Variability of spike timing is a common phenomenon in biological neurons. Variability can be quantified by the $C_{V}$ value of interval distributions; by the Fano factor of the spike count; or by the repeatability of spike timings between one trial and the next. Whether the variability represents noise or uncontrolled components of a signal which is not well characterized, is a topic of debate. Experiments show that a neuron in vitro, or in one of the sensory areas in vivo, shows highly reliable spike timings if driven by a strong stimulus with large-amplitude fluctuations of the signal. Spontaneous activity in vivo, however, is unreliable and exhibits large variability of interspike intervals and spike counts.
The simplest stochastic description of neuronal firing is a Poisson process. However, since each spike firing in a Poisson process is independent from earlier spikes, Poisson firing cannot account for refractoriness. In renewal processes, the probability of firing depends on the time since the last spike. Therefore refractoriness is taken care of. The independent events are the interspike-intervals which are drawn from a interval distribution $P_{0}(s)$. Knowledge of $P_{0}(s)$ is equivalent to knowing the survivor function $S_{0}(s)$ or the hazard $\rho_{0}(s)$. In neurons showing strong adaptation, interspike intervals are not independent so that renewal theory is not sufficient. Moreover, standard renewal theory is limited to stationary stimuli, whereas real-world stimuli have a strong temporal component – the solution is then a time-dependent generalization of renewal theory which we will encounter in Chapter 14.
A description of neuronal spike trains in terms of firing rates or interval distributions does not imply that neurons use the firing rate (or interval distribution) to transmit signals. In fact, neither the spike count (averaging over time) nor the time-dependent rate of the PSTH (averaging over trials) can be the neural code of sensory processing because they are too slow given known reaction times. A firing rate in the sense of a population activity, defined as the instantaneous average of spikes across a population of neurons with similar properties, is, however, a candidate neural code. Other candidate codes, with some experimental support are a latency code (time-to-first-spike), or a phase code.
In models, noise is usually added ad hoc to account for the observed variability of neural spike trains: two standard ways of adding noise to neuron models will be presented in the next two chapters. But even without explicit noise source, neural activity may look noisy if the neuron is embedded in a large deterministic network with fixed random connectivity. The analysis of such networks will be the topic of Part III.
A review of noise in the nervous system with a focus on internal noise sources can be found in (145). Analysis of spike trains in terms of stochastic point processes has a long tradition (392; 174) and often involves concepts from renewal theory (105). Some principles of spike-train analysis with an emphasis on modern results have been reviewed by Gabbiani and Koch (165) and Rieke et al. (436). For a discussion of the variability of interspike intervals see the debate of Shadlen and Newsome (470), Softky (486), and Bair and Koch (37); these papers also give a critical discussion of the concept of temporal averaging. An accessible mathematical treatment of the inhomogeneous Poisson model in the context of neuronal signals is given in Rieke et al. (436). The same book can also be recommended for its excellent discussion of rate codes, and their limits, as well as the method of stimulus reconstruction (436).
Poisson process in discrete and continuous time . We consider a Poisson neuron model in discrete time. In every small time interval $\Delta t$ , the probability that the neuron fires is given by $\nu\,\Delta t$ . Firing in different time intervals is independent. The limit $\Delta t\to 0$ will be taken only at the end.
(a) What is the probability that the neuron does not fire during a time of arbitrarily large length $t=N\,\Delta t$ ?
Hint: Consider first the probability of not firing during a single short interval $\Delta t$ , and then extend your reasoning to $N$ time steps.
(b) Suppose that the neuron has fired at time $t=0$ . Calculate the distribution of intervals $P(t)$ , i.e., the probability density that the neuron fires its next spike at a time $t=N\,\Delta t$ .
(c) Start from your results in (a) and (b) and take the limit $N\to\infty$ , $\Delta t\to 0$ , while keeping $t$ fixed. What is the resulting survivor function $S_{0}(t)$ and the interval distribution $P_{0}(s)$ in continuous time?
(d) Suppose that the neuron is driven by some input. For $t<t_{0}$ , the input is weak, so that its firing rate is $\nu$ = 2Hz. For $t_{0}<t<t_{1}=t_{0}+100$ ms, the input is strong and the neuron fires at $\nu=20$ Hz. Unfortunately, however, the onset time $t_{0}$ of the strong input is unknown; can an observer, who is looking at the neuronâs output, detect the period of strong input? How reliably?
Hint: Calculate the interval distributions for weak and strong stimuli. What is the probability of having a âburstâ consisting of two intervals of less than 20 ms each if the input is weak/strong?
Autocorrelation of a Poisson process in discrete time . The autocorrelation
$C_{ii}(s)=\langle S_{i}(t)\,S_{i}(t+s)\rangle_{t}\,,$ | (7.58) |
is defined as the joint probability density of finding a spike at time $t$ and a spike at time $t+s$ . In Eq. ( 7.46 ) we have stated the autocorrelation of the homogeneous Poisson process in continuous time. Derive this result by starting with a Poisson process in discrete time where the probability of firing in a small time interval $\Delta t$ , is given by $\nu\,\Delta t$ . To do so, take the following steps:
(a) What is the joint probability to find a spike in the bin $[t,t+\Delta t]$ AND in the bin $[t^{\prime},t^{\prime}+\Delta t]$ where $t\neq t^{\prime}$ ?
