Past Differential Equations and Applications Seminar

2 March 2006
Stephen Coombes
<span class="326354111-16012006">&nbsp;</span>I will discuss the dynamics of synaptically coupled model neurons that undergo a form of accommodation in the presence of sustained activity. The basic model is an integral equation for synaptic activity that depends upon the non-local network connectivity, synaptic response, and firing rate of a single neuron. A phenomenological model of accommodation is examined whereby the firing rate is taken to be a simple state-dependent threshold function. As in the case without threshold accommodation classical Mexican-Hat connectivity is shown to allow for the existence of spatially localised states (bumps). Importantly an analysis of bump stability (in both one and two spatial dimensions) using recent Evans function techniques shows that bumps may undergo instabilities leading to the emergence of both breathers and travelling waves. Numerical simulations show that bifurcations in this model have the same generic properties as those seen in many other dissipative systems that support localised structures, and in particular those of coupled cubic complex Ginzburg-Landau equations, and three component reaction diffusion equations. Interestingly, travelling pulses in this model truly have a discrete character in the sense that they scatter as auto-solitons. <code>/notices/events/abstracts/differential-equations/ht06/Coombes.shtml</code> &nbsp; &nbsp;
  • Differential Equations and Applications Seminar