7.2 Transients

How quickly can a population of neurons respond to a rapid change in the
input? We know from reaction time experiments that the response of humans and
animals to new stimuli can be very fast (Thorpe et al., 1996). We therefore expect
that the elementary processing units, i.e., neurons or neuronal populations
should also show a rapid response. In this section we concentrate on one
element of the problem of rapid reaction time and study the response of the
population activity to a rapid change in the input. To keep the arguments as
simple as possible, we consider an input which has a constant value *I*_{0}
for *t* < *t*_{0} and changes then abruptly to a new value
*I*_{0} + *I*.
Thus

For the sake of simplicity, we consider a population of independent
integrate-and-fire or SRM_{0} neurons without lateral coupling. Given the
current
*I*^{ext}(*t*), the input potential can be determined from
*h*(*t*) = (*s*) *I*^{ext}(*t* - *s*) d*s*. For *t**t*_{0},
the input potential has then a value
*h*_{0} = *R* *I*_{0} where we have used
(*s*)d*s* = *R*. For *t* > *t*_{0}, the input potential *h* changes due to
the additional current *I* so that

Given the input potential

Let us suppose that for *t* < *t*_{0} the network is in a state of *asynchronous* firing so that the population activity is constant,
*A*(*t*) = *A*_{0} for *t**t*_{0}; cf. Chapter 6.4. As soon as
the input is switched on at time *t* = *t*_{0}, the population activity will change

A(t) = A_{0} + A(t) for t > t_{0} . |
(7.40) |

In this section we will use the linear population equation,

in order to calculate the linear response

7.2.1 Transients in a Noise-Free Network

In the noiseless case, neurons which receive a constant input *I*_{0} fire
regularly with some period *T*_{0}. For *t* < *t*_{0}, the mean activity is simply
*A*_{0} = 1/*T*_{0}. The reason is that, for a constant activity, averaging over
time and averaging over the population are equivalent;
cf. Chapter 6.4.

Let us consider a neuron which has fired exactly at *t*_{0}. Its next spike
occurs at *t*_{0} + *T* where *T* is given by the threshold condition
*u*_{i}(*t*_{0} + *T*) = . We focus on the initial phase of the transient and apply the
noise-free kernel
(*x*) (*x*);
cf. Tab. 7.1.
If we insert the function into Eq. (7.46)
we find

A(t) h(t) for t_{0} < t < t_{0} + T . |
(7.43) |

For both SRM

with a constant

In this example we apply Eq. (7.48) to integrate-and-fire neurons. The response kernel is

The response of the input

which has the characteristics of a low-pass filter with time constant . The population activity, however, reacts

where (

A similar result holds for a population of SRM_{0} neurons.
The initial transient of
SRM_{0} is identical to that of integrate-and-fire neurons;
cf. Fig. 7.5.
A subtle difference, however, occurs during the late
phase of the transient.
For
integrate-and-fire neurons the transient is over as soon as each neuron has
fired once.
After the next reset, all neurons fire periodically
with a new period *T* that corresponds to
the constant input
*I*_{0} + *I*.
A population of SRM_{0} neurons, however, reaches a periodic state
only asymptotically.
The reason is that the interspike interval *T* of
SRM_{0} neurons [which is given by the threshold condition
*h*(*t*) = - (*T*)] depends on the momentary
value of the input potential *h*(*t*).

7.2.2 Transients with Noise

So far, we have considered noiseless neurons. We have seen that after an
initial sharp transient the population activity approaches a new periodic
state where the activity oscillates with period *T*. In the presence of
noise, we expect that the network approaches - after a transient - a new
asynchronous state with stationary activity
*A*_{0} = *g*(*I*_{0} + *I*);
cf. Chapter 6.4.

In Fig. 7.6A illustrates the response of a population of noisy
neurons to a step current input. The population activity responds
instantaneously as soon as the additional input is switched on.
Can we understand the sharply peaked
transient? Before the abrupt change the input was stationary and the
population in a state of *asynchronous* firing. Asynchronous firing was
defined as a state with constant activity so that at any point in time
some of the neurons fire, others are in the refractory period, again others
approach the threshold. There is always a group of neuron whose potential is
just below threshold. An increase in the input causes those neurons to fire
immediately - and this accounts for the strong population response during the
initial phase of the transient.

As we will see in the example below, the above consideration is strictly valid only for neurons with slow noise in the parameters, e.g., noisy reset as introduced in Chapter 5.4. In models based on the Wilson-Cowan differential equation the transient does not exhibit such a sharp initial peak; cf. Fig. 7.6B.

For diffusive noise models the picture is more complicated. A rapid response occurs if the current step is sufficiently large and the noise level not too high. On the other hand, for high noise and small current steps the response is slow. The question of whether neuronal populations react rapidly or slowly depends therefore on many aspects, in particular on the type of noise and the type of stimulation. It can be shown that for diffusive noise that is low-pass filtered by a slow synaptic time constant (i.e., cut-off frequency of the noise lower than the neuronal firing rate) the response is sharp, independent of the noise amplitude. On the other hand, for white noise the response depends on the noise amplitude and the membrane time constant (Brunel et al., 2001).

