Dynamics of Neuronal Populations: The Equilibrium Solution

The behavior of an aggregate of neurons is followed by means of a population equation which describes the probability density of neurons as a function of membrane potential. The model is based on integrate-and-fire membrane dynamics and a synaptic dynamics which produce a fixed potential jump in response to stimulation. In spite of the simplicity of the model, it gives rise to a rich variety of behaviors. Here only the equilibrium problem is considered in detail. Expressions for the population density and firing rate over a range of parameters are obtained and compared with like forms obtained from the diffusion approximation. The introduction of the jump response to stimulation produces a delay term in the equations, which in turn leads to analytical challenges. A variety of asymptotic techniques render the problem solvable. The asymptotic resultsshow excellent agreement with direct numerical simulations.