I am wondering (and searching with no success) if there are any examples of differential equations in psychology?

I mean, no tutorial explaining what is differenetial equations or even partial differential equations, but a research (like in emotions or social psychology, or cognitive psychology) by psychologists that use these equations to model the data?

[update, 29.05.2016] Stack Community always reminds me that the quality of the answers depends on the question asked. Just as a good question is half of the problem.
I'll try to make myself clear with an example. I thought about an article in psychology, say about cognitive dissonance, emotions, coping or else, that use differential equations to decribe phenomenon without invoking physics or chemistry or other hard science. Article by Deboeck & Bergeman introduces pendulum model(Christiaan's comment). But I don't have to refer to physics when doing ordinary squares regression, so that's why I thought it is possible to use differential equations without explicite references to physical models.

June 2, 2016: I never suggested we could "use differential equations without explicit reference to physical models". I have always said the same: the mathematical interpretation of cellular processes has lost prominence (owing to further research). The biological processes obviously remain; the interpretation has not been able to hold. Teresa Pelka

  • 1
    $\begingroup$ a google search yields quite a bunch of hits - first hit (on my pc that is): ncbi.nlm.nih.gov/pubmed/23527605 - search terms - your title :-) 'differential equations in psychology' $\endgroup$
    – AliceD
    May 23, 2016 at 8:42
  • $\begingroup$ @Christiaan as far as I know Internet browsers can provide divergent search answers based on your previous activity, which is unique. But as I said I did my own search and research and found this article before heading to Stack. $\endgroup$ May 29, 2016 at 9:40
  • $\begingroup$ @huh I think you should post that as an answer. $\endgroup$ Jun 2, 2016 at 8:46

3 Answers 3


The Hodgkin and Huxley model of neuronal firing is based on non-linear differential equations. A significant portion of research on sensation and perception is based on such models.


A large and notable area within psychology to which differential equations are central is dynamical systems theory. It is widely applied to cognitive science. Differential equations are used to describe (model) the states on one or more dimensions that a system can take which changes over time. Dynamical systems have been widely used to model cognition.

Many applications of dynamical systems theory in psychology accordingly concern questions classically within the scope of cognitive psychology, but there are examples from other areas like social and developmental psychology.

Mind that dynamical systems theory originated within physics, specifically mechanics, and that accordingly it has been widely used to model material processes. However the use of models from physics is not uncommon in psychology (think of conceiving the circadian rhythm as a harmonic oscillator (pendulum) or Lorenz' psychohydraulic model of motivation) and the states of some modelled system can be highly abstract and ›non-physical‹.

Since dynamical systems have been discussed here to some extent, I am referring you to this and this question instead of providing more information here.

Let me however second the recommendation of Schöner's »Dynamical Systems Approaches to Cognition« in Sun (ed.): Cambridge Handbook of Computational Psychology (2008) there for a decent overview, and point out a passage on nonlinear dynamical system modeling in R.W.J. Neufeld's chapter »Mathematical and Computational Modeling in Clinical Psychology« in Busemeyer et al. (eds.): The Oxford Handbook of Computational and Mathematical Psychology (2015).


The Hodgkin-Huxley is an observation on cellular processes. It cannot embrace structures of just a few neurons, as a neural schema. For action potentials, we could say they are all-or-none options. Singular potentials yet are ignored by the nervous system by biological default, and we cannot discard the role of graded potentials, in the work of the system.

Of the Hodgkin-Huxley, it is the hypothesis that cell membranes had ion channels to remain valid and affirmed (Erwin Neher, Bert Sakmann, and Roderick MacKinnon). However, singular cell incitation may test inconclusive (Vander et al., 1985), therefore mathematical models have been mostly abandoned, for life sciences.

Mathematical calculation on the human mind is not considered likely either, see Damasio, A. 2000. The Feeling of What Happens: Body, Emotion and the Making of Consciousness. MIT Press.


There are no equations in psychology, unless you take an equation for a figure of speech. :)

"Mathematical psychology" or psychometrics would reflect on data, and data presentations may be affected by the way measurements are gathered and analyzed.



Ability to sum up, make an integer, or a differential equation on machine output does not mean capableness to perceive a human being mathematically. (Further, I really do not know why try to picture humans as sets of mathematical figures.)

"Mathematical predictability" about a person could be only a manipulated result, via incentives or stimuli. Obviously, such manipulation would be ethically disagreeable as well as scientifically useless.

Update, June 5 2016: Artem Kaznatcheev's marking the question as answered with artificial network and hologram semantics theory is a misunderstanding. This question is about psychology as concerned with emotions, social psychology, or cognitive psychology:

"I mean (...) research (like in emotions or social psychology, or cognitive psychology)", is the wording.

For example, the holoword theory as indicated among the "answers" cannot apply to natural information processing:

A major stumbling block on the road to a unified account of lexical processing has been the lack of a unified representation for words that incorporates information about orthography, phonology, semantics, and syntax.


No such unified account is likely about humans. Human information processing is feedback-oriented. The observation does not encourage pursuits for a uniform neural model for language. Human speech and language are a natural faculty by a human person.

Personality factors matter already with language cerebral patterns. The patterns and their networks are not innate. They are biological connections everyone needs to make on his or her own. In consequence, there cannot be a universal (unified, uniform) neural format for language.


Likewise, "mind modularity”, “language universals”, or “notional matrices” cannot explain the language information pool phenomena of the Warrens experiment. The fact does not make room for mathematics or physics. Not only with generative phonology, mathematical values or symbolics that pertain with physics would be an arbitrary ascription, to think about live human brain tissue.


The question wording also is,

"I am wondering (and searching with no success)"

This is because there are no such equations. I'm fine with that, and now I've done all I could to explain: if you make a computer, you naturally mind this is a computer you make, not a human brain. Here's the bibliography :)


  • $\begingroup$ I think this takes a very narrow view of Psychology and "life sciences". Roughly a 1/4 of my former Psychology department made regular use of equations in their research and maybe 1/10 used differential equations. $\endgroup$
    – StrongBad
    Jun 2, 2016 at 1:02
  • $\begingroup$ How would I take a "narrow view": this means 3/4 of your former department did not make regular use of equations in research, and 9/10 did not use differential equations. It is important to tell statistics and metadata from other specialisms and data. For example, if we use equations to present statistics on students and their progress, it does not mean we have equations in grammar. :) $\endgroup$ Jun 2, 2016 at 4:47
  • $\begingroup$ Incidentally, about 3/4 of published psychological research now appears to be irreproducible garbage science. $\endgroup$ Jun 2, 2016 at 8:28
  • $\begingroup$ @Christian, I happen to be critical about psychology, yet I'd never go into your model for evaluation (you say 3/4 published psychological research is "garbage science"). You say in your profile you have "unsavory love for Bayes factors". Do you mean you would apply posterior probability to yourself, historical events, or writing poetry? I have a joke about language and calculation here, find the word "intelligence" on page travelingrammar.com/2015/09/19/language-mapping-integration :) $\endgroup$ Jun 2, 2016 at 10:59
  • $\begingroup$ Let me be concise: You don't know seem to know what statistics, mathematical models, or differential equations are. You may also be confusing "naive skepticism" with "critical thinking". These look similar at a glance, but are actually quite different in that the latter requires one to know a great deal about what one is thinking about, whereas the former requires only that one remain entirely ignorant about the subject. And while I appreciate the attempt to connect with me on a personal level, Bayes factors do not produce posterior probabilities. $\endgroup$ Jun 2, 2016 at 11:48

Not the answer you're looking for? Browse other questions tagged or ask your own question.