Noise Exploitation and Adaptation in Neuromorphic Sensors - Engineered Biomimicry - page 42

design of neuromorphic sensors [9] and is
sufficient to explain the noise exploitation tech-
niques described in this chapter.
We first define a spike train ρ ( t ) using a
sequence of time-shifted Kronecker delta func-
tions as

The simplest form of neural coding is the
rate-based encoding [13] that computes the
instantaneous spiking rate of the i th neuron R i ( t )
according to

t + T

R i ( t ) = 1
T

ρ i ( t ) dt ,

(2.3)

t

where ρ i ( t ) denotes the spike train generated by
the i ith neuron and is given by Eq. (2.1) , and T
is the observation interval over which the inte-
gral or spike count is computed. Note that the
instantaneous spiking rate R ( t ) does not capture
any information related to the relative phase of
the individual spikes, and hence it embeds sig-
nificant redundancy in encoding. However, at
the sensory layer, this redundancy plays a criti-
cal role because the stimuli need to be precisely
encoded and the encoding have to be robust to
the loss or temporal variability of the indivi-
dual spikes.
Another mechanism by which neurons
improve reliability and transmission of spikes is
through the use of bursting, which refers to
trains of repetitive spikes followed by periods
of silence . This method of encoding has been
shown to improve the reliability of information
transmission across unreliable synapses [14]
and, in some cases, to enhance the SNR of the
encoded signal. Modulating the bursting pat-
tern also provides the neuron with more ways
to encode different properties of the stimulus.
For instance, in the case of the electric fish, a
change in bursting signifies a change in the
states (or modes) of the input stimuli, which
could distinguish different types of prey in the
fish's environment [14] .
Whether bursting is used or not, the main
disadvantage of rate-based encoding is that it is
intrinsically slow. The averaging operation in
Eq. (2.3) requires that a sufficient number of
spikes be generated within T to reliably
compute R i ( t ) . One possible approach to improve
the reliability of rate-based encoding is to
compute the rate across a population of neurons
where each neuron is encoding the same stimuli.

∞

ρ( t ) = δ( t − t m ),

(2.1)

m =1

+∞
−∞ δ(τ ) d τ = 1 . In
the above Eq. (2.1) , the spike is generated when
t is equal to the firing time of the neuron t m . If
the somatic (or membrane) potential of the neu-
ron is denoted by v ( t ) , then the dynamics of the
integrate-and-fire model can be summarized
using the following first-order differential
equation:

where δ ( t ) = 0 for t ≠ 0 and

N

d
dt v ( t ) =− v ( t )/τ m −

W j [ h ( t )∗ρ j ( t )]+ x ( t ),

j =1

(2.2)

where N denotes the number of presynaptic
neurons, W j is a scalar transconductance rep-
resenting the strength of the synaptic connec-
tion between the j th presynaptic neuron and
the postsynaptic neuron, τ m is the time constant
that determines the maximum firing rate, h ( t )
is a presynaptic filtering function that filters
the spike train ρ j ( t ) before it is integrated at the
soma, and * denotes a convolution operator.
The variable x ( t ) in Eq. (2.2) denotes an extrin-
sic contribution to the membrane current, which
could be an external stimulation current.
When the membrane potential v ( t ) reaches a
certain threshold, the neuron generates a spike
or a train of spikes. Again, different chaotic
models have been proposed that can capture
different types of spike dynamics. For the sake
of brevity, specific details of the dynamical
models can be found in Ref. 13 . We next briefly
describe different methods by which neuronal
spikes encode information.