Mean-Field Theory of Irregularly Spiking Neuronal Populations and Working Memory in Recurrent Cortical Networks - Computational Neuroscience: A Comprehensive Approach - page 461

down a hilly landscape. The valleys (or minima) correspond to the steady-states, and
the class of stimuli which are attracted to each steady-state (called its basin of attrac-
tion) are the set of all locations in the landscape which roll down to the same minima.
Indeed, the dynamics (15.92) can be re-written as

d ¯ s
dt = −

dU ( s )

. (15.97)

d ¯ s

This dynamics describes the movement of a point particle at location ¯ s sliding down
the landscape defined by the function U is

also called the potential of the dynamics, and it is such that the speed of the particle at
location s is equal to minus the slope of U at that location. In Figure 15.5D we show
three examples of the U

( s ) in the presence of high friction. U ( s )

for values of the gain at the recurrent connections g tot such
that the network has either a single high ( g tot

( s )

g High

g Low

> <

tot ) activity
steady-state, and for an intermediate value of g tot , where the network is bistable. For
low enough gain, U

)orlow( g tot

tot

has a single minimum, and as the gain increases, a second
minimum at a higher value of s appears. These two minima coexist for a range of
values of the gain, but if the gain is high enough, the low activity minimum disap-
pears (see inset in Figure 15.5D).

( s )

Several features deserve comments: First, in contrast to the network of linear
synapses of Section 15.3.2, the firing rate in the high activity fixed point is about 40
Hz (Figure 15.5B), much less than saturation rates, even in the absence of inhibi-
tion. This rate is in the upper range of the available physiological data for persistent
activity(20-50 Hz). This relatively low rate is due to the saturation properties of the
NMDA receptor. Second, the low activity state has a low firing rate of about 1Hz.
This is due to the presence of noise in the system. In the absence of noise, i.e.,
when the synaptic current is constant in time, the input-output function of the neuron
becomes a sigmoid with a 'hard' threshold. For currents below this threshold the
output rate is identically zero (see trend for decreasing noise levels in Figure 15.2)
and in the supra-threshold regime it increases as a sub-linear function of the input
current until saturation at 1

t re f is reached. In these conditions, when the network
is bistable, the low activity state is necessarily zero. When noise is included in the
description, the firing rate can be non-zero even in the sub-threshold regime: the
membrane potential, which hovers around its steady state below threshold, crosses
this threshold once in a while as a result of the random fluctuations in the input cur-
rent [8, 118]. Such a state of low, fluctuation-driven activity has been suggested to
correspond to the background or spontaneous activity state found in the cortex [11].
The fact that the low activity state is stable in Figure 15.5 is due to the fact that
the excitatory feedback is unrealistically weak (see caption of Figure 15.5). When
excitatory feedback is stronger, the non-zero low rate state becomes unstable, and
inhibition becomes necessary to achieve stability at low rates [11]. Hence, a more
detailed model is required.

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