- 6.4.1 Stationary Activity and Mean Firing Rate

- 6.4.2 Gain Function and Fixed Points of the Activity
- 6.4.2.1 Example: SRM
_{0}neurons with escape noise - 6.4.2.2 Example: Integrate-and-fire model with diffusive noise

- 6.4.2.1 Example: SRM
- 6.4.3 Low-Connectivity Networks

6.4 Asynchronous firing

We define
asynchronous firing of a neuronal population as
a macroscopic firing state with constant activity
*A*(*t*) = *A*_{0}.
In this section we use the population activity
equations (6.73) and (6.75)
to study the existence
of asynchronous firing states
in a homogeneous population of spiking neurons.
We will see that the neuronal
gain function plays an important role.
More specifically, we will show
that the knowledge of the single-neuron gain function *g*(*I*_{0})
and the coupling parameter
*J*_{0} is sufficient
to determine the activity *A*_{0} during asynchronous firing.

In this section we will show that
during asynchronous firing the population activity *A*_{0}
is equal to the mean firing rate of a single neuron
in the population. To do so, we
search for a stationary solution *A*(*t*) = *A*_{0} of the
population equation (6.73). Given constant activity
*A*_{0}
and constant external input
*I*^{ext}_{0},
the total input *I*_{0} to each neuron is constant.
In this case, the state of each neuron
depends only on *t* - , i.e.,
the time since its last output spike.
We are thus in the situation
of *stationary* renewal theory.

In the stationary state, the survivor function
and the interval distribution
can not depend explicitly upon
the absolute time, but only on
the time difference
*s* = *t* - .
Hence we set

S_{I}( + s | ) |
S_{0}(s) |
(6.92) | |

P_{I}( + s | ) |
P_{0}(s) |
(6.93) |

The value of the stationary activity

We use d

1 = A_{0}1 S_{0}(s) ds = A_{0}s P_{0}(s) ds , |
(6.95) |

where we have exploited the fact that

s P_{0}(s)ds = T |
(6.96) |

is the mean interspike interval. Hence

This equation has an intuitive interpretation: If everything is constant, then averaging over time (for a single neuron) is the same as averaging over a population of identical neurons.

How can we compare the population activity *A*_{0} calculated in
Eq. (6.97) with simulation results? In a simulation of a
population containing a finite number *N* of spiking neurons, the observed
activity fluctuates. Formally, the (observable) activity *A*(*t*) has been
defined in Eq. (6.1) as a sum over functions. The
activity *A*_{0} predicted by the theory is the *expectation* value of
the observed activity. Mathematically speaking, the observed activity *A*
converges for
*N* in the weak topology to its expectation value.
More practically this implies that we should convolve the observed activity
with a continuous test function (*s*) before comparing with *A*_{0}. We
take a function with the normalization
(*s*) d*s* = 1. For the sake of simplicity we assume furthermore that
has finite support so that
(*s*) = 0 for *s* < 0 or
*s* > *s*^{max}. We define

(t) = (s) A(t - s) ds . |
(6.98) |

The firing is asynchronous if the averaged fluctuations |(

For the purpose of illustration, we have plotted
in Fig. 6.10A the spikes of
eight neurons of the network simulation shown in
Fig. 6.9. The mean
interspike-interval for a single neuron is
*T* = 20ms
which corresponds to a population activity
of *A*_{0} = 50Hz.

6.4.2 Gain Function and Fixed Points of the Activity

The gain function of a neuron is the firing rate
*T* as a
function of its input current *I*. In the previous subsection, we have seen
that the firing rate is equivalent to the population activity *A*_{0} in the
state of asynchronous firing. We thus have

Recall that the total input

The constant factor (

This, however, is rather an input

This is the central result of this section, which is not only valid for SRM

Figure 6.11 shows a graphical solution of Eq. (6.102) in terms
of the mean interval
*T* as a function of the input *I*_{0}
(i.e., the gain function) and the total input *I*_{0} as a function of the
activity *A*_{0}. The intersections of the two functions yield fixed points of
the activity *A*_{0}.

As an aside we note that the graphical construction is identical to that of
the Curie-Weiss theory of ferromagnetism which can be found in any physics
textbook. More generally, the structure of the equations corresponds to the
mean-field solution of a system with feedback. As shown in
Fig. 6.11, several solutions may coexist. We cannot conclude
from the figure, whether one or several solutions are stable. In fact, it is
possible that *all* solutions are unstable. In the latter case, the
network leaves the state of asynchronous firing and evolves towards an
oscillatory or quiescent state. The stability analysis of the asynchronous
state is deferred to Chapter 8.