(b) What is the joint probability to find a spike in the bin $[t,t+\Delta t]$ AND in the bin $[t^{\prime},t^{\prime}+\Delta t]$ where $t=t^{\prime}$ ?
(c) What is the probability to find two spikes in the bin $[t,t+\Delta t]$ ? Why can this term be neglected in the limit $\Delta t\to 0$ ?
(d) Take the limit $\Delta t\to 0$ while keeping $t$ and $t^{\prime}$ fixed so as to find the autocorrelation function $C_{0}(s)$ in continuous time.
Repeatability and random coincidences . Suppose that a Poisson neuron with a constant rate of 20 Hz emits, in a trial of 5-second duration, 100 spikes at times $t^{(1)};t^{(2)};\dots t^{(100)}$ . Afterward, the experiment is repeated and a second spike train with a duration of 5 seconds is observed. How many spikes in the first trial can be expected to coincide with a spike in the second trial? More generally, what percentage of spikes coincide between two trials of a Poisson neuron with arbitrary rate $\nu_{0}$ under the assumption that trials are sufficiently long?
Spike count and Fano Factor . A homogeneous Poisson process has a probability to fire in a very small interval $\Delta t$ equal to $\nu\,\Delta t$ .
(a) Show that the probability to observe exactly $k$ spikes in the time interval $T=N\,\Delta t$ is $P_{k}(T)=[1/k!]\,(\nu\,T)^{k}\,\exp(-\nu\,T)\,$ .
Hint: Start in discrete time and write the probability to observe $k$ events in $N$ slots using the binomial distribution: $P(k;N)=[N!/k!(N-k)!]\,p^{k}\,(1-p)^{N-k}$ where $p$ is the probability of firing in a time bin of duration $\Delta t$ . Take the continuous time limit with Stirling’s formula $N!\approx(N/e)^{N}$ .
(b) Repeat the above argument for an inhomogeneous Poisson process.
(c) Show for the inhomogeneous Poisson process that the mean spike count in an interval of duration $T$ is $\langle k\rangle=\int_{0}^{T}\nu(t)\,dt$ .
(d) Calculate the variance of the spike count and the Fano factor for the inhomogeneous Poisson process.
From interval distribution to hazard . During stimulation with a stationary stimulus, interspike intervals in a long spike train are found to be independent and given by the distribution
$P(t|t^{\prime})={(t-t^{\prime})\over\tau^{2}}\,\exp\left(-{t-t^{\prime}\over% \tau}\right)$ | (7.59) |
for $t>t^{\prime}$ .
(a) Calculate the survivor function $S(t|t^{\prime})$ , i.e. the probability that the neuron survives from time $t^{\prime}$ to $t$ without firing.
Hint: You can use $\int_{0}^{y}x\,e^{ax}dx=e^{ay}[ay-1]/a^{2}$ .
(b) Calculate the hazard function $\rho(t|t^{\prime})$ , that is, the stochastic intensity that the neuron fires, given that its last spike was at $t^{\prime}$ and interpret the result: what are the signs of refractoriness?
(c) A spike train starts at time $0$ and we have observed a first spike at time $t_{1}$ . We are interested in the probability that the $n$ th spike occurs around time $t_{n}=t_{1}+s$ . With this definition of spike labels, calculate the probability density $P(t_{3}|t_{1})$ that the third spike occurs around time $t_{3}$ .
Gamma-distribution . Stationary interval distributions can often be fitted by a Gamma distribution (for $s>0$ )
$P(s)={1\over(k-1)!}\,{s^{k-1}\over\tau^{k}}\,e^{-s/\tau},$ | (7.60) |
where $k$ is a positive natural number. We consider in the following $k=1$ .
(a) Calculate the mean interval $\langle s\rangle$ and the mean firing rate.
(b) Assume that intervals are independent and calculate the power spectrum.
Hint: Use Eq. ( 7.45 ).
$C_{V}$ value of Gamma distribution . Stationary interval distributions can often be fitted by a Gamma distribution
$P(s)={1\over(k-1)!}\,{s^{k-1}\over\tau^{k}}\,e^{-s/\tau},$ | (7.61) |
where $k$ is a positive natural number.
Calculate the coefficient of variation $C_{V}$ for $k=1,2,3$ . Interpret your result.
Poisson with dead time as a renewal process . Consider a process where spikes are generated with rate $\rho_{0}$ , but after each spike there is a dead time of duration $\Delta^{\rm abs}$ . More precisely, we have a renewal process
$\rho(t|\hat{t})=\rho_{0}\quad{\rm for}\,t>\hat{t}+\Delta^{\rm abs},$ | (7.62) |
and zero otherwise.
(b) Calculate the Fano factor.
(c) If a first spike occurred at time $t=0$ , what is the probability that a further spike (there could be other spikes in between) occurs at $t=x\,\Delta^{\rm abs}$ where $x=0.5;1.5;2.5$ .
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