# Correlation Analysis of Parallel Spike Trains

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**DOI:**https://doi.org/10.1007/978-1-4614-7320-6_390-2

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## Synonyms

## Definition

Cross-correlation is a measure of the similarity of two signals as a function of the time lag or lead applied to one of the signals. In case the two signals are simultaneously recorded spike trains, the cross-correlation becomes a count of the number of coincidences of firing for the two spike trains as a function of the time delay between them. If one considers one spike train as the input to a system and the other spike train as the output, then the cross-correlation function between the input and output spike trains normalized on the input autocorrelation function is equal to the impulse response of the linear part of the system. Cross-correlation can also, and in the nervous system more generally, result from common input to the two spike trains, i.e., from providing a sensory stimulus or from rhythmic or other spontaneous activity in the brain.

## Detailed Description

_{x}and μ

_{y}from the two signals (I omit the sampled form here):

_{x}

^{2}and σ

_{y}

^{2}, of the two spike trains (again the sampled form is omitted):

The expressions in the above equations are defined empirical estimates of the underlying statistics. Any deviation from a value of ±1 for the correlation coefficient is due to (1) nonlinearity of the system, (2) the presence of noise in the spike generation of the system, and (3) the presence of other spike train inputs to the system that contribute to the output spike train y(t) but are not accounted for by the observed input spike train x(t).

## Representation of the Cross-Correlogram in Terms of Coincidences

_{x}and N

_{y}are the number of spikes in spike trains x(t) and y(t), respectively, and N is the number of bins in the total spike train duration. \( \frac{N_x{N}_y}{N} \) is the expected number of coincidences in case of independence. For small numbers of spikes compared to the number of bins (N) in the record, i.e., low firing rates, this can be approximated by.

Statistical significance should be set conservatively, e.g., at z ≥ 4.

The standard normal distribution that is the basis for the interpretation of z, as the number of standard deviations between a value x and the mean μ of the distribution, is based on the z-score z = (x − μ)/σ. For neural cross-correlograms, we generally do not have normal distributions, but we assume that the bin filling with coincidences follows a Poisson process (the bin size should be so small that the probability of >1 coincidence is vanishingly small). For a Poisson process the standard deviation is the square root of the mean, leading to the standard deviation of the number of coincidences \( {\sigma}_{xy}=\sqrt{\frac{N_x{N}_y}{N}} \).

If one wants to assess significance based on the cross-correlation coefficient, the standard deviation thereof is equal to \( SD\left({\rho}_{xy}\right)=\sqrt{\frac{1}{N}\left(1-\frac{N_x{N}_y}{N^2}\right)} \). For large numbers of bins in the record and relatively low firing rates, this can be approximated as \( SD\left({\rho}_{xy}\right)=\sqrt{\frac{1}{N}} \). So for a 900 s record length and 2 ms bin width, Δ, the SD = 0.0015. So taking 4 SD would make ρ_{xy} values above 0.006 significant, but one can wonder whether values close to 0.006 are relevant. For larger numbers of spikes, the SD decreases compared to this estimate.

## Representations of the Cross-Correlogram in Terms of Firing Rate

The cross-correlogram is sometimes defined as a first-order conditional rate function R_{y|x}(τ) = R_{xy} (τ)/μ_{x} (where μ_{x} is the mean firing rate for spike train x) that measures, for one cell y and close to any particular time t, the average instantaneous rate or the likelihood of generating a spike, conditional on an x spike τ time units away (Brillinger et al. 1976). If the trains are independent, then R_{y|x}(τ) = μ_{y}, where μ_{y} is the mean firing rate for spike train y, for all τ. If y spikes are independent of “later” x spikes (causality), then R_{y|x}(τ) = μ_{y} for all τ < 0. This “causality” may apparently be violated if both x and y spikes are caused with some jitter by a third, unobserved, input. Deviation of R_{y|x}(τ) from μ_{y} is suggestive of dependency of the y-train on what happened in the x-train τ time units earlier. For all natural spike trains, R_{y|x}(τ) will be flat and essentially equal to μ_{y} at large |τ| values because inevitably (with the exception for oscillatory correlograms) any influence of x on y will have vanished. Assuming a Poisson distribution, one can graph \( \sqrt{R_{y\mid x}\left(\tau \right)} \). The variance of \( \sqrt{R_{y\mid x}\left(\tau \right)} \) is approximately constant for all τ at a level of (4Δ Tμ_{x})^{−1}, so SD lines can be drawn at ± (4Δ Tμ_{x})^{−2} and become smaller in proportion to √ T. This provides a good approximation if Δ is not too large and the correlation width of the process is not too large (Voigt and Young 1990).

