- 5.3.1 Escape rate and hazard function
- 5.3.1.1 Example: Hard and soft threshold
- 5.3.1.2 Example: Motivating a sigmoidal escape rate
- 5.3.1.3 Example: Transition from continuous to discrete time

- 5.3.2 Interval distribution and mean firing rate

5.3 Escape noise

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.

5.3.1 Escape rate and hazard function

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,

that depends on the momentary distance between the (noiseless) membrane potential and the threshold; see Fig. 5.5. We can think of

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

where (

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

The choice of the escape function

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

Thus, the neuron never fires if

How can we `soften' the sharp threshold? A simple choice for a soft threshold is an exponential dependence,

where and are parameters. For , we return to the noiseless model of Eq. (5.44). Alternatively, we could introduce a piecewise linear escape rate,

with slope for

Finally, we can also use a sigmoidal escape rate (Wilson and Cowan, 1972; Abeles, 1982),

with time constant and noise parameter . The error function is defined as

with

We want to motivate the sigmoidal escape rate by a model with stochastic
threshold in *discrete* time
*t*_{n} = *n* . After each time step
of length , a new value of the threshold is chosen from a Gaussian
distribution of threshold values
with mean ,

The probability of firing at time step

cf. Fig. 5.7A. The firing probability divided by the step size can be interpreted as a firing intensity (Wilson and Cowan, 1972)

which is the sigmoidal escape rate introduced in Eq. (5.47).

Instead of a model with stochastic threshold, we can also consider a model
fixed threshold, but a membrane potential
*u*(*t*_{n}) + *u*(*t*_{n}) 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'*|) d*t'* (*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

Even if

5.3.2 Interval distribution and mean firing rate

In this section, we combine the escape rate model with the concepts of
renewal theory and calculate the input-dependent interval distribution
*P*_{I}(*t*|) for escape rate models.

We recall Eq. (5.9) and express the interval distribution in terms of the hazard ,

This expression can be compared to the interval distribution of the stationary Poisson process in Eq. (5.18). The main difference to the simple Poisson model is that 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

In order to make the role of refractoriness explicit, we consider the version SRM

with

Fig. 5.9 shows the interval distribution (5.56) for constant input current

We study the model SRM_{0} defined in Eq. (5.55) for absolute
refractoriness

The hazard is (

For stationary input

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

cf. Eq. (5.2). Let us set

For constant input current

For a leaky integrate-and-fire neuron with constant input *I*_{0}, the
membrane potential is

u(t|) = R I_{0} 1 - e^{} |
(5.61) |

where we have assumed

(t - ) = 1 - e^{- (t-)} , |
(5.62) |

with =

We study the model SRM_{0} as defined in Eq. (5.55) with periodic
input
*I*(*t*) = *I*_{0} + *I*_{1}cos( *t*). This leads to an input
potential
*h*(*t*) = *h*_{0} + *h*_{1} cos( *t* + ) with bias *h*_{0} = *I*_{0} and a periodic component with a certain amplitude *h*_{1} and phase
. We choose a refractory kernel with absolute and relative
refractoriness defined as

and adopt the exponential escape rate (5.45).

Suppose that a spike has occurred at = 0. The probability
density that the next spike occurs at time *t* is given by
*P*_{I}(*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.

Cambridge University Press, 2002

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