Is Computational Neuroscience the Key to Understanding the Mind?

[StJohnRiver]

I've always want to understand the mind and its inner working. After lots of consideration I've narrowed my choice down to computational neuroscience. I know it's a hardcore science but I'd like to know more about it. How much math is required? What disciplines would suit it best?

Every suggestions count

 

[Ryan_m_b]

Mind or brain? If you want to understand the mind then something like cognitive neuroscience or psychology would probably be better for you.

 

[Number Nine]

There are two points here. The first, which is rather blunt (but important, I think), is that you really shouldn't be narrowing down your interests to this extent without being familiar with the field already, in which case you would have decent understanding of the prerequisites. Myself, I became interested in computational neuroscience as a result of my interest in the biological underpinnings of reinforcement learning and working memory processes (largely, basal ganglia and PFC circuitry). Having immersed myself in the literature already, I found the mathematical approach to subject to be the most intuitive and informative, so I naturally gravitated to computational neuroscience (having already had a background in mathematics helped).

As far as the math goes, there are a few prerequisites that are absolutely non-negotiable, and a few that depend on what level of abstraction you're interested in (neuron modelling, systems/network modelling, cognitive modelling, etc).

The essential prerequisites are a solid understanding of calculus (up to multivariable and vector calculus), basic linear algebra, ODE's (you can go quite a ways in neuroscience without PDE's), and probability/statistics. For all practical purposes, some knowledge of dynamical systems is essential (in many branches of computational neuroscience, you can't even get your foot in the door without it). Additionally, you'll want to be very comfortable with at least one programming language (Matlab and python are very common for "general purpose", though there are software packages dedicated specifically to certain types of neuronal modelling).

In certain areas like computational vision research, where the goal is to understand exactly how a population of neurons are coding for a specific stimulus, the prerequisites for statistics are much higher, and you'll want all the experience with generalized linear models and bayesian statistics that you can get. Biologically realistic neuron modelling, in turn, usually involves much more sophisticated theory in differential equations and dynamical systems (though you should learn all you can about both regardless of what you're doing).

In computational cognitive neuroscience (my interest), the prerequisites get a little trickier because the models are much more abstract, and so you can draw upon some pretty surprising areas of mathematics (I read a paper recently that applied some fairly deep concepts in differential geometry to the modelling of visual processes...and was published in a journal of physiology, go figure). Some people (often in cognitive psychology) are only interested in throwing together the occasional neural network, in which case some basic calculus and linear algebra is enough. More rigorous work tends to draw very strongly from dynamical systems, and references biophysically realistic modelling enough that you'll need to understand every that was said about it above.

 

[madness]

Interesting post Number Nine. I've found the amount of maths required in computational neuroscience to be disappointingly low, but then I've come from a strong mathematical background. More correctly, I'd probably say it's been disappointingly hand-wavy, like you would expect from an engineer rather than a mathematician. It's certainly at the level that people with backgrounds in engineering or computer science can get straight into it.

To be fair, some of the maths is quite technical (e.g. weiner series for cell responses), but I've never once seen it explained properly in a computational neuroscience setting.

 

[Number Nine]

Interesting post Number Nine. I've found the amount of maths required in computational neuroscience to be disappointingly low, but then I've come from a strong mathematical background. More correctly, I'd probably say it's been disappointingly hand-wavy, like you would expect from an engineer rather than a mathematician. It's certainly at the level that people with backgrounds in engineering or computer science can get straight into it.

To be fair, some of the maths is quite technical (e.g. weiner series for cell responses), but I've never once seen it explained properly in a computational neuroscience setting.

This is, unfortunately, true. If you look in the right areas, the mathematics can be far more abstract and complex than you would expect, but, fundamentally, the authors are interested in doing neuroscience, not mathematics.

 

[atyy]

To be fair, some of the maths is quite technical (e.g. weiner series for cell responses), but I've never once seen it explained properly in a computational neuroscience setting.

