Variability of spike timing, quantified for example by the *C*_{V} values of
interval distributions, is a common phenomenon in biological neurons. In
models, noise is usually added *ad hoc* to account for this variability.
There are at least two different ways of adding noise. First, a noisy
threshold can be defined by an instantaneous escape rate which depends on the
distance of the membrane potential from the threshold. Escape rate models can
be solved for arbitrary input currents in the context of renewal theory.
Second, stochastic arrival of excitatory and inhibitory input pulses leads to
a diffusion of the membrane potential. The interval distribution of an
integrate-and-fire model with diffusive noise is equivalent to the first
passage time problem of the Ornstein-Uhlenbeck process and difficult to solve.
Both noise models are (approximately) equivalent in the subthreshold regime.
The critical variable for firing is
*x*(*t*) = [*u*_{0}(*t*) - ]/, that
is the momentary distance between the noise-free membrane potential and the
threshold in units of the membrane potential fluctuations,
= .

The subthreshold regime has several interesting properties. First, constant input current plus noise leads to a distribution of interspike intervals with a large coefficient of variation, similar to what is found in cortical neurons. Second, in the subthreshold regime the neuron is most sensitive to temporal variations in the input. Stochastic resonance is an example of this phenomenon.

In rate models, the neuron is fully characterized by its nonlinear transfer function. If inputs are constant and all neurons are in a stationary state, then the static rate model provides a useful description. Dynamic versions of rate models are possible, but problematic. Stochastic rate models, finally, form the borderline to stochastic spiking neuron models.

Analysis of spike trains in terms of stochastic point processes has a long tradition (Perkel et al., 1967a; Gerstein and Perkel, 1972) and often involves concepts from renewal theory (Cox and Lewis, 1966). In formal spiking neuron models, stochasticity was introduced early on by adding a stochastic component to the membrane potential (Geisler and Goldberg, 1966; Weiss, 1966). If the correlation time of the noise is short, such an approach is closely related to an escape rate or hazard model (Weiss, 1966). Stochastic spike arrival as an important source of noise has been discussed by Stein in the context of integrate-and-fire models (Stein, 1967b,1965). Some principles of spike-train analysis with an emphasis on modern results have been reviewed by Gabbiani and Koch (1998) and Rieke et al. (1996). For a discussion of the variability of interspike intervals see the debate of Shadlen and Newsome (1994), Softky (1995), and Bair and Koch (1996). In this context, the role of subthreshold versus superthreshold stimuli has been summarized in the review of König et al. (1996).

The intimate relation between stochastic spike arrival and diffusive noise has been known for a long time (Johannesma, 1968; Gluss, 1967). Mathematical results of diffusive noise in the integrate-and-fire neuron (i.e., the Ornstein-Uhlenbeck model) are reviewed in many texts (Tuckwell, 1988; van Kampen, 1992). The mathematical aspects of stochastic resonance have been reviewed by Gammaitoni et al. (1998); applications of stochastic resonance in biology have been summarized by Wiesenfeld and Jaramillo (1998).

Rate models are widely used in the formal theory of neural networks. Excellent introductions to the Theory of Neural Networks are the books of Hertz et al. (1991) and Haykin (1994). The history of neural networks is highlighted in the nice collection of original papers by Anderson and Rosenfeld (1988) which contains for example a reprint of the seminal article of McCulloch and Pitts (1943).

Cambridge University Press, 2002

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