The total input current which drives a neuron can often be separated into a deterministic component , which is known and repeats between one trial and the next; and a stochastic component which is unknown and potentially changes between trials:
In this section we show that the noisy, unknown part in the input can be approximated to a high degree of accuracy by an appropriately chosen escape function.
In the subthreshold regime, the leaky integrate-and-fire model with stochastic input (white noise) can be mapped approximately onto an escape noise model with a certain escape rate (403). In this section, we motivate the mapping and the choice of .
In the absence of a threshold, the membrane potential of an integrate-and-fire model has a Gaussian probability distribution around the noise-free reference trajectory . If we take the threshold into account, the probability density at of the exact solution vanishes, since the threshold acts as an absorbing boundary; see Eq. (8.44). Nevertheless, in a phenomenological model, we can approximate the probability density near by the ‘free’ distribution (i.e., without the threshold)
where is the noise-free reference trajectory. The idea is illustrated in Fig. 9.10. We note that in a leaky integrate-and-fire model with colored noise in the input (as opposed to white noise) the density at threshold is not zero.
We have seen in Eq. (8.13) that the variance of the free distribution rapidly approaches a constant value where scales the strength of the diffusive input. We therefore replace the time dependent variance by its stationary value . The right-hand side of Eq. (9.28) is then a function of the noise-free reference trajectory only. In order to transform the left-hand side of Eq. (9.28) into an escape rate, we divide both sides by . The firing intensity is thus
The factor in front of the exponential has been split into a constant parameter and the time constant of the neuron in order to show that the escape rate has units of one over time. Equation (9.29) is the well-known Arrhenius formula for escape across a barrier of height in the presence of thermal energy (529).
Let us now suppose that the neuron receives, at , an input current pulse which causes a jump of the membrane trajectory by an amount ; see Fig. (9.10). In this case the Gaussian distribution of membrane potentials is shifted instantaneously across the threshold so that there is a nonzero probability that the neuron fires exactly at . To say it differently, the firing intensity has a peak at . The escape rate of Eq. (9.29), however, cannot reproduce this peak. More generally, whenever the noise free reference trajectory increases with slope , we expect an increase of the instantaneous rate proportional to , because the tail of the Gaussian distribution drifts across the threshold; cf. Eq. (8.35). In order to take the drift into account, we generalize Eq. (9.29) and study
We emphasize that the right-hand side of Eq. (9.30) depends only on the dimensionless variable
and its derivative . Thus the amplitude of the fluctuations define a ‘natural’ voltage scale. The only relevant variable is the momentary distance of the noise-free trajectory from the threshold in units of the noise amplitude . A value of implies that the membrane potential is one below threshold. A distance of mV at high noise (e.g., mV) is as effective in firing a cell as a distance of 1 mV at low noise ( mV).
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 to a neuron and the corresponding output spike train . In the absence of noise, a subthreshold stimulus 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. (9.30), spike firing is most likely at moments when the normalized distance between the membrane potential and the threshold is small. Since the escape rate in Eq. (9.30) depends exponentially on , any variation in the membrane potential that is generated by the temporal structure of the input is enhanced; cf. Fig. (9.8). On the other hand, for very large noise (), we have , 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. Even though stochastic resonance does not require periodicity (see, e.g., Collins et al. (101)), it is typically studied with a periodic input signal such as
For , the membrane potential of the noise-free reference trajectory has the form
where and 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 is measured as the amplitude of the power spectral density of the spike train evaluated at frequency , i.e., . The noise level is usually estimated from the noise power of a Poisson process with the same number of spikes as the measured spike train, i.e., . Figure 9.12 shows the signal-to-noise ratio of a periodically stimulated integrate-and-fire neuron as a function of the noise level . 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
Since , signal transmission is optimal if the stochastic fluctuations of the membrane potential have an amplitude
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