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We discuss how to characterize long-time dynamics of non-smooth dynamical systems, such as integrate-and-fire (I&F) like neuronal network, using Lyapunov exponents and present a stable numerical method for the accurate evaluation of the spectrum of Lyapunov exponents for this large class of dynamics. These dynamics contain (i) jump conditions as in the firing-reset dynamics and (ii) degeneracy such as in the refractory period in which voltage-like variables of the network collapse to a single constant value. Using the networks of linear I&F neurons, exponential I&F neurons, and I&F neurons with adaptive threshold, we illustrate our method and discuss the rich dynamics of these networks.

Original publication

DOI

10.1007/s10827-009-0201-3

Type

Journal article

Journal

Journal of computational neuroscience

Publication Date

04/2010

Volume

28

Pages

229 - 245

Addresses

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA. zdz@cims.nyu.edu

Keywords

Nerve Net, Neurons, Membrane Potentials, Action Potentials, Algorithms, Nonlinear Dynamics, Models, Neurological, Computer Simulation