International Journal of Bifurcation and Chaos (2000), 10:1171--1266.
Eugene M. Izhikevich
Bifurcation mechanisms involved in the generation of action potentials
(spikes) by neurons are reviewed here. We show how the type of bifurcation
affects the neuro-computational properties of the cells. For example, when
the rest state is near a saddle-node bifurcation, the cell can fire all-or-none
spikes with an arbitrary low frequency, it has a well-defined threshold
manifold, and it acts as an integrator; i.e., the higher the frequency
of incoming pulses, the sooner it fires. In contrast, when the rest state
is near an Andronov-Hopf bifurcation, the cell fires in a certain frequency
range, its spikes are not all-or-none, it does not have a well-defined
threshold manifold, it can fire in response to an inhibitory pulse, and
it acts as a resonator; i.e., it responds preferentially to a certain
(resonant) frequency of the input. Increasing the input frequency may actually
delay or terminate its firing.
We also describe the phenomenon of neural bursting, and we use geometric bifurcation theory to extend the existing classification of bursters, including many new types. We discuss how the type of burster defines its neuro-computational properties, and we show that different bursters can interact, synchronize, and process information differently.
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