Mean-field analysis of neuronal spike dynamics
Abstract
I consider a mean-field description of the dynamics of interacting intergrate-and-fire neuron-like units. The basic dynamical variables are the membrane potential of each (point-like) ‘cell’ and the conductance associated with each synaptic connection, both of which evolve discontinuously in time. In addition, an intrinsic potassium conductance, also evolving discontinuously in time, can be associated to each cell in order to model firing frequency adaptation in real neurons. The mean-field theory is exact if the units can be grouped into N C classes, each comprising infinitely many identical, and identically coupled, units; and can be used as an approximation if, instead, a class comprises few or just one unit. The formalism yields both the stationary asynchronous solutions and the transients leading to those solutions. The full spectrum of time-constants for the transients associated with one particular steady state is given by a single equation, imposing the vanishing of the determinant of an N C ×N C matrix. In the case of an associative memory, this equation can be manipulated into a simple form, using standard replica methods. An analysis of the spectrum indicates that the major role in determining the transients time constants is played by the effective decay times of postsynaptic currents, which can be quite short. This suggests that local recurrent neocortical circuits may produce a very rapid dynamics, consistent with such circuits participating in the rapid course of information processing, evidenced by new experimental data recorded in primate temporal cortex.