5.4 Slow noise in the parameters

In one of the previous examples (`Motivating sigmoidal escape rates' in Section 5.3.1), a new value of the threshold was chosen at each time step; cf. Eq. (5.49). If time steps are short enough, such an approach is closely related to escape rate models. A completely different class of noise models can be constructed if the value of a parameter is changed after each spike. Thus between two spikes the noise is `frozen' so that the value of the fluctuating parameter does not change. In other words, the noise is slow compared to the fast neuronal dynamics. In principle, any of the neuronal parameters such as threshold, membrane time constant, or length of the refractory period, can be subject to this type of noise (Gerstner, 2000b; Lansky and Smith, 1989; Gestri, 1978; Knight, 1972a). In this section we want to show how to analyze such slow variations and calculate the interval distribution. We emphasize that these `slow' noise models cannot be mapped onto an escape rate formalism.

Figure 5.11: Slow noise in the parameters. After each spike either the reset value (A) or the firing threshold (B) is set to a new randomly chosen value.

To keep the arguments simple, we will concentrate on noise in the formulation of reset and refractoriness. We assume an exponential refractory kernel,

$$\eta \eta_{0}^{} \tau$$ (5.64)

with time constant $\tau$. In order to introduce noise, we suppose that the amount $\eta_{0}^{}$ of the reset depends on a stochastic variable r,

$$\eta_{0}^{} \tilde{{\eta}}_{0}^{} \tau$$ (5.65)

where $\tilde{{\eta_0}}$ < 0 is a fixed parameter and r is a random variable with zero mean. In the language of the integrate-and-fire neuron, we can describe the effect of r as a stochastic component in the value of the reset potential.

In the `noisy reset' model, firing is given by the threshold condition

$$\vartheta \hat{{t}} \eta_{r}^{} \hat{{t}} \int_{0}^{\infty} \kappa \hat{{t}}$$ (5.66)

where $\eta_{r}^{}$(t - $\hat{{t}}$) = $\eta_{0}^{}$(r) exp[- (t - $\hat{{t}}$)/$\tau$]. Since u depends on the current value of r, we have written u(t|$\hat{{t}}$, r) instead of u(t|$\hat{{t}}$). Let us write T($\hat{{t}}$, r) for the next interval of a neuron which has fired at $\hat{{t}}$ and was reset with a stochastic value r, i.e.,

$$\hat{{t}} \left\{\vphantom{ t-\hat{t}\,\vert\, u(t\vert\hat{t},r) = \vartheta }\right. \hat{{t}} \hat{{t}} \vartheta \left.\vphantom{ t-\hat{t}\,\vert\, u(t\vert\hat{t},r) = \vartheta }\right\}$$ (5.67)

If r is drawn from a Gaussian distribution $\mathcal {G}$_$\sigma$(r) with variance $\sigma$ $\ll$ $\tau$, the interval distribution is

$$\hat{{t}} \int \delta \hat{{t}} \hat{{t}} \mathcal {G} \sigma$$ (5.68)

Let us now evaluate the interval distribution (5.68) for the variant SRM₀ of the Spike Response Model,

$$\hat{{t}} \eta_{r}^{} \hat{{t}}$$ (5.69)

with constant input potential h(t) = h₀. First we show that a stochastic reset according to (5.65) with r$\ne$ 0 shifts the refractory kernel horizontally along the time axis. To see this, let us consider a neuron that has fired its last spike at $\hat{{t}}$ and has been reset with a certain value r. The refractory term is

$$\eta_{r}^{} \hat{{t}} \tilde{{\eta}}_{0}^{} \hat{{t}} \tau$$ (5.70)

which is identical to that of a noiseless neuron that has fired its last spike at t' = $\hat{{t}}$ + r. Given the constant input potential h₀, a noise-free SRM₀ neuron would fire regularly with period T₀. A noisy neuron that was reset with value r is delayed by a time r and fires therefore after an interval T($\hat{{t}}$, r) = T₀ + r. Integration of Eq. (5.68) yields the interval distribution

$$\hat{{t}} \mathcal {G} \sigma \hat{{t}}$$ (5.71)

Thus, the Gaussian distribution $\mathcal {G}$_$\sigma$(r) of the noise variable r maps directly to a Gaussian distribution of the intervals around the mean T₀. For a detailed discussion of the relation between the distribution of reset values and the interval distribution P₀(t - $\hat{{t}}$) of leaky integrate-and-fire neurons, see Lansky and Smith (1989).

Even though stochastic reset is not a realistic noise model for individual neurons, noise in the parameter values can approximate inhomogeneous populations of neurons where parameters vary from one neuron to the next (Wilson and Cowan, 1972; Knight, 1972a). Similarly, a fluctuating background input that changes slowly compared to the typical interspike interval can be considered as a slow change in the value of the firing threshold. More generally, noise with a cut-off frequency smaller than the typical firing rate can be described as slow noise in the parameters.