Biomedical Engineering Reference
In-Depth Information
= 0.9 in Figure 3A), there is one intersection, but at a high activity level. Con-
nectivity in the network is so high that activity is self-sustained. This is expected
from an excitatory network: for very low connectivity the network is inactive,
while for high connectivity the activity is maintained through positive feedback.
It is also known that for intermediate connectivity the low and high activity
states can both exist, as described below.
For intermediate values of
n
(
n
= 0.5 in Figure 3A), we find three intersec-
tions. There are steady states at low, high, and intermediate levels. Note that the
middle steady state is
unstable
. This can be easily seen since the slope of
a
(
n
#
a
) is greater than 1 at this point, therefore if
a
is slightly increased (respectively,
decreased) its derivative
(
)
ΒΈ will become positive (resp., nega-
tive), which will tend to further increase (resp., decrease)
a
. The slightest
movement away from this steady state will therefore be amplified. This is illus-
trated in Figure 3B. If the network activity is perturbed from the low state to just
below the middle state level (dashed line), activity will decrease back to its low
level (i). On the other hand, if the network is kicked to just above the middle
state, activity will jump to the high steady level (ii). The middle steady state is
thus a
network threshold
, separating the low (inactive) and high (active) states.
We can summarize these results by plotting the activity levels (steady
states) calculated when
n
is varied continuously. We obtain the important dia-
gram shown in Figure 3C. The resulting "S-shaped" curve has 3
branches
: the
lower branch (solid) corresponds to the low activity states, the middle branch
(dashed) corresponds to the unstable states, and the upper branch (solid) corre-
sponds to the high activity states. The S-curve defines 2 domains in the (
a
-
n
)
plane. For any value of
n
, if the activity is such that the point (
n
,
a
) is on the right
of the curve, then activity will increase until the system reaches the upper branch
(high state). Conversely, if (
n
,
a
) is to the left of the S-curve, activity will de-
crease until it reaches the low state.
We can see that for a range of values of
n
(approximately between 0.31 and
0.73) there are two possible stable states. The network is
bistable
. As we have
seen for
n
= 0.5, a perturbation strong enough to cross the middle branch allows
switching between the two stable states. This bistability is the basis for the oscil-
latory and episodic behavior described below. Imagine the network is in the
high-activity state and we slowly decrease the connectivity. The state of the sys-
tem, defined by a point in the (
a
-
n
) plane, will be on the upper branch and
slowly move to the left, with a minimal decrease of activity. However, when
n
passes a critical value around 0.31 where the upper and middle states coincide
(the "left knee" of the S-curve), the only remaining state is the low-activity state
and the network crashes to that state. Now, we slowly increase
n
, so the state of
the system tracks the lower branch, going to the right. Similarly, activity is only
going to increase slightly until the "right knee," where the middle and lower
states coalesce. Once
n
is above that point, only the high state remains and the
network will jump to its high activity state, terminating the cycle.
aana a
d
=
