Encyclopedia of Computational Neuroscience

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Correlation Analysis of Parallel Spike Trains

  • Jos J. EggermontEmail author
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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

I will use the definition of cross-correlation as standard in the field of linear systems analysis as applied to the interaction between two neurons (Marmarelis and Marmarelis 1978). I will (except when noted) assume stationarity of the spike trains. The cross-correlation function reflects the similarity of two spike trains x(t) and y(t), or in sampled form x(n) and y(n), as a function of the delay between them, expressed as a time delay τ or as the number of bins k. T is the duration of the spike trains, and N is the number of bins in the spike train. Note that the time delay can be positive and negative.
$$ {\displaystyle \begin{array}{ll}{R}_{xy}\left(\tau \right)& =\frac{1}{T}\underset{0}{\overset{T}{\int }}\Big(x(t)y\left(t+\tau \right)\\ {}& \times dt\, \mathrm{or}\, \mathrm{in}\, \mathrm{sampled}\, \mathrm{form}:{R}_{xy}(k)\\ {}& =\frac{1}{N}\sum \limits_{n=1}^Nx(n)y\left(n+k\right)\end{array}} $$
(1)
The cross-covariance function is obtained by subtracting the means μx and μy from the two signals (I omit the sampled form here):
$$ {C}_{xy}\left(\tau \right)=\frac{1}{T}\underset{0}{\overset{T}{\int }}\left(\Big(x(t)-{\mu}_x\right)\left(y\left(t+\tau \right)-{\mu}_y\right) dt. $$
(2)
The cross-correlation coefficient function is obtained by dividing the cross-covariance function by the time-average variances, σx2 and σy2, of the two spike trains (again the sampled form is omitted):
$$ {\rho}_{xy}\left(\tau \right)=\frac{1}{T}\underset{0}{\overset{T}{\int }}\left(\Big(x(t)-{\mu}_x\right)\left(y\left(t+\tau \right)-{\mu}_y\right)\times dt/\sqrt{\sigma_x^2{\sigma}_y^2}. $$
(3)

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

In case of Poisson spike trains, the cross-correlation coefficient becomes (Eggermont 1992).
$$ {\rho}_{xy}\left(\tau \right)=\left[{R}_{xy}\left(\tau \right)-\frac{N_x{N}_y}{N}\right]/\sqrt{\left({N}_x-\frac{N_x^2}{N}\right)\left({N}_y-\frac{N_y^2}{N}\right)}. $$
(4)
Here, Nx and Ny 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.
$$ {\rho}_{xy}\left(\tau \right)=\left[{R}_{xy}\left(\tau \right)-\frac{N_x{N}_y}{N}\right]/\sqrt{\left({N}_x{N}_y\right)}. $$
(5)
For statistical purposes this can also be converted (holds for all firing rates) to a z-score, i.e., the number of standard deviations above the mean-corrected R, which is equal to zero:
$$ {z}_{xy}\left(\tau \right)=\left[{R}_{xy}\left(\tau \right)-\frac{N_x{N}_y}{N}\right]/\sqrt{\frac{N_x{N}_y}{N}}. $$
(6)

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 ρxy 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 Ry|x(τ) = Rxy (τ)/μx (where μx 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 Ry|x(τ) = μy, where μy is the mean firing rate for spike train y, for all τ. If y spikes are independent of “later” x spikes (causality), then Ry|x(τ) = μy 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 Ry|x(τ) from μy is suggestive of dependency of the y-train on what happened in the x-train τ time units earlier. For all natural spike trains, Ry|x(τ) will be flat and essentially equal to μy 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 Tμx)−1, so SD lines can be drawn at ± (4Δ Tμx)−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

Visual inspection of the time-dependent cross-correlation function, Rxy(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. Rxy(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 Rxy(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 Rxy(t,τ), whereas the right-hand side shows Ryx(t,τ), i.e., the reference unit has changed.
Fig. 1