## Time-Dependent Cross-Correlation Functions

_{xy}(t,τ), with t as the running time in the spike trains and τ as the lag-lead time between spikes in trains x and y, shown here in an example (Fig. 1), is often the best first step in judging stationarity. R

_{xy}(t,τ) without any averaging (panels a and b) represents the recurrence times (Van Stokkum et al. 1986) between the two spike trains x(t) and y(t) as a function of τ. Panels c and d represent the time-averaged R

_{xy}(τ). The example reflects simultaneous recordings from two neurons in the auditory midbrain of the European grass frog in response to species-specific vocalizations. The left-hand and right-hand columns refer to the same data, but on the left-hand side, the correlation shown is R

_{xy}(t,τ), whereas the right-hand side shows R

_{yx}(t,τ), i.e., the reference unit has changed.

Vertically, the time over the first 150 s since stimulus onset, t, is shown and horizontally the lag-lead times, τ, between the x and y spikes. Different symbols indicate different order recurrence times. One observes that with the exception of the first 10 s or so, the occurrence of the recurrence times is stationary. The most important events that determine the various order cross-correlograms here (shown in part c, d) are the first-order recurrences, i.e., there are hardly any bursts (indicated by ■ and × symbols) in the y-train (a) but quite a few in the x-train (b), which had the higher firing rate. For calculation of the correlograms, we assume periodic stationarity: this means that for each stimulus presentation at intervals of P sec, the response of the neurons is statistically the same, albeit it is nonstationary in firing rate. By using ensemble averaging, i.e., in Eqs. (1, 2, and 3, the time variable is now t modulo P, and stimulus-dependent cross-correlations can be obtained.

The top correlogram is that based on only first-order recurrence times, the second one on second-order recurrence times, and the third one on third-order recurrence times; the fourth shows the all-order cross-correlogram as commonly used. All correlograms are on the same scale. The shaded correlograms are the shift predictors (see next section), calculated from the time-dependent shift correlograms shown in the bottom row that show only modest stimulus locking to the stimulus. The shape of the cross-correlogram suggests common input as the main source of correlated spike times (see later in this entry).

## Pair Correlation Under Stimulus Conditions

_{i}and t

_{j}are the spike times. The cross-correlation function is given by.

_{x}μ

_{y}of R

_{xy}(τ) under independence of firing of the two spike trains, a cross-covariance histogram is defined:

_{1}(t) and y

_{1}(t) for the first stimulus presentation, respectively, and x

_{2}(t) and y

_{2}(t) for the second one. Presenting the stimulus twice allows us to estimate the correlation between the firing of the two spike trains based on their time locking to the stimulus. In the text I assume a long stimulus consisting of various elements. The predictor for that is to correlate spike train x for stimulus presentation 1 with spike train y for stimulus presentation 2, under the assumption that memory effects have long subsided between the two stimulus presentations. This result is called the shift predictor. For those four spike trains, one can compute ten covariance histograms (or correlograms) as shown in Table 1.

Overview of possible covariance histograms for a double-unit recording with two presentations of the stimulus ensemble

x | y | x | y | |
---|---|---|---|---|

x | C | C | C | C |

y | C | C | C | |

x | C | C | ||

y | C |

On the main diagonal two estimates, each of the units’ autocovariance histograms is found. Under long-term stationary firing conditions, C_{x1x1} = C_{x2x2}, and one commonly uses C_{xx} = (C_{x1x1} + C_{x2x2})/2 as an estimator of the autocovariance function for spike train x(t). For the off-diagonal elements, it is noted that, e.g., C_{x1y1}(τ) = C_{y1x1}(−τ), therefore the table is symmetric provided that the sign of τ is adjusted. We define C_{x1y1}(τ) and C_{x2y2}(τ) as simultaneous cross-covariance histograms, and again we may use their average as an estimate for the cross-covariance function. Following our previously introduced terminology (Eggermont et al. 1983; Epping and Eggermont 1987), cross-correlations obtained under stimulus conditions will be called neural synchrony; they will be a mix of signal and noise correlations (e.g., Gawne and Richmond 1993). Cross-correlations obtained under spontaneous firing conditions or after appropriate correction for correlation resulting from stimulus locking will be called neural correlation; they exclusively are noise correlations. During stimulation, the “total” neural synchrony consists of a component due to stimulus synchrony and one that is a result of neural connectivity, the neural correlation.

The nonsimultaneous cross-covariance histograms between spike trains x_{1}(t) and y_{2}(t), respectively, and x_{2}(t) and y_{1}(t) represent correlation between spike trains x(t) and y(t) as a result of stimulus coupling. These stimulus-induced correlations are known as the shift predictors. Another nonsimultaneous cross-covariance histogram is obtained by taking both spike trains for the same unit to obtain C_{x1x2} and C_{y1y2}. These histograms represent the amount of coupling of the units to the stimulus individually and have been called “existence functions” (Aertsen et al. 1979) because any significant peak around τ = 0 indicates a stimulus coupling effect and hence the existence of a stimulus-response relation. This shuffled autocorrelogram technique was rediscovered by Joris et al. (2006).

