Pattern formation in biological neural networks with rebound currents

Waves and patterns in the brain are well known to subserve natural computation. Much attention in the theoretical neuroscience community has been devoted to analysing networks of relatively simple spiking neurons (IF type) or firing rate models (Wilson-Cowan type) and to great effect!  Indeed, the understanding of how spatio-temporal patterns of neural activity may arise in the cortex has advanced significantly with the development and analysis of such models. To replicate this success for sub-cortical tissues requires an extension to include relevant ionic currents that can further shape firing response. Here I will advocate for two complementary approaches: i) that augments the approach for IF networks to include piecewise linear caricatures of gating dynamics for nonlinear ionic current models, ii) firing rate reductions for systems where the nonlinear ionic currents are slow.  By way of illustration, I will show how to construct spatially periodic waves and patterns in i) a simple spiking tissue model of medial enthorinal cortex (with an I_h current), ii) a firing rate model of thalamus (with an I_T current). The biological commonality between these two models is that both express local 'rebound' currents that can usefully shape global tissue response.  The mathematical commonality is the use of tools from non-smooth dynamical systems theory to make analytical progress in determining patterns and their stability.