Do you consider http://cbcl.mit.edu/people/poggio/journals/palm-poggio-SIAM-JApplMath-1977.pdf or http://www.stanford.edu/~boyd/papers/pdf/fading_volterra.pdf rigourous or not? The former is written by neurobiologists, and the latter is cited in neurobiology papers such as http://www.ncbi.nlm.nih.gov/pubmed/12433288.

 

[madness]

Do you consider http://cbcl.mit.edu/people/poggio/journals/palm-poggio-SIAM-JApplMath-1977.pdf or http://www.stanford.edu/~boyd/papers/pdf/fading_volterra.pdf rigourous or not? The former is written by neurobiologists, and the latter is cited in neurobiology papers such as http://www.ncbi.nlm.nih.gov/pubmed/12433288.

Thanks for the links. Looks fairly rigourous to me! At the time I was learning this stuff, I spent a while looking for proper references, but they all seemed to link back to the unavailable original papers. There certainly are some decent mathematical treatments out there in computational neuroscience, but generally they come from a slightly separate field called mathematical neuroscience (http://www.mathematical-neuroscience.com/). Mathematical neuroscience generally amounts to the application of dynamical systems theory to neurons and neural networks.

 

[jackmell]

. How much math is required? What disciplines would suit it best?

Every suggestions count

Non-linear differential equations and integral equations in my opinion best model the brain. Unfortunately that's tough, you have to love math to do well in the field, pracitially be a math major because other fields of math affect the subject. Therefore, I believe to excel in computational neuroscience, I believe you need to love math, major in it, then take interest in neuroscience.

 

[madness]

Non-linear differential equations and integral equations in my opinion best model the brain. Unfortunately that's tough, you have to love math to do well in the field, pracitially be a math major because other fields of math affect the subject. Therefore, I believe to excel in computational neuroscience, I believe you need to love math, major in it, then take interest in neuroscience.

I have to disagree slightly. It is only a small portion of computational neuroscience which you need such a high level of maths for. In general, the field is full of people trained in engineering and computer science, as well as some with physics and maths backgrounds. Even psychologists sometimes enter the field. Of course it is helpful to have a strong maths background. But it is helpful to have a huge variety of different backgrounds that no one person actually has when they enter the field. Pretty much anyone starting the field is going to have to learn some biology, machine learning, statistics, computational modelling etc. I would hate for someone reading your post to be put off into thinking they can't enter the field because they don't have a degree in maths, because it isn't true.

 

[Pythagorean]

What Jack speaks of is often the Physicist's route to comp neuro. But physicists don't need to be mathematicians to do nonlinear science.

 

[jackmell]

Note I said "excel". That's the difference. Oh, you can study computational neuroscience without a lot of math, without a passion for math, and just do meodicre work but that's not excelling, not ground-breaking, not rocking the foundation of neuroscience. There is in my opinion only one thing in the entire Universe that can create the apparent unlimited diversity of the human mind and that is non-linear dynamics. It has to be the key to understanding the brain, consciousness, and mind. But it will take more than just a cursory understanding to turn this key. Something more is needed and that is where the passion not for neuroscience but math will come in.

So I'm not a science advisor in here, just an ordinary Joe and my ordinary Joe opinion is in order to rock the foundation of neuroscience, have a passion and love for math and neuroscience, then major in math.

 

[Pythagorean]

I find that to be a fairly narrow-minded view...

 

[madness]

What Jack speaks of is often the Physicist's route to comp neuro. But physicists don't need to be mathematicians to do nonlinear science.

It was my route too. I studied physics first, and in a strongly mathematical physics programme where I took topology and whole range of other maths courses on the side. So far I haven't had much of a chance to use any advanced maths in computational neuroscience. In fact it has been shown that using equations in the biology field massively reduces the number of citations you will receive.

http://www.pnas.org/content/109/29/11735.full

I think to some extent you have to make a decision in computational neuroscience as to whether you will actually work with biological data and sacrifice using too much mathematics or stick with mathematics and risk having no biologists take you seriously.

