DYNAMICS OF A COMPARTMENTAL MODEL INTEGRATE-AND-FIRE NEURON WITH SOMATIC POTENTIAL RESET

An integral equation approach to analysing the dynamics of a compartmental model integrate-and-fire neuron with partial reset is presented. The neuron is taken to consist of a somatic compartment coupled to a single dendritic compartment; only the former resets when the neuron fires. The compartmental model equations are shown to reduce to a scalar Volterra integral equation that takes into account the feedback between the reset region and the dendrites. An iterative solution to the integral equation is constructed in the form of a second order nonlinear difference equation for the firing-times. In the case of constant inputs, this becomes a first order equation in the inter-spike intervals, which has a unique fixed point that is globally asymptotically stable. The corresponding steady-state firing-frequency of the neuron is determined as a function of the coupling between soma and dendrites. Finally, the response of the two-compartment model to oscillatory inputs is studied and, in particular, the firing-times are shown to evolve according to a map of the cylinder.