The estimation of interspike interval (ISI) distributions from experimental data is a common method to study neuronal variability given a certain stationary input. In a typical experiment, the spike train of a single neuron (e.g., a neuron in visual cortex) is recorded while driven by a constant stimulus. The stimulus might be an external input applied to the system (e.g., a visual contrast grating moving at constant speed); or it may be an intracellularly applied constant driving current. The spike train is analyzed and the distribution of intervals $s_{k}$ between two subsequent spikes is plotted in a histogram. For a sufficiently long spike train, the histogram provides a good estimate of the ISI distribution which we denote as $P_{0}(s)$; cf. Fig. 7.9. The interval distribution can be interpreted as a conditional probability density

$P_{0}(s)=P(t^{(f)}+s|t^{(f)})$ | (7.12) |

where $\int_{t}^{t+\Delta t}P(t^{\prime}|t^{(f)})\,{\text{d}}t^{\prime}$ is the probability that the next spike occurs in the interval $[t,t+\Delta t]$ given that the last spike occurred at time $t^{(f)}$.

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In order to extract the mean firing rate from a stationary interval distribution $P_{0}(s)$, we start with the definition of the mean interval,

$\langle s\rangle=\int_{0}^{\infty}s\,P_{0}(s)\,{\text{d}}s\,.$ | (7.13) |

The mean firing rate is the inverse of the mean interval

$\nu={1\over\langle s\rangle}=\left[\int_{0}^{\infty}s\,P_{0}(s)\,{\text{d}}s\,% \right]^{-1}$ | (7.14) |

Interspike interval distributions $P_{0}(s)$ derived from a spike train under stationary conditions can be broad or sharply peaked. To quantify the width of the interval distribution, neuroscientists often evaluate the coefficient of variation, short $C_{V}$, defined as the ratio of the standard deviation and the mean. Therefore the square of the $C_{V}$ is

$C_{V}^{2}={\langle\Delta s^{2}\rangle\over\langle s\rangle^{2}}\,,$ | (7.15) |

where $\langle s\rangle=\int_{0}^{\infty}sP_{0}(s)\,{\text{d}}s$ and $\langle\Delta s^{2}\rangle=\int_{0}^{\infty}s^{2}\,P_{0}(s)\,{\text{d}}s-% \langle s\rangle^{2}$. A Poisson process produces distributions with $C_{V}=1$. A value of $C_{V}>1$, implies that a given spike train is less regular than a Poisson process with the same firing rate. If $C_{V}<1$, then the spike train is more regular. Most deterministic integrate-and-fire neurons fire periodically when driven by a constant stimulus and therefore have $C_{V}=0$. Intrinsically bursting neurons, however, can have $C_{V}>1$.

Example: Poisson process with absolute refractoriness

We study a Poisson neuron with absolute refractory period $\Delta^{\rm abs}$. For times since last spike larger than $\Delta^{\rm abs}$, the neuron is supposed to fire stochastically with rate $r$. The interval distribution of a Poisson process with absolute refractoriness (Fig. 7.10A) is given by

$P_{0}(s)=\left\{\begin{array}[]{*{2}{c@{\qquad}}c}0\qquad&{\rm for}\qquad&s<% \Delta^{\rm abs}\\ r\,\exp\left[-r\,(s-\Delta^{\rm abs})\right]\qquad&{\rm for}\qquad&s>\Delta^{% \rm abs}\end{array}\right.\,;$ | (7.16) |

and has a mean $\langle s\rangle=\Delta^{\rm abs}+1/r$ and variance $\langle\Delta s^{2}\rangle=1/r^{2}$. The coefficient of variation is therefore

$C_{V}=1-{\Delta^{\rm abs}\over\langle s\rangle}\,.$ | (7.17) |

Let us compare the $C_{V}$ of Eq. (7.17) with that of a homogeneous Poisson process of the same mean rate $\nu=\langle s\rangle^{-1}$. As we have seen, a Poisson process has $C_{V}=1$. A refractory period $\Delta^{\rm abs}>0$ lowers the $C_{V}$, because a neuron with absolute refractoriness fires more regularly than a Poisson neuron. If we increase $\Delta^{\rm abs}$, we must increase the instantaneous rate $r$ in order to keep the same mean rate $\nu$. In the limit of $\Delta^{\rm abs}\to\langle s\rangle$, the $C_{V}$ approaches zero, since the only possible spike train is regular firing with period $\langle s\rangle$.

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