There are various ways to introduce noise in formal spiking neuron models. In this section we focus on a `noisy threshold' (also called escape or hazard model). In section 5.5 we will discuss `noisy integration' (also called stochastic spike arrival or diffusion model). In both cases, we are interested in the effect of the noise on the distribution of interspike intervals.
In the escape model, we imagine that the neuron can fire even though the formal threshold has not been reached or may stay quiescent even though the formal threshold has been passed. To do this consistently, we introduce an `escape rate' or `firing intensity' which depends on the momentary state of the neuron.
Given the input I and the firing time of the last spike, we can calculate the membrane potential of the Spike Response Model or the integrate-and-fire neuron from Eq. (5.1) or (5.2), respectively. In the deterministic model the next spike occurs when u reaches the threshold . In order to introduce some variability into the neuronal spike generator, we replace the strict threshold by a stochastic firing criterion. In the noisy threshold model, spikes can occur at any time with a probability density,
Since u on the right-hand side of Eq. (5.41) is a function of time, the firing intensity is time-dependent as well. In view of Eqs. (5.1) and (5.2), we write
Is Eq. (5.42) a sufficiently general noise model? We have seen in Chapter 2.2 that the concept of a pure voltage threshold is questionable. More generally, the spike trigger process could, for example, also depend on the slope = du/dt with which the `threshold' is approached. In the noisy threshold model, we may therefore also consider an escape rate (or hazard) which depends not only on u but also on its derivative
Note that the hazard in Eq. (5.43) is implicitly time-dependent, via the membrane potential u(t|). In an even more general model, we could in addition include an explicit time dependence, e.g., to account for a reduced spiking probability immediately after the spike at . In the following examples we will stick to the hazard function as defined by Eq. (5.42).
We have motivated the escape model by a noisy version of the threshold process. In order to explore the relation between noisy and deterministic threshold models, we consider an escape function f defined as
How can we `soften' the sharp threshold? A simple choice for a soft threshold is an exponential dependence,
Finally, we can also use a sigmoidal escape rate (Wilson and Cowan, 1972; Abeles, 1982),
We want to motivate the sigmoidal escape rate by a model with stochastic threshold in discrete time tn = n . After each time step of length , a new value of the threshold is chosen from a Gaussian distribution of threshold values with mean ,
Instead of a model with stochastic threshold, we can also consider a model fixed threshold, but a membrane potential u(tn) + u(tn) with a stochastic component u. If u is chosen at each time step independently from a Gaussian distribution with variance and vanishing mean, we arrive again at formula (5.47) (Abeles, 1982; Geisler and Goldberg, 1966; Weiss, 1966).
The sigmoidal escape rate (5.51) has been motivated here for models in discrete time. There are two potential problems. First, if we keep fixed and take 0 we do not recover the deterministic threshold model. Thus the low-noise limit is problematic. Second, since the firing intensity diverges for 0, simulations will necessarily depend on the discretization . This is due to the fact that the bandwidth of the noise is limited by because a new value of or u is chosen at intervals . For 0, the bandwidth and hence the noise power diverge. Despite its problems, the sigmoidal escape rate is also used in neuronal models in continuous time and either motivated by a Gaussian distribution of threshold values (Wilson and Cowan, 1972) or else for fixed threshold by a Gaussian distribution of membrane potentials with band-limited noise (Abeles, 1982; Weiss, 1966). If we use Eq. (5.47) in continuous time, the time scale becomes a free parameter and should be taken proportional to the correlation time of the noise u in the membrane potential (i.e., proportional to the inverse of the noise bandwidth). If the correlation time is short, the model becomes closely related to continuous-time escape rate models (Weiss, 1966). A `natural' correlation time of the membrane potential will be calculated in Section 5.5 in the context of stochastic spike arrival.
In the previous example, we have started from a model in discrete time and found that the limit of continuous time is not without problems. Here we want to start from a model in continuous time and discretize time as it is often done in simulations. In a straightforward discretization scheme, we calculate the probability of firing during a time step t of a neuron that has fired the last time at as (t'|) dt' (t|) t. For u , the hazard (t|) = f[u(t|) - ] can take large values; see, e.g., Eq. (5.45). Thus t must be taken extremely short so as to guarantee (t|) t < 1.
In order to arrive at an improved discretization scheme, we calculate the probability that a neuron does not fire in a time step t. Since the integration of Eq. (5.6) over a finite time t yields an exponential factor analogous to Eq. (5.7), we arrive at a firing probability
In this section, we combine the escape rate model with the concepts of renewal theory and calculate the input-dependent interval distribution PI(t|) for escape rate models.
We recall Eq. (5.9) and express the interval distribution in terms of the hazard ,
To do so, we express by the escape rate. In order to keep the notation simple, we suppose that the escape rate f is a function of u only. We insert Eq. (5.42) into Eq. (5.53) and obtain
We study the model SRM0 defined in Eq. (5.55) for absolute refractoriness
In this example we show that interval distributions are particularly simple if a linear escape rate is adopted. We start with the non-leaky integrate-and-fire model. In the limit of , the membrane potential of an integrate-and-fire neuron is
For a leaky integrate-and-fire neuron with constant input I0, the membrane potential is
|u(t|) = R I0 1 - e||(5.61)|
|(t - ) = 1 - e- (t-) ,||(5.62)|
We study the model SRM0 as defined in Eq. (5.55) with periodic input I(t) = I0 + I1cos( t). This leads to an input potential h(t) = h0 + h1 cos( t + ) with bias h0 = I0 and a periodic component with a certain amplitude h1 and phase . We choose a refractory kernel with absolute and relative refractoriness defined as
Suppose that a spike has occurred at = 0. The probability density that the next spike occurs at time t is given by PI(t|) and can be calculated from Eq. (5.53). The result is shown in Fig. 5.10. We note that the periodic component of the input is well represented in the response of the neuron. This example illustrates how neurons in the auditory system can transmit stimuli of frequencies higher than the mean firing rate of the neuron; see Chapter 12.5. We emphasize that the threshold in Fig. 5.10 is at =1. Without noise there would be no output spike. On the other hand, at very high noise levels, the modulation of the interval distribution would be much weaker. Thus a certain amount of noise is beneficial for signal transmission. The existence of a optimal noise level is a phenomenon called stochastic resonance and will be discussed below in Section 5.8.
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