Coggr

Just recently been thinking about “computational modeling”, and I’ve been thinking more and more that the important thing is not so much the “computational” part of modeling (the fact that you’ve simulated a model on a computer), but rather the mathematical model that is so implemented.  I think that ultimately we’re interested in having models of how things work.  In this case, I think most of the “understanding” is in the models themselves, as stated mathematically, not in a particular implementation of the model.  Yes, computational implementation is important because: 1) it keeps us honest by forcing us to make our models physically realizable (and therefore explicit and consistent), and 2) it can reveal aspects of our model that might not be apparent to us initially.  But my sense is that the mathematical model is more important than its “computational” realization.

I just finished “fast reading” Christof Koch’s book “Biophysics of Computation”, i.e. reading it but not stopping to understand concepts I didn’t get on the first pass, making sure to understand equations, etc.  The book gives a great description of how various aspects of the brain can be modeled mathematically, and how the brain might carry out various “computations” in doing its job.

I’m personally interested in building mathematical models of the brain, and due to the vast complexity of the brain we are forced to make major simplifications in our models of it.  This book gives great information on potentially relevant aspect of neural functioning, which helps a modeler make informed decisions about which simplifications can / should be made.

In addition to a mathematical / computational perspective, the book also more generally gives a great description of what we know about how the brain works at a low level.

Just a quick post on typesetting mathematics with LaTex.  This summer, in the process of brushing up on a few things, I wanted to make a math study guide, and found that Microsoft Word doesn’t do a great job at math typesetting.  I looked at several other programs that do this (many of them require purchasing a license), and finally came upon the standard, “professional” (and free) solution – LaTex.

LaTex converts text markup (which is relatively simple) into beautifully-formatted formulas, and there are many great, free implementations of it.  You can learn about it generally at http://www.latex-project.org/ and there is a good free implementation at http://miktex.org/.  As an example I’ve also included the LaTex source and resulting pdf for my math study guide.

Two posts ago I asked a question about neural synchronization – if a group of neurons are firing synchronously, don’t we lose information?  In other words, if many neurons are all saying the same thing, do we really gain anything by having them all say it?  Additionally, if a group of neurons is “busy” firing synchronously, wouldn’t that preclude them from doing anything else or being responsive to other inputs?  Synchronization appears to be an important property of the brain, so to try to get a better understanding of how it might work, I read several articles (Fries, 2005; Schnitzler & Gross, 2005; Singer, 1993), along with Gyorgy Buzsaki’s book ”Rhythms of the Brain” (2006) and I now have a better idea of how neural synchronization might work, and the purposes it might serve.

The most critical piece I was missing is that neural synchronization does not necessarily mean that a group of neurons are all firing in sychrony (i.e. every neuron in the group firing in every cycle of the oscillation).  Instead, it could be that the membrane potential of the synchronized neurons are oscillating together.  This would effectively result in a synchronized group varying in how excitable (likely to fire) they were, together - giving windows when the neurons were all easily excitable, and periods where they were not very excitable at all.  This would allow neurons to be synchronized and still have the information that gets transmitted be interesting – a spike could still mean something beyond “we are all synchronized”.

Another mistaken idea I thought I had heard was that “synchronization allows neurons to communicate / transmit information.”  This didn’t make sense to me because it doesn’t seem like you need synchronization to communicate – a single neuron can fire and send out an action potential to other neurons regardless of what other neurons are doing.  The missing piece was that synchronization isn’t needed for communication per se, but rather synchronization might allow for selective communication.  There is a vast anatomical connectivity between neurons, and it seems likely that only certain subsets of neurons need to be communicating at a given time.  If a “sending” and “receiving” group are oscillating together, then there will be periodic “windows” when the receiving neurons are excitable enough to receive information.  At other points in the cycle, the receiving neurons would likely be unresponsive to input because they would be at a “trough” in their excitability (they would be hyperpolarized).  Thus, only neurons which were synchronized with the receiving neurons would be able to send to them.

From the above sources, a few of the possible functions of synchronization might be:

• “Binding” disparate information together.  If neurons in separated areas representing related information are synchronized, and if another group of neurons which is “interested” in this information is also synchronized with them, then the “interested” area can receive information only from the related, relevant areas, and not receive information from other currently unrelated neurons which have anatomical connections to it.
• Selective communication.  As mentioned above (and in some ways similar to the first point), synchronization could generally allow groups of neurons to selectively communicate and filter out the “noise” of unrelated but anatomically connected other neurons.
• Greater impact.  Since in many cases it takes many post-synaptic potentials to cause a neuron to fire, the effect of one or a few neurons firing an action potential at a target neuron may have little effect on the target.  However, if many neurons are synchronized and fire at the same time, they can have a much greater chance of pushing the target over firing threshold.
• Facilitating synaptic changes / plasticity.  There is evidence to suggest that some / many aspects of synaptic plasticity require many incoming post-synaptic potentials in a very short time window in order to occur.  Many “upstream” synchronized neurons firing at the same time would likely have a much greater chance of effecting synaptic changes than more spread out, unsynchronized firing.

