Dynamics of neuronal populations: The equilibrium solution

The behavior of an aggregate of neurons is followed by means of a populatio n equation which describes the probability density of neurons as a function of membrane potential. The model is based on integrate-and-fire membrane d ynamics and a synaptic dynamics which produce a fixed potential jump in res ponse to stimulation. In spite of the simplicity of the model, it gives ris e to a rich variety of behaviors. Here only the equilibrium problem is cons idered in detail. Expressions for the population density and ring rate over a range of parameters are obtained and compared with like forms obtained f rom the diffusion approximation. The introduction of the jump response to s timulation produces a delay term in the equations, which in turn leads to a nalytical challenges. A variety of asymptotic techniques render the problem solvable. The asymptotic results show excellent agreement with direct nume rical simulations.