Biomedical Engineering Reference
In-Depth Information
4.2. Episodic and Rhythmic Behavior Due to Activity-Dependent
Depression of Network Excitability
The network could switch spontaneously between the high and low states
according to the above mechanism if we add an activity-dependent mechanism
to modulate the connectivity. Therefore, we now let the slow variable
s
vary
according to Eq. [3] (for now, the fast depression variable is still frozen:
d
= 1).
Because
s
is a decreasing function of
a
, the new variable
s
and therefore effec-
tive connectivity
n
#
s
will tend to decrease when the network is in the high state
and increase when the network is in the low state. This may lead to slow oscilla-
tions between the high and low states as explained in the previous paragraph and
illustrated in Figure 4A.
We can explain the oscillatory behavior of the system geometrically as
shown in Figure 4B. This treatment is similar to the phase plane analysis of sin-
gle neuron excitability pioneered by Fitzhugh (12,30). The variations of the
variables
a
and
s
(synaptic activity and fraction are not affected by slow depres-
sion) define a trajectory in the (
a
-
s
) plane, called the
phase plane
. The solid
gray S-curve in Figure 4B defines the states of the system for which
(cf.
Figure 3C) and is called the
a
-nullcline. The dashed curve defines the states of
the system for which
a
=
0
and is called the
s
-nullcline (it is simply the curve
s
=
s
(
a
)). For any value of
s
, if the activity is below that, curve
s
will be increas-
ing, while
s
will be decreasing if
a
is above the
s
-nullcline. The steady states of
the system (comprised of Eqs. [1] and [3]) are the intersections of the two null-
clines. In the case of Figure 4B, there is only one steady state and it is unstable.
A necessary condition for the steady state to be unstable is that the intersection
occurs on the middle branch of the
a
-nullcline. If there was an intersection
on the upper or lower branch, that intersection would define a stable steady
state at high or low activity, which would prevent the episodic behavior. This
immediately imposes a constraint on the parameters of the model if episodes are
to occur.
Imagine the system is in a state (
s
,
a
) on the right of the
a
-nullcline. The
trajectory will quickly go up as if
s
was constant because U
a
<< U
s
, until it
reaches the upper branch of the
a
-nullcline. Then
a
will remain constant, while
s
will decrease, since the system is now above the
s
-nullcline. The trajectory will
thus track the upper branch, going left, until it reaches the left knee of the
a
-
nullcline. Here, a further decrease in
s
forces the system to leave the
a
-nullcline,
and, being on the left of the nullcline, activity decreases so that the trajectory
quickly goes down to the lower branch of the
a
-nullcline. During this transition,
the trajectory crosses the
s
-nullcline, so
s
increases. The trajectory will thus
track the lower branch going right, until it passes the right knee, causing a new
transition upward to the upper state.
So far, the combination of the bistability of the activity (Eq. [1]) and the
slow, activity-dependent variations of the effective connectivity (Eq. [3]) creates
s
=
0
