimal impact when modeling long data sequences. Each pair of superscripted parameters (for example, A ′(1) and A ′(2) ) in Eq. ( 2 ) is a dissociated version of the corresponding nonsuperscripted parameter in Eq. ( 1 ) (for example, A ′). The computation graph for Eq. ( 2 ) is provided in Fig. 1b (and Supplementary Fig. 1a ). In Eq. ( 2 ), the recursions for computing \({x}_{k}^{\left(1\right)}\) are not dependent on \({x}_{k}^{\left(2\right)}\) , thus allowing the former to be computed without the latter. By contrast, \({x}_{k}^{\left(2\right)}\) can depend on \({x}_{k}^{\left(1\right)}\) , and this dependence is modeled via K (2) (see Supplementary Note 2 ). Note that such dependence of \({x}_{k}^{\left(2\right)}\) on \({x}_{k}^{\left(1\right)}\) via K (2) does not introduce new dynamics to \({x}_{k}^{\left(2\right)}\) because it does not involve the recursion parameter A ′(2) , which describes the dynamics of \({x}_{k}^{\left(2\right)}\) . This two-section RNN formulation is mathematically motivated by equivalent representations of a dynamical system model in different bases and by the relation between the predictor and stochastic forms of dynamical systems (Supplementary Notes 1 and 2 ).

For the RNN formulated in Eq. ( 1 ) or ( 2 ), neural activity y k constitutes the input, and predictions of neural and behavioral signals are the outputs (Fig. 1b ) given by

Note that each x k is estimated purely using all past y k (that is, y 1 , …, y k   –  1 ), so the predictions in Eq. ( 3 ) are one-step-ahead predictions of y k and z k using past neural observations (Supplementary Note 1 ). Once the model parameters are learned, the extraction of latent states x k involves iteratively applying the first line from Eq. ( 2 ), and predicting behavior or neural activity involves applying Eq. ( 3 ) to the extracted x k . As such, by writing the nonlinear model in predictor form 67 , 68 (Supplementary Note 1 ), we enable causal and computationally efficient prediction.

Learning: four-step numerical optimization approach

Unlike nondynamic models 1 , 34 , 35 , 36 , 69 , dynamical models explicitly model temporal evolution in time series data. Recent dynamical models have gone beyond linear or generalized linear dynamical models 2 , 3 , 4 , 5 , 6 , 7 , 70 , 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 , 79 , 80 , 81 to incorporate switching linear 10 , 11 , 12 , 13 , locally linear 37 or nonlinear 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 23 , 24 , 26 , 27 , 38 , 61 , 82 , 83 , 84 , 85 , 86 , 87 , 88 , 89 , 90 dynamics, often using deep learning methods 25 , 91 , 92 , 93 , 94 . But these recent nonlinear/switching works do not aim to localize nonlinearity or allow for flexible nonlinearity and do not enable fully prioritized dissociation of behaviorally relevant neural dynamics because they either do not consider behavior in their learning objective at all 14 , 16 , 37 , 38 , 61 , 95 , 96 or incorporate it with a mixed neural–behavioral objective 9 , 18 , 35 , 61 (Extended Data Table 1 ).

In DPAD, we develop a four-step learning method for training our two-section RNN in Eq. ( 1 ) and extracting the latent states that (1) enables dissociation and prioritized learning of the behaviorally relevant neural dynamics in the nonlinear model, (2) allows for flexible modeling and localization of nonlinearities, (3) extends to data with diverse distributions and (4) does all this while also achieving causal decoding and being applicable to data both with and without a trial structure. DPAD is for nonlinear modeling, and its multistep learning approach, in each step, uses numerical optimization tools that are rooted in deep learning. Thus, DPAD is mathematically distinct from our prior PSID work for linear models, which is an analytical and linear technique. PSID is based on analytical linear algebraic projections rooted in control theory 6 , which are thus not extendable to nonlinear modeling or to non-Gaussian, noncontinuous or intermittently sampled data. Thus, even when we restrict DPAD to linear modeling as a special case, it is still mathematically different from PSID 6 .

To dissociate and prioritize the behaviorally relevant neural dynamics, we devise a four-step optimization approach for learning the two-section RNN model parameters (Supplementary Fig. 1a ). This approach prioritizes the extraction and learning of the behaviorally relevant dynamics in the first two steps with states \({x}_{k}^{\left(1\right)}\in {{\mathbb{R}}}^{{n}_{1}}\) while also learning the rest of the neural dynamics in the last two steps with states \({x}_{k}^{\left(2\right)}\in {{\mathbb{R}}}^{{n}_{2}}\) and dissociating the two subtypes of dynamics. This prioritization is important for accurate learning of behaviorally relevant neural dynamics and is achieved because of the multistep learning approach; the earlier steps learn the behaviorally relevant dynamics first, that is, with priority, and then the subsequent steps learn the other neural dynamics later so that they do not mask or confound the behaviorally relevant dynamics. Importantly, each optimization step is independent of subsequent steps so all steps can be performed in order, with no need to iteratively repeat any step. We define the neural and behavioral prediction losses that are used in the optimization steps based on the negative log-likelihoods (NLLs) associated with the neural and behavior distributions, respectively. This approach benefits from the statistical foundation of maximum likelihood estimation and facilitates generalizability across behavioral distributions. We now expand on each of the four optimization steps for RNN training.

Optimization step 1

In the first two optimization steps (Supplementary Fig. 1a ), the objective is to learn the behaviorally relevant latent states \({x}_{k}^{\left(1\right)}\) and their associated parameters. In the first optimization step, we learn the parameters A ′(1) , \({C}_{z}^{\,\left(1\right)}\) and K (1) of the RNN

and estimate its latent state \({x}_{k}^{\left(1\right)}\) while minimizing the NLL of the behavior z k given by \({x}_{k}^{\left(1\right)}\) . For continuous-valued (Gaussian) behavioral data, we minimize the following sum of squared prediction error 69 , 97 given by

where the sum is over all available samples of behavior z k , and \({\Vert .\Vert }_{2}\) indicates the two-norm operator. This objective, which is typically used when fitting models to continuous-valued data 69 , 97 , is proportional to the Gaussian NLL if we assume isotropic Gaussian residuals (that is, ∑ 𝜖  = σ 𝜖 I ) 69 , 97 . If desired, a general nonisotropic residual covariance ∑ 𝜖 can be empirically computed from model residuals after the above optimization is solved (see Learning noise statistics ), although having ∑ 𝜖 is mainly useful for simulating new data and is not needed when using the learned model for inference. Similarly, in the subsequent optimization steps detailed later, the same points hold regarding how the appropriate mean squared error used for continuous-valued data is proportional to the Gaussian NLL if we assume isotropic Gaussian residuals and how the residual covariance can be computed empirically after the optimization if desired.