A few other interesting points I noticed:

• Relativley high frequency oscillations seem to be used for small groups of synchronized neurons, and lower frequency / slower oscillations may be used for larger synchronized groups.  This may be due to the mechanics of synchronization such as longer axonal transmission times over longer distances, more “links in the chain”, etc.
• One of the potential challenges to synchronization is that, in order to stay “in phase”, the transmission time from each sending neuron to each receiving neuron needs to be very close to the same.  Amazingly, some evidence suggests that some networks seem to be tuned so that longer-distance axons are more heavily myelinated, resulting in faster conduction speeds for longer distances, and thus very close latency between closer and further connections (such as from the thalamus to the cortex) (Salami et al., 2003).
• Magnetoencephalography is one useful tool for measuring synchronization, since it has the time resolution needed to detect the rapid voltage changes.

References

Buzsaki, G. (2006). Rhythms of the brain. New York: Oxford University Press.

Fries, P. (2005). A mechanism for cognitive dynamics: Neuronal communication through neuronal coherence. TRENDS in Cognitive Sciences, 9, 474-480.

Salami, M. et al. (2003). Change of conduction velocity by regional myelination yields constant latency irrespective of distance between thalamus and cortex. Proceedings of the National Academy of Sciences of the United States of America, 100, 6174-6179.

Schnitzler, A., & Gross, J. (2005). Normal and pathological oscillatory communication in the brain. Nature Reviews Neuroscience, 6, 285-296.

Singer, W. (1993). Synchronizatoin of cortical activity and its putative role in information processing and learning. Annual Review of Physiology, 55, 349-374.

Are there emotions which are “incompatible”?  In other words, could experiencing one emotion make it harder to experience another emotion?  It seems like we have anecdotal evidence on both sides.  On one hand, it can seem very difficult to get someone who is feeling sad or depressed to enjoy anything – feeling one emotion (sadness or depression) seems to make it harder to feel another emotion (happiness).  On the other hand, we’ve probably all had experiences where we felt multiple emotions at the same time – such as feeling sad and happy at the same time about something (and we even have the word “bittersweet”).

Emotions are complicated, and both of these views are probably right to a degree – we can experience more than one emotion at a time, but there are also likely to be situations where feeling certain emotions makes it less likely to feel other emotions.

This is more than just a theoretical question – it is relevant to understanding ways of regulating our emotions, ways of changing our emotions when they are unhealthy.  For example, when we are stressed out and worried about something, it would be nice to be able to get rid of that anxiety.  Similarly, if we are angry or depressed, we might like to change those feelings.

Could the intensity of a negative emotion be reduced by getting ourselves to feel a different, “incompatible” emotion?  In particular, could sadness be incompatible with anxiety?  Recently, when I have been especially anxious about something, I’ve been trying to imagine that “the worst possible outcome” (my fear) has already happened, and then to get myself to feel sad about it.  Sometimes this has had little effect on the anxiety, but more than a few times the resulting sadness has been accompanied by my anxiety going away and a sense of calm.

My guess about this is that there are differences between the emotional “circuits” responsible for dealing with things you can do something about vs. things you can’t do anything about.  Once there is “nothing that can be done”, the “sadness circuits” take over, and the “anxiety circuits” stop being active.  The truth is probably more complicated than this, but there might be something to it.

Neural synchronization occurs when many neurons fire together, with their timings aligned, often periodically with a given frequency.  Synchronization is clearly very pervasive in the brain (in EEG recordings of “brain waves” and different brain “rhythms”, for example).  However, I imagine we are still a ways away from having an authoritative explanation of the purpose of this synchronization.

In some cases the need for synchronizatoin seems clear – for example rhythmic movements (such as walking) seem like they would require rhythmic, synchronized neural firing.  But synchronization seems to be much more prevalent than that, and likely plays a critical role in many / most aspects of brain functioning.  As I’m writing this I haven’t read much on synchronization, and am about to read “Rhythms of the Brain” by Gyorgy Buzsaki, which looks to be a great introduction to neural synchronization, to try to get some sense of what we know about this.

