Biomedical Engineering Reference
In-Depth Information
was shown that adding noise to a weak stimuli
improves the timing precision in neuronal acti-
vity and that the cells are able to
adapt
their
intrinsic threshold values to the overall input
signal power. We defer our discussion of the
role of
adaptation
in SR until Section
2.4
.
An organism in which stochastic resonance is
exploited for enhanced electrical sensing is the
paddlefish (
P. spathula
)
.
Its electro-sensory recep-
tors use stochastic resonance to detect and loca-
lize low-frequency electrical fields (0.5-20 Hz)
emanated by planktons (
Daphnia
). In this case,
the source of noise is due to the prey themselves,
which in turn increases the sensitivity of the
paddlefish electro-sensory receptors
[6]
.
Each neuron is characterized by its intrinsic
voltage
v
i
(
t
)
, i
∈
[
1,
N
], and the neuron fires
whenever
v
i
exceeds a threshold
V
th
.
Between
consecutive firings, the dynamics of the mem-
brane potential can be expressed using the inte-
grate-and-fire model as
[30]
N
d
dt
v
i
(
t
)
=−
v
i
(
t
)/τ
m
−
W
ij
exp(
−
(
t
−
t
j
)/τ
s
)
j
=
1
+
α
x
i
(
t
).
(2.5)
Here,
t
j
are the set of firing times of the
j
th neu-
ron and
τ
m
denotes the time constant of the neu-
ron, capturing the leaky nature of integration
denoted by the leaky potential of the membrane
v
i
(
t
)
/τ
m
. The exponential term and the related
time constant
τ
s
model the presynaptic filter
h
(
t
)
in Eq.
(2.2)
and the time constant of the presyn-
aptic spike train. The parameter set
W
ij
denotes
the synaptic weights between the
i
th and
j
th
neurons and denotes the set of learning param-
eters for this integrate-and-fire neural network.
To show how the synaptic weights
W
ij
influ-
ence noise shaping, consider two specific cases
as described in Ref.
30
: (a) when
W
ij
=
0, implying
there is no coupling between the neurons and
each neuron fires independently of the other; and
(b) when
W
ij
=
W
, implying that the coupling
between the neurons is inhibitory and constant.
For a simple demonstration,
τ
m
is set to 1 ms and
N
is set to 50 neurons.
For the case in which the input
x
i
(
t
) is con-
stant, the raster plots indicating the firing of the
50 neurons for the uncoupled case and for the
coupled case, as in
Figures 2.6
a and b, respec-
tively. The bottom trace in each panel shows the
firing pattern of the neuronal population that
has been obtained by combining the firings of all
the neurons. For the uncoupled case, the popula-
tion firing shows clustered behavior where mul-
tiple neurons fire could fire in close proximity,
whereas for the coupled case, the firing rates are
uniform, indicating that the inhibitory coupling
reduces the correlation between the neuronal
firings.
2.3.2 Noise Shaping
In Section
2.2
, we described population rate
encoding or averaging the firing activity across
multiple neurons as a method for achieving
higher dynamic range. Unfortunately, the sim-
ple averaging of noise across independent neu-
rons in the network is suboptimal because the
SNR improves only as a square root of the num-
ber of neurons
[30]
. It would therefore require
an extraordinary number of neurons to achieve
the SNR values (greater than 120 dB) typically
observed in biological systems.
It has been proposed that a possible mecha-
nism behind the remarkable processing acuity
achieved by neuronal networks is
noise shaping
,
a term that refers to the mechanism of shifting
the energy contained in noise and interference
out of the regions (spectral or spatial) where the
desired information is present. It has been
argued by Mar
et al
.
[30]
that inhibitory connec-
tions between neurons could lead to noise-shap-
ing behavior and that the SNR improves directly
as the number of neurons, a significant improve-
ment over simple averaging techniques.
In this section, we describe the noise-shaping
mechanism using the integrate-and-fire model
described
via
Eq.
(2.2)
. Consider a neuronal net-
work consisting of
N
integrate-and-fire neurons.
