Interface approaches for studying travelling waves in spiking neural networks

Presented by Kyle Wedgwood (Exeter, UK)

Certain neural systems show computation through patterned activity: persistent localised activity, in the form of bumps, has been linked to working memory, whilst the propagation of activity in the form of waves has been associated with binocular rivalry tasks. Individual neurons typically exhibit an all-or-nothing response, dependent on the summation of signals they receive from the rest of the network. This fact, coupled with the desire to understand coherent patterns of activity across the network has resulted in the widespread use of non-smooth neural models that greatly simplify the complex dynamics of individual cells. Whilst these descriptions often provide tractable models of neural tissue, their non-smooth nature presents its own mathematical challenges.

We will show how localised bumps of activity and travelling waves are generated in a synaptically coupled neural network and how they lose stability through bifurcations of both smooth and non-smooth type. This will be done exemplified using two different model approaches. The first, taking the form of a discrete time Markov chain model, the second forming a continuum approximation of a discrete neural network. In both cases, analysis will be facilitated via the construction of interface equations that take advantage of the non-smooth nature of the model. Armed with this framework, we compute existence and linear stability properties of waves and bumps in such networks and further show how numerical coarse-graining procedures can be used to assess the impact of spatial and temporal noise. Finally, we will examine how this can be used to explain the experimentally observed activity in grid cells, which are a significant component of the brain's representation of spatial location within an environment.