Optimization step 2

The second optimization step uses the extracted latent state \({x}_{k}^{\left(1\right)}\) from the RNN and fits the parameter \({C}_{y}^{\left(1\right)}\) in

while minimizing the NLL of the neural activity y k given by \({x}_{k}^{(1)}\) . For continuous-valued (Gaussian) neural activity y k , we minimize the following sum of squared prediction error 69 :

where the sum is over all available samples of y k . Optimization steps 1 and 2 conclude the prioritized extraction and modeling of behaviorally relevant latent states \({x}_{k}^{(1)}\) (Fig. 1b ) and the learning of the first section of the RNN model (Supplementary Fig. 1a ).

Optimization step 3

In optimization steps 3 and 4 (Supplementary Fig. 1a ), the objective is to learn any additional dynamics in neural activity that are not learned in the first two optimization steps, that is, \({x}_{k}^{\left(2\right)}\) and the associated parameters. To do so, in the third optimization step, we learn the parameters A ′(2) , \({C}_{y}^{\,\left(2\right)}\) and K (2) of the RNN

and estimate its latent state \({x}_{k}^{\left(2\right)}\) while minimizing the aggregate NLL of y k given both latent states, that is, by also taking into account the NLL obtained from step 2 via the \({C}_{y}^{\,\left(1\right)}\left({x}_{k}^{\left(1\right)}\right)\) term in Eq. ( 6 ). The notations \({y}_{k}^{{\prime} }\) and \({e}_{k}^{{\prime} }\) in the second line of Eq. ( 8 ) signify the fact that it is not y k that is predicted by the RNN of Eq. ( 8 ), rather it is the yet unpredicted parts of y k (that is, unpredicted after extracting \({x}_{k}^{(1)}\) ) that are being predicted. In the case of continuous-valued (Gaussian) neural activity y k , we minimize the following loss:

where the sum is over all available samples of y k . Note that in the continuous-valued (Gaussian) case, this loss is equivalent to minimizing the error in predicting the residual neural activity given by \({y}_{k}-{C}_{y}^{\,\left(1\right)}\left({x}_{k}^{\left(1\right)}\right)\) and is computed using the previously learned parameter \({C}_{y}^{\,\left(1\right)}\) and the previously extracted states \({x}_{k}^{\left(1\right)}\) in steps 1 and 2. Also, the input to the RNN in Eq. ( 8 ) includes both y k and the extracted \({x}_{k+1}^{\left(1\right)}\) from optimization step 1. The above shows how the optimization steps are appropriately linked together to compute the aggregate likelihoods.

Optimization step 4

If we assume that the second set of states \({x}_{k}^{\left(2\right)}\) do not contain any information about behavior, we could stop the modeling. However, this may not be the case if the dimension of the states extracted in the first optimization step (that is, n 1 ) is selected to be very small such that some behaviorally relevant neural dynamics are not learned in the first step. To be robust to such selections of n 1 , we can use another final numerical optimization to determine based on the data whether and how \({x}_{k}^{\left(2\right)}\) should affect behavior prediction. Thus, a fourth optimization step uses the extracted latent state in optimization steps 1 and 3 and fits C z in

while minimizing the negative log-likelihood of behavior given both latent states. In the case of continuous-valued (Gaussian) behavior z k , we minimize the following loss:

The parameter C z that is learned in this optimization step will replace both \({C}_{z}^{\,\left(1\right)}\) and \({C}_{z}^{\,\left(2\right)}\) in Eq. ( 2 ). Optionally, in a final optimization step, a similar nonlinear mapping from \({x}_{k}^{\left(1\right)}\) and \({x}_{k}^{\left(2\right)}\) can also be learned, this time to predict y k , which allows DPAD to support nonlinear interactions of \({x}_{k}^{\left(1\right)}\) and \({x}_{k}^{\left(2\right)}\) in predicting neural activity. In this case, the resulting learned C y parameter will replace both \({C}_{y}^{\,\left(1\right)}\) and \({C}_{y}^{\,\left(2\right)}\) in Eq. ( 2 ). This concludes the learning of both model sections (Supplementary Fig. 1a ) and all model parameters in Eq. ( 2 ).

In this work, when optimization steps 1 and 3 are both used to extract the latent states (that is, when 0 <  n 1  <  n x ), we do not perform the additional fourth optimization step in Eq. ( 10 ), and the prediction of behavior is done solely using the \({x}_{k}^{\left(1\right)}\) states extracted in the first optimization step. Note that DPAD can also cover NDM as a special case if we only use the third optimization step to extract the states (that is, n 1  = 0, in which case the first two steps are not needed). In this case, we use the fourth optimization step to learn C z , which is the mapping from the latent states to behavior. Also, in this case, we simply have a unified state x k as there is no dissociation in NDM, and the only goal is to extract states that predict neural activity accurately.

Additional generalizations of state dynamics

Finally, the first lines of Eqs. ( 4 ) and ( 8 ) can also be written more generally as

where instead of an additive relation between the two terms of the righthand side, both terms are combined in nonlinear functions \({{A}^{{\prime} {\prime} }}^{\left(1\right)}\) and \({{A}^{{\prime} {\prime} }}^{\left(2\right)}\) , which as a special case can still learn the additive relation in Eqs. ( 4 ) and ( 8 ). Whenever both the state recursion A and neural input K parameters (with the appropriate superscripts) are specified to be nonlinear, we use the more general architecture in Eqs. ( 12 ) and ( 13 ), and if any one of A or K or both are linear, we use Eqs. ( 4 ) and ( 8 ).

As another option, both RNN sections can be made bidirectional, which enables noncausal prediction for DPAD by using future data in addition to past data, with the goal of improving prediction, especially in datasets with stereotypical trials. Although this option is not reported in this work, it is implemented and available for use in DPAD’s public code library.

Learning noise statistics

Once the learning is complete, we also compute the covariances of the neural and behavior residual time series e k and 𝜖 k as ∑ e and ∑ 𝜖 , respectively. This allows the learned model in Eq. ( 1 ) to be usable for generating new simulated data. This application is not the focus of this work, but an explanation of it is provided in Numerical simulations .

Regularization

Adding norm 1 or norm 2 regularization for any set of parameters and the option to automatically select the regularization weight with inner cross-validation is implemented in the DPAD code. However, we did not use regularization in any of the analyses presented here.

Forecasting

DPAD also enables the capability to predict neural–behavioral data more than one time step into the future. To obtain two-step-ahead prediction, we pass the one-step-ahead neural predictions of the model as neural observations into it. This allows us to perform one state update iteration, that is, line 1 of Eq. ( 2 ), with y k being replaced with \({\hat{y}}_{k}\) from Eq. ( 3 ). Repeating this procedure m times gives the ( m  + 1)-step-ahead prediction of the latent state and neural–behavioral data.

Extending to intermittently measured behaviors

We also extend DPAD to modeling intermittently measured behavior time series (Extended Data Figs. 8 and 9 and Supplementary Fig. 8 ). To do so, when forming the behavior loss (Eqs. ( 5 ) and ( 11 )), we only compute the loss on samples where the behavior is measured and solve the optimization with this loss.

