# Chapter 5 Nonlinear Integrate-and-Fire 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, also called integrate-and-fire models.

The shape of the action potential of a given neuron is rather stereotyped with very little change between one spike and the next. Thus, the shape of the action potential which travels along the axon to a postsynaptic neuron cannot be used to transmit information; rather, from the point of view of the receiving neuron, action potentials are ’events’ which are fully characterized by the arrival time of the spike at the synapse. Note that spikes from different neuron types can have different shapes and the duration and shape of the spike does influence neurotransmitter release; but the spikes that arrive at a given synapse all come from the same presynaptic neuron and – if we neglect effects of fatigue of ionic channels in the axon – we can assume that its time course is always the same. Therefore we make no effort to model the exact shape of an action potential. Rather, spikes are treated as events characterized by their firing time - and the task consists in finding a model so as to reliably predict spike timings.

In generalized integrated-and-fire models, spikes are generated whenever the membrane potential $u$ crosses some threshold $\theta_{\rm reset}$ from below. The moment of threshold crossing defines the firing time $t^{(f)}$,

 $t^{(f)}:\quad u(t^{(f)})=\theta_{\rm reset}\quad\text{and}\quad\left.{{\text{d% }}u(t)\over{\text{d}}t}\right|_{t=t^{(f)}}>0\,.$ (5.1)

In contrast to the two-dimensional neuron models, encountered in Chapter 4, we don’t have a relaxation variable that enables us to describe the return of the membrane potential to rest. In the integrate-and-fire models, discussed in this and the following chapters, the downswing of the action potential is replaced by an algorithmic reset of the membrane potential to a new value $u_{r}$ each time the threshold $\theta_{\rm reset}$ is reached. The duration of an action potential is sometimes, but not always, replaced by a dead-time $\Delta^{\rm abs}$ after each spike, before the voltage dynamics restarts with $u=u_{r}$ as initial condition.

In this chapter, we focus on integrate-and-fire models with a single variable $u$ which describes the time course of the membrane potential. In Chapter 6, we extend the models developed in this chapter so as to include adaptation of neuronal firing during extended strong stimulation. In Chapters 711 we consider questions of coding, noise, and reliability of spike-time prediction — using the generalized integrate-and-fire model which we introduce now.