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.