Neuronal Dynamics (7)

Variability of Spike Trains and Neural Codes

The online version of this chapter:

---

Chapter 7 Variability of Spike Trains and Neural Codes https://neuronaldynamics.epfl.ch/online/Ch7.html

---

Spiking train variability

Are neurons noisy?

Termal noise: literally omnipresent. Due to the discrete nature of electric charge carriers, the voltage \(u\) across any electrical resistor \(R\) fluctuates at finite temperature (Johnson noise). The variance of the fluctuations at rest is \(\langle \Delta u^{2} \rangle \propto RkTB\) where \(k\) is the Boltzmann constant, \(T\) the temperature and \(B\) the bandwidth of the system. Fluctuations due to Johnson noise are of minor importance compared to other noise sources in neurons.

Another source of noise arises from the finite number of ion channels in a patch of neuronal membrane. For a given constant membrane potential \(u\), a fraction \(P_i(u)\) of ion channel of type \(i\) is open on average. The actual number of open channels fluctuates around \(N_iP_i(u)\) where \(N_i\) is the total number of ion channels of type \(i\) in that patch of membrane.

Noise from the Network

Extrinsic noise: noise that are due to signal transmission and network effects (extrinsic noise).

Synaptic transmission failure imposes a substantial limitation to signal transmisson with a neuronal network.

Networks of excitatory and inhibitory neurons with fixed random connectivity can produce highly irregular spike trains - even in the absence of any source of noise.

Mean Firing Rate

Rate as a Spike Count and Fano Factor

An experimentalist observes in trial \(k\) the spikes of a given neuron. The firing rate in trial \(k\) is the spike count \(n_k^{sp}\) in an interval of duration \(T\) divided by \(T\). \[ \nu_k=\frac{n_k^{sp}}{T}. \tag{7.1} \]

\(n_k^{sp}\) mean by \(\langle n^{sp}\rangle\) and deviations from the mean as \(\Delta n_{k}^{sp}=n_k^{sp}-\langle n^{sp}\rangle\). Variability of the spike count measure is characterized by the Fano Factor, defined as the variance of the spike count \(\langle (\Delta n^{sp})^{2}\rangle\) divided by its mean \[ F=\frac{\langle (\Delta n^{sp})^{2}\rangle}{\langle n^{sp}\rangle} \tag{7.2} \]

（Fano因子定义为方差除以均值） In experiments, the mean and variance are estimated by averaging over \(K\) trials \(\langle n^{sp}\rangle=(1/K) \sum_{k=1}^{K} n_k^{sp}\) and \(\langle (\Delta n^{sp})^{2}\rangle=(1/K) \sum_{k=1}^{K} (\Delta n_k^{sp})^{2}\).

The firing rate defined here as spike count divided by the measurement time \(T\) is identical to the inverse of the mean interspike interval. Since \(1/\langle x\rangle \neq \langle (1/x)\rangle\), if we assign a variable \(\tilde{\nu}(t)=1/(t^{(k+1)}-t^{(k)})\) for all times \(t^{(k)}<t\leqslant t^{(k+1)}\), the temporal average of \(\tilde{\nu}(t)\) over a much longer time \(T\) is not the same as the mean rate \(\nu\) defined here as spike count divided by \(T\).

Shortback: Too slow for animal!!!

Example: Homogeneous Poisson Process

Since the exact firing time of a spike does not matter (as we only focus on \(\nu\)), it is tempting to describe spiking as a Poisson process where spikes occur independently and stochastically with a constant rate \(\nu\).

In a homogeneous Poisson process, the probability to find a spike in a short segment of duration \(\Delta t\) is \[ P_{F}(t;t+\Delta t)=\nu \Delta t. \tag{7.3} \]

In other words, spike events are independent of each other and occur with a constant rate (also called stochastic intensity) defined as \[ \nu= \lim_{\Delta t \to 0} \frac{P_{F}(t;t+\Delta t)}{\Delta t}. \tag{7.4} \]

The expected number of spikes to occur in the measuremet interval \(T\) is therefore \[ \langle n^{sp}\rangle=\nu T, \tag{7.5} \]

For a Poisson process, the Fano factor is exactly one.