_{Sx}(σ) and R

_{Sy}(σ), and calculates their correlation integral modulo stimulus period P (here σ is the poststimulus time):

One can also calculate the cross-correlogram between one spike train and a shifted (by 1 stimulus period) version of the second spike train. This latter procedure, resulting in the shift predictor, has been the most popular and has the advantage that it can be applied to very long stimuli (such as a set of animal vocalizations) that allow only one or a few repetitions; in such a case, calculating the PSTH would be useless. The shift predictor was assumed to represent the amount of correlation that could be attributed to the locking of the neuron’s firings to the stimulus. Subtracting the shift predictor from the neural synchrony then would result in the neural correlation.

The assumption of additivity of stimulus-induced correlation and neural correlation may well be the weakest link in the chain of reasoning that leads to an interpretation of stimulus-dependent neural correlations. A strong warning against the use of the shift predictor in estimates of the neural interaction strength came from studies using realistic model neurons (Melssen and Epping 1987). Because stimulus effects and neural connectivity effects on the neural synchrony are generally nonadditive, it was concluded that the use of the shift predictor to separate stimulus and neural effects was of limited value. Yang and Shamma (1990) underlined these objections to the use of correlation methods in identifying neural connectivity and arrived at conclusions similar to those of Melssen and Epping (1987) using simulations with similar model neurons. In case of strong stimulus locking, even all synchrony can be due to stimulus correlation (Eggermont 1994). Under “adequate” stimulus conditions that evoke strongly stimulus-driven and reproducible spike trains, especially for auditory stimuli, the neural synchrony becomes identical to the expected correlation based on the shift predictor, and consequently no inference can be made about possible underlying connectivity (changes). One may avoid this overcorrection effect due to stimulus locking by applying suboptimal stimulation. Thus, a sufficient amount of “neural noise” in the record is required to allow conclusions about changes in functional neural connectivity. Thus, when stimulus-dependent neural correlations are obtained, there are a number of possibilities: the interneuronal coupling itself changes due to application of a stimulus (facilitation, depression, short-term learning), the finding is an artifact due to the selection of the wrong model, or the finding can be explained on the basis of a pool of neurons affecting the firing behavior of the pair under study in a stimulus-dependent way.

_{i}caused by incoming spikes z

_{i}possibly originating from different projecting neurons. Actually g(z|u) describes the well-known S-shaped curve. For relatively small u values, i.e., in the purely exponential region of the curve and temporal overlap of u

_{1}and u

_{2}, we have:

_{1}and u

_{2}are nonoverlapping (low spike rates), averaging over time gives.

Thus, simply on basis of the size u and duration of the EPSP, the firing probability of the output neuron can be multiplicative (Eq. 13) or additive (Eq. 14) with respect to the input spike trains. In fact u = u(t) and additional conditions regarding the decay time constant may enter the description.

Let us now consider that a multiplicative effect of the stimulus on spike train activity is present, e.g., by two input spike trains both depending on the stimulus and fulfilling the conditions for the EPSP but only one of these input trains is observed together with the output train. In such a case, the degree of multiplicative action is strongly dependent on the type of stimulus. Stimuli such as continuous noise that cause the input spike trains to have a relatively large time jitter, i.e., have a broad existence function, C_{ii’}(z), (Aertsen et al. 1979), produce results that better fit a multiplicative model than stimuli that tend to lock the spikes to a considerable degree (e.g., tone pips) as suggested by the Srinivasan and Bernard (1976) model results. Thus, also multiplicative interaction effects may be stimulus dependent. In case they are not, the application of a correction procedure based on an additive model, however, erroneously will indicate a stimulus dependence of the neural correlation (Eggermont 1994). Eggermont (2006) introduced a method to correct for these multiplicative effects, which are substantial when cortical network oscillations are present and will be discussed later on in this entry.

## Practical Estimation of Stimulus Correlation

During spontaneous activity, one estimates the expected value of the cross-correlogram under the assumption of independence and subtracts this in order to obtain the neural correlation. During stimulation, the effect of a common input to both spike trains as a result of a correlation of the neuronal firing to the stimulus has to be taken into account if one wants an estimate of the true neural correlation. If the stimulus is periodic, one can, for instance, use the shift predictor obtained by calculating the cross-correlation between spike trains x(t) and y(t + P), i.e., between the original x-train and the y-train shifted over one stimulus period. This method is the only one that can be used if a long stimulus sequence, e.g., a sequence of vocalizations or other natural sounds, is only repeated once, as discussed above.

