A Review of the Integrate-and-fire Neuron Model: I. Homogeneous Synaptic Input

Abstract

The integrate-and-fire neuron model is one of the most widely used models for analyzing the behavior of neural systems. It describes the membrane potential of a neuron in terms of the synaptic inputs and the injected current that it receives. An action potential (spike) is generated when the membrane potential reaches a threshold, but the actual changes associated with the membrane voltage and conductances driving the action potential do not form part of the model. The synaptic inputs to the neuron are considered to be stochastic and are described as a temporally homogeneous Poisson process. Methods and results for both current synapses and conductance synapses are examined in the diffusion approximation, where the individual contributions to the postsynaptic potential are small. The focus of this review is upon the mathematical techniques that give the time distribution of output spikes, namely stochastic differential equations and the Fokker-Planck equation. The integrate-and-fire neuron model has become established as a canonical model for the description of spiking neurons because it is capable of being analyzed mathematically while at the same time being sufficiently complex to capture many of the essential features of neural processing. A number of variations of the model are discussed, together with the relationship with the Hodgkin-Huxley neuron model and the comparison with electrophysiological data. A brief overview is given of two issues in neural information processing that the integrate-and-fire neuron model has contributed to - the irregular nature of spiking in cortical neurons and neural gain modulation.

Temporal Considerations for Stimulating Spiral Ganglion Neurons with Cochlear Implants

“…2B), is through capacitive membrane charging. Conceptually, this can be understood with the venerable integrate-and-fire model (Lapicque 1907;Brunel and van Rossum 2007;Knight 1972;Burkitt 2006). If depolarizing monophasic pulses are separated by some time which is smaller than the membrane time constant, then the charge accumulated on the membrane from the first pulse will be too great to completely discharge before the membrane begins to integrate the next pulse.…”

Source Separation and Clustering of Phase-Locked Subspaces

“…If we define a matrix A where the (i, t)-th entry is A i (t), C ≡ [C 1 (1) (13) and (14), and by eliminating the common term e iω(t ) , the equation for y 1 yields C T e iα 1 = A T Z( ) ⇔ C T = A T Z( − α 1 ) ⇔ C T = A T Z(α 1 − ), where the last equation was obtained by taking 10 The y j 's must be of this form to have a PLF of 1 with the sources: the PLF between two signals is 1 if and only if their phase difference is constant. the complex conjugate of both sides (note that C and A are real).…”

“…The value of this threshold can be analytically found from the parameters of an "integrate-and-fire model." Integrate-and-fire oscillators, also known as relaxation oscillators, are described in more detail in the context of synchrony in [1], [13], and [14].…”

Statistical Models of Spike Trains

“…The most direct, and probably best-explored, view of the IF model is as a linear diffusion process (Ricciardi, 1977;Tuckwell, 1989;Burkitt, 2006). This connection allows us to apply powerful tools from stochastic calculus (Karatzas and Shreve, 1997) to understand the behavior of this model.…”