Extending to noncontinuous-valued data observations

We can also extend DPAD to noncontinuous-valued (non-Gaussian) observations by devising modified loss functions and observation models. Here, we demonstrate this extension for categorical behavioral observations, for example, discrete choices or epochs/phases during a task (Fig. 7 ). A similar approach could be used in the future to model other non-Gaussian behaviors and non-Gaussian (for example, Poisson) neural modalities, as shown in a thesis 56 .

To model categorical behaviors, we devise a new behavior observation model for DPAD by making three changes. First, we change the behavior loss (Eqs. ( 5 ) and ( 11 )) to the NLL of a categorical distribution, which we implement using the dedicated class in the TensorFlow library (that is, tf.keras.losses.CategoricalCrossentropy). Second, we change the behavior readout parameter C z to have an output dimension of n z  ×  n c instead of n z , where n c denotes the number of behavior categories or classes. Third, we apply Softmax normalization (Eq. ( 14 )) to the output of the behavior readout parameter C z to ensure that for each of the n z behavior dimensions, the predicted probabilities for all the n c classes add up to 1 so that they represent valid probability mass functions. Softmax normalization can be written as

where \({l}_{k}\in {{\mathbb{R}}}^{{n}_{z}\times {n}_{c}}\) is the output of C z at time k , and the superscript ( m , n ) denotes the element of l k associated with the behavior dimension m and the class/category number n . With these changes, we obtain a new RNN architecture with categorical behavioral outputs. We then learn this new RNN architecture with DPAD’s four-step prioritized optimization approach as before but now incorporating the modified NLL losses for categorical data. Together, with these changes, DPAD extends to modeling categorical behavioral measurements.

Behavior decoding and neural self-prediction metrics and performance frontier

Cross-validation.

To evaluate the learning, we perform a cross-validation with five folds (unless otherwise noted). We cut the data from the recording session into five equal continuous segments, leave these segments out one by one as the test data and train the model only using the data in the remaining segments. Once the model is trained using the neural and behavior training data, we pass the neural test data to the model to get the latent states in the test data using the first line of Eq. ( 1 ) (or Eq. ( 2 ), equivalently). We then pass the extracted latent states to Eq. ( 3 ) to get the one-step-ahead prediction of the behavior and neural test data, which we refer to as behavior decoding and neural self-prediction, respectively. Note that only past neural data are used to get the behavior and neural predictions. Also, the behavior test data are never used in predictions. Given the predicted behavior and neural time series, we compute the CC between each dimension of these time series and the actual behavior and neural test time series. We then take the mean of CC across dimensions of behavior and neural data to get one final cross-validated CC value for behavior decoding and one final CC value for neural self-prediction in each cross-validation fold.

Selection of the latent state dimension

We often need to select a latent state dimension to report an overall behavior decoding and/or neural self-prediction accuracy for each model/method (for example, Figs. 2 – 7 ). By latent state dimension, we always refer to the total latent state dimension of the model, that is, n x . For DPAD, unless otherwise noted, we always used n 1  = 16 to extract the first 16 latent state dimensions (or all latent state dimensions when n x  ≤ 16) using steps 1 and 2 and any remaining dimensions using steps 3 and 4. We chose n 1  = 16 because dedicating more, even all, latent state dimensions to behavior prediction only minimally improved it across datasets and neural modalities. For all methods, to select a state dimension n x , in each cross-validation fold, we fit models with latent state dimensions 1, 2, 4, 16,…and 128 (powers of 2 from 1 to 128) and select one of these models based on their decoding and neural self-prediction accuracies within the training data of that fold. We then report the decoding/self-prediction of this selected model computed in the test data of that fold. Our goal is often to select a model that simultaneously explains behavior and neural data well. For this goal, we pick the state dimension that reaches the peak neural self-prediction in the training data or the state dimension that reaches the peak behavior decoding in the training data, whichever is larger; we then report both the neural self-prediction and the corresponding behavior decoding accuracy of the same model with the selected state dimension in the test data (Figs. 3 – 4 , 6 and 7f , Extended Data Figs. 3 and 4 and Supplementary Figs. 4 – 7 and 9 ). Alternatively, for all methods, when our goal is to find models that solely aim to optimize behavior prediction, we report the cross-validated prediction performances for the smallest state dimension that reaches peak behavior decoding in training data (Figs. 2 , 5 and 7d , Extended Data Fig. 8 and Supplementary Fig. 3 ). We emphasize that in all cases, the reported performances are always computed in the test data of the cross-validation fold, which is not used for any other purpose such as model fitting or selection of the state dimension.

Performance frontier

When comparing a group of alternative models, we use the term ‘performance frontier’ to describe the best performances reached by models that in every comparison with any alternative model are in some sense better than or at least comparable to the alternative model. More precisely, when comparing a group \({\mathcal{M}}\) of models, model \({\mathcal{A}}\in {\mathcal{M}}\) will be described as reaching the best performance frontier when compared to every other model \({\mathcal{B}}{\mathscr{\in }}{\mathcal{M}}\) , \({\mathcal{A}}\) is significantly better than \({\mathcal{B}}\) in behavior decoding or in neural self-prediction or is comparable to \({\mathcal{B}}\) in both. Note that \({\mathcal{A}}\) may be better than some model \({{\mathcal{B}}}_{1}\in {\mathcal{M}}\) in decoding while being better than another model \({{\mathcal{B}}}_{2}\in {\mathcal{M}}\) in self-prediction; nevertheless \({\mathcal{A}}\) will be on the frontier as long as in every comparison one of the following conditions hold: (1) there is at least one measure for which \({\mathcal{A}}\) is more performant and (2) \({\mathcal{A}}\) is at least equally performant in both measures. To avoid exclusion of models from the best performance frontier due to very minimal performance differences, in this analysis, we only declare a difference in performance significant if in addition to resulting in P  ≤ 0.05 in a one-sided signed-rank test there is also at least 1% relative difference in the mean performance measures.

DPAD with flexible nonlinearity: automatic determination of appropriate nonlinearity

Fine-grained control over nonlinearities.