Time-dependent cross-correlation diagrams of spike trains. The different symbols used indicate the order of the spikes following the reference spike: “|” indicates the first-order recurrence time (defined in Van Stokkum et al. 1986), i.e., the first spike in the y-train following or preceding a spike in the x-train (for the left-hand column); “■” indicates the second-order recurrence time, i.e., the second spike in the y-train following or preceding a spike in the x-train; and finally “×” indicates the third-order recurrence time. (a) Simultaneous time-dependent cross-correlation diagram (no averaging involved), spikes from unit x (2,188 events) are used as triggers; (b) as (a) but now with spikes from unit y (528 events) as triggers; (c) simultaneous and nonsimultaneous (shaded) recurrence time histograms of orders 1, 2, and 3 and cross-correlation histogram derived from (a) and (e), respectively; (d) as (c) but now derived from (b) and (f); (e) nonsimultaneous time-dependent cross-correlation diagram (defined as the correlation between the 1 s duration spike train x to stimulus presentation at t and spike train y to the next stimulus presentation at time t + 1 s), spikes from unit x are used as triggers; (f) as (e), but now with spikes from unit y as triggers. (With kind permission from Springer Science + Business Media: Biological Cybernetics, Representation of Time-Dependent Correlation and Recurrence Time Functions, volume 55, 1986, page 17–24, I. H. M. van Stokkum, P. I. M. Johannesma, and J. J. Eggermont, Fig. 6)

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

A peak in the cross-correlogram can be the result of a neural interaction between units (“noise correlation”) but also the result of a coupling of the firings of the neurons to a stimulus (“signal correlation”) (Gawne and Richmond 1993). As we will see, the interpretation of the cross-correlogram will require both the autocorrelograms of the two units and an estimate of the correlation due to stimulus coupling. Following our earlier approach (Eggermont et al. 1983), we consider two spike trains x(t) and y(t) of duration T and represented as a sequence of δ-functions:
$$ x(t)=\sum \limits_{i=1}^N\delta \left(t-{t}_i\right)\, \mathrm{and}\, y(t)=\sum \limits_{j=1}^M\delta \left(t-{t}_j\right). $$
(7)
where ti and tj are the spike times. The cross-correlation function is given by.
$$ {R}_{xy}\left(\tau \right)=\frac{1}{T}\sum \limits_{i=1}^N\sum \limits_{j=1}^M\delta \left[\tau -\left({t}_i-{t}_j\right)\right]. $$
(8)
For a bin width Δ and expectation E = μx μy of Rxy (τ) under independence of firing of the two spike trains, a cross-covariance histogram is defined:
$$ {C}_{xy}\left(\tau, \Delta \right)=\frac{1}{\Delta}\underset{\tau -\Delta /2}{\overset{\tau +\Delta /2}{\int \limits }}\left({R}_{xy}\left(\sigma \right)-E\right) d\sigma . $$
(9)
If one presents the stimulus twice, then the resulting spike trains are x1(t) and y1(t) for the first stimulus presentation, respectively, and x2(t) and y2(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.
Table 1

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

 

x1(t)

y1(t)

x2(t)

y2(t)

x1(t)

Cx1x1

Cx1y1

Cx1x2

Cx1y2

y1(t)

 

Cy1y1

Cx2y1

Cy1y2

x2(t)

  

Cx2x2

Cx2y2

y2(t)

   

Cy2y2

On the main diagonal two estimates, each of the units’ autocovariance histograms is found. Under long-term stationary firing conditions, Cx1x1 = Cx2x2, and one commonly uses Cxx = (Cx1x1 + Cx2x2)/2 as an estimator of the autocovariance function for spike train x(t). For the off-diagonal elements, it is noted that, e.g., Cx1y1(τ) = Cy1x1(−τ), therefore the table is symmetric provided that the sign of τ is adjusted. We define Cx1y1(τ) and Cx2y2(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 x1(t) and y2(t), respectively, and x2(t) and y1(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 Cx1x2 and Cy1y2. 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).

There is no model-free approach for the separation of correlations produced by direct neural interaction and those caused by stimulus coupling. Under the assumption that the effects of an external stimulus and the effects of neural connectivity to the neural synchrony are additive, Perkel et al. (1967) proposed a correction for the effects of the stimulus. Two formally identical stimulus predictors were suggested, one resulting from a cross-correlation of the two single-unit poststimulus time histograms (PSTHs), which can be used when the stimulus is periodic. One constructs the PSTHs, i.e., the cross-correlograms between stimulus onsets and spikes, RSx(σ) and RSy(σ), and calculates their correlation integral modulo stimulus period P (here σ is the poststimulus time):
$$ {R}_{S\mathrm{corr}}\left(\tau \right)=\underset{0}{\overset{T}{\int }}{R}_{Sx}\left(\sigma \right){R}_{Sy}\left(\sigma +\tau \right) d\sigma . $$
(10)

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.

