4. Formal Spiking Neuron Models

Detailed conductance-based neuron models can reproduce electrophysiological measurements to a high degree of accuracy, but because of their intrinsic complexity these models are difficult to analyze. For this reason, simple phenomenological spiking neuron models are highly popular for studies of neural coding, memory, and network dynamics. In this chapter we discuss formal threshold models of neuronal firing. Spikes are generated whenever the membrane potential u crosses some threshold $\vartheta$ from below. The moment of threshold crossing defines the firing time t^(f),

$$\vartheta \left.\vphantom{ {{\text{d}}u(t) \over {\text{d}}t} }\right. {{\text{d}}u(t) \over {\text{d}}t} \left.\vphantom{ {{\text{d}}u(t) \over {\text{d}}t} }\right\vert _{{t=t^{(f)}}}^{}$$ (4.1)

Since spikes are stereotyped events they are fully characterized by their firing time. We focus on models that are based on a single variable u. Some well-known instances of spiking neuron models differ in the specific way the dynamics of the variable u is defined. We start our discussion with the integrate-and-fire neuron (Section 4.1) and turn then to the Spike Response Model (Section 4.2). In Section 4.3 we illustrate the relation of spiking neuron models to conductance-based models. Section 4.4 outlines an analytical approach for a study of integrate-and-fire neurons with passive dendrites. As a first application of spiking neuron models we reconsider in Section 4.5 the problem of neuronal coding. The spiking neuron models introduced in this chapter form the basis for the analysis of network dynamics and learning in the following chapters.