Each parameter in the DPAD model represents an operation in the computation graph of DPAD (Fig. 1b and Supplementary Fig. 1a ). We solve the numerical optimizations involved in model learning in each step of our multistep learning via standard stochastic gradient descent 43 , which remains applicable for any modification of the computation graph that remains acyclic. Thus, the operation associated with each model parameter (for example, A ′, K , C y and C z ) can be replaced with any multilayer neural network with an arbitrary number of hidden units and layers (Supplementary Fig. 1c ), and the model remains trainable with the same approach. Having no hidden layers implements the special case of a linear mapping (Supplementary Fig. 1b ). Of course, given that the training data are finite, the typical trade-off between model capacity and generalization error remains 69 . Given that neural networks can approximate any continuous function (with a compact domain) 98 , replacing model parameters with neural networks should have the capacity to learn any nonlinear function in their place 99 , 100 , 101 . The resulting RNN in Eq. ( 1 ) can in turn approximate any state-space dynamics (under mild conditions) 102 . In this work, for nonlinear parameters, we use multilayer feed-forward networks with one or two hidden layers, each with 64 or 128 units. For all hidden layers, we always use a rectified linear unit (ReLU) nonlinear activation (Supplementary Fig. 1c ). Finally, when making a parameter (for example, C z ) nonlinear, we always do so for that parameter in both sections of the RNN (for example, both \({C}_{z}^{\,\left(1\right)}\) and \({C}_{z}^{\,\left(2\right)}\) ; see Supplementary Fig. 1a ) and using the same feed-forward network structure. Given that no existing RNN implementation allowed individual RNN elements to be independently set to arbitrary multilayer neural networks, we developed a custom TensorFlow RNN cell to implement the RNNs in DPAD (Eqs. ( 4 ) and ( 8 )). We used the Adam optimizer to implement gradient descent for all optimization steps 43 . We continued each optimization for up to 2,500 epochs but stopped earlier if the objective function did not improve in three consecutive epochs (convergence criteria).

Automatic selection of nonlinearity settings

We devise a procedure for automatically determining the most suitable combination of nonlinearities for the data, which we refer to as DPAD with flexible nonlinearity. In this procedure, for each cross-validation fold in each recording session of each dataset, we try a series of nonlinearities within the training data and select one based on an inner cross-validation within the training data (Fig. 1d ). Specifically, we consider the following options for the nonlinearity. First, each of the four main parameters (that is, A ′, K , C y and C z ) can be linear or nonlinear, resulting in 16 cases (that is, 2 4 ). In cases with nonlinearity, we consider four network structures for the parameters, that is, having one or two hidden layers and having 64 or 128 units in each hidden layer (Supplementary Fig. 1c ), resulting in 61 cases (that is, 15 × 4 + 1, where 1 is for the fully linear model) overall. Finally, specifically for the recursion parameter A ′, we also consider modeling it as an LSTM, with the other parameters still having the same nonlinearity options as before, resulting in another 29 cases for when this LSTM recursion is used (that is, 7 × 4 + 1, where 1 is for the case where the other three model parameters are all linear), bringing the total number of considered cases to 90. For each of these 90 considered linear or nonlinear architectures, we perform a twofold inner cross-validation within the training data to compute an estimate of the behavior decoding and neural self-prediction of each architecture using the training data. Note that although this process for automatic selection of nonlinearities is computationally expensive, it is parallelizable because each candidate model can be fitted independently on a different processor. Once all candidate architectures are fitted and evaluated within the training data, we select one final architecture purely based on training data to be used for that cross-validation fold based on one of the following two criteria: (1) decoding focused: pick the architecture with the best neural self-prediction in training data among all those that reach within 1 s.e.m. of the best behavior decoding; or (2) self-prediction focused: pick the architecture with the best behavior decoding in training data among all those that reach within 1 s.e.m. of the best neural self-prediction. The first criterion prioritizes good behavior decoding in the selection, and the second criterion prioritizes good neural self-prediction. Note that these two criteria are used when selecting among different already-learned models with different nonlinearities and thus are independent of the four internal objective functions used in learning the parameters for a given model with the four-step optimization approach (Supplementary Fig. 1a ). For example, in the first optimization step of DPAD, model parameters are always learned to optimize behavior decoding (Eq. ( 5 )). But once the four-step optimization is concluded and different models (with different combinations of nonlinearities) are learned, we can then select among these already-learned models based on either neural self-prediction or behavior decoding. Thus, whenever neural self-prediction is also of interest, we report the results for flexible nonlinearity based on both model selection criteria (for example, Figs. 3 , 4 and 6 ).

Localization of nonlinearities

DPAD enables an inspection of where nonlinearities can be localized to by providing two capabilities, without either of which the origin of nonlinearities may be incorrectly found. As the first capability, DPAD can train alternative models with different individual nonlinearities and then compare these alternative nonlinear models not only with a fully linear model but also with each other and with fully nonlinear models (that is, flexible nonlinearity). Indeed, our simulations showed that simply comparing a linear model to a model with nonlinearity in a given parameter may incorrectly identify the origin of nonlinearity (Extended Data Fig. 2b and Fig. 6a ). For example, in Fig. 6a , although the nonlinearity is just in the neural input parameter, a linear model does worse than a model with a nonlinear behavior readout parameter. Thus, just a comparison of the latter model to a linear model would incorrectly find the origin of nonlinearity to be the behavior readout. This issue is avoided in DPAD because it can also train a model with the neural input being nonlinear, thus finding it to be more predictive than models with any other individual nonlinearity and as predictive as a fully nonlinear model (Fig. 6a ). As the second capability, DPAD can compare alternative nonlinear models in terms of overall neural–behavioral prediction rather than either behavior decoding or neural prediction alone. Indeed, our simulations showed that comparing the models in terms of just behavior decoding (Extended Data Fig. 2d,f ) or just neural self-prediction (Extended Data Fig. 2d,h ) may lead to incorrect conclusions about the origin of nonlinearities; this is because a model with the incorrect origin may be equivalent in one of these metrics to the one with the correct origin. DPAD avoids this problem by jointly evaluating both neural–behavioral metrics. Here, when comparing models with nonlinearity in different individual parameters for localization purposes (for example, Fig. 6 ), we only consider one network architecture for the nonlinearity, that is, having one hidden layer with 64 units.

Numerical simulations

To validate DPAD in numerical simulations, we perform two sets of simulations. One set validates linear modeling to show the correctness of the four-step numerical optimization for learning. The other set validates nonlinear modeling. In the linear simulation, we randomly generate 100 linear models with various dimensionality and noise statistics, as described in our prior work 6 . Briefly, the neural and behavior dimensions are selected from 5 ≤  n y , n z  ≤ 10 randomly with uniform probability. The state dimension is selected as n x  = 16, of which n 1  = 4 latent state dimensions are selected to drive behavior. Eigenvalues of the state transition matrix are selected randomly as complex conjugate pairs with uniform probability within the unit disk. Each element in the behavior and neural readout matrices is generated as a random Gaussian variable. State and neural observation noise covariances are generated as random positive definite matrices and scaled randomly with a number between 0.003 and 0.3 or between 0.01 and 100, respectively, to obtain a wide range of relative noises across random models. A separate random linear state-space model with four latent state dimensions is generated to produce the behavior readout noise 𝜖 k , representing the behavior dynamics that are not encoded in the recorded neural activity. Finally, the behavior readout matrix is scaled to set the ratio of the signal standard deviation to noise standard deviation in each behavior dimension to a random number from 0.5 to 50. We perform model learning and evaluation with twofold cross-validation (Extended Data Fig. 1 ).

