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T
1
1
3
3
1
1
2
3
T
2
I<
T
v
2
I>
T
0
2
2
0
0
0
1
0
−1
1
v
1
Fig. 3.
Dynamical regions and fixed points for the feedback circuit. Numbers in circles
represent the fixed point which tend towards the points in the region with the same
number when there is no external input. Numbers in diamonds represent the fixed
point which tend towards the points in the region with the same number when there
is external input.
This way we can further predict the behaviour of our network stating that
the two neurons of the excitatory-inhibitory loop will oscillate between spiking
and resting periods when the
T
1
,T
2
pair lies inside the rectangle defined by the
four fixed points both in the case of no external input or when we inject an
external input to the excitatory neuron. As the output of both neurons cross the
respective thresholds in their orbits this produces different rhythmical patterns.
This behaviour can be also seen in [8]. Outside of that rectangle the system
converges to a fixed point. As expected, high values of the thresholds produce
that the corresponding neuron tend to remain resting, meanwhile low thresholds
produce tonic bursting.
The injection of external input essentially produces a shift to higher values of
the
v
1
coordinate of the fixed points. The effect of this shift is that the network
can change its regime from resting to oscillations about the threshold or from
oscillations to tonic spiking once the external input is applied.
The inhibition role can be seen clearly in the fact that, for a fixed value of the
excitatory neuron threshold
T
1
, this neuron changes his regime from spiking to
resting as the inhibitory coupling
W
21
becomes more negative.
The output neuron is fully controlled by the feedback circuit. Its behaviour de-
pends essentially of its spiking threshold value
T
3
. The neuron will oscillate (but
not necessarily about its threshold value) only when the excitatory-inhibitory
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