## Existence and stability of standing pulses in neural networks

### Guo Yixin Mathematics Department, University of Pittsburgh

#### Abstract

We consider the existence and the stability of standing pulse solutions of an integro-differential equation used to describe the activity of neuronal networks. The network consists of a single-layer of neurons with non-saturating piecewise linear gain function, synaptically coupled by lateral inhibition. The existence condition for pulses can be reduced to the solution of an algebraic system and using this condition we map out the shape of the pulses for different weight kernels and gains. We also find conditions for the existence of pulse with a 'dimple' on top and for a double-pulse. For a fixed gain and connectivity, we find two single-pulse solutions-a large'' one and a small'' one. We derive conditions to show that the large one is stable and the small one is unstable. Using the same conditions, we also show that the dimple-pulse is stable. More importantly, the large single-pulse and the dimple pulse are bistable with the all-off state. This bistable localized activity may have strong implications for the mechanism underlying of working memory.