5.7 From diffusive noise to escape noise

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 *u*_{0}(*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)

where

We have seen in Eq. (5.97) that the variance
*u*^{2}(*t*) of the free distribution rapidly approaches a constant value
/2. We therefore replace the time dependent variance
2*u*(*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

The factor in front of the exponential has been split into a constant parameter

Let us now suppose that the neuron receives, at *t* = *t*_{0}, 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 *t*_{0}.
To say it differently, the firing intensity
(*t*) = *f*[*u*_{0}(*t*) - ]
has a peak at *t* = *t*_{0}. 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

where = d

We emphasize that the right-hand side of Eq. (5.116) depends only on the dimensionless variable

x(t) = , |
(5.117) |

and its derivative . Thus the amplitude of the fluctuations define a `natural' voltage scale. The only relevant variable is the momentary distance of the noise-free trajectory from the threshold in units of the noise amplitude . A value of

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 *u*_{0}(*t*) as in Fig. 5.18. We find that the interval
distributions
*P*_{I}^{diff} for the diffusive noise model and
*P*_{I}^{A&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

For the example shown in Fig. 5.22 we find

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(- *x*^{2}) by
2 exp(- *x*^{2})/[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.

Cambridge University Press, 2002

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