In one of the previous examples (`Motivating sigmoidal escape rates' in Section 5.3.1), a new value of the threshold was chosen at each time step; cf. Eq. (5.49). If time steps are short enough, such an approach is closely related to escape rate models. A completely different class of noise models can be constructed if the value of a parameter is changed after each spike. Thus between two spikes the noise is `frozen' so that the value of the fluctuating parameter does not change. In other words, the noise is slow compared to the fast neuronal dynamics. In principle, any of the neuronal parameters such as threshold, membrane time constant, or length of the refractory period, can be subject to this type of noise (Gerstner, 2000b; Lansky and Smith, 1989; Gestri, 1978; Knight, 1972a). In this section we want to show how to analyze such slow variations and calculate the interval distribution. We emphasize that these `slow' noise models cannot be mapped onto an escape rate formalism.
To keep the arguments simple, we will concentrate on noise in the formulation of reset and refractoriness. We assume an exponential refractory kernel,
In the `noisy reset' model, firing is given by the threshold condition
|= u(t|, r) = (t - ) + (t - , s) I(t - s) ds ,||(5.66)|
|T(, r) = mint - | u(t|, r) = .||(5.67)|
Let us now evaluate the interval distribution (5.68) for the variant SRM0 of the Spike Response Model,
|u(t|, r) = (t - ) + h(t) ,||(5.69)|
|(t - ) = exp[- (t - - r)/] ,||(5.70)|
Even though stochastic reset is not a realistic noise model for individual neurons, noise in the parameter values can approximate inhomogeneous populations of neurons where parameters vary from one neuron to the next (Wilson and Cowan, 1972; Knight, 1972a). Similarly, a fluctuating background input that changes slowly compared to the typical interspike interval can be considered as a slow change in the value of the firing threshold. More generally, noise with a cut-off frequency smaller than the typical firing rate can be described as slow noise in the parameters.
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