Consider a population of (noisy) SRM_{0} neurons with escape rate *f*,
e.g.
*f* (*u* - ) = exp[ (*u* - )];
cf. Chapter 5.3. The stationary activity *A*_{0} in the presence
of a constant input potential
*h*_{0} = *R* *I*_{0} is given by

where

In the limit of diffusive noise the stationary activity is

where is the variance of the noise; cf. Eq. (6.33). In a asynchronously firing population of

I_{0} = I^{ext}_{0} + J_{0} A_{0} ; |
(6.105) |

cf. Eq. (6.5). The fixed points for the population activity are once more determined by the intersections of these two functions; cf. Fig. 6.12B.

6.4.3 Low-Connectivity Networks

In the preceding subsections we have studied the stationary state of a
population of neurons for a given noise level. The noise was modeled either as
diffusive noise mimicking stochastic spike arrival or as escape noise
mimicking a noisy threshold. In both cases noise was added *explicitly*
to the model. In this section we discuss how a network of *deterministic*
neurons with fixed random connectivity can generate its own noise. In
particular, we will focus on spontaneous activity and argue that there exist
stationary states of asynchronous firing at low firing rates which have broad
distributions of interspike intervals even though individual neurons are
deterministic.
This point has been emphasized by van Vreeswijk and Sompolinsky (1996,1998)
who used a network of binary neurons to demonstrate
broad interval distribution in deterministic networks.
Amit and Brunel (1997a,b) where the first to analyze a network
of integrate-and-fire neurons with fixed random connectivity.
While they allowed for
an additional fluctuating input current, the major part
of the fluctuations were in fact generated by the network itself.
The theory of randomly connected
integrate-and-fire neurons has been further developped by
Brunel and Hakim (1999). In a recent study, Brunel (2000)
confirmed that asynchronous highly irregular firing
can be a stable solution of the network dynamics
in a completely deterministic network consisting
of excitatory and inhibitory integrate-and-fire neurons.
The
analysis of randomly connected networks of integrate-and-fire neurons is
closely related to
earlier theories for random nets of formal analog or binary neurons
(Nützel, 1991; Kree and Zippelius, 1991; Amari, 1977b,1972,1974; Crisanti and Sompolinsky, 1988; Cessac et al., 1994).

The network structure plays a central role in the arguments. While we assume
that all neurons in the population are of the same type, the connectivity
between the neurons in the population is not homogeneous. Rather it is random,
but fixed. Each neuron in the population of *N* neurons receives input from
*C* randomly selected neurons in the population. Sparse connectivity means
that the ratio

is a small number. Is this realistic? A typical pyramidal neuron in the cortex receives several thousand synapses from presynaptic neurons while the total number of neurons in the cortex is much higher. Thus globally the cortical connectivity

As a consequence of the sparse random network connectivity two neurons *i* and
*j* share only a small number of common inputs. In the limit of *C*/*N* 0
the probability that neurons *i* and *j* have a common presynaptic neuron
vanishes. Thus, *if* the presynaptic neurons fire stochastically, then
the input spike trains that arrive at neuron *i* and *j* are independent
(Kree and Zippelius, 1991; Derrida et al., 1987). In that case, the input of neuron *i* and *j* can be
described as stochastic spike arrival which, as we have seen, can be described
by a diffusive noise model.

The above reasoning, however, is based on the assumption that the presynaptic neurons (that are part of the population) fire stochastically. To make the argument self-consistent, we have to show that the firing of the postsynaptic neuron is, to a good approximation, also stochastic. The self-consistent argument will be outlined in the following.

We have seen in Chapter 5 that integrate-and-fire neurons with diffusive noise generate spike trains with a broad distribution of interspike intervals when they are driven in the sub-threshold regime. We will use this observation to construct a self-consistent solution for the stationary states of asynchronous firing.