Rate as a Spike Density and the Peri-Stimulus-Time Histogram

Stimulate the neuron with some input sequence, repeated the same sequence several times and the neuronal response is reported in a Peri-Stimulus-Time Histogram (PSTH) with bin width \(\Delta t\). \(t\) is the start of the stimulation and \(\Delta t\) defines the time bin for generating the histogram.

The number of occurrences of spikes \(n_{K}(t;t+\Delta t)\) summed over all repetitions. The spike density \[ \rho(t)=\frac{1}{\Delta t}\frac{n_{K}(t;t+\Delta t)}{K}. \tag{7.6} \]

Sometimes the result is smoothed to get a continuous rate variable, usually reported in units of Hz. We call the PSTH the time-dependent firing rate.

We have defined the spike train \[ S(t)=\sum_{f}^{} \delta(t-t^{(f)}) \tag{7.7} \]

If each stimulation can be considered as an independent sample from the identical stochastic process, we can define an instantaneous firing rate as an expectation over trials \[ \nu(t)=\langle S(t)\rangle = \frac{1}{K \Delta t}\sum_{k=1}^{K} \int_{t}^{t+\Delta t} S_k(t') \mathrm{d}t' \tag{7.8-7.9} \]

The PSTH (the right-hand side of (7.9)) provides therefore an empirical estimate of the instantaneous firing rate (the left-hand side).

Shortback: Not possible for animal!!!

Example: Inhomogeneous Poisson process

An inhomogeneous Poisson process can be used to describe the spike density measured in a PSTH. In an inhomogeneous Poisson process, spike events are independent of each other and occur with an instantaneous firing rate \[ \nu(t)=\lim_{\Delta t \to 0} \frac{P_{F}(t;t+\Delta t)}{\Delta t}. \tag{7.10} \]

Therefore, the probability to find a spike in a short segment of duration \(\Delta t\) is \(P_{F}(t;t+\Delta t)=\nu(t)\Delta(t)\). More generally, the expected number of spikes in an interval of finite duration \(T\) is \(\langle n^{sp}\rangle=\int_{0}^{T} \nu(t) \mathrm{d}t\) and the Fano factor is one, as was the case for the homogeneous Poisson process.

If a bunch of neurons fire at \(\hat{t}\), then the probability to 'survive' without firing for \(t\) (denoted as \(S\)) satisfies \[ \frac{\mathrm{d}S}{\mathrm{d}t}=-\nu (t) \]

so \[ S(t|\hat{t})=\exp \biggl(-\int_{\hat{t}}^{t} \nu(t') \mathrm{d}t'\biggr) \]

The probability for a neuron that fires at \(t\) (the first spike after the spike at \(\hat{t}\)) is \[ P(t|\hat{t})=\nu(t)S(t|\hat{t}) \]

Rate as a Population Activity (Average over Several Neurons)

Suppose we have a population of neurons with identical properties. The spikes of the neurons in a population \(m\) are sent off to another population \(n\). The relevant quantity, from the point of view of the receiving neuron, is the proportion of active neurons in the presynaptic population \(m\). Formally, we define the population activity \[ A(t)=\frac{1}{\Delta t}\frac{n_{act}(t;t+\Delta t)}{N}=\frac{1}{\Delta t}\frac{\int_{t}^{t+\delta t} \sum_{j}^{} \sum_{f}^{} \delta(t-t_j^{(f)}) \mathrm{d}t}{N} \] (7.11)

where \(N\) is the size of the population, \(n_{act}(t;t+\Delta t)\) is the number of spikes (summed over all neurons in the population) that occur between \(t\) and \(t+\Delta t\) where \(\Delta t\) is a small time interval.

