The interval distribution $P(t|\hat{t})$ as defined above is a probability density. Thus, integration of $P(t|\hat{t})$ over time yields a probability. For example, $\int_{\hat{t}}^{t}P(t^{\prime}|\hat{t})\,{\text{d}}t^{\prime}$ is the probability that a neuron which has emitted a spike at $\hat{t}$ fires the next action potential between $\hat{t}$ and $t$. Thus

is the probability that the neuron stays quiescent between $\hat{t}$ and $t$. $S(t|\hat{t})$ is called the survivor function: it gives the probability that the neuron ‘survives’ from $\hat{t}$ to $t$ without firing.

The survivor function $S(t|\hat{t})$ has an initial value $S(\hat{t}|\hat{t})=1$ and decreases to zero for $t\to\infty$. The rate of decay of $S(t|\hat{t})$ will be denoted by $\rho(t|\hat{t})$ and is defined by

In the language of renewal theory, $\rho(t|\hat{t})$ is called the ‘age-dependent death rate’ or ‘hazard’ (106; 105).

Integration of the differential equation $dS/dt=-\rho\,S$ [cf. the first identity in Eq. (7.25)] yields the survivor function

According to the definition of the survivor function in Eq. (7.24), the interval distribution is given by

which has a nice intuitive interpretation: In order to emit its next spike at $t$, the neuron has to survive the interval $(\hat{t},t)$ without firing and then fire at $t$. The survival probability is $S(t|\hat{t})$ and the hazard of firing a spike at time $t$ is $\rho(t|\hat{t})$. Multiplication of the survivor function $S$ with the momentary hazard $\rho$ gives the two factors on the right-hand side of Eq. (7.27). Inserting Eq. (7.26) in (7.27), we obtain an explicit expression for the interval distribution in terms of the hazard:

On the other hand, given the interval distribution we can derive the hazard from Eq. (7.25). Thus, each of the three quantities $\rho(t|\hat{t})$, $P(t|\hat{t})$, and $S(t|\hat{t})$ is sufficient to describe the statistical properties of a renewal system. Since we focus on stationary renewal systems, the notation can be simplified and Eqs. (7.24)–(7.28) hold with the replacement

Eqs. (7.24) - (7.28) are standard results of renewal theory. The notation that we have chosen in Eqs. (7.24) - (7.28) will turn out to be useful in later chapters and highlights the fact that these quantities are conditional probabilities, probability densities, or rates.

Renewal theory is usually associated with stationary input conditions. The interval distribution $P_{0}$ can then be estimated experimentally from a single long spike train. The applicability of renewal theory relies on the hypothesis that a memory back to the last spike suffices to describe the spike statistics. In particular, there should be no correlation between one interval and the next. In experiments, the renewal hypothesis, can be tested by measuring the correlation between subsequent intervals. Under some experimental conditions, correlations are small indicating that a description of spiking as a stationary renewal process is a good approximation (192); however, under experimental conditions where neuronal adaptation is strong, intervals are not independent (Fig. 7.11). Given a time series of events with variable intervals $s_{j}$, a common measure of memory effects is the serial correlation coefficients

Spike-frequency adaptation causes a negative correlation between subsequent intervals ($c_{1}<0$). Long intervals are most likely followed by short ones, and vice versa, so that the assumption of renewal theory does not hold (463; 423; 93).

The notion of stationary input conditions is a mathematical concept that cannot be easily translated into experiments. With intracellular recordings under in vitro conditions, constant input current can be imposed and thus the renewal hypothesis can be tested directly. Under in vivo conditions, the assumption that the input current to a neuron embedded in a large neural system is constant (or has stationary statistics) is questionable; see (392; 393) for a discussion. While the externally controlled stimulus can be made stationary (e.g., a grating drifting at constant speed), the input to an individual neuron is out of control.

Let us suppose that, for a given experiment, we have checked that the renewal hypothesis holds to a reasonable degree of accuracy. From the experimental interval distribution $P_{0}$ we can then calculate the survivor function $S_{0}$ and the hazard $\rho_{0}$ via Eqs. (7.24) and  (7.25). If some additional assumptions regarding the nature of the noise are made, the form of the hazard $\rho_{0}(t|\hat{t})$ can be interpreted in terms of neuronal dynamics. In particular, a reduced hazard immediately after a spike is a signature of neuronal refractoriness (192; 52).

In case of a stationary renewal process, the interval distribution $P_{0}$ contains all the statistical information so that the autocorrelation function and noise spectrum can be derived. In this section we calculate the noise spectrum of a stationary renewal process. As we have seen above, the noise spectrum of a neuron is directly related to the autocorrelation function of its spike train. Both noise spectrum and autocorrelation function are experimentally accessible.

