How is sensory information encoded in spike trains?

The brain is fast: compared to the timescales determined by the dynamics of cell membranes, responses to stimuli are remarkably rapid. The brain is also efficient; it runs at only 25 watts. Furthermore, along with the many less quantifiable glories of human thought, the brain is able to perform quite perplexing feats of recall and recognition. It is not known how all this is achieved, but it is clear that neuronal signalling, in the form of spike trains, is at the heart of this mystery. Our mental abilities, our ability to reliably perceive external stimuli, together with all those other mental processes which contribute to the diverse qualia that constitute our thinking selves, rely on information being communicated as spike trains. Questions about neuronal signalling that are well understood intuitively are not easily phrased in a meaningful and addressable form. This is a mathematical problem: there is no good mathematical description of spike trains.

Spike trains need to be described in a way that is mathematically natural and informed by the biology of data processing in neuronal networks. I have a background in mathematical physics. I had an established research programme on integrable solitons in three dimensions particle physics, a subject which combines topology, geometry and computation. A few years ago, numerical calculations involving simulated annealing lead me to an interest in complexity and, ultimately, to the realization that there is hugely exciting new mathematics to be discovered in neuroscience. My background informs the work of my laboratory: often experimental results are shoehorned into an established, but inappropriate, mathematical framework. We believe our strength in mathematics give us the confidence and ability to discover the appropriate mathematics for the problem.

The unifying theme of our research is the idea that the important features of spike trains, their information content and temporal structure, will be easy to calculate and study when a natural mathematical framework has been constructed. We have discovered a new metric, based on the biology of synaptic transmission. This metric seems to be useful for describing sparseness in neuronal firing rates. We have a novel, better founded, approach to information theory which is native to the metric space of spike train. We have begun exciting new work on the persistent homology of electrophysiological data.