Subsections

• • 5.8.0.1 Example: Extracting oscillations

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) = $\sum_{{f}}^{}$$\delta$(t - t^(f)). In the absence of noise, a subthreshold stimulus I(t) does not generate action potentials so that no information on the temporal structure of the stimulus can be transmitted. In the presence of noise, however, spikes do occur. As we have seen in Eq. (5.116), spike firing is most likely at moments when the normalized distance | x| = |(u - $\vartheta$)/$\sigma$| between the membrane potential and the threshold is small. Since the escape rate in Eq. (5.116) depends exponentially on x², any variation in the membrane potential u₀(t) that is generated by the temporal structure of the input is enhanced; cf. Fig. (5.10). On the other hand, for very large noise ( $\sigma$$\to$$\infty$), we have x²$\to$ 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

$$\Omega$$ (5.119)

For t - $\hat{{t}}$ $\gg$ $\tau_{m}^{}$, the membrane potential of the noise-free reference trajectory has the form

$$\infty \Omega \varphi_{1}^{}$$ (5.120)

where u₁ and $\varphi_{1}^{}$ are amplitude and phase of its periodic component. To quantify the signal transmission properties, a long spike train is studied and the signal-to-noise ratio (SNR) is computed. The signal $\mathcal {S}$ is measured as the amplitude of the power spectral density of the spike train evaluated at frequency $\Omega$, i.e., $\mathcal {S}$ = $\mathcal {P}$($\Omega$). The noise level $\mathcal {N}$ is usually estimated from the noise power $\mathcal {P}$_Poisson of a Poisson process with the same number of spikes as the measured spike train, i.e., $\mathcal {N}$ = $\mathcal {P}$_Poisson. Figure 5.23 shows the signal-to-noise ratio $\mathcal {S}$/$\mathcal {N}$ of a periodically stimulated integrate-and-fire neuron as a function of the noise level $\sigma$. Two models are shown, viz., diffusive noise (solid line) and escape noise with the Arrhenius&Current escape rate (dashed line). The two curves are rather similar and exhibit a peak at

$$\sigma^{{\rm opt}}_{} \approx {2\over 3} \vartheta \infty$$ (5.121)

Since $\sigma^{2}_{}$ = 2$\langle$$\Delta$u²$\rangle$, signal transmission is optimal if the stochastic fluctuations of the membrane potential have an amplitude

$$\sqrt{{\langle \Delta u^2 \rangle }} \approx \vartheta \infty$$ (5.122)

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.

Figure 5.23: Signal-to-noise ratio (SNR) for the transmission of a periodic signal as a function of the noise level $\sigma$/($\vartheta$ - u₀). Solid line: Diffusion model. Dashed line: Arrhenius&Current escape model. Taken from (Plesser and Gerstner, 2000).

5.8.0.1 Example: Extracting oscillations

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 $\vartheta$, or synaptic weights could be increased or decreased so that the mean potential u_$\infty$ 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,

$$\nu^{{\rm in}}_{} \nu_{0}^{} \nu_{1}^{} \Omega$$ (5.123)

with 0 < $\nu_{1}^{}$ < $\nu_{0}^{}$. Thus, even though input spikes arrive stochastically, they have some inherent temporal structure, since they are generated by an inhomogeneous Poisson process. In the second scenario input spikes are generated by a homogeneous (that is, stationary) Poisson process with constant rate $\nu_{0}^{}$. In a large interval T $\gg$ $\Omega^{{-1}}_{}$, however, we expect in both cases a total number of $\nu_{0}^{}$ T input spikes.

Stochastic spike arrival leads to a fluctuating membrane potential with variance $\Delta^{2}_{}$ = $\langle$$\Delta$u²$\rangle$. 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

$$\infty \sqrt{{\langle \Delta u^2 \rangle }} \vartheta \infty \sqrt{{\langle \Delta u^2 \rangle }}$$ (5.124)

We conclude that a neuron in the subthreshold regime is capable of transforming a temporal code (amplitude $\nu_{1}^{}$ 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.