In the nonlinear simulations that are used to validate both DPAD and the hypothesis testing procedure it enables to find the origin of nonlinearity, we start by generating 20 random linear models ( n y  =  n z  = 1) either with n x  =  n z  =  n y (Extended Data Fig. 2 ) or n x  = 2 latent states, only one of which drives behavior (Supplementary Fig. 2 ). We then introduce nonlinearity in one of the four model parameters (that is, A ′, K , C y or C z ) by replacing that parameter with a nonlinear trigonometric function, such that roughly one period of the trigonometric function is visited by the model (while keeping the rest of the parameters linear). To do this, we first scale each latent state in the initial random linear model to find a similarity transform for it where the latent state has a 95% confidence interval range of 2 π . We then add a sine function to the original parameter that is to be changed to nonlinear and scale the amplitude of the sine such that its output reaches roughly 0.25 of the range of the outputs from the original linear parameter. This was done to reduce the chance of generating unrealistic unstable nonlinear models that produce outputs with infinite energy, which is likely when A ′ is nonlinear. Changing one parameter to nonlinear can change the range of the statistics of the latent states in the model; thus, we generate some simulated data from the model and redo the scaling of the nonlinearity until ratio conditions are met.

To generate data from any nonlinear model in Eq. ( 1 ), we first generate a neural noise time series e k based on its covariance ∑ e in the model and initialize the state as x 0  = 0. We then iteratively apply the second and first lines of Eq. ( 1 ) to get the simulated neural activity y k from line 2 and then the next state \({x}_{k+1}\) from line 1, respectively. Finally, once the state time series is produced, we generate a behavior noise time series 𝜖 k based on its covariance ∑ 𝜖 in the model and apply the third line of Eq. ( 1 ) to get the simulated behavior z k . Similar to linear simulations, we perform the modeling and evaluation of nonlinear simulations with twofold cross-validation (Extended Data Fig. 2 and Supplementary Fig. 2 ).

Neural datasets and behavioral tasks

We evaluate DPAD in five datasets with different behavioral tasks, brain regions and neural recording modalities to show the generality of our conclusions. For each dataset, all animal procedures were performed in compliance with the National Research Council Guide for Care and Use of Laboratory Animals and were approved by the Institutional Animal Care and Use Committee at the respective institution, namely New York University (datasets 1 and 2) 6 , 45 , 46 , Northwestern University (datasets 3 and 5) 47 , 48 , 54 and University of California San Francisco (dataset 4) 21 , 49 .

Across all four main datasets (datasets 1 to 4), the spiking activity was binned with 10-ms nonoverlapping bins, smoothed with a Gaussian kernel with standard deviation of 50 ms (refs. 6 , 14 , 34 , 103 , 104 ) and downsampled to 50 ms to be used as the neural signal to be modeled. The behavior time series was also downsampled to a matching 50 ms before modeling. In the three datasets where LFP activity was also available, we also studied two types of features extracted from LFP. As the first LFP feature, we considered raw LFP activity itself, which was high-pass filtered above 0.5 Hz to remove the baseline, low-pass filtered below 10 Hz (that is, antialiasing) and downsampled to the behavior sampling rate of a 50-ms time step (that is, 20 Hz). Note that in the context of the motor cortex, low-pass-filtered raw LFP is also referred to as the local motor potential 50 , 51 , 52 , 105 , 106 and has been used to decode behavior 6 , 50 , 51 , 52 , 53 , 105 , 106 , 107 . As the second feature, we computed the LFP log-powers 5 , 6 , 7 , 40 , 77 , 79 , 106 , 108 , 109 in eight standard frequency bands (delta: 0.1–4 Hz, theta: 4–8 Hz, alpha: 8–12 Hz, low beta: 12–24 Hz, mid-beta: 24–34 Hz, high beta: 34–55 Hz, low gamma: 65–95 Hz and high gamma: 130–170 Hz) in sliding 300-ms windows at a time step of 50 ms using Welch’s method (using eight subwindows with 50% overlap) 6 . The median analyzed data length for each session across the datasets ranged from 4.6 to 9.9 min.

First dataset: 3D reaches to random targets

In the first dataset, the animal (named J) performed reaches to a target randomly positioned in 3D space within the reach of the animal, grasped the target and returned its hand to resting position 6 , 45 . Kinematic data were acquired using the Cortex software package (version 5.3) to track retroreflective markers in 3D (Motion Analysis) 6 , 45 . Joint angles were solved from the 3D marker data using a Rhesus macaque musculoskeletal model via the SIMM toolkit (version 4.0, MusculoGraphics) 6 , 45 . Angles of 27 joints in the shoulder, elbow, wrist and fingers in the active hand (right hand) were taken as the behavior signal 6 , 45 . Neural activity was recorded with a 137-electrode microdrive (Gray Matter Research), of which 28 electrodes were in the contralateral primary motor cortex M1. The multiunit spiking activity in these M1 electrodes was used as the neural signal. For LFP analyses, LFP features were also extracted from the same M1 electrodes. We analyzed the data from seven recording sessions.

To visualize the low-dimensional latent state trajectories for each behavioral condition (Extended Data Fig. 6 ), we determined the periods of reach and return movements in the data (Fig. 7a ), resampled them to have similar number of time samples and averaged the latent states across those resampled trials. Given the redundancy in latent descriptions (that is, any scaling, rotation and so on on the latent states still gives an equivalent model), before averaging trials across cross-validation folds and sessions, we devised the following procedure to standardize the latent states for each fold in the case of 2D latent states (Extended Data Fig. 6 ). (1) We z score all state dimensions to have zero mean and unit variance. (2) We rotate the 2D latent states such that the average 2D state trajectory for the first condition (here, the reach epochs) starts from an angle of 0. (3) We estimate the direction of the rotation for the average 2D state trajectory of the first condition, and if it is not counterclockwise, we multiply the second state dimension by –1 to make it so. Note that in each step, the same mapping is applied to the latent states during the whole test data, regardless of condition, so this procedure does not alter the relative differences in the state trajectory across different conditions. The procedure also does not change the learned model and simply corresponds to a similarity transform that changes the basis of the model. This procedure only removes the redundancies for describing a 2D latent state-space model and standardizes the extracted latent states so that trials across different test sets can be averaged together.

Second dataset: saccadic eye movements

In the second dataset, the animal (named A) performed saccadic eye movements to one of eight targets on a display 6 , 46 . The visual stimuli in the task with saccadic eye movements were controlled via custom LabVIEW (version 9.0, National Instruments) software executed on a real-time embedded system (NI PXI-8184, National Instruments) 46 . The 2D position of the eye was tracked and taken as the behavior signal. Neural activity was recorded with a 32-electrode microdrive (Gray Matter Research) covering the prefrontal cortex 6 , 46 . Single-unit activity from these electrodes, ranging from 34 to 43 units across different recording sessions, was used as the neural signal. For LFP analyses, LFP features were also extracted from the same 32 electrodes. We analyzed the data from the first 7 days of recordings. We only included data from successful trials where the animal performed the task correctly by making a saccadic eye movement to the specified target. To visualize the low-dimensional latent state trajectories for each behavioral condition (Extended Data Fig. 6 ), we grouped the trials based on their target position. Standardization across folds before averaging was done as in the first dataset.

