Chaos in integrate-and-fire dynamical systems

Coombes, S.
February 2000
AIP Conference Proceedings;2000, Vol. 502 Issue 1, p88
Academic Journal
Integrate-and-fire (IF) mechanisms are often studied within the context of neural dynamics. From a mathematical perspective they represent a minimal yet biologically realistic model of a spiking neuron. The non-smooth nature of the dynamics leads to extremely rich spike train behavior capable of explaining a variety of biological phenomenon including phase-locked states, mode-locking, bursting and pattern formation. The conditions under which chaotic spike trains may be generated in synaptically interacting networks of neural oscillators is an important open question. Using techniques originally introduced for the study of impact oscillators we develop the notion of a Liapunov exponent for IF systems. In the strong coupling regime a network may undergo a discrete Turing-Hopf bifurcation of the firing times from a synchronous state to a state with periodic or quasiperiodic variations of the interspike intervals on closed orbits. Away from the bifurcation point these invariant circles may break up. We establish numerically that in this case the largest IF Liapunov exponent becomes positive. Hence, one route to chaos in networks of synaptically coupled IF neurons is via the breakup of invariant circles. © 2000 American Institute of Physics.


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