Interval distribution and coefficient of variation

Define the estimation of interspike interval (ISI) distributions and interpreted it as a conditional probability density: \[ P_0(s)=P(t^{(f)}+s|t^{(f)}) \tag{7.12} \] where \(\int_{t}^{t+\Delta t} P(t'|t^{(f)}) \mathrm{d}t'\) is the probability that the next spike occurs in the interval \([t,t+\Delta t]\) given that the last spike occured at time \(t^{(f)}\).

In ordet to extract the mean firing rate from a stationary interval distribution \(P_0(s)\), we start with the definition of the mean interval, \[ \langle s \rangle=\int_{0}^{s} sP_0(s) \mathrm{d}s. \tag{7.13} \]

The mean firing rate is the inverse of the mean interval \[ \nu=\frac{1}{\langle s\rangle}=\left[ \int_{0}^{\infty} sP_0(s) \mathrm{d}s \right] ^{-1} \]

Coefficient of variation \(C_{V}\)

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 \[ C_{V}^{2}=\frac{\langle \Delta s ^{2}\rangle}{\langle s \rangle ^{2}} \] where \(\langle s\rangle=\int_{0}^{\infty} sP_0(s) \mathrm{d}s\) and \(\langle \Delta s^{2}\rangle=\int_{0}^{\infty} s^{2}P_0(s) \mathrm{d}s-\langle s \rangle ^{2}\).

A Poisson process produces distributions with \(C_{V}=1\). A value of \(C_{V}>1\), implies that a given spike train is less regular that a Poisson process with the same firing rate. If \(C_{V}<1\), then the spike 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 can have \(C_{V}>1\).

Example: Poisson process with absolute refractoriness

We study a Poisson neuron with absolute refractory period \(\Delta^{abs}\). For times since last spike larger than $ ^{abs}$, the neuron is supposed to fire stochastically with rate \(r\). The interval distribution of a Poisson process with absolute refractoriness is given by \[ P_0(s)= \begin{cases} 0, \quad s<\Delta^{abs} \\ r\exp [-r(s-\Delta_{abs})], \quad s>\Delta^{abs} \end{cases} \] (7.16)

(Notice that \(\int_{0}^{\infty} P_0(s) \mathrm{d}s=1\))

and has a mean \(\langle s \rangle =\Delta^{abs}+1/r\) and variance \(\langle \Delta s^{2}\rangle=1/r^{2}\). The coefficient of variation is therefore \[ C_{V}=1-\frac{\Delta^{abs}}{\langle s \rangle} \] (7.17)

Autocorrelation function and noise spectrum

Consider a spike train \(S_i(t)=\sum_{f}^{} \delta(t-t_i^{(f)})\) of length \(T\). We suppose that \(T\) is sufficiently long so that we can formally consider the limit \(T \to \infty\). The autocorrelation function \(C_{ii}(s)\) of the spike train is a measure for the probability to find two spikes at a time interval \(s\), i.e. \[ C_{ii}(s)=\langle S_i(t)S_i(t+s)\rangle _{t}, \tag{7.18} \] where \[ \langle f(t)\rangle _{t}=\lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} f(t) \mathrm{d}t. \tag{7.19} \]

By symmetry, \(C_{ii}(-s)=C_{ii}(s)\).

Define the power spectrum (or power spectral density) of a spike train \[ \mathscr{P}(\omega)=\lim_{T \to \infty}\mathscr{P}_{T}(\omega), \] where \(\mathscr{P}_{T}\) is the power of a segment of length \(T\) of the spike train, \[ \mathscr{P}_{T}(\omega)=\frac{1}{T}\biggl\lvert \int_{-T/2}^{T/2} S_i(t)\mathrm{e}^{-i\omega t} \mathrm{d}t \biggr\rvert^{2} \tag{7.20} \] The power spectrum \(\mathscr{P}(\omega)\) of a spike train is equal to the Fourier transform \(\hat{C}_{ii}(\omega)\) of its autocorrelation function (Wiener-Khinchin Theorem): \[ \begin{aligned} \hat{C}_{ii}(\omega) &= \int_{-\infty}^{\infty} \langle S_i(t)S_i(t+s)\rangle \mathrm{e}^{-i\omega s} \mathrm{d}s \\ &= \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} S_i(t)\int_{-\infty}^{\infty} S_i(t+s)\mathrm{e}^{-i\omega s} \mathrm{d}s \mathrm{d}t \\ &=\lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} S_i(t) \mathrm{e}^{i\omega t}\mathrm{d}t \int_{-\infty}^{\infty} S_i(s') \mathrm{e}^{-i\omega s'} \mathrm{d}s' \\ &= \lim_{T \to \infty}\frac{1}{T}\biggl\lvert \int_{-T/2}^{T/2} S_i(t)\mathrm{e}^{-i\omega t} \mathrm{d}t \biggr\rvert^{2}. \end{aligned} \] (7.21)

