- 7.1.1 Noise-free Population Dynamics (*)

- 7.1.2 Escape noise (*)
- 7.1.2.1 The kernel
(
*x*) for escape noise (*) - 7.1.2.2 Example: Step-function escape rate (*)
- 7.1.2.3 Example: Absolute refractoriness (*)

- 7.1.2.1 The kernel
(
- 7.1.3 Noisy reset (*)

7.1 Linearized Population Equation

We consider a homogeneous population of independent neurons. All neurons
receive the same current *I*(*t*) fluctuating about the mean *I*_{0}.
More specifically we set

For small fluctuations, |

with |

where

is the input potential generated by the time-dependent part of the input current. The first term of the right-hand side of Eq. (7.3) takes into account that previous perturbations

Here we give an overview of the main results that we will obtain in the
present chapter; explicit expressions for the kernel
(*x*) are presented in
Tab. 7.1.

- (i)
- In the low-noise limit, the kernel
(
*x*) is a Dirac function. The dynamics of the population activity*A*has therefore a term proportional to the*derivative*of the input potential; cf. Eq. (7.3). We will see that this result implies a fast response*A*to any change in the input. - (ii)
- For high noise, the kernel
(
*x*) depends critically on the noise model. For noise that is slow compared to the intrinsic neuronal dynamics (e.g., noise in the reset or stochastic spike arrival in combination with a slow synaptic time constant) the kernel (*x*) is similar to that in the noise-free case. Thus the dynamics of*A*is proportional to the*derivative*of the input potential and therefore fast. - (iii)
- For a large amount of `fast' noise (e.g., escape noise), the
kernel
(
*x*) is broad so that the dynamics of the population activity is rather proportional to the input potential than to its derivative; cf. Eq. (7.3). As we will see, this implies that the response to a change in the input is slow.

Results for escape noise and reset noise have been derived by
Gerstner (2000b) while results for diffusive noise have been presented by
Brunel et al. (2001) based on a linearization of the membrane potential density
equation (Brunel and Hakim, 1999).
The effect of slow noise in parameters has already been discussed
in Knight (1972a).
Apart from the approach discussed in this section,
a fast response of a population of integrate-and-fire neurons with diffusive
noise can also be induced if the *variance* of the diffusive noise is
changed (Bethge et al., 2001; Lindner and Schimansky-Geier, 2001).

Before we turn to the general case, we will focus in Section 7.1.1
on a noise-free population. We will see why the dynamics of
*A*(*t*) has
a contribution proportional to the *derivative* of the input potential.
In Section 7.1.2 we derive the general expression for the
kernel
(*x*) and apply it to different situations. Readers not interested
in the mathematical details may skip the remainder of this section and move
directly to Section 7.2.

7.1.1 Noise-free Population Dynamics (*)

We start with a reduction of the population integral equation (6.75)
to the noise-free case. In the limit of no noise, the input-dependent
interval distribution
*P*_{I}(*t* | ) reduces to a Dirac
function, i.e.,

where

The interval

Note that

We recall from the rules for functions that

[f (x)] g(x) dx = |
(7.8) |

if

whenever a solution of =

To evaluate
*T'*() we use the threshold condition (7.7).
From
= *u*[ + *T*()] = [*T*()] + *h*[ + *T*()|] we find by taking the derivative with respect
to

0 = [T()] T'() + h[ + T()|] [1 + T'()] + h[ + T()|] . |
(7.10) |

The prime denotes the derivative with respect to the argument. We have introduced a short-hand notation for the partial derivatives, viz.,

T' = - , |
(7.11) |

where we have suppressed the arguments for brevity. A simple algebraic transformation yields

which we insert into Eq. (7.9). The result is

where

Let us consider a fluctuating input current that generates small perturbations
in the population activity
*A*(*t*) and the input potential
*h*(*t*)
as outlined at the beginning of this section. If we substitute
*A*(*t*) = *A*_{0} + *A*(*T*) and
*h*(*t*|) = *h*_{0} + *h*(*t*|) into
Eq. (7.13) and linearize in *A* and *h* we obtain an
expression of the form

where

For SRM_{0} neurons we have
*h*(*t*|) = *h*(*t*) so that the partial
derivative with respect to vanishes. The factor in square brackets in
Eq. (7.13) reduces therefore to
[1 + (*h'*/)]. If we linearize
Eq. (7.13) we find
the compression factor

For integrate-and-fire neurons we have a similar result. To evaluate the
partial derivatives that we need in Eq. (7.13) we write
*u*(*t*) = (*t* - ) + *h*(*t*|) with

cf. Eqs. (4.34) and (4.60). Here

Taking the derivative of and the partial derivatives of *h* yields

which we now insert in Eq. (7.13). Since we are interested in the

Here

In order to motivate the name `compression factor' and to give an
interpretation of Eq. (7.14) we consider SRM_{0}
neurons with an exponential refractory kernel
(*s*) = - exp(- *s*/). We want to show graphically that the population
activity *A* has a contribution that is proportional to the *derivative* of the input potential.

We consider Fig. 7.2. A neuron which has fired at will
fire again at
*t* = + *T*(). Another neuron which has fired
slightly later at
+ fires its next spike at
*t* + *t*.
If the input potential is constant between *t* and
*t* + *t*, then
*t* = . If, however, *h* increases between *t* and
*t* + *t*
as is the case in Fig. 7.2, then the firing time difference is
reduced. The compression of firing time differences is directly related to an
increase in the activity *A*. To see this, we note that all neurons which
fire between and
+ , must fire again between
*t* and
*t* + *t*. This is due to the fact that the network is homogeneous
and the mapping
*t* = + *T*() is monotonous. If firing
time differences are compressed, the population activity increases.