One of the initial questions I have is (from an information perspective), if many neurons are firing in the same pattern, don’t we lose information?  In other words, isn’t much of the firing redundant?  I’m interested to find out how synchronization fits into an information processing framework.

How much math do you need to know for a career in psychology?  The stereotype seems to be that psychology and math are unrelated, but “surprisingly” I’ve found that many areas of psychology can involve quite a bit of math.  This of course can be good or bad news, depending on your love (or hate!) of mathematics.  Here’s my take, as a former undergrad Computer Science major and now Ph.D. student in Psychology, on a few of the roles one could pursue in psychology, and the probable math involvement of each:

• Therapist:  As a therapist, relatively little math seems required, although some basic statistics might be helpful to be able to understand the results of research articles.
• Clinical / social / experimental psychology researcher:  These areas require some knowledge of statistics in order to correctly design and analyze experiments.  In some cases more involved statistical knowledge is necessary for building more complex statistical models, such as structural equation modeling or mediation / moderation relationship modeling.
• Computational brain modeling:  This is an area I’ve gotten more interested in / involved with lately, and I’ve found that it can require extensive math, as it involves modeling complex physical systems.
• Neuroscientist:  Potentially extensive math, depending on type of research.  At some level, models involve, for example, complex differential equations governing the different mechanisms involved.  Additionally, math can be involved in analyzing brain imaging data, such as data from fMRI.
• General cognitive science researcher:  This is a broad area, but may involve mathematical models of perceptual and cognitive processes, with or without explicit modeling of the brain.

Analysis of variance (ANOVA) and linear regression are two of the most popular statistical techniques used in behavioral research.  I’ve often come across statements to the effect that ANOVA and linear regression are really “the same thing” – that in some sense they are special cases of something called “the general linear model”.  And yet, in many statistics texts there are few details given of exactly how this works – exactly how, say, ANOVA is part of a general linear model.

I recently came across a classic article that addresses this point directly, Jacob Cohen’s “Multiple Regression as a General Data-Analytic System” (1968).

Cohen has written many articles on the proper use of statistics in psychological research (and in research in general).  Statistics clearly plays an important role in correctly designing and analyzing experiments, but many researchers quite naturally are not statisticians, and there is often a gap between statistical knowledge (of statisticians) and the everyday application of statistics by researchers in other fields.  Cohen has made many contributions to address this divide, by bringing a knowledge of mathematical statistics to bear on research practices.

In this article Cohen shows that analysis of variance, analysis of covariance, and other statistical techniques can be converted into equivalent cases of multiple regression.  Once he has given details about how this can be done (including how to carry out the requisite hypothesis tests, etc.), he goes on to discuss why multiple regression is ultimately a more flexible and powerful technique, and thus can be viewed as a “general linear model”.

One of the main advantages of multiple regression is that it can accommodate a wide range of situations easily – interval/ratio variables, categorical variables, interactions, and so forth – all in a single model.  ANOVA and other specific techniques (such as t tests) are specialized in that they can only handle a fairly limited range of situations.  Multiple regression, on the other hand, can handle many different cases, all at the same time in a single model.

Reference

Cohen, J. (1968). Multiple regression as a general data-analytic system. Psychological Bulletin, 70, 426-443.

Last month I wrote about recreating a brain simulation model in an article I read about (Hamker, 2005).  As I discussed there, the Hamker article describes a computational firing-rate model of several brain areas thought to play a role in spatial and non-spatial attention.  As I’m very interested in building computational brain models myself and am just getting started, I am attempting to recreate their simulation using Matlab.

This is still a work in progress, but I now have an initial version that models input and two of the brain areas, V4 and IT.  The simulation also displays graphs of the resulting firing rates over time for each simulated neuron.  The Matlab files are available here.

The biggest issue I’ve come across so far has been optimizing performance.  In my initial version of the simulation, solving the equation system took a prohibitive amount of time – simulating 5 neurons for 3 milliseconds of model time took several minutes, and the system ultimately needs to model around 250 neurons for 3000 milliseconds.  Ultimately the problem turned out to be the differential equation solver algorithm.  I initially used Matlab’s default “ode45″ solver, since the model appeared to be simple and I didn’t expect the system to be “stiff”, but switching to the “ode15s” solver (a “stiff” solver) drastically reduced the execution time.

Reference

Hamker, F. H. (2005). The reentry hypothesis: The putative interaction of the frontal eye field, ventrolateral prefrontal cortex, and areas V4, IT for attention and eye movement. Cerebral Cortex, 15, 431-447.