Biomedical Engineering Reference
In-Depth Information
Table 1
. Values of the parameters used for the models,
unless mentioned otherwise in the text or figures
Parameter
s
-model R-model
n
, connectivity
1
1
U
a
, network time constant
1
1
R, network threshold (half activation)
0.18
variable
k
a
, inverse of slope of
a
at half activation
-0.05
-0.05
H
d
, time constant of fast synaptic depression
d
2
2
R
d
, half activation of
d
0.5
0.5
k
d
, inverse of slope of
d
at half activation
0.2
0.2
H
s
, time constant of slow synapic depression
s
500
R
s
, half activation of
s
0.14
k
s
, inverse of slope of
s
at half activation
0.02
U
R
, time constant of slow cellular adaptation R
1000
R
R
, half activation of R
0.15
k
R
, inverse of slope of R
at half activation
-0.05
the solution states of the fast subsystem and at critical (bifurcation) points exe-
cuting rapid transitions between states. This analysis involves numerical bifurca-
tion and branch-tracking methods and forward-in-time integrations, which were
carried out using the XPPAUT package (freely available software written by
G.B. Ermentrout, http://www.pitt.edu/~phase/). A fourth-order Runge-Kutta
scheme with a time step of 0.1 was used to numerically integrate the differential
equations. Parameter values used in the simulations are given in Table 1.
4.
PROPERTIES AND APPLICATIONS OF THE MODEL
4.1. Bistability of the Excitatory Network with Fixed Synaptic Efficacy
Let us first analyze the properties of the network without depression, that is,
we study Eq. [1] and freeze the depression variables (
s
=
d
= 1). Such a sys-
tem will reach a steady state ( (
for which the activity is defined by
a
=
a
(
n
#
a
) (from Eq. [1]). The steady states can be determined graphically as the
intersections of the straight line and the curve of
a
(
n
#
a
) shown in Figure 3A
for any value of
n
. When
n
is too small (
n
= 0.3 in Figure 3A), there is only one
a
=
)
