Stochastic mean-field formulation of the dynamics of diluted neural networks

David Angulo Garcia

We consider pulse-coupled leaky integrate-and-fire neural networks with
randomly distributed synaptic couplings.  This random dilution induces
fluctuations in the evolution of the macroscopic variables and deterministic
chaos at the microscopic level.  We mimic the effect of the dilution as a
noise source acting on the dynamics of a globally coupled nonchaotic system.
Indeed, the evolution of a diluted neural network can be well approximated as
a fully pulse-coupled network, where each neuron is driven by a mean synaptic
current plus additive noise.  These terms represent the average and the
fluctuations of the synaptic currents  acting on the single neurons in the
diluted system.  The main microscopic and macroscopic dynamical features can
be retrieved with this stochastic approximation.  Furthermore, the microscopic
stability of the diluted network can be also reproduced, as demonstrated from
the almost coincidence of the measured Lyapunov exponents in the deterministic
and stochastic cases for an ample range of system sizes.  Our results suggest
that the fluctuations in the synaptic currents are responsible for the
emergence of chaos in this class of pulse-coupled networks.