It must be remarked that the additive model (Perkel et al. 1967) is the only one routinely used in stimulus-correction procedures (e.g., Voigt and Young 1990; Eggermont 1994). However, Srinivasan and Bernard (1976) have shown that coincidence-detecting neurons can fire at a rate proportional to the product of the rates of two input neurons, provided that their spike trains are statistically independent. The criteria determining whether the action is additive or multiplicative are the size of the excitatory postsynaptic potential (EPSP) relative to threshold and the time constant of decay for the EPSP. Specifically, EPSP values smaller than the threshold of firing and with a duration less than the average interspike interval of the individual spike trains cause the action to be multiplicative. Larger EPSP values or those with longer duration make the action additive. An alternative formulation from which this effect can be understood is that we consider a spike generator with probability of firing, g, which depends sigmoidally on the EPSP value u (Eggermont et al. 1983).
$$ g\left(z|u\right)=\exp .\beta u/\left(1+\exp .\beta u\right) $$
(11)
Here u is simply considered to be the linear summation of contributions ui caused by incoming spikes zi 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 u1 and u2, we have:
$$ g\left(z|{u}_1,{u}_2\right)=\, \exp .\beta \left({u}_1+{u}_2\right)\times /\left(1+\exp .\beta \left({u}_1+{u}_2\right)\right)\approx \exp {bu}_1\bullet \exp {bu}_2\approx g\left(z|{u}_1\right)\times \bullet g\left(z|{u}_2\right) $$
(12)
If the u values are large enough to be in the linear, intermediate, part of the S-curve, then after expansion of the exponential and long division:
$$ g\left(z|u\right)\, \approx \frac{1}{2}+1/4 bu $$
(13)
and in case u1 and u2 are nonoverlapping (low spike rates), averaging over time gives.
$$ g\left(z|{u}_1,{u}_2\right)\approx g\left(z|{u}_1\right)+g\left(z|{u}_2\right). $$
(14)

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, Cii’(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.

As an example of this latter stimulus-correction procedure, we show recordings from the auditory midbrain to 500 ms long 30 Hz amplitude-modulated noise-burst stimulation repeated every 3 s (Fig. 2). The shift predictor is shaded and shows the 30 Hz modulation in the firing rate that obviously is stimulus locked. The triangular shape of the predictor is the result of the approximately rectangular envelope for the 500 ms long noise burst. The neural correlation consists of a narrow zero-lag centered peak approximately 5 ms in width (W) at half peak amplitude.
Fig. 2

(a) Simultaneous and nonsimultaneous (i.e., the shift predictor; shaded) cross-correlation histograms for two units recorded from the auditory midbrain in the grass frog under amplitude-modulated noise-burst stimulation of 500 ms duration. The triangular shift predictor results from the approximately rectangular stimulus envelope. A 30 Hz stimulus-locked oscillation is visible as well. (b) and (c) difference histogram shown on different time scales; W is the width of the peak at half amplitude. Bin width was 4 ms in (a) and (b) and 1 ms in (c). The number of spikes for each unit is indicated in the top right of panel A. (From Epping and Eggermont 1987)

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

An example, exhibiting stimulus-dependent correlation, is provided in Fig. 3. As discussed above, this may potentially result from using the invalid assumption of “neural synchrony equals the sum of stimulus correlation and neural correlation.” The two spontaneously active units shown were recorded in the auditory midbrain of the grass frog on electrodes with an estimated tip separation of 130 μm. The spontaneous rates were 0.34 (A) and 2.5 (B) spikes/s, respectively. Their spectro-temporal sensitivities, represented as dot displays of neural activity, as determined with tone pips presented once per second, are shown in Fig. 3a, b. The spontaneous activity of unit a was suppressed by frequencies below 1000 Hz (Fig. 3a), and the very low spontaneous activity did not result in many spikes in the 256 ms time window, whereas unit b was activated in this frequency range with a latency of 67 ms (Fig. 3b). Neural synchrony histograms and shift predictors (shaded) of spontaneous and stimulus-evoked activity are shown in Fig. 3cf. For spontaneous firing, the neural synchrony and shift predictor are flat and hardly different (Fig. 3c). For tone-pip stimulation, the shift predictor (Fig. 3d) shows a decrement due to the antagonistic (suppression vs. activation) stimulus influence on the two units. Just visible on top of the shift predictor is a small, rather symmetric peak around the origin, likely indicating shared neural input. The shift predictor to sinusoidally amplitude-modulated (AM) 200 ms tone bursts is broad (Fig. 3e) with a sharp (W = 6 ms) peak of the neural synchrony visible on top. Figure 3f shows results for a low-pass noise-modulated tone. One observes an asymmetrical peak with flanking valleys due to the autocorrelation structure of the individual spike trains. The two units responded antagonistically to the tone-pip stimulus, but more alike to the other stimulus ensembles; e.g., when stimulated with