5.8 Stochastic resonance

Noise can - under certain circumstances - improve the signal transmission
properties of neuronal systems. In most cases there is an optimum for the
noise amplitude which has motivated the name *stochastic resonance* for
this rather counterintuitive phenomenon. In this section we discuss stochastic
resonance in the context of noisy spiking neurons.

We study the relation between an input *I*(*t*) to a neuron and the corresponding output
spike train
*S*(*t*) = (*t* - *t*^{(f)}). In the absence of noise, a
subthreshold stimulus *I*(*t*) does not generate action potentials so that no
information on the temporal structure of the stimulus can be transmitted. In
the presence of noise, however, spikes do occur. As we have seen in
Eq. (5.116), spike firing is most likely at moments when the normalized
distance
| *x*| = |(*u* - )/| between the membrane potential and the
threshold is small. Since the escape rate in Eq. (5.116) depends
exponentially on *x*^{2}, any variation in the membrane potential *u*_{0}(*t*) that
is generated by the temporal structure of the input is enhanced; cf. Fig. (5.10). On the other hand, for very large noise (
), we have *x*^{2} 0, and spike firing occurs at a constant rate,
irrespective of the temporal structure of the input. We conclude that there
is some intermediate noise level where signal transmission is optimal.

The optimal noise level can be found by plotting the signal-to-noise ratio as a function of noise (McNamara and Wiesenfeld, 1989; Collins et al., 1996; Cordo et al., 1996; Longtin, 1993; Wiesenfeld and Jaramillo, 1998; Stemmler, 1996; Levin and Miller, 1996; Douglass et al., 1993); for a review see Gammaitoni et al. (1998). Even though stochastic resonance does not require periodicity (see, e.g., Collins et al. (1996)), it is typically studied with a periodic input signal such as

For

where

Since = 2

An optimality condition similar to (5.121) holds over a wide variety of stimulation parameters (Plesser, 1999). We will come back to the signal transmission properties of noisy spiking neurons in Chapter 7.3.

The optimality condition (5.121) can be fulfilled by adapting either
the
left-hand side or the right-hand side of the equation. Even though it cannot
be
excluded that a neuron changes its noise level so as to optimize the
left-hand side of Eq. (5.121) this does not seem very likely. On
the other hand, it is easy to imagine a mechanism that optimizes the
right-hand side of Eq. (5.121). For example, an adaptation current
could change the value of , or synaptic weights could be
increased or decreased so that the mean potential *u*_{} is in the
appropriate regime.

We apply the idea of an optimal threshold to a problem of neural coding. More
specifically, we study the question whether an integrate-and-fire or Spike
Response Model neuron is only sensitive to the total number of spikes that
arrive in some time window *T*, or also to the relative timing of the input
spikes. In contrast to Chapter 4.5 where we have discussed
this question in the deterministic case, we will explore it here in the
context of stochastic spike arrival. We consider two
different scenarios of stimulation. In the first scenario input spikes
arrive with a periodically modulated rate,

with 0 < < . Thus, even though input spikes arrive stochastically, they have some inherent temporal structure, since they are generated by an

Stochastic spike arrival leads to a fluctuating membrane potential with
variance
= *u*^{2}. If the membrane potential
hits the threshold an output spike is emitted. If stimulus 1 is applied
during the time *T*, the neuron emits emit a certain number of action
potentials, say *n*^{(1)}. If stimulus 2 is applied it emits *n*^{(2)}
spikes. It is found that the
spike count numbers *n*^{(1)} and *n*^{(2)}
are significantly different
if the threshold is in the range

u_{} + < < u_{} +3 . |
(5.124) |

We conclude that a neuron in the subthreshold regime is capable of transforming a temporal code (amplitude of the variations in the input) into a spike count code. Such a transformation plays an important role in the auditory pathway; see Chapter 12.5.

Cambridge University Press, 2002

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