Mean-Field Theory of Irregularly Spiking Neuronal Populations and Working Memory in Recurrent Cortical Networks - Computational Neuroscience: A Comprehensive Approach - page 460

A B

80

g Low
tot 0.8

60

0.6

g High
tot

40

0.4

20

0.2

0

10 20 30 40 10 20 30 40

g tot s NMDA (nS) g tot s NMDA (nS)

C

D

60

2

gain < g Low
tot

40

1

0 0.05 0.1

20

0

0 −1

0.025

−2

gain > g High
tot

−0.025

0 2 4 6 8 0 0.25 0.5 0.75 1

s NMDA

Time (s)

Figure 15.5
Bistability in a simple, one-population recurrent network. (A) . Mean synaptic ac-
tivity as a function of the total recurrent synaptic input ˜ s g tot s NMDA (sigmoid; thick

solid line) and same quantity when s is interpreted as the total recurrent network
feedback (thin dashed lines). g High

=

50 nS and g Low 34 nS correspond to the

highest and lowest values of the network gain in which the network is bistable for
this network parameters. The crossings of a straight line corresponding to an in-
termediate value of g tot

= =

tot tot

=

40 nS with the sigmoid, correspond to the 3 steady state
solutions in the bistable regime. Stable (unstable) solutions are marked with a filled
(open) circle. (B) . Mean firing rate of the neurons as a function of the total recur-
rent synaptic input. The fixed point solutions in A are shown. Note the low rate
( ∼ 40 Hz) of the stable high activity state, and the non-zero rate ( ∼ 1Hz)ofthe

7 KHz, g ext
AMPA

stable low activity state. Network parameters are: C ext n ext = 1

nS. Parameters for the single cells are like in Figure 15.1. Synaptic parameters are
given in Section 15.2.6.2. (C) . Time course of activity (top) in the network when
g tot =

0 . = 3 .

40 nS. Brief current pulses (bottom) switch the network between its two stable
states. (D) . Potential function associated to the dynamics of Equation (15.92) for
three different values of g tot , corresponding to the situations where there is only one
low activity steady-state (upper curve), one high activity steady-state (lower curve)
and for a bistable network (middle curve). The values of s NMDA at the steady states
are given by the minima of the potential function. Inset : blown-up version of the
low activity region of U shows the disappearance of the low activity minimum of the
potential as g tot increases.