Very long transients, irregular firing, and chaotic dynamics in networks of randomly connected inhibitory integrate-and-fire neurons

Rüdiger Zillmer, Nicolas Brunel, David Hansel
Physical Review E. 2009-03-18; 79(3):
DOI: 10.1103/physreve.79.031909

1. Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Mar;79(3 Pt 1):031909. doi:
10.1103/PhysRevE.79.031909. Epub 2009 Mar 18.

Very long transients, irregular firing, and chaotic dynamics in networks of
randomly connected inhibitory integrate-and-fire neurons.

Zillmer R(1), Brunel N, Hansel D.

Author information:
(1)Laboratoire de Neurophysique et Physiologie, Université Paris Descartes,
75270 Paris Cedex 06, France.

We present results of an extensive numerical study of the dynamics of networks
of integrate-and-fire neurons connected randomly through inhibitory
interactions. We first consider delayed interactions with infinitely fast rise
and decay. Depending on the parameters, the network displays transients which
are short or exponentially long in the network size. At the end of these
transients, the dynamics settle on a periodic attractor. If the number of
connections per neuron is large ( approximately 1000) , this attractor is a
cluster state with a short period. In contrast, if the number of connections per
neuron is small ( approximately 100) , the attractor has complex dynamics and
very long period. During the long transients the neurons fire in a highly
irregular manner. They can be viewed as quasistationary states in which,
depending on the coupling strength, the pattern of activity is asynchronous or
displays population oscillations. In the first case, the average firing rates
and the variability of the single-neuron activity are well described by a
mean-field theory valid in the thermodynamic limit. Bifurcations of the long
transient dynamics from asynchronous to synchronous activity are also well
predicted by this theory. The transient dynamics display features reminiscent of
stable chaos. In particular, despite being linearly stable, the trajectories of
the transient dynamics are destabilized by finite perturbations as small as
O(1/N) . We further show that stable chaos is also observed for postsynaptic
currents with finite decay time. However, we report in this type of network that
chaotic dynamics characterized by positive Lyapunov exponents can also be
observed. We show in fact that chaos occurs when the decay time of the synaptic
currents is long compared to the synaptic delay, provided that the network is
sufficiently large.