We consider two populations, an excitatory population with *N*_{E} neurons and
an inhibitory population with *N*_{I} neurons. We assume that excitatory and
inhibitory neurons have the same parameters , , *R*, and
*u*_{r}. In addition all neurons are driven a common external current
*I*^{ext}. Each neuron in the population receives *C*_{E} synapses from excitatory
neurons with weight *w*_{E} > 0 and *C*_{I} synapses from inhibitory neurons with
weight *w*_{I} < 0. If an input spike arrives at the synapses of neuron *i* from
a presynaptic neuron *j*, its membrane potential changes by an amount
*u*_{i} = *w*_{j} where *w*_{j} = *w*_{E} if *j* is excitatory and *w*_{j} = *w*_{I} if *j* is
inhibitory. We set

Since excitatory and inhibitory neurons receive
the same number of inputs in our model, we assume
that they fire with a common firing rate .
The total input potential generated by
the external current and by the lateral couplings
is

The variance of the input is given by Eq. (6.24), i.e.,

The stationary firing rate

In a stationary state we must have

The arguments that have been developed above for low-connectivity networks can
be generalized to fully connected networks with *asymmetric random*
connectivity
(Sompolinsky et al., 1988; van Vreeswijk and Sompolinsky, 1996; Ben Arous and Guionnet, 1995; Amari, 1972; Cessac et al., 1994).

6.4.3.1 Example: Balanced excitation and inhibition

In the preceding sections, we have often considered neurons driven by a mean
input potential *h*_{0} = 0.8 and a variance
= 0.2. Let us find
connectivity parameters of our network so that
= 0.2 is the result of
stochastic spike arrivals from presynaptic neurons within the network. As
always we set
*R* = = 1 and
= 10ms.

Figure 6.13A shows that *h*_{0} = 0.8 and
= 0.2
correspond to a firing rate of
*A*_{0} = 16 Hz.
We set *w*_{E} = 0.025, i.e., 40 simultaneous spikes
are necessary to make a neuron fire.
Inhibition has the same strength
*w*_{I} = - *w*_{E}
so that *g* = 1.
We constrain our search to solutions with
*C*_{E} = *C*_{I} so that
= 1.
Thus, on the average, excitation and inhibition
balance each other. To get an average
input potential of *h*_{0} = 0.8 we need
therefore a constant driving current
*I*^{ext} = 0.8.

To arrive at
= 0.2 we solve Eq. (6.109) for
*C*_{E} and find
*C*_{E} = *C*_{I} = 200. Thus for this choice of the parameters
the network generates enough noise to allow a stationary solution of
asynchronous firing at 16Hz.

Note that, for the same parameter, the inactive state where all neurons are silent is also a solution. Using the methods discussed in this section we cannot say anything about the stability of these states. For the stability analysis see (Brunel, 2000) and Chapter 7.

6.4.3.2 Example: Spontaneous cortical activity

About eighty percent of the neurons in the cerebral cortex
are excitatory and twenty percent inhibitory.
Let us suppose that we have *N*_{E} = 8000
excitatory and *N*_{I} = 2000 inhibitory neurons
in a cortical column. We assume random connectivity
and take
*C*_{E} = 800,
*C*_{I} = 200
so that
= 1/4.
As before, excitatory synapses have
a weight
*w*_{E} = 0.025, i.e,
an action potential can be triggered by the simultaneous arrival
of 40 presynaptic spikes.
If neurons are driven in the regime
close to threshold, inhibition
is rather strong and we take
*w*_{I} = - 0.125 so that *g* = 5.
Even though we have less inhibitory than excitatory neurons,
the mean feedback is then dominated by
inhibition since
*g* > 1.
We search for a consistent solution
of Eqs. (6.108) - (6.110)
with a spontaneous activity of = 8Hz.

Given the above parameters, the variance is
0.54;
cf. Eq. (6.109). The gain function of
integrate-and-fire neurons gives us for = 8Hz a corresponding total
potential of
*h*_{0} 0.2; cf. Fig. 6.13B. To attain
*h*_{0} we have to apply an external stimulus
*h*_{0}^{ext} = *R* *I*^{ext}
which is slightly larger than *h*_{0} since the net effect of the lateral
coupling is inhibitory. Let us introduce the effective coupling
*J*^{eff} = *C*_{E} *w*_{E} (1 - *g*). Using the above parameters we find
from Eq. (6.108)
*h*_{0}^{ext} = *h*_{0} - *J*^{eff} *A*_{0} 0.6.

The external input could, of course, be provided by (stochastic) spike arrival from other columns in the same or other areas of the brain. In this case Eq. (6.108) is to be replaced by

with

The equations (6.110), (6.111) and (6.112) can be solved numerically (Amit and Brunel, 1997a,b). The analysis of the stability of the solution is slighlty more involved but can be done (Brunel, 2000; Brunel and Hakim, 1999).

Cambridge University Press, 2002

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