Third dataset: sequential reaches with a 2D cursor controlled with a manipulandum

In the third dataset, which was collected and made publicly available by the laboratory of L. E. Miller 47 , 48 , the animal (named T) controlled a cursor on a 2D screen using a manipulandum and performed a sequential reach task 47 , 48 . The 2D cursor position and velocity were taken as the behavior signal. Neural activity was recorded using a 100-electrode microelectrode array (Blackrock Microsystems) in the dorsal premotor cortex 47 , 48 . Single-unit activity, recorded from 37 to 49 units across recording sessions, was used as the neural signal. This dataset did not include any LFP recordings, so LFP features could not be considered. We analyzed the data from all three recording sessions. To visualize the low-dimensional latent state trajectories for each behavioral condition (Extended Data Fig. 6 ), we grouped the trials into eight different conditions based on the angle of the direction of movement (that is, end position minus starting position) during the trial, with each condition covering movement directions within a 45° (that is, 360/8) range. Standardization across folds before averaging was performed as in the first dataset.

Fourth dataset: virtual reality random reaches with a 2D cursor controlled with the fingertip

In the fourth dataset, which was collected and made publicly available by the laboratory of P. N. Sabes 49 , the animal (named I) controlled a cursor based on the fingertip position on a 2D surface within a 3D virtual reality environment 21 , 49 . The 2D cursor position and velocity were taken as the behavior signal. Neural activity was recorded with a 96-electrode microelectrode array (Blackrock Microsystems) 21 , 49 covering M1. We selected a random subset of 32 of these electrodes, which had 77 to 99 single units across the recording sessions, as the neural signal. LFP features were also extracted from the same 32 electrodes. We analyzed the data for the first seven sessions for which the wideband activity was also available (sessions 20160622/01 to 20160921/01). Grouping into conditions for visualization of low-dimensional latent state trajectories (Extended Data Fig. 6 ) was done as in the third dataset. Standardization across folds before averaging was done as in the first dataset.

Fifth dataset: center-out cursor control reaching task

In the fifth dataset, which was collected and made publicly available by the laboratory of L. E. Miller 54 , the animal (named H) controlled a cursor on a 2D screen using a manipulandum and performed reaches from a center point to one of eight peripheral targets (Fig. 4i ). The 2D cursor position was taken as the behavior signal. Neural activity was recorded with a 96-electrode microelectrode array (Blackrock Microsystems) covering area 2 of the somatosensory cortex 54 . Preprocessing for this dataset was done as in ref. 36 . Specifically, the spiking activity was binned with 1-ms nonoverlapping bins and smoothed with a Gaussian kernel with a standard deviation of 40 ms (ref. 110 ), with the behavior also being sampled with the same 1-ms sampling rate. Trials were also aligned as in the same prior work 110 with data from –100 to 500 ms around movement onset of each trial being used for modeling 36 .

Additional details for baseline methods

For the fifth dataset, which has been analyzed in ref. 36 and introduces CEBRA, we used the exact same CEBRA hyperparameters as those reported in ref. 36 (Fig. 4i,j ). For each of the other four datasets (Fig. 4a–h ), when learning a CEBRA-Behavior or CEBRA-Time model for each session, fold and latent dimension, we also performed an extensive search over CEBRA hyperparameters and picked the best value with the same inner cross-validation approach as we use for the automatic selection of nonlinearities in DPAD. We considered 30 different sets of hyperparameters: 3 options for the ‘time-offset’ hyperparameter (1, 2 or 10) and 10 options for the ‘temperature’ hyperparameter (from 0.0001 to 0.01), which were designed to include all sets of hyperparameters reported for primate data in ref. 36 . We swept the CEBRA latent dimension over the same values as DPAD, that is, powers of 2 up to 128. In all cases, we used a k -nearest neighbors regression to map the CEBRA-extracted latent embeddings to behavior and neural data as done in ref. 36 because CEBRA itself does not learn a reconstruction model 36 (Extended Data Table 1 ).

It is important to note that CEBRA and DPAD have fundamentally different architectures and goals (Extended Data Table 1 ). CEBRA uses a small ten-sample window (when ‘model_architecture’ is ‘offset10-model’) around each datapoint to extract a latent embedding via a series of convolutions. By contrast, DPAD learns a dynamical model that recursively aggregates all past neural data to extract an embedding. Also, in contrast to CEBRA-Behavior, DPAD’s embedding includes and dissociates both behaviorally relevant neural dimensions and other neural dimensions to predict not only the behavior but also the neural data well. Finally, CEBRA does not automatically map its latent embeddings back to neural data or to behavior during learning but does so post hoc, whereas DPAD learns these mappings for all its latent states. Given these differences, several use-cases of DPAD are not targeted by CEBRA, including explicit dynamical modeling of neural–behavioral data (use-case 1), flexible nonlinearity, hypothesis testing regarding the origin of nonlinearity (use-case 4) and forecasting.

We used the Wilcoxon signed-rank test for all paired statistical tests.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Data availability

Three of the datasets used in this work are publicly available 47 , 48 , 49 , 54 . The other two datasets used to support the results are available upon reasonable request from the corresponding author. Source data are provided with this paper.

Code availability

The code for DPAD is available at https://github.com/ShanechiLab/DPAD .

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Acknowledgements

This work was supported, in part, by the following organizations and grants: the Office of Naval Research (ONR) Young Investigator Program under contract N00014-19-1-2128, National Institutes of Health (NIH) Director’s New Innovator Award DP2-MH126378, NIH R01MH123770, NIH BRAIN Initiative R61MH135407 and the Army Research Office (ARO) under contract W911NF-16-1-0368 as part of the collaboration between the US DOD, the UK MOD and the UK Engineering and Physical Research Council (EPSRC) under the Multidisciplinary University Research Initiative (MURI) (to M.M.S.) and a University of Southern California Annenberg Fellowship (to O.G.S.).

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Ming Hsieh Department of Electrical and Computer Engineering, Viterbi School of Engineering, University of Southern California, Los Angeles, CA, USA

Omid G. Sani & Maryam M. Shanechi

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Bijan Pesaran

Thomas Lord Department of Computer Science, University of Southern California, Los Angeles, CA, USA

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O.G.S. and M.M.S. conceived the study, developed the DPAD algorithm and wrote the manuscript, and O.G.S. performed all the analyses. B.P. designed and performed the experiments for two of the NHP datasets and provided feedback on the manuscript. M.M.S. supervised the work.

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Correspondence to Maryam M. Shanechi .

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University of Southern California has a patent related to modeling and decoding of shared dynamics between signals in which M.M.S. and O.G.S. are inventors. The other author declares no competing interests.