## Rate Correlation and Event Correlation

Cross-correlograms frequently show a narrow peak on a broader pedestal (e.g., Fig. 2a) especially when units are recorded on the same electrode (Eggermont 1992; Nowak et al. 1995). The broader pedestal is typically the result of covariance in firing rate between the two spike trains, e.g., caused by the stimulus, whereas the sharp peak indicates event synchrony, i.e., firing coincidences (Neven and Aertsen 1992). The rate covariance under spontaneous conditions can easily be estimated by calculating a cross-correlogram for 50 ms bins that slide in 10 ms steps (Eggermont and Smith 1995). Subtracting the rate covariance from the overall correlogram will result in the event correlation.

This generally assumed model, even if correct, may still result in an inaccurate estimation of the true event correlation. Staude et al. (2008) decomposed the raw cross-correlation as the sum of the modulation rate covariance and spike coordination based on the mean conditional covariance of the processes given the rates. Both, rate covariation and spike coordination alike, contribute to the raw correlation function by triangular peaks. This separation does not always have to result in a broad triangular peak for rate correlation and a narrow one for spike correlation; at least in their procedure, the reverse can be possible as well. Staude et al. (2008) stated “without prior knowledge of the underlying model class, rate estimation will most likely be suboptimal. The resulting predictor leaves a residual central peak in the corrected correlation function, which is likely to be wrongly interpreted as spike coordination.”

## Stimulus-Dependent Neural Correlation

## Correcting for Common Input Firing Periodicities and Firing Rates

According to de la Rocha et al. (2007), the stimulus-corrected cross-correlation coefficient for common input still depends on the geometric mean of the firing rates of the two neurons, both in cortical slice recordings and in their simulated data. This effect was also visible for in vivo recordings from visual cortex (Krüger 1991), and it was used to argue that nearly all of the cortical cell correlations resulted from common retinal input. This suggests that additivity-corrected procedures are inadequate and that multiplicative interactions of stimulus and spontaneous firing activity occur. This can be corrected for by a deconvolution procedure as described below.

_{xy}(τ), resulting in the cross spectrum S

_{xy}(f) and dividing by the square root of the product of the two Fourier-transformed autocorrelation functions, i.e., the power spectra. This gives the complex coherence:

An inverse Fourier transform results in the corrected “correlation coefficient” (Eggermont 2006).

## Correlation and Connectivity

In general there is no unique relationship between known connectivity and observed cross-correlation (Meyer and van Vreeswijk 2002). Correlations are in principle determined uniquely by connectivity, including the strengths and delay dependence of the connections. From the neural correlation one can, under certain assumptions about the integration of neural input and the shape on the neuron’s response curve, estimate the strength of the neural interaction. Because the estimate of the peak neural correlation is very sensitive about the shape of and working point on the neuron’s response curve (Melssen and Epping 1987), it is generally not permitted to equate the strength of the neural correlation estimated from extracellular recordings with the strength of the neural interaction. When an actual stimulus is applied, one of its effects, especially for cortical neurons, will be to shift the neuron’s working point, and apparent changes in neural correlation may be found without a concomitant change in the strength of the neural interaction (Aertsen and Gerstein 1985; Melssen and Epping 1987).

The simplest interactions shown by a pair of neurons are that one neuron makes a direct excitatory or inhibitory connection with the other or that both neurons share a common excitatory or inhibitory input. It has been shown in simulation studies that common excitatory input and common inhibitory input result in a peak in the cross-correlogram around zero-lag time. Reciprocal excitatory and inhibitory input to the neurons provided by the same source result in a central dip in the cross-correlogram (Moore et al. 1970). Unilateral excitation shows a peak shifted to positive lag times, and unilateral inhibition shows a dip at positive lag times.

## Effects of Spike Sorting on Pair Correlations

Multiunit (MU) recording is common in structures with high cell densities and especially using multielectrode arrays. Spike sorting can result in well-sorted units, but by no means can one be sure that one is dealing with single units (SU), that is, only the case when doing intracellular or patch-clamp recordings. Evidence from neural modeling suggests that interpretation of neural correlations from multiunit recordings may be ambiguous as it is not a linear combination of correlations for the various single-unit pairs (Bedenbauch and Gerstein 1997; Gerstein 2000). However, changes in single-unit correlation strengths will be accompanied by comparable changes in the correlation between multiunit activities as shown in Eggermont (2000). Thus, if a number of single units contribute to each multiple single-unit recording, the changes in the MU values will be positively, but nonlinearly, related to those for the corresponding SU ones. So if SU correlations go up, so will the MU correlation. The only difference is that the R-values are larger for MU than for the corresponding SU ones but smaller than the sum of all the relevant SU pair R-values. Trying to estimate functional connectivity from sorted multiunit activity is not encouraged.

## Correlations and the Brain

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