Biomedical Engineering Reference
In-Depth Information
The first equation describes how activity evolves in a recurrent excitatory
network. Basically,
a
tends to
a
with a time constant U
a
. The function
a
repre-
sents the input-output function of the network. This function depends on the
input-output functions of the individual neurons, their distribution across the
population, and the dynamics of the synaptic signals. For simplicity we have
chosen a sigmoidal function
a
(
i
) = 1/(1 +
e
(
i
-
R
)/
ka
), as illustrated in Figure 2C. For
low inputs to the network, there is very little output (
a
0), until a threshold
(R) is reached; the output then quickly reaches its maximal value (
a
1); R can
be seen as an average firing threshold in the neuronal population. Note that the
activity
a
is itself the input to the network—modulated by the effective connec-
tivity factor
n
#
s
#
d
—because of the recurrent excitatory connections. The pa-
rameter
n
is the network connectivity, a composite measure of the number of
connections per neuron and synaptic strength, which determines the maximal
gain of the positive feedback loop created by excitatory connections. As de-
scribed below, the activity defined by this equation is bistable over a wide range
of parameters.
The second equation describes the evolution of the synaptic variable
d
,
which represents a fast depression of the effective connectivity—U
d
(U
a
) is on
the order of 100 ms, as in cortical networks (5). When
d
= 0 all synapses are
totally depressed while synapses have full strength when
d
= 1;
d
is a decreas-
ing function of
a
, also chosen to be sigmoidal for convenience. The interplay
between
a
and
d
can create oscillations of the activity. Finally, Eq. [3] describes
the variations of the slow (U
s
>> U
a
) synaptic variable
s
. This variable also de-
creases when
a
is large and increases for low activity, but on a much slower time
scale. A possible biophysical mechanism for this slow (time scale minutes) syn-
aptic depression involves the loss of chloride ions by the neurons during an epi-
sode, decreasing the excitatory action of gabaergic and glycinergic connections
(8). Alternatively, the slow depression could be due to a cellular (not synaptic)
process increasing the cellular threshold R for high levels of activity (Eq. [3']; R
is an increasing function of
a
). The slow type of depression, whether synaptic or
cellular, is responsible for the episodic nature of the activity (see below).
By reducing the network to a system of three differential equations, we lose
the complexity of a population of neurons and only study the mean-field (deter-
ministic) interactions between the fast positive feedback and the slower negative
feedback in generating this activity. The advantage is that we are able to qualita-
tively analyze the dynamics of the system with fast/slow dissection techniques
(see, e.g. (30)). We present this dissection in the next section, but in brief the
methodology is as follows. The slow variable
s
is first considered as a parameter,
and the dynamic states of the one-variable (
a
) (or two-variable (
a,d
), as shown
by Tabak et al. (36)) fast subsystem are fully described; the steady state (and
oscillatory, in the case of a two-variable fast subsystem) solutions are obtained
as a function of
s
. Then when
s
is free to follow its autonomous dynamics the
full system has solutions that evolve on the slow time scale of
s
, sampling
