title: Spectrum of Lyapunov exponents of non-smooth dynamical systems of integrate-and-fire type.
abstract: 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.
authors.affiliation: Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA. zdz@cims.nyu.edu
authors.forename: Douglas
authors.initials: D
authors.lastname: Zhou
authors.forename: Yi
authors.initials: Y
authors.lastname: Sun
authors.forename: Aaditya V
authors.initials: AV
authors.lastname: Rangan
authors.forename: David
authors.initials: D
authors.lastname: Cai
mesh_terms: D000200:Action Potentials / Q000502:physiology; D000465:Algorithms; D003198:Computer Simulation; D008564:Membrane Potentials / Q000502:physiology; D008959:Models, Neurological; D009415:Nerve Net / Q000502:physiology*; D009474:Neurons / Q000502:physiology*; D017711:Nonlinear Dynamics