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The effects of various spatial distributions of weak noise on rhythmic spiking
Henry Tuckwell and Jürgen Jost
We consider the response of the classical Hodgkin-Huxley (HH) spatial system in the weak to intermediate noise regime near the bifurcation to repetitive spiking. The deterministic component of the input (signal) is restricted to a small segment near the origin whereas noise, with parameter $\sigma$, occurs either only in the signal region or throughout the whole neuron. In both cases small noise inhibits the spiking and there is a minimum in the spike counts at $\sigma \approx 0.15$. At the same value of $\sigma$, the variance of the spike counts undergoes a pronounced maximum. For spatially restricted noise, the spike count continues to increase beyond the minimum until $\sigma=0.5$, but in the case of spatially extended noise the spike count begins to decline around $\sigma$=0.35 to give a local maximum. For both spatial distributions of noise, the variance of the spike count is found to also have a local minimum at about $\sigma=0.4$. Examples are given of the probability distributions of the spike counts and the spatial distributions of spikes with varying noise level.
The differences in behaviours of the spike counts as noise increases beyond 0.3 are attributable to noise-induced spiking outside the signal region, which has a larger probability of occurrence when the noise is over an extended region. This aspect is investigated by ascertaining the probability of noise-induced spiking as a function of noise level and examination of the corresponding latency distributions. These findings prompt a definition of weak noise in the standard HH model as that for which the probability of secondary phenomena is negligible, which occurs when $\sigma$ is less than about 0.3.
Finally, if signal and weak ($\sigma < 0.3$) noise are applied on disjoint intervals, then the noise has no effect on the instigation or propagation of spikes, no matter how large its region of application. These results are expected to apply to Type 2 neurons in general, including the majority of cortical pyramidal cells.