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I
as the value of the external input,
W
iI
and
T
I
are respectively the (excitatory)
coupling strength and the threshold of the external input. For normalization
purposes we ask without loss of generality the condition
a
i
+
i
=
j
|
+
W
iI
=
1. This condition ensures the bounding of the neuron output in the interval
[
W
ji
|
1
,
1]. In this model we consider that neuron
j
is spiking at time instant
t
if the
value of neuron
j
,
v
j
, is over the spike threshold
T
j
, so a constant value of neuron
i
over the spike threshold means a tonic spiking, meanwhile an oscillating value
around the spike threshold means that the neuron alternates between spiking
and resting regimes.
The network object of study is composed by a feedback loop of one excitatory
and one inhibitory neuron. The excitatory neuron also receives an external input.
This excitatory-inhibitory loop controls the behaviour of a third neuron. Simi-
lar circuits have been explored by means on differential models and computer
simulations by [11] and [8,9]. The whole network can be seen in figure 1.
−
3
+
_
+
+
1
2
I
−
Fig. 1.
A network with a feedback loop that controls a third neuron
The dynamical evolution of the network is modelled by a coupled system of
piecewise linear iterated maps that vary in time for each neuron, namely
u
1
(
t
+1)=
a
1
u
1
(
t
)+
W
21
H
(
u
2
(
t
)
−
T
2
)+
W
I
1
H
(
I
−
T
I
)
u
2
(
t
+1)=
a
2
u
2
(
t
)+
W
12
H
(
u
1
(
t
)
−
T
1
)
u
3
(
t
+1)=
a
3
u
3
(
t
)+
W
13
H
(
u
1
(
t
)
−
T
1
)+
W
23
H
(
v
2
(
t
)
−
T
2
)
(3)
3 Dynamic Behaviour
The feedback circuit is modelled by the coupled maps
v
1
(
t
+1)=
a
1
v
1
(
t
)+
W
21
H
(
v
2
(
t
)
−
T
2
)+
W
I
1
H
(
I
−
T
I
)
v
2
(
t
+1)=
a
2
v
2
(
t
)+
W
12
H
(
v
1
(
t
)
−
T
1
)
(4)
with
W
21
<
0,
W
I
1
>
0and
W
12
>
0 representing the excitatory-inhibitory loop
with the normalization conditions
a
1
−
W
21
+
W
I
1
=1and
a
2
+
W
12
=1.This
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