In order to establish the relation between
Fig. 7.2 and
Eq. (7.15),
we note that the compression faction is
equal to *h'*/.
For a SRM_{0} neuron with exponential refractory kernel,
(*s*) > 0 holds for all *s* > 0.
An input with *h'* > 0 implies then, because of
Eq. (7.14),
an increase of the activity:

h' > 0 A(t) > A(t - T) . |
(7.19) |

7.1.2 Escape noise (*)

In this section we focus on a population of neurons
with escape noise.
The aim of this section is two-fold. First, we want to show how to derive the
linearized population equation (7.3) that has
already been stated at the beginning of Section 7.1.
Second, we will show that in the case of high noise the population activity
follows the input potential *h*(*t*), whereas for low noise the activity follows
the derivative *h'*(*t*). These results will be used in the following three
sections for a discussion of signal transmission and coding properties.

In order to derive the linearized response *A* of the population
activity to a change in the input we start from the conservation law,

cf. (6.73). As we have seen in Chapter 6.3 the population equation (6.75) can be obtained by taking the derivative of Eq. (7.20) with respect to

For constant input

h(t|) = h_{0}(t|) + h(t|) , |
(7.22) |

where

is the change of the postsynaptic potential generated by

We have used the notation

We note that the first term on the right-hand side of Eq. (7.25) has the same form as the population integral equation (6.75), except that

To make some progress in the treatment of the second term on the right-hand
side of Eq. (7.25), we restrict the choice of neuron model and focus on
SRM_{0} or integrate-and-fire neurons. For SRM_{0} neurons, we may drop the
dependence of the potential and set
*h*(*t*|) = *h*(*t*) where *h* is the input potential caused by the
time-dependent current *I*; compare Eqs. (7.4) and
(7.23). This allows us to pull the variable
*h*(*s*) in
front of the integral over and write Eq. (7.25) in the form

with a kernel

(x) = - d ^{SRM}(x) ; |
(7.25) |

cf. Tab. 7.1.

For integrate-and-fire neurons we set
*h*(*t*|) = *h*(*t*) - *h*() exp[- (*t* - )/];
cf. Eq. (7.16). After some rearrangements of
the terms, Eq. (7.25) becomes identical to Eq. (7.26) with a
kernel

(x) = - d + d e^{-/} ^{IF}(x) ; |
(7.26) |

cf. Tab. 7.1.

Let us discuss Eq. (7.26). The first term on the right-hand side of
Eq. (7.26) is of the same form as the dynamic equation (6.75) and
describes how perturbations
*A*() in the past influence the
present activity
*A*(*t*). The second term gives an additional
contribution which is proportional to the derivative of a *filtered*
version of the potential *h*.

We see from Fig. 7.3
that the width of the kernel
depends on the noise level.
For low noise, it is significantly sharper than for high noise.
For a further discussion of Eq. (7.26)
we approximate the kernel by an exponential
*low-pass* filter

where

The noise-free
threshold process can be retrieved from Eq. (7.29) for
.
In this limit
^{SRM}(*x*) = *a* (*x*) and the initial transient
is proportional to *h'* as discussed above. For small , however, the
behavior is different. We use Eq. (7.29) and rewrite the last term in
Eq. (7.26) in the form

where (

In the escape noise model, the survivor function is given by

where

where

as noted in Tab. 7.1.

We take
*f* (*u*) = (*u* - ), i.e.,
a step-function escape rate. For
neurons fire
immediately as soon as
*u*(*t*) > and we are back to the noise-free
sharp threshold. For finite , neurons respond stochastically with time
constant .
We will show that the kernel
(*x*) for neurons
with step-function escape rate is an exponential function;
cf. Eq. (7.29).

Let us denote by *T*_{0} the time between the last firing time and the
formal threshold crossing,
*T*_{0} = *min**s* | (*s*) + *h*_{0} = . The derivative of *f* is a
-function,

where = |

as claimed above.

We take an arbitrary escape rate *f* (*u*) 0 with
*lim*_{u-}*f* (*u*) = 0 = *lim*_{u-}*f'*(*u*). Absolute refractoriness is defined
by a refractory kernel
(*s*) = - for
0 < *s* < and zero
otherwise. This yields
*f*[(*t* - ) + *h*_{0}] = *f* (*h*_{0}) (*t* - - ) and hence

The survivor function

Note that for neurons with absolute refractoriness the transition to the noiseless case is not meaningful. We have seen in Chapter 6 that absolute refractoriness leads to the Wilson-Cowan integral equation (6.76). Thus defined in (7.37) is the kernel relating to Eq. (6.76); it could have been derived directly from the linearization of the Wilson-Cowan integral equation. We note that it is a low-pass filter with cut-off frequency

We consider SRM_{0}-neurons with noisy reset as introduced in
Chapter 5.4. After each spike the membrane potential is
reset to a randomly chosen value
parameterized by the reset variable *r*. This is an example of a `slow' noise model,
since a new value of the stochastic variable *r* is chosen
only once per inter-spike interval. The
interval distribution of the noisy reset model is

where

A neuron that has been reset at time with value

where

To simplify the expression, we write
*A*(*t*) = *A*_{0} + *A*(*t*) and expand
Eq. (7.40) to first order in *A*. The result is

A comparison of Eqs. (7.41) and (7.3) yields the kernel (

Cambridge University Press, 2002

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