The power spectral density of a spike train during spontaneous activity is called the noise spectrum of the neuron.

---

Spectral density - Wikipedia https://en.wikipedia.org/wiki/Spectral_density

---

Renewal statistics

（相关内容可以看Ross的随机过程的1.6节）

Poisson processes cannot be used to describe realistic interspike interval distributions. Spikes are generated in a renewal process, with a stochastic intensity. \[ \rho(t|\hat{t})=\rho_0(t-\hat{t}) \tag{7.22} \] where \(\hat{t}\) is the time since the last spike. The central assumption of renewal theory is that the state does not depend on earlier events. Renewal theory allows to calculate the interval distribution \[ P_0(s)=P(t^{(f)}+s|t^{(f)}) \tag{7.23} \]

Survivor function and hazard

Since \(\int_{\hat{t}}^{t} P(t'|\hat{t}) \mathrm{d}t'\) is the probability for a second action potential between \(\hat{t}\) and \(t\). Thus \[ S(t|\hat{t})=1-\int_{\hat{t}}^{t} P(t'|\hat{t}) \mathrm{d}t' \] is the probability that the neuron stays quiescent between \(\hat{t}\) and \(t\). \(S(t|\hat{t})\) is called the survivor function. It has an initial value \(S(\hat{t}|\hat{t})=1\). The rate of decay of \(S(t|\hat{t})\) is defined by \[ \rho(t|\hat{t})=-\frac{\frac{\mathrm{d}}{\mathrm{d}t}S(t|\hat{t})}{S(t|\hat{t})}=\frac{P(t|\hat{t})}{1-\int_{\hat{t}}^{t} P(t'|\hat{t}) \mathrm{d}t'}. \] (7.25)

\(\rho(t|\hat{t})\) is called the 'age-dependent death rate' or 'hazard' in renewal theory.

Integrating \(\mathrm{d}S/\mathrm{d}t=-\rho S\) yields the survivor function \[ S(t|\hat{t})=\exp \left[ -\int_{\hat{t}}^{t} \rho(t'|\hat{t}) \mathrm{d}t' \right] \tag{7.26} \]

The interval distribution is given by \[ P(t|\hat{t})=-\frac{\mathrm{d}}{\mathrm{d}t}S(t|\hat{t})=\rho(t|\hat{t})S(t|\hat{t}), \tag{7.27} \]

(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 \(t\).)

\[ P(t|\hat{t})=\rho(t|\hat{t}) \exp \left[-\int_{\hat{t}}^{t} \rho(t'|\hat{t}) \mathrm{d}t' \right]. \tag{7.28} \]

We focus on stationary renewal systems: \[ P(t|\hat{t})=P_0(t-\hat{t}) \tag{7.29} \] \[ S(t|\hat{t})=S_0(t-\hat{t}) \tag{7.30} \] \[ \rho(t|\hat{t})=\rho_0(t-\hat{t}) \tag{7.31} \]

Renewal theory and experiments

Under experimental conditions where neuronal adaptation is strong, intervals are not independent. A common measure of memory effects in a time series of events with variable intervals \(s_j\) is the serial correlation coefficients \[ c_k=\frac{\langle s_{j+k}s_j\rangle _j-\langle s_j\rangle _j^{2}}{\langle s_j^{2}\rangle- \langle s_j\rangle^{2}} \]

Spike-frequency adaptation causes a negative correlation between subsequent intervals \((c_1<0)\).

