A framework for spiking neuron models - the spike response model

A description of neuronal activity on the level of ion channels, as in the Hodgkin-Huxley model, leads to a set of coupled nonlinear differential equations which are difficult to analyze. In this paper, we present a conceptual framework for a reduction of the nonlinear spike dynamics to a threshold process. Spikes occur if the membrane potential $u(t)$ reaches a threshold $\vartheta$. The voltage response to spike input is described by the postsynaptic potential $\epsilon$. Postsynaptic potentials of several input spikes are added linearly until $u$ reaches $\vartheta$. The output pulse itself and the reset/refractory period which follow the pulse are described by a function $\eta$. Since $\epsilon$ and $\eta$ can be interpreted as response kernels, the resulting model is called the Spike Response Model (SRM). After a short review of the Hodgkin-Huxley model we show that (i) Hodgkin-Huxley dynamics with time-dependent input can be reproduced to a high degree of accuracy by the SRM; (ii) the simple integrate-and-fire neuron is a special case of the Spike Response Model; (iii) compartmental neurons with a passive dendritic tree and a threshold process for spike generation can be treated in SRM-framework; (iv) small nonlinearities lead to interactions between spikes to be described by higher-order kernels.