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way, adaptive neural networks can be implemented
in which proper plasticity rules are active. In
presence of input signals, the simulation of the
whole network shows the selection of neural groups
in which the activity appears very high, while it
remains at a quite low level in the regions among
different groups. This auto-confinement property of
the network activity seems to remain stable, even
when the considered input is terminated.
The paper is organized as follows: description
and simulation of some basic properties of the
neuron, such as threshold and latency; introduction
of the simplified model on the basis of the previous
analysis, description of the network structure,
plasticity algorithm, input structure, simulation
results and performance tests.
2 LATENCY
CHARACTERIZATION
Different kinds of neurons can be considered in
nature, with special and peculiar properties (S.
Ramon y Cajal, 1909-1911). On the other hand,
many models have been introduced and compared in
terms of biological plausibility and computational
cost. Antipodes are the Integrated and Fire (L.
Lapicque, 1907) and the Hodgkin-Huxley Model
(A.L. Hodgkin, A.F. Huxley, 1952), the first one
characterized by low computational cost and low
fidelity, while the second is a quite complete
representation of the real case.
In the latter case, the model consists of four
differential equations describing membrane
potential, activation of Na+ and K+ currents, and
inactivation of Na+ current (E. M. Izhikevich, 2007).
From an electrochemical point of view, the neuron
can be characterized by its membrane potential V
m
.
In the simulation starting case, the neuron lies in
its resting state, i.e. V
m
= V
rest
(Resting Potential),
until an external excitation is received.
The membrane potential varies by integrating the
input excitations. Since contributions from outside
are constantly added inside the neuron, a significant
accumulation of excitements may lead the neuron to
cross a threshold, called firing threshold TF (E. M.
Izhikevich, 2007), so that an output spike can be
generated.
However, the output spike is not immediately
produced, but after a proper delay time called
latency (R. FitzHugh, 1955). Thus, the latency is the
delay time between exceeding the membrane
potential threshold and the actual spike generation.
From a physiological point of view, such a delay
time is usually attributed to the slow charging of the
dendritic tree, as well as to the action of the A-
current, namely the voltage-gated transient K+
current with fast activation and slow inactivation.
The current activates quickly in response to a
depolarization and prevents the neuron from
immediate firing. With time, however, the A-current
inactivates and eventually allows firing (E. M.
Izhikevich, 2007). This phenomenon is affected by
the amplitudes and widths of the input stimuli and
thus rich dynamics of latency can be observed,
making it very interesting for the global network
desynchronization.
It is quite evident that latency concept strictly
depends on an exact definition of the threshold level.
However, strictly speaking, the true threshold is not
a fixed value, as it depends on the previous activities
of the neuron, as shown by the Hodgkin-Huxley
equations (A.L. Hodgkin, A.F. Huxley, 1952).
Indeed, a neuron is similar to a dynamical system, in
which any actual state depends on the previous ones.
The first work addressing the threshold from a
mathematical point of view was FitzHugh (R.
FitzHugh, 1955), who defined the Quasi Threshold
Phenomenon (QTP). A finite maximum latency is
defined, but neither a true discontinuity in response
nor an exact threshold level are considered. Indeed,
with reference to the squid giant axon model, it has
been pointed out that the membrane fluctuations for
experimental observations or insufficient accuracy
for the simulators, make not possible to establish an
exact value of the threshold. To this purpose, in
figures 1a and 1b, it is shown that the neuron
behaviour is very sensitive with respect to small
variations of the excitation current.
Nevertheless, in the present work, an appreciable
maximum value of latency will be used. This value
is determined by simulation and applied to establish
a reference threshold point. When the membrane
potential becomes greater than the threshold, the
latency appears as a function of V
m
. To this purpose,
proper simulations have been carried out for single
neurons.
3 SINGLE NEURON
SIMULATIONS
Significant latency properties will be analysed in this
section. To this purpose, the NEURON Simulation
Environment (http://www.neuron.yale. edu/neuron/)
has been used, a tool for quick development of
ACCURATE LATENCY CHARACTERIZATION FOR VERY LARGE ASYNCHRONOUS SPIKING NEURAL
NETWORKS
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