BRAINEMATICS

Every second these spikes are detected more than 100 billion times in your brain. These spikes represent sudden electrical impulses, shot through one brain cell on its way to the next. Spikes are the currency of information and they drive everything we think and do. Unravelling how neurons spike is a crown jewel of twentieth century, neuroscience and mathematics was central to this solution. Questions have gripped the scientific community ever since spikes were first seen more than 100 years ago. Hodgkin and Huxley, two physiologists, showed how mathematics could solve all of them at once.

Computational neuroscience is the mathematical and physical modelling of neural processes.  The computational neuroscience literature is full of beautiful mathematically constructed models that have had minimal impact on main stream neuroscience and our understanding of brain function. Biological processes can be understood with the help of mathematics.

To decipher the connectome of the mammalian brain at cellular scale, no one would disagree that the connections between cells represented by the vast spaghetti of processes that make up the neuropil are a complex intermingling of curves. We can translate this agreed upon statement into a mathematical statement. For example, we can say that the set of edges that connects the vertices that make up the network of interest are not represented by Euclidean geodesics but by curves that can be described geometrically as Jordan arcs. We may decide to characterize the turning numbers of similar curves in a set. The point is that we have taken a simple agreed upon experimental neurobiological statement of fact and have translated it into a mathematical statement. We have captured some desirable aspect or property about this experimental axiom within the language of mathematics.

Mathematical neuroscience by its very formulation breaks free of differential equations and rewards creativity and imagination. There is no template or rule book to this process. One is free to write down a set of axioms and construct and prove a conjecture from those axioms using whatever mathematics is deemed appropriate.

Of course, this makes it an inherently difficult process, since it is not clear priori. There in lies the challenge but also the reward. The importance of mathematics in every field can be summed up by a quote by a famous physicist Richard Feynman- “People who wish to analyse nature without using mathematics must settle for a reduced understanding.”