Let $\nu_{i}=\langle S_{i}\rangle$ denote the mean firing rate (expected number of spikes per unit time) of the spike train. Thus the probability of finding a spike in a short segment $[t,t+\Delta t]$ of the spike train is $\nu\,\Delta t$. For large intervals $s$, firing at time $t+s$ is independent from whether or not there was a spike at time $t$. Therefore, the expectation to find a spike at $t$ and another spike at $t+s$ approaches for $s\to\infty$ a limiting value $\lim_{s\to\infty}\langle S_{i}(t)\,S_{i}(t+s)\rangle=\lim_{s\to\infty}C_{ii}(s)=\nu_{i}^{2}$. It is convenient to subtract this baseline value and introduce a ‘normalized’ autocorrelation,

with $\lim_{s\to\infty}C_{ii}^{0}(s)=0$. The Fourier transform of Eq. (7.37) yields

Thus $\hat{C}_{ii}(\omega)$ diverges at $\omega=0$; the divergence is removed by switching to the normalized autocorrelation. In the following we will calculate the noise spectrum $\hat{C}_{ii}(\omega)$ for $\omega\neq 0$.

In the case of a stationary renewal process, the autocorrelation function is closely related to the interval distribution $P_{0}(s)$. This relation will now be derived. Let us suppose that we have found a first spike at $t$. To calculate the autocorrelation we need the probability density for a spike at $t+s$. Let us construct an expression for $C_{ii}(s)$ for $s>0$. The correlation function for positive $s$ will be denoted by $\nu_{i}\,C_{+}(s)$ or

The factor $\nu_{i}$ in Eq. (7.39) takes care of the fact that we expect a first spike at $t$ with rate $\nu_{i}$. $C_{+}(s)$ gives the conditional probability density that, given a spike at $t$, we will find another spike at $t+s>t$. The spike at $t+s$ can be the first spike after $t$, or the second one, or the $n$th one; see Fig. 7.12. Thus for $s>0$

or

as can be seen by inserting Eq. (7.40) on the right-hand side of (7.41); see Fig. 7.13.

Due to the symmetry of $C_{ii}$, we have $C_{ii}(s)=\nu\,C_{+}(-s)$ for $s<0$. Finally, for $s=0$, the autocorrelation has a $\delta$ peak reflecting the trivial autocorrelation of each spike with itself. Hence,

In order to solve Eq. (7.41) for $C_{+}$ we take the Fourier transform of Eq. (7.41) and find

Together with the Fourier transform of Eq. (7.42), $\hat{C}_{ii}=\nu_{i}\,[1+2\,\text{Re}\{\hat{C}_{+}(\omega)\}]$, we obtain

For $\omega=0$, the Fourier integral over the right-hand side of Eq. (7.40) diverges, since $\int_{0}^{\infty}P_{0}(s){\text{d}}s=1$. If we add the diverging term from Eq. (7.38), we arrive at

This is a standard result of stationary renewal theory (105) which has been repeatedly applied to neuronal spike trains (137; 36).

It is possible to use the renewal concept in a broader sense and define a renewal process as a system where the state at time $t$ (and hence the probability of generating an event at $t$), depends both on the time that has passed since the last event (i.e., the firing time $\hat{t}$) and the input $I(t^{\prime})$, $\hat{t}<t^{\prime}<t$, that the system received since the last event. Input-dependent renewal systems are also called modulated renewal processes (425), non-stationary renewal systems (182; 183), or inhomogeneous Markov interval processes (251). The aim of a theory of input-dependent renewal systems is to predict the probability density

of the next event to occur at time $t$, given the timing $\hat{t}$ of the last event and the input $I(t^{\prime})$ for $\hat{t}<t^{\prime}<t$; see Fig. 7.14. The relation between hazard, survivor function, and interval distribution for the input-dependent case is the same as the one given in Equations (7.25) – (7.28). The generalization to a time-dependent renewal theory will be useful later on in Chapter 9.

The lower index $I$ of $P_{I}(t|\hat{t})$ is intended to remind the reader that the probability density $P_{I}(t|\hat{t})$ depends on the time course of the input $I(t^{\prime})$ for $t^{\prime}<t$. Since $P_{I}(t|\hat{t})$ is conditioned on the spike at $\hat{t}$, it can be called a spike-triggered spike density. We interpret $P_{I}(t\,|\,\hat{t})$ as the distribution of interspike intervals in the presence of an input current $I$ or as the input-dependent interval distribution. For stationary input, $P_{I}(t|\hat{t})$ reduces to $P_{0}(t-\hat{t})$.