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Table 3.
Neuron parameters for each experiment
Cases
a
1
a
2
a
3
W
I
1
W
21
W
12
W
13
W
23
Case 1 0.4 0.4 0.2 0.4
0.2
0.6
0.6
0.2
Case 2 0.4 0.4 0.2 0.2
0.4
0.6
0.6
0.2
Case 3 0.4 0.4 0.2 0.4
0.2
0.6
0.2
0.6
Excitatory Neuron
Inhibitory Neuron
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
0
10
20
30
40
50
0
10
20
30
40
50
Time
Time
Output Neuron
E xcitatory- I nhibitory
1
1
1
0.5
0.5
0.5
0
0
0
-0.5
-0.5
-0.5
-1
-1
-1
0
10
20
30
40
50
-1
-1
-0.5
-0.5
0
0
0.5
0.5
1
1
Time
Output (Excitatory)
Fig. 4.
Representation of the first case
(
v
i
=0)
,i
=1
,
2
,
3, with the input
I
set at 0
.
5 and thresholds
T
i
,i
=1
,
2
,
3set
to 0
.
3. Neuron
N
1 is the excitatory neuron,
N
2 is the inhibitory neuron and
N
3
is the output neuron. The three neurons use similar values, shown on table 3.
Case 1 will represent a spiking model defined when
T
1
<
W
I
1
+
W
21
1
−a
1
,thisis,
oscillations about the spiking threshold. Case 2 is similar to Case 1 with oscil-
lations but no spiking. Case 3 is the case where the network reaches a stable
point.
4.1 Case 1
As can be seen in figure 4, the output neuron value graph is oscillatory. Ob-
serving the comparative graph (N1 vs N2), the two neurons start at the origin
were both neurons are inactive. This is region 0 and the fixed point expected is
(
W
I
1
1
,
0
,
0) = (0
.
33
,
0
,
0). That's the point we reach in the next three steps, on
region 1 where N1 is firing.
On this region the fixed point is (
W
I
1
1
−a
1
,
W
12
1
,
W
13
1
)=(0
.
33
,
1
,
0
.
75) so our new
point is in that direction. Ascending almost vertically, we go over the N2 thresh-
old and it starts firing, the fixed point is not reached and the direction changes
to the fixed point of region 3 (
W
I
1
+
W
21
1
−a
1
−a
2
−a
3
,
W
12
1
,
W
13
+
W
23
1
0
.
33
,
1
,
0
.
5).
Following that direction, N1 stops firing as we enter region 2 with fixed point
(
W
I
1
+
W
21
1
−a
1
)=(
−
−a
1
−a
2
−a
3
,
0
,
W
23
1
−a
3
)=(
−
0
.
33
,
0
,
−
0
.
25). To go under N2 threshold and enter
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