Kyle C. A. Wedgwood,

Non-smooth models, such as the leaky integrate-and-fire model, forego biophysical realism in favour of being analytically tractable. These simple models instead focus on reproducing key features of neural response, such as spike times. Since qualitative features of neural response are often preserved between the non-smooth variants and their smooth counterparts, we can hope to learn something about the biophysical models by studying the simpler ones.

This is, however, an overly simplistic view, since non-smooth models come with their own set of difficulties. Even at the single cell level, one must take care when performing stability analysis since the field of non-smooth bifurcation analysis is in its infancy. The severity of the problem is realised when we consider that there currently exists no equivalent of the centre manifold theorem for non-smooth systems, meaning that dimension reduction near instabilities is, in general, difficult. Even in the case where systems pass through smooth instabilities, the presence of manifolds in which the vector field is non-smooth can drastically alter what solutions are ultimately observed.

Our research focusses on a specific neural model, which can be solved analytically. This fact means that we can be sure that the behaviour we observe is true to the model, rather than being some artefact of numerical integration of such a system. We demonstrate how the non-smoothness can be 'tamed' in a variety of ways and show how these smooth and non-smooth bifurcations are relevent to neural processing. We are also applying similar techniques to large networks of neurons to see how subtle effects at the single cell level impact on those at the network level.