 

[Pythagorean]

My route too. You don't have to appeal to biologists though. Think of them as the ones who are providing experimental data for you. We have journals like Chaos and Physics Review E. Our audience really is nonlinear scientists.

 

[jackmell]

I think to some extent you have to make a decision in computational neuroscience as to whether you will actually work with biological data and sacrifice using too much mathematics or stick with mathematics and risk having no biologists take you seriously.

You mean risk having no biologist understand you. Biologist in general do not like math. I know because I use to be a biology major but changed to Chemisty. Biology students in general are frightened to death of DEs and yet the math does such a wonderful job of explaining many of the puzzling phenomena in biology. For example, why are humans so different from apes but share 98% of DNA? You not going to have any chance of understanding why without understanding Catastrophe Theory and that involves non-linear differential equations. Ok, how about neurons. They have history you know. Their current behavior is dependent on their past behavior. Well, integro-differential equations have history too.

 

[madness]

You mean risk having no biologist understand you. Biologist in general do not like math. I know because I use to be a biology major but changed to Chemisty. Biology students in general are frightened to death of DEs and yet the math does such a wonderful job of explaining many of the puzzling phenomena in biology. For example, why are humans so different from apes but share 98% of DNA? You not going to have any chance of understanding why without understanding Catastrophe Theory and that involves non-linear differential equations. Ok, how about neurons. They have history you know. Their current behavior is dependent on their past behavior. Well, integro-differential equations have history too.

No I really do mean risk no biologist take you seriously. There is a strong feeling amongst biologists that people coming from physics and maths just don't understand the complexity of biology and that the simplifications they make to model biological systems are a result of their ignorance or arrogance. Many seem to take the point of view that the use of advanced maths in biology is just a waste of time.

 

[Pythagorean]

You not going to have any chance of understanding why without understanding Catastrophe Theory and that involves non-linear differential equations.

It's this kind of attitude that creates this kind of response:

There is a strong feeling amongst biologists that people coming from physics and maths just don't understand the complexity of biology and that the simplifications they make to model biological systems are a result of their ignorance or arrogance.

But more importantly, it's false that you need to know catastrophe theory to understand the dynamics of genetic expression. The nonlinear perspective is only one perspective; understanding comes form grasping multiple perspectives. Confirmation bias comes from favoring one approach as the approach. But there is no panacea.

 

[madness]

But more importantly, it's false. The nonlinear perspective is only one perspective; understanding comes form grasping multiple perspectives. Confirmation bias comes from favoring one approach as the approach. But there is no panacea.

Yes, I agree it's false. I think it comes from a mistaken view that everything in physics is simple and linear and can be solved exactly. In other words, it come from a lack of understanding of what mathematical modelling is actually useful for.

Without concrete quantitative predictions, there's not much to verify or falsify in an experiment. Sometimes I feel like biology papers are just stabbing around in the dark with no real hypothesis to test (or at least no hypothesis that is strictly defined before you get the results of the experiment). I definitely believe mathematical models are an important part of understanding the brain.

 

[Pythagorean]

Sorry, I meant jack's statement was false.

 

[Pythagorean]

For example, why are humans so different from apes but share 98% of DNA?

Additionally, there's other problems with this kind of statement:

"You share 98% DNA with monkeys"
"You share 50% DNA with your siblings"

no catastrophe theory needed to explain the problem with intuition here... the problem is that the statements are ambiguous. The are both true in their original context. A biologist does not need to know much mathematics (let alone catastrophe theory) to demonstrate to a student that there's two different ways to count groups of things.

 

[Pythagorean]

OP, I think madness answered your question best:

Interesting post Number Nine. I've found the amount of maths required in computational neuroscience to be disappointingly low, but then I've come from a strong mathematical background. More correctly, I'd probably say it's been disappointingly hand-wavy, like you would expect from an engineer rather than a mathematician. It's certainly at the level that people with backgrounds in engineering or computer science can get straight into it.