For a mathematical discussion of the transient behavior, it is sufficient to
consider the equation that describes the initial phase of the linear response
to a sudden onset of the input potential; cf. Eq. (7.46).
Table 7.1 summarizes the kernel
(*x*) that is at the heart of
Eq. (7.46) for several noise models. In the limit of low noise,
the choice of noise model is irrelevant - the transient response is
proportional to the *derivative* of the potential,
*A* *h'*. If the level of noise is increased, a population of neurons with slow
noise (e.g., with noisy reset) retains its sharp transients since the kernel
is proportional to *h'*,

Neurons with escape noise turn in the high-noise limit to a different regime
where the transients follow *h* rather than *h'*. To see why, we recall that
the kernel
essentially describes a low-pass filter;
cf. Fig. 7.3.
The time constant
of the filter increases with the noise level and hence the response switches
from a behavior proportional to *h'* to a behavior proportional to *h*.

The width of the kernel
(*x*)
in Eq. (7.46)
depends on the noise level.
For low noise, the kernel is
sharply peaked at *x* = 0 and
can be approximated by a Dirac function.
The response *A*
of the population activity is sharp
since it is proportional to the *derivative*
of the input potential.

For high noise, the kernel is broad and the response becomes proportional to the input potential; cf. Fig. 7.7.

In Chapter 6.3, we have introduced the Wilson-Cowan
differential equations which are summarized here for a population of
independent neurons,

cf. Eq. (6.87). A step current input, causes a potential

h(t) = h_{0} + R I 1 - exp - (t - t_{0}) . |
(7.49) |

The response of the population activity is therefore

where

For neurons with noisy reset, the kernel
is a Dirac function;
cf. Tab. 7.1. As in the noiseless case, the initial transient
is therefore proportional to the derivative of *h*. After this initial phase
the reset noise leads to a smoothing of subsequent oscillations so that the
population activity approaches rapidly a new asynchronous state;
cf. Fig. 7.6A. The initial transient, however, is sharp.

In this example, we present qualitative arguments to show that, in the limit
of low noise, a population of spiking neurons with diffusive noise will
exhibit an immediate response to a *strong* step current input. We have
seen in the noise-free case, that the rapid response is due the derivative
*h'* in the compression factor. In order to understand, why the derivative of
*h* comes into play, let us consider, for the moment, a finite step in the
input *potential*
*h*(*t*) = *h*_{0} + *h* (*t* - *t*_{0}). All neurons *i*
which are hovering below threshold so that their potential *u*_{i}(*t*_{0}) is
between
- *h* and will be put above threshold and
fire synchronously at *t*_{0}. Thus, a step in the potential causes a
-pulse in the activity
*A*(*t*) (*t* - *t*_{0}) *h'*(*t*_{0}). In Fig. 7.8a we have used a *current* step
(7.42) [the same step current as in Fig. 7.5]. The
response at low noise (top) has roughly the form
*A*(*t*) *h'*(*t*) (*t* - *t*_{0}) as expected. The rapid transient is slightly less
pronounced than for noisy reset, but nevertheless clearly visible; compare
Figs. 7.6A and 7.8A. As the amplitude of the noise
grows, the transient becomes less sharp. Thus there is a transition from a
regime where the transient is proportional to *h'* (Fig. 7.8A) to
another regime where the transient is proportional to *h*
(Fig. 7.8B). What are the reasons for the change of behavior?

The simple argument from above based
on a potential step
*h* > 0 only holds
for a *finite* step size which is
at least of the order of the noise amplitude .
With diffusive noise, the threshold acts as an absorbing
boundary. Therefore the density of neurons with potential
*u*_{i} vanishes for
*u*_{i};
cf. Chapter 6.2.
Thence, for
*h* 0 the proportion of neurons
which are instantaneously put across threshold is 0.
In a stationary state, the 'boundary layer' with low density
is of the order ; e.g.,
cf. Eq. (6.28).
A potential step
*h* > puts a significant
proportion of neurons above threshold and leads to a
-pulse in the activity. Thus the result that
the response is proportional to the derivative of the potential
is essentially valid in the low-noise regime.

On the other hand, we may also consider diffusive noise
with large noise amplitude
in the sub-threshold regime.
In the limit of high noise,
a step in the potential raises the
instantaneous rate of the neurons, but does
not force them to fire immediately.
The response to a *current* step is therefore smooth
and follows the potential *h*(*t*); cf.
Fig. 7.8B.
A comparison of Figs. 7.8 and 7.7 shows that
the escape noise model exhibits a similar transition
form sharp to smooth responses with increasing noise level.
In fact, we have seen in Chapter 5 that
diffusive noise can be well approximated
by escape noise (Plesser and Gerstner, 2000).
For the analysis of
response properties with diffusive
noise see Brunel et al. (2001).

Cambridge University Press, 2002

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