In my previous post, I talked about how we probably needed a new worldview before we would be prepared to understand the brain. What I thought I would do here is to introduce what I think forms the worldviews of people who do dynamical systems (which forms a sizable contingent of the mathematical neuroscience community) and physics (in particular statistical mechanics and field theory). Having trained in physics and applied mathematics, served on the faculty in a math department and worked on biology, I've gotten a chance to see how these different groups view science. The interesting thing is that in many ways biologists and mathematicians can sometimes understand each other better than physicists and mathematicians. I was led to this belief after hearing Alla Borisyuk, who is an applied mathematician, exclaiming at a conference I helped organize in 2000 for young researchers, that she had no trouble talking to biologists but had no idea what the physicists were talking about.

The reason why biologists may have more in common with mathematicians than physicists is because unlike physics, biology has no guiding laws other than evolution, which is not quantitative. They rarely will say, "Oh, that can't be true because it would violate conservation of momentum," which was how Pauli predicted the neutrino. Given that there are no sweeping generalizations to make they are forced to pay attention to all the details. They apply deductive logic to form hypotheses and try to prove their hypotheses are true by constructing new experiments. Pure mathematicians are trained to take some axiomatic framework and prove things are true based on them. Except for a small set of mathematicians and logicians, most mathematicians don't take a stance on the "moral value" of their axioms. They just deduce conclusions within some well defined framework. Hence, in a collaboration with a biologist, a mathematician may take everything a biologist says with equal weight and then go on from there. On the other hand, a physicist may bring a lot of preconceived notions to the table (Applied mathematicians are a heterogeneous group and their world views lie on a continuum between physicists and pure mathematicians.) Physicists also don't need to depend as much on deductive logic since they have laws and equations to rely on. This may be what frustrates biologists (and mathematicians) when they talk to physicists. They can't understand why the physicists can be so cavalier with the details and be so confident about it.

However, when physicists (and applied mathematicians) are cavalier with details, it is not because of Newton or Maxwell or even Einstein. The reason they feel that they can sometimes dispense with details is because their worldviews are shaped by Poincare, Landau and Ken Wilson. What do I mean by this? I'll cover Poincare (and here I use Poincare to represent several mathematicians near the turn of the penultimate century) in this post and get to Landau and Wilson in the next one. Poincare, among his many contributions, showed how dynamical systems can be understood in terms of geometry and topology. Prior to Poincare, dynamical systems were treated using the tools of analysis. The question was: Given an initial condition, what are the analytical properties of the solutions as a function of time? Poincare said, let's not focus on the notion of movement with respect to time but look at the shape of trajectories in phase space. For a dynamical system with smooth enough properties, the families of solutions map out a surface in phase space with tiny arrows pointing in the direction the solutions would move on this surface (i.e. vector field). The study of dynamical systems becomes the study of differential geometry and topology.

Hence, any time dependent system including those in biology that can be described by a (nice enough) system of differential equations, is represented by a surface in some high dimensional space. Now, given some differential equation, we can always make some change of variables and if this variable transformation is smooth then the result will just be a smooth change of shape of the surface. Thus, what is really important is the topology of the surface, i.e. how many singularities or holes are in them. The singularities are defined by places where the vector field vanishes, in other words the fixed points. Given that the vector field is smooth outside of the fixed points then the global dynamics can be reconstructed by carefully examining the dynamics near to the fixed points. The important thing to keep track of when changing parameters is the appearance and disappearance of fixed points or the change of dynamics (stability) of fixed points. These discrete changes are called bifurcations. The dynamics near fixed points and bifurcations can be classified systematically in terms of normal form equations. Even for some very complicated dynamical system, the action is focused at the bifurcations. These bifurcations and the equations describing them are standardized (e.g. pitchfork, transcritical, saddle node, Hopf, homoclinic) and do not depend on all the details of the original system. Thus, when a dynamical systems person comes to a problem, she immediately views things geometrically. She also believes that there may be underlying structures that capture the essential dynamics of the system. This is what gives her confidence that some details are more important than others. Statistical mechanics and field theory takes this idea to another level and I'll get to that in the next post.

## Saturday, September 06, 2008

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