To be fair, some of the maths is quite technical (e.g. weiner series for cell responses), but I've never once seen it explained properly in a computational neuroscience setting.

Computational Neuroscience is a really new field and it's multidisciplinary as well as interdisciplinary. Consider the University of Waterloo in Canada. Their Computational Neuroscience institute is supported by several classic departments. It is directed by a computer science PhD and has research contributions from faculty/students in physics, mathematics, electrical engineering, and neurobiology.

All that it takes to be computational neuroscience right now is that you apply computational approaches to neuroscience problems. One approach is the deterministic nonlinear approach. It will depend on your university, though, so you should actually look at what kind of research people are doing, maybe even try looking at their papers or their websites. "Computational Neuroscience" is a huge, complex field, with lots of interacting disciplines but also little pockets of interest.

 

[mazinse]

You mean risk having no biologist understand you. Biologist in general do not like math. I know because I use to be a biology major but changed to Chemisty. Biology students in general are frightened to death of DEs and yet the math does such a wonderful job of explaining many of the puzzling phenomena in biology. For example, why are humans so different from apes but share 98% of DNA? You not going to have any chance of understanding why without understanding Catastrophe Theory and that involves non-linear differential equations. Ok, how about neurons. They have history you know. Their current behavior is dependent on their past behavior. Well, integro-differential equations have history too.

You can't make assumptions that biologists don't like math. When you have multidisciplinary fields like biochem, biophysics and bioengineer.

 

[atyy]

I do think that nonlinear differential equations are the only way to understand the brain. However, it must also be understood that formalisms that seem not related to nonlinear differential equations are intimately related. For example, the Volterra approximation is used when the dynamical system "resets" to the same state in the absence of a stimulus, ie. when the stimulus can be considered a "disturbance" about a fixed point. Similarly, although at the microscopic level, there may be a fundamental distinction between fixed points and limit cycles, if the experimentalist only measures a coarse-grained variable in which he is unable to distinguish the separate states in a limit cycle, the resulting coarse grained variable may be described by a probabilistic dynamical system. The basic message is that because the deterministic nonlinear dynamics of the brain are very high dimensional, if one is working in a regime in which an effective theory with fewer degrees of freedom provides sufficient accuracy for one's purposes, then one should use the effective theory. This in general requires guesswork and luck. Mostly luck - the systematic hardwork prepares you to take advantage of luck when it comes your way.

 

[jackmell]

Additionally, there's other problems with this kind of statement:

"You share 98% DNA with monkeys"
"You share 50% DNA with your siblings"

That I believe is misleading. When I say 98%, I mean 98% of the proteins coded by ape DNA can be found in humans. However (close to) 100% of the proteins coded by my brother's DNA can be found in me.

Now, I'm not sure that's the way to interpret that. If not, I'd appreciate someone saying so. But I believe that's what it means when we say 98% of the DNA found in apes can be found in humans. But I haven't researched it yet so you guys can jump on Jack if that's not right.

no catastrophe theory needed to explain the problem with intuition here... the problem is that the statements are ambiguous. The are both true in their original context. A biologist does not need to know much mathematics (let alone catastrophe theory) to demonstrate to a student that there's two different ways to count groups of things.

Ok, maybe don't need to know much about Catastrophe Theory. Couldn't hurt. Sides, I think you're missing my point entirely: the vast difference between humans and apes appear to be due to just a small number of genes. There was a paper recently that suggested three major changes in humans (musscle in jaw bone, a different brain protein, don't remember the other) had a tremendous affect in shaping human evolution 2 million years ago.

