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 x2, any variation in the membrane potential u0(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 x2 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
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,
Stochastic spike arrival leads to a fluctuating membrane potential with variance = u2. 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)|
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