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Extended data

Extended data fig. 1 dpad dissociates and prioritizes the behaviorally relevant neural dynamics while also learning the other neural dynamics in numerical simulations of linear models..

a , Example data generated from one of 100 random models ( Methods ). These random models do not emulate real data but for terminological consistency, we still refer to the primary signal (that is, y k in Eq. ( 1 )) as the ‘neural activity’ and to the secondary signal (that is, z k in Eq. ( 1 )) as the ‘behavior’. b , Cross-validated behavior decoding accuracy (correlation coefficient, CC) for each method as a function of the number of training samples when we use a state dimension equal to the total state dimension of the true model. The performance measures for each random model are normalized by their ideal values that were achieved by the true model itself. Performance for the true model is shown in black. Solid lines and shaded areas are defined as in Fig. 5b ( N  = 100 random models). c , Same as b but when learned models have low-dimensional latent states with enough dimensions just for the behaviorally relevant latent states (that is, n x  =  n 1 ). d - e , Same as b - c showing the cross-validated normalized neural self-prediction accuracy. Linear NDM, which learns the parameters using a numerical optimization, performs similarly to a linear algebraic subspace-based implementation of linear NDM 67 , thus validating NDM’s numerical optimization implementation. Linear DPAD, just like PSID 6 , achieves almost ideal behavior decoding even with low-dimensional latent states ( c ); this shows that DPAD correctly dissociates and prioritizes behaviorally relevant dynamics, as opposed to aiming to simply explain the most neural variance as non-prioritized methods such as NDM do. For this reason, with a low-dimensional state, non-prioritized NDM methods can explain neural activity well ( e ) but prioritized methods can explain behavior much better ( c ). Nevertheless, using the second stage of PSID and the last two optimization steps in DPAD, these two prioritized techniques are still able to learn the overall neural dynamics accurately if state dimension is high enough ( d ). Overall, the performance of linear DPAD and PSID 6 are similar for the special case of linear modeling.

Extended Data Fig. 2 DPAD successfully identifies the origin of nonlinearity and learns it in numerical simulations.

DPAD can perform hypothesis testing regarding the origin of nonlinearity by considering both behavior decoding (vertical axis) and neural self-prediction (horizontal axis). a , True value for nonlinear neural input parameter K in an example random model with nonlinearity only in K and the nonlinear value that DPAD learned for this parameter when only K in the learned model was set to be nonlinear. The true and learned mappings match and almost exactly overlap. b , Behavior decoding and neural self-prediction accuracy achieved by DPAD models with different locations of nonlinearities. These accuracies are for data generated from 20 random models that only had nonlinearity in the neural input parameter K . Performance measures for each random model are normalized by their ideal values that were achieved by the true model itself. Pluses and whiskers are defined as in Fig. 3 ( N  = 20 random models). c , d , Same as a , b for data simulated from models that only have nonlinearity in the recursion parameter A ′. e - f , Same as a , b for data simulated from models that only have nonlinearity in the neural readout parameter C y . g , h , Same as a , b for data simulated from models that only have nonlinearity in the behavior readout parameter C z . In each case ( b , d , f , h ), the nonlinearity option that reaches closest to the upper-rightmost corner of the plot, that is, has both the best behavior decoding and the best neural self-prediction, is chosen as the model that specifies the origin of nonlinearity. Regardless of the true location of nonlinearity ( b , d , f , h ), always the correct location (for example, K in b ) achieves the best performance overall compared with all other locations of nonlinearities. These results provide evidence that by fitting and comparing DPAD models with different nonlinearities, we can correctly find the origin of nonlinearity in simulated data.

Extended Data Fig. 3 Across spiking and LFP neural modalities, DPAD is on the best performance frontier for neural-behavioral prediction unlike LSTMs, which are fitted to explain neural data or behavioral data.

a , The 3D reach task. b , Cross-validated neural self-prediction accuracy achieved by each method versus the corresponding behavior decoding accuracy on the vertical axis. Latent state dimension for each method in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). Pluses and whiskers are defined as in Fig. 3 ( N  = 35 session-folds). Note that DPAD considers an LSTM as a special case ( Methods ). Nevertheless, results are also shown for LSTM networks fitted to decode behavior from neural activity (that is, RNN decoders in Extended Data Table 1 ) or to predict the next time step of neural activity (self-prediction). Also, note that LSTM for behavior decoding (denoted by H) and DPAD when only using the first two optimization steps (denoted by G) dedicate all their latent states to behavior prediction, whereas other methods dedicate some or all latent states to neural self-prediction. Compared with all methods including these LSTM networks, DPAD always reaches the best performance frontier for predicting the neural-behavioral data whereas LSTM does not; this is partly due to the four-step optimization algorithm in DPAD that allows for overall neural-behavioral description rather than one or the other, and that prioritizes the learning of the behaviorally relevant neural dynamics. c , Same as b for raw LFP activity ( N  = 35 session-folds). d , Same as b for LFP band power activity ( N  = 35 session-folds). e - h , Same as a - d for the second dataset, with saccadic eye movements ( N  = 35 session-folds). i , j , Same as a and b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). k - n , Same as a - d for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). Results and conclusions are consistent across all datasets.

Extended Data Fig. 4 DPAD can also be used for multi-step-ahead forecasting of behavior.

a , The 3D reach task. b , Cross-validated behavior decoding accuracy for various numbers of steps into the future. For m -step-ahead prediction, behavior at time step k is predicted using neural activity up to time step k − m . All models are taken from Fig. 3 , without any retraining or finetuning, with m -step-ahead forecasting done by repeatedly ( m −1 times) passing the neural predictions of the model as its neural observation in the next time step ( Methods ). Solid lines and shaded areas are defined as in Fig. 5b ( N  = 35 session-folds). Across the number of steps ahead, the statistical significance of a one-sided pairwise comparison between nonlinear DPAD vs nonlinear NDM is shown with the orange top horizontal line with p-value indicated by asterisks next to the line as defined in Fig. 2b (N = 35 session-folds). Similar pairwise comparison between nonlinear DPAD vs linear dynamical system (LDS) modeling is shown with the purple top horizontal line. c - d , Same as a - b for the second dataset, with saccadic eye movements ( N  = session-folds). e - f , Same as a - b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). g - h , Same as a - b for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds).

Extended Data Fig. 5 Neural self-prediction accuracy of nonlinear DPAD across recording electrodes for low-dimensional behaviorally relevant latent states.

a , The 3D reach task. b , Average neural self-prediction correlation coefficient (CC) achieved by nonlinear DPAD for analyzed smoothed spiking activity is shown for each recording electrode ( N  = 35 session-folds; best nonlinearity for decoding). c , Same as b for modeling of raw LFP activity. d , Same as b for modeling of LFP band power activity. Here, prediction accuracy averaged across all 8 band powers ( Methods ) of a given recording electrode is shown for that electrode. e-h , Same a - d for the second dataset, with saccadic eye movements ( N  = 35 session-folds). For datasets with single-unit activity ( Methods ), spiking self-prediction of each electrode is averaged across the units associated with that electrode. i - j , Same as a , b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). White areas are due to electrodes that did not have a neuron associated with them in the data. k - n , Same as a - d for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all results, the latent state dimension was 16, and all these dimensions were learned using the first optimization step (that is, n 1  = 16).