Autocorrelation and noise spectrum of a renewal process

Let \(\nu_i=\langle S_i \rangle\) denote the mean firing rate (expected number of spikes per unit time) of the spike train. 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 \(\lim_{s \to \infty}\langle S_i(t)S_i(t+s)\rangle=\lim_{s \to \infty}C_{ii}(s)=\nu_i^{2}\). Substract this baseline value and we get a 'normalized' autocorrelation, \[ C_{ii}^{0}(s)=C_{ii}(s)-\nu_i^{2}, \tag{7.37} \] with \(\lim_{s \to \infty}C_{ii}^{0}(s)=0\). The Fourier transform of (7.37) yields \[ \hat{C}_{ii}(\omega)=\hat{C}_{ii}^{0}(\omega)+2\pi \nu_i^{2} \delta(\omega). \tag{7.38} \] Thus \(\hat{C}_{ii}(\omega)\) diverges at \(\omega=0\). the divergence is removed by switching to the normalized autocorrelation.

Let us suppose that we have found a first spike at \(t\). The correlation function for positive \(s\) will be denoted by \(\nu_i C_{+}(s)\) or \[ C_{+}(s)=\frac{1}{\nu_i} C_{ii}(s) \Theta(s) \tag{7.39} \]

The factor \(\nu_i\) in (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, thus for \(s>0\) \[ C_{+}(s)=P_0(s)+\int_{0}^{\infty} P_0(s')P_0(s-s') \mathrm{d}s'+\int_{0}^{\infty} \int_{0}^{\infty} P_0(s')P_0(s'')P_0(s-s'-s'') \mathrm{d}s' \mathrm{d}s''+\cdots \] (7.40) or \[ C_{+}(s)=P_0(s)+\int_{0}^{\infty} P_0(s')C_{+}(s-s') \mathrm{d}s' \tag{7.41} \]

Due to the symmetry of \(C_{ii}\), we have \(C_{ii}(s)=\nu C_{+}(-s)\) for \(s<0\). So

\[ C_{ii}(s)=\nu_i[\delta(s)+C_{+}(s)+C_{+}(-s)]. \tag{7.42} \] (the autocorrelation has a \(\delta\) peak reflecting the trivial autocorrelation of each spike with itself.

Take the Fourier transform of (7.41) and find \[ \hat{C}_{+}(\omega)=\frac{\hat{P}_0(\omega)}{1-\hat{P}_0(\omega)}. \tag{7.43} \] Together with the Fourier transform of (7.42), we obtain \[ \hat{C}_{ii}(\omega)=\nu_i \Re \left\{\frac{1+\hat{P}_0(\omega)}{1-\hat{P}_0(\omega)}\right\} +2\pi \nu_i^{2}\delta(\omega) \tag{7.45} \]

Example: Stationary Poisson process

For a Poisson process, \[ C_{+}(s)=\nu \mathrm{e}^{-\nu s} [1+\nu s+\frac{1}{2}(\nu s)^{2}+\cdots ]=\nu \tag{7.46} \]

So the autocorrelation of a Poisson process is \[ C_{ii}(s)=\nu \delta(s)+\nu^{2} \tag{7.47} \]

The Fourier transform of (7.47) yields a flat spectrum with a \(\delta\) peak at zero: \[ \hat{C}_{ii}(\omega)=\nu+2\pi\nu^{2}\delta(\omega). \tag{7.48} \]

Example: Poisson process with absolute refractoriness

For a Poisson proccess with absolute refractoriness defined in (7.16). The neuron fires with rate \(r\). For \(\omega \neq 0\), (7.45) yields the noise spectrum \[ \hat{C}_{ii}(\omega)=\nu\left\{ 1+2 \frac{\nu^{2}}{\omega^{2}}[1-\cos (\omega \Delta^{abs})]+2 \frac{\nu}{\omega}\sin (\omega \Delta^{abs})\right\}^{-1}, \] (7.49)

For \(\omega \to 0\), the noise spectrum is decreased by a factor \([1+2(\nu \Delta^{abs})+(\nu \Delta^{abs})^{2}]^{-1}\). Explanation: the mean interval of a Poisson neuron with absolute refractoriness is \(\langle s\rangle=\Delta^{abs}+r^{-1}\). Hence the mean firing rate is \[ \nu=\frac{r}{1+\Delta^{abs}r}. \tag{7.50} \]

For finite \(\Delta^{abs}\) the firing is more regular than that of a Poisson process with the same mean rate \(\nu\), and hence the spectrum for \(\omega \to 0\) is less noisy.

