In the subthreshold regime, the integrate-and-fire model with stochastic input (diffusive noise) can be mapped approximately onto an escape noise model with a certain escape rate f (Plesser and Gerstner, 2000). In this section, we motivate the mapping and the choice of f.
In the absence of a threshold, the membrane potential of an integrate-and-fire model has a Gaussian probability distribution, around the noise-free reference trajectory u0(t). If we take the threshold into account, the probability density at u = of the exact solution vanishes, since the threshold acts as an absorbing boundary; see Eq. (5.92). Nevertheless, in a phenomenological model, we can approximate the probability density near u = by the `free' distribution (i.e., without the threshold)
We have seen in Eq. (5.97) that the variance u2(t) of the free distribution rapidly approaches a constant value /2. We therefore replace the time dependent variance 2u(t)2 by its stationary value . The right-hand side of Eq. (5.114) is then a function of the noise-free reference trajectory only. In order to transform the left-hand side of Eq. (5.114) into an escape rate, we divide both sides by t. The firing intensity is thus
Let us now suppose that the neuron receives, at t = t0, an input current pulse which causes a jump of the membrane trajectory by an amount u > 0; see Fig. (5.21). In this case the Gaussian distribution of membrane potentials is shifted instantaneously across the threshold so that there is a nonzero probability that the neuron fires exactly at t0. To say it differently, the firing intensity (t) = f[u0(t) - ] has a peak at t = t0. The escape rate of Eq. (5.115), however, cannot reproduce this peak. More generally, whenever the noise free reference trajectory increases with slope > 0, we expect an increase of the instantaneous rate proportional to , because the tail of the Gaussian distribution drifts across the threshold; cf. Eq. (5.111). In order to take the drift into account, we generalize Eq. (5.115) and study
We emphasize that the right-hand side of Eq. (5.116) depends only on the dimensionless variable
|x(t) = ,||(5.117)|
To check the validity of the arguments that led to Eq. (5.116), let us compare the interval distribution generated by the diffusion model with that generated by the Arrhenius&Current escape model. We use the same input potential u0(t) as in Fig. 5.18. We find that the interval distributions PIdiff for the diffusive noise model and PIA&C for the Arrhenius&Current escape model are nearly identical; cf. Fig. (5.22). Thus the Arrhenius&Current escape model yields an excellent approximation to the diffusive noise model. We quantify the error of the approximation by the measure
Even though the Arrhenius&Current model has been designed for sub-threshold stimuli, it also works remarkably well for super-threshold stimuli with typical errors around E = 0.04. An obvious shortcoming of the escape rate (5.116) is that the instantaneous rate decreases with u for u > . The superthreshold behavior can be corrected if we replace the Gaussian exp(- x2) by 2 exp(- x2)/[1 + erf(- x)] (Herrmann and Gerstner, 2001a). The subthreshold behavior remains unchanged compared to Eq. (5.116) but the superthreshold behavior of the escape rate f becomes linear. With this new escape rate the typical error E in the super-threshold regime is as small as that in the subthreshold regime.
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