## Global (Local) Invariant Manifolds as Organizing Centers for Excitability

### Saeed Farjami

Department of Physiology, McGill University

One of the most important features of neurons is their ability to produce action potentials (AP), a fast change in membrane voltage associated with other processes that evolve slowly. These action potentials can be in the forms of a single transient spike, a transient burst or a periodic burst. In this talk, I will consider global and local invariant manifolds as the main organizing topological structure that explains the underlying mechanisms for the generation of such activities. First, I will explain how a stable manifold of a saddle equilibrium causes a switching phenomenon in a single transient AP of a cerebellar stellate cell model. Then, I will introduce an algorithm for computing stable manifolds of a saddle slow manifold using two-point boundary value problem and continuation methods. Finally, I will show the role of this manifold in determining the dynamics of transient and periodic bursting.

This is joint work with Anmar Khadra, Vivien Kirk and Hinke M. Osinga.