Extended Data Fig. 6 Nonlinear DPAD extracted distinct low dimensional latent states from neural activity for all datasets, which were more behaviorally relevant than those extracted using nonlinear NDM.

a , The 3D reach task. b , The latent state trajectory for 2D states extracted from spiking activity using nonlinear DPAD, averaged across all reach and return epochs across sessions and folds. Here only optimization steps 1-2 of DPAD are used to just extract 2D behaviorally relevant states. c , Same as b for 2D states extracted using nonlinear NDM (special case of using just DPAD optimization steps 3-4). d , Saccadic eye movement task. Trials are averaged depending on the eye movement direction. e , The latent state trajectory for 2D states extracted using DPAD (extracted using optimizations steps 1-2), averaged across all trials of the same movement direction condition across sessions and folds. f , Same as d for 2D states extracted using nonlinear NDM. g-i , Same as d - f for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum. j - l , Same as d - f for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position. Overall, in each dataset, latent states extracted by DPAD were clearly different for different behavior conditions in that dataset ( b , e , h , k ), whereas NDM’s extracted latent states did not as clearly dissociate different conditions ( c , f , i , l ). Of note, in the first dataset, DPAD revealed latent states with rotational dynamics that reversed direction during reach versus return epochs, which is consistent with the behavior roughly reversing direction. In contrast, NDM’s latent states showed rotational dynamics that did not reverse direction, thus were less congruent with behavior. In this first dataset, in our earlier work 6 , we had compared PSID and a subspace-based linear NDM method and, similar to b and c here, had found that only PSID uncovers reverse-directional rotational patterns across reach and return movement conditions. These results thus also complement our prior work 6 by showing that even nonlinear NDM models may not uncover the distinct reverse-directional dynamics in this dataset, thus highlighting the need for dissociative and prioritized learning even in nonlinear modeling, as enabled by DPAD.

Extended Data Fig. 7 Neural self-prediction across latent state dimensions.

a , The 3D reach task. b , Cross-validated neural self-prediction accuracy (CC) achieved by variations of nonlinear and linear DPAD/NDM, for different latent state dimensions. Solid lines and shaded areas are defined as in Fig. 5b ( N  = 35 session-folds). Across latent state dimensions, the statistical significance of a one-sided pairwise comparison between nonlinear DPAD/NDM (with best nonlinearity for self-prediction) vs linear DPAD/NDM is shown with a horizontal green/orange line with p-value indicated by asterisks next to the line as defined in Fig. 2b ( N  = 35 session-folds). c , d , Same as a , b for the second dataset, with saccadic eye movements ( N  = 35 session-folds). e , f , Same as a , b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). g , h Same as a , b for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, the first 16 latent state dimensions are learned using the first two optimization steps and the remaining dimensions are learned using the last two optimization steps (that is, n 1  = 16). As expected, at low state dimensions, DPAD’s latent states achieve higher behavior decoding (Fig. 5 ) but lower neural self-prediction than NDM because DPAD prioritizes the behaviorally relevant neural dynamics in these dimensions. However, by increasing the state dimension and utilizing optimization steps 3-4, DPAD can reach similar neural self-prediction to NDM while doing better in terms of behavior decoding (Fig. 3 ). Also, for low dimensional latent states, nonlinear DPAD/NDM consistently result in significantly more accurate neural self-prediction than linear DPAD/NDM. For high enough state dimensions, linear DPAD/NDM eventually reach similar neural self-prediction accuracy to nonlinear DPAD/NDM. Given that NDM solely aims to optimize neural self-prediction (irrespective of the relevance of neural dynamics to behavior), the latter result suggests that the overall neural dynamics can be approximated with linear dynamical models but only with high-dimensional latent states. Note that in contrast to neural self-prediction, behavior decoding of nonlinear DPAD is higher than linear DPAD even at high state dimensions (Fig. 3 ).

Extended Data Fig. 8 DPAD accurately learns the mapping from neural activity to behavior dynamics in all datasets even if behavioral samples are intermittently available in the training data.

Nonlinear DPAD can perform accurately and better than linear DPAD even when as little as 20% of training behavior samples are kept. a , The 3D reach task. b , Examples are shown from one of the joints in the original behavior time series (light gray) and intermittently subsampled versions of it (cyan) where a subset of the time samples of the behavior time series are randomly chosen to be kept for use in training. In each subsampling, all dimensions of the behavior data are sampled together at the same time steps; this means that at any given time step, either all behavior dimensions are kept or all are dropped to emulate the realistic case with intermittent measurements. c , Cross-validated behavior decoding accuracy (CC) achieved by linear DPAD and by nonlinear DPAD with nonlinearity in the behavior readout parameter C z . For this nonlinear DPAD, we show the CC when trained with different percentage of behavior samples kept (that is, we emulate different rates of intermittent sampling). The state dimension in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak decoding in training data. Bars, whiskers, dots, and asterisks are defined as in Fig. 2b ( N  = 35 session-folds). d , e , Same as a , c for the second dataset, with saccadic eye movements ( N  = 35 session-folds). f , g , Same as a , c for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). h , i , Same as a , c for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, the first 16 latent state dimensions are learned using the first two optimization steps and the remaining dimensions are learned using the last two optimization steps (that is, n 1  = 16).

Extended Data Fig. 9 Simulations suggest that DPAD may be applicable with sparse sampling of behavior, for example with behavior being a self-reported mood survey value collected once per day.

a , We simulated the application of decoding self-reported mood variations from neural signals 40 , 41 . Neural data is simulated based on linear models fitted to intracranial neural data recorded from epilepsy subjects. Each recorded region in each subject is simulated as a linear state-space model with a 3-dimensional latent state, with the same parameters as those fitted to neural recordings from that region. Simulated latent states from a subset of regions were linearly combined to generate a simulated mood signal (that is, biomarker). As the simulated models were linear, we used the linear versions of DPAD and NDM (NDM used the subspace identification method that we found does similarly to numerical optimization for linear models in Extended Data Fig. 1 ). We generated the equivalent of 3 weeks of intracranial recordings, which is on the order the time-duration of the real intracranial recordings. We then subsampled the simulated mood signal (behavior) to emulate intermittent behavioral measures such as mood surveys. b , Behavior decoding results in unseen simulated test data, across N  = 87 simulated models, for different sampling rates of behavior in the training data. Box edges show the 25 th and 75 th percentiles, solid horizontal lines show the median, whiskers show the range of data, and dots show all data points ( N  = 87 simulated models). Asterisks are defined as in Fig. 2b . DPAD consistently outperformed NDM regardless of how sparse behavior measures were, even when these measures were available just once per day ( P  < 0.0005, one-sided signed-rank, N  = 87).

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Sani, O.G., Pesaran, B. & Shanechi, M.M. Dissociative and prioritized modeling of behaviorally relevant neural dynamics using recurrent neural networks. Nat Neurosci (2024). https://doi.org/10.1038/s41593-024-01731-2

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DOI : https://doi.org/10.1038/s41593-024-01731-2

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