This means that Poisson neurons with absolute refractoriness can transmit slow signals more reliably than a simple Poisson process.

Input dependent renewal theory

The Problem of Neural Coding

Limits of rate codes

Limitations of the spike count code. Averaging over a large number of spikes takes a long time. In a changing environment, a postsynaptic neuron does not have the time to perform a temporal average over many (noisy) spikes.

Limitations of the PSTH. It needs several trials to build up. Nevertheless, the PSTH measure of the instantaneous firing rate can make sense if there are large populations of similar neurons that receive the same stimulus.

Limitations of rate as a population average. (7.11) requires a homogeneous population of neurons with identical connections which is hardly realistic. Real populations will always have a certain degree of heterogeneity both in their internal parameters and in their connectivity pattern.

For inhomogeneous populations, the definition (7.11) may be replaced by a weighted average over the population. Suppose that we are studying a population of neurons which respond to a stimulus \(\mathbf{x}\). We may think of \(\mathbf{x}\) as the location of the stimulus in input space. Neuron \(i\) respond best to stimulus \(\mathbf{x}_i\). We may say that the spikes of a neuron \(i\) 'represent' an input vector \(\mathbf{x}_i\).

In a large population, many neurons will be active simultaneously when a new stimulus \(\mathbf{x}\) is represented. The location of this stimulus can then be estimated from the weighted population average \[ \mathbf{x}^{est}(t)=\frac{\int_{t}^{t+\Delta t} \sum_{j}^{} \sum_{f}^{} \mathbf{x}_j\delta(t-t_j^{(f)}) \mathrm{d}t}{\int_{t}^{t+\Delta t} \sum_{j} \sum_{f}^{} \delta(t-t_j^{(f)}) \mathrm{d}t} \] (7.52)

Candidate temporal codes

Time-to-first-spike: Latency code

Take saccading as an example. After each saccade, the photo receptors in the retina receive a new visual input. Information about the onset of a saccades should easily be available in the brain and could serve as an internal reference signal.

Experimental evidences indicate that a coding scheme based on the latency of the first spike transmit a large amount of information.

Phase

We can apply a code by 'time ot first spike' also in the situation where the reference signal is not a single event, but a periodic signal. Oscillations of some global variable (for example the population activity) are common in the hippocampus, the olfactory system, and other areas of the brain. These oscillations could serve as an internal reference signal.

Correlations and Synchrony

We can also use spikes from other neurons as the reference signal for a spike code. Neurons which represent the same object in a complex scene consisting of several objects could be 'labeled' by the fact that they fire synchronously.

Not only synchrony but any precise spatio-temporal pulse pattern could be a meaningful event.

Stimulus Reconstruction and Reverse Correlation

Reverse correlation: Every time a spike occurs, we note the time course of the stimulus in a time window of about 100 milliseconds immediately before the spike. Averaging the results over several spikes yields the typical time course of the stimulus just before a spike.

Rate versus temporal codes

The stimulus reconstruction with kernels can also be considered as a rate code based on spike counts. To see this, consider a spike count measure with a running time window \(K(.)\). We can estimate the rate \(\nu\) at time \(t\) by \[ \nu(t)=\frac{\int_{-\infty}^{\infty} K(\tau)S(t-\tau) \mathrm{d}\tau}{\int_{-\infty}^{\infty} K(\tau) \mathrm{d}\tau} \tag{7.54} \] where \(S(t)=\sum_{f=1}^{n} \delta(t-t^{(f)})\) is the spike train. For a rectangular time window \(K(\tau)=1\) for \(-T/2<\tau<T/2\) and zero otherwise, reduces exactly to (7.1). Perform the integration over the \(\delta\) function and we yield \[ \nu(t)=c \sum_{f=1}^{n} K(t-t^{(f)}) \tag{7.55} \] where \(c=[\int_{}^{} K(s) \mathrm{d}s]^{-1}\) is a constant.

