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Each time a spike occurs f.t/ increases by .1 f/and q.t/ decreases by a
d
d.
There are also more detailed models about the change of f;q, given by
C
P
j
ı.t
j
/a
f
.1 f/
df
dt
f
0
f
f
D
P
j
ı.t
j
/a
d
q
(4.22)
dq
dt
d
0
q
d
D
Moreover, there are different models to describe the variation of f and q, which do
not include the spiking times. For example,
dq
dt
d
0
q
d
D
a
d
.V/q
(4.23)
a
1Ce
k.V
V
thr
/
ad
.V/ D
Neurons plasticity affects memory. A basic assumption in neuroscience is that
memories are stored in the strength of the synaptic connections (Hebbian Learning).
Hebb rules enforce or weaken a synapse, depending on whether presynaptic or post-
synaptic neurons are active. If both neurons are active, then one has enforcement of
the neural synapse, otherwise the synapse is weakened (for both neurons to be active
it is necessary the potential of each neuron's membrane to have a high value. The
potential can be computed by the solution of the differential equations describing
the propagation of the membrane's potential).
4.7
Synchronization of Coupled FitzHugh-Nagumo Neurons
Using Differential Flatness Theory
4.7.1
The Problem of Synchronization of Coupled Neural
Oscillators
The voltage of the neuron's membrane exhibits oscillatory variations after receiving
suitable external excitation either when the neuron is independent from neighboring
neural cells or when the neuron is coupled to neighboring neural cells through
synapses or gap junctions [
16
,
98
,
182
]. In the latter case it is significant to analyze
conditions under which synchronization between coupled neural oscillators takes
place, which means that the neurons generate the same voltage variation pattern
subject to a phase difference. The loss of synchronism between neurons can
cause several neuro-degenerative disorders. Moreover, it can affect several basic
functions of the body such as gait, respiration, and heart's rhythm. For this reason
synchronization of coupled neural oscillators has become a subject of significant
research during the last years. The associated results have been also used in several
engineering applications, such as biomedical engineering and robotics. For example,