Keep in mind, I do not believe the answer to this question has been firmly established. I'm only proposing a possible answer based on dynamics:

Why or more precisely, how can a small change create a very large difference? One possibility is that perhaps the genetic dynamics trajected through a critical point. It could have just as eassily been 99.9% similarity and still effected a very big change because of the qualitative change that often occurs when passing through a critical point. My proposal is that maybe humans emerged from the ape lineage through a critical-point transition in genetic dynamics. But just saying that is not easily understood. But if you know non-linear dynamics, it then becomes much easier to understand. Likewise, the entire principle of "Punctuated Equilibrium" is beautifully explanable by critical-point non-linear dynamics.

 

[jackmell]

You can't make assumptions that biologists don't like math. When you have multidisciplinary fields like biochem, biophysics and bioengineer.

You know I'm not a smart-ellic. But if I was I could, but woundn't, set up a poll: How many biologist in here like DEs?

 

[Pythagorean]

Ok, maybe don't need to know much about Catastrophe Theory. Couldn't hurt. Sides, I think you're missing my point entirely: the vast difference between humans and apes appear to be due to just a small number of genes. There was a paper recently that suggested three major changes in humans (musscle in jaw bone, a different brain protein, don't remember the other) had a tremendous affect in shaping human evolution 2 million years ago.

I get your point; I just wanted you to concede that it doesn't require catastrophe theory to understand, but furthermore, I want you to acknowledge that it might give geometers a bad name in the eyes of biologists when they hear that talk. Especially since, you know, the biologists are the ones that figured out gene expression...

In fact, the amount of genetics we share with all vertebrae (let alone other hominids) is an amazing example of diversity that can occur with small gene sets. One interesting example is the hox genes, which decide vertebrae flavor. In snakes, during development, their genetic dynamics get "stuck" in a "thorax loop"

One important thing to note in [a snake's] anterior vertebrae is that they have both characters of cervical and thoracic indicating information required for thoracic formation have extended into neck.Hence the whole trunk resembles as one extended thorax.As mentioned earlier Hox genes are involved in specifying type of vertebrae

http://www.ncbi.nlm.nih.gov/pubmed/10365960?dopt=Citation

We have the evo-devo story:

for fun (it's a fish! no a reptile! no it's a panda bear! no a human!):
http://www.youtube.com/watch?list=SP0537ADF3F3372F8A&v=wFY_KPFS3LA&feature=player_detailpage#t=10s

 

[Pythagorean]

I do think that nonlinear differential equations are the only way to understand the brain. However, it must also be understood that formalisms that seem not related to nonlinear differential equations are intimately related. For example, the Volterra approximation is used when the dynamical system "resets" to the same state in the absence of a stimulus, ie. when the stimulus can be considered a "disturbance" about a fixed point. Similarly, although at the microscopic level, there may be a fundamental distinction between fixed points and limit cycles, if the experimentalist only measures a coarse-grained variable in which he is unable to distinguish the separate states in a limit cycle, the resulting coarse grained variable may be described by a probabilistic dynamical system. The basic message is that because the deterministic nonlinear dynamics of the brain are very high dimensional, if one is working in a regime in which an effective theory with fewer degrees of freedom provides sufficient accuracy for one's purposes, then one should use the effective theory. This in general requires guesswork and luck. Mostly luck - the systematic hardwork prepares you to take advantage of luck when it comes your way.

I completely agree that many of the formalisms in dynamical systems (especially those that don't obey the principle of superposition) highlight an important cognitive flaw in thinking about problems (the tendency to dichotomize things into linear trends).

When I can, I see the whole world in terms of nonlinear dynamics: from my social relationships to the physics of the atmosphere when I'm sailing, to my biophysical signatures when I'm being mindful. Superposition is the special case. I just wanted to remove the elitist tone is all.

 

[atyy]

I completely agree that many of the formalisms in dynamical systems (especially those that don't obey the principle of superposition) highlight an important cognitive flaw in thinking about problems (the tendency to dichotomize things into linear trends).

When I can, I see the whole world in terms of nonlinear dynamics: from my social relationships to the physics of the atmosphere when I'm sailing, to my biophysical signatures when I'm being mindful. Superposition is the special case. I just wanted to remove the elitist tone is all.

Good thought!