Mathematics in cognitive sciences

Question

Which areas of Maths can be formally studied in a cognitive sc major? Or: which areas of maths can support the study of brain. Some areas that seem relevant would be: mathematical logic, graph theory, linear algebra. Is any element of topology relevant? Please direct to links/sources if possible. Is network science being used in these fields?

Answer 1

The first and main area of mathematics that stands out to me is Nonlinear Dynamics, and Chaos theory. I did research into modeling cardiac action potentials with systems of nonlinear differential equations, in particular with Van Der Pol type equations (Master's Thesis). The idea of extending that to neuronal activity in the brain is pretty natural. In fact, one of the signatures on my thesis was a Biologist (needed one outside department signature), and the subject of neuronal activity in the brain was definitely discussed for future research.

Here is a nice reference to dynamical systems in cognitive sciences. There are many papers written on the field of nonlinear dynamics where the focus is on the dynamics of neuronal activity in the brain. For example, Phase-Coupled Oscillations in the Brain is an example of applying coupled systems of oscillators to a small model of the interaction of brain neurons. There are several more like this. It is a natural idea to try to model the interactive dynamics of the spike-burst action potential mechanism of any type of neuron, and certainly brain neurons with systems of relaxation oscillators, and other types of oscillators is no longer an original idea, but there is alot of research yet to be done in this field.

Another nice reference to support my response here is ...dynamical systems ... neuronal..., and this is an example of trying to model some phenomena in the interaction of neurons in a brain.

Here is some modeling of EEG signals with coupled oscillators. The list just keeps going on.

So in conclusion, I would say that the mathematical field of nonlinear dynamics lends itself well to the "study of the brain", in particular modeling the dynamics and interactions of brain neurons with coupled systems of nonlinear differential equations. One could say that nonlinear dynamics, and chaos is a frontier field of mathematics, and biological models are likely going to be a really popular application for years to come.