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
Mind or brain? If you want to understand the mind then something like cognitive neuroscience or psychology would probably be better for you.
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.
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.
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.
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.
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.
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.
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.
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.
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.
It's this kind of attitude that creates this kind of response: 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.
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.
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.