Example: Towards a definition of rate codes

We have seen in (7.55) that stimulus reconstruction with a linear kernel can be seen as a special instance of a rate code. This suggests a formal definition of a rate code via the reconstruction procedure: if all information contained in a spike train can be recovered by the linear reconstruction procedure of (7.53), the the neuron is using a rate code.

Exercises

1. Poisson process in continuous time. We consider a Poisson neuron model. In every small time interval \(\Delta t\), the probability that the neuron fires is given by \(\nu \Delta t\).

1. The probability that the neuron does not fire during a time of arbitrarily large length \(t\) (survivor function \(S_0(t)\)) is \[ S_0(t)=\mathrm{e}^{-\nu t}. \]
2. Suppose that the neuron has fired at time \(t=0\), then the distribution of intervals \(P(t)\), i.e., the probability density that the neuron fires its next spike at a time \(t\), is \[ P_0(s)=\nu \mathrm{e}^{-\nu t}. \]

2. Autocorrelation of a Poisson process. Find the autocorrelation function \(C_0(s)=\langle S_i(t)S_i(t+s)\rangle _{t}\) of the homogeneous Poisson process in continuous time.
3. Repeatability and random coincidences. 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?
4. Spike count and Fano Factor. A homogeneous Poisson process has a probability to fire in a very small interval \(\Delta t\) equal to \(\nu \Delta t\).

1. The probability to observe exactly \(k\) spikes in the time interval \(T\) is \(P_{k}(T)=[1/k!](\nu T)^{k}\exp (-\nu T)\).
2. For the inhomogeneous Poisson process the mean spike count in an interval of duration \(T\) is \(\langle k \rangle=\int_{0}^{T} \nu(t) \mathrm{d}t\).
3. Calculate the variance of the spike count and the Fano factor for the inhomogeneous Poisson process.

5. 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 \[ P(t|t')=\frac{(t-t')}{\tau^{2}}\exp \left(-\frac{t-t'}{\tau}\right) \] for \(t>t'\).

1. Calculate the survivor function \(S(t|t')\), the hazard function \(\rho(t|t')\). \[ S(t|t')=\left(\frac{t-t'}{\tau}+1\right)\exp\left(-\frac{t-t'}{\tau}\right) \] \[ \rho(t|t')=\frac{t-t'}{\tau(t-t')+\tau^{2}} \]
2. A spike train starts at time \(0\) and we observed a first spike at time \(t_1\). Calculate the probability density \(P(t_{n}|t_1)\) that the \(n\)th spike occurs around \(t_n\). (self-convolution for \(n-2\) times)

6. Gamma-distribution Stationary interval distributions can often be fitted by a Gamma distrubution (for \(s>0\)) \[ P(s)=\frac{1}{\Gamma(k)}\frac{s^{k-1}}{\tau^{k}}\mathrm{e}^{-s/\tau} \]

1. Assume that intervals are independent and calculate the power spectrum.
2. Calculate the coefficient of variation \(C_{V}\) \[ C_{V}=\frac{1}{k} \]

7. 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 \(\Delta^{abs}\). More precisely, we have a renewal process \[ \rho(t|\hat{t})=\rho_0 \quad \text{for}\ t>\hat{t}+\Delta^{abs} \] and zero otherwise. Calculate the interval distribution and the Fano factor. \[ P(t|\hat{t})= \begin{cases} 0, \hat{t}<t<\hat{t}+\Delta^{abs} \\ \exp (-t-\hat{t}-\Delta^{abs}), t>\hat{t}+\Delta^{abs} \end{cases} \]