Biomedical Engineering Reference
In-Depth Information
since our knowledge of the coding mechanisms involved is far from com-
plete. In particular, Troyer and Doupe used to represent the brain's neural
activity a variable that represents the average activity of a large array of
neurons. Models at this level of description are known as “rate models”
[Troyer and Doupe 2000a, Troyer and Doupe 2000b].
In general, the “activity” of a neuron is measured in terms of the average
number of action potentials generated per unit of time, or
firing rate
.Oneof
the simplest continuous-time models used to describe the dynamics of such
a variable is the additive network model, which reads
x
i
+
S
ρ
i
+
c
ij
x
j
,
dx
i
dt
=
−
(8.30)
where
x
i
denotes the activity of the
i
th neuron within a network,
ρ
i
stands
for the external inputs, and the coe
cients
c
ij
describe the connections be-
tween the
i
th and
j
th neurons. The function
S
(
x
) is a continuous, monoton-
ically increasing function of its real argument which tends to a saturation
value for large arguments, and tends rapidly to zero for arguments smaller
than zero. For example, we can use the model
S
(
x
)=1
/
(1 +
e
−x
). The
dynamics of this equation are simple enough to understand qualitatively:
depending on whether the argument of
S
(
x
) is large enough or not, the ac-
tivity of the
i
th neuron will converge to the saturation value of
S
or to zero
[Hoppensteadt and Izhikevich 1997].
Under certain conditions of the coupling between the neurons within a nu-
cleus, it is possible to attempt to write down a simple average rate model for
the activity of a population of neurons. This was the program of Schuster and
Wagner [Schuster et al. 1990], who studied a neural circuit of model neurons
whose efferent synapses were either excitatory or inhibitory, in a configura-
tion such that the neurons were densely interconnected on a local scale, but
only sparsely connected on a larger scale. Under these conditions, Schuster
and Wagner showed that it was possible to derive macroscopic mean-field
equations for clusters of neurons.
This simplification of the problem would allow us to write, for example,
sets of equations governing the behavior of different populations of neurons
within some of the nuclei of the motor pathway. For example, we could de-
scribe the average activities
E
1
and
E
2
of the subpopulations of the RA
nucleus projecting to the respiratory center and to nXIIts (controlling the
syringeal muscles), respectively, and the activity
I
of the inhibitory interneu-
rons:
dE
1
dt
τ
1
=
−
E
1
+
S
(
ρ
1
+
c
11
E
1
+
c
13
I
)
,
(8.31)
dE
2
dt
τ
2
=
−
E
2
+
S
(
ρ
2
+
c
22
E
2
+
c
23
I
)
,
(8.32)
dI
dt
τ
3
=
−
I
+
S
(
ρ
3
+
c
31
E
1
+
c
32
E
2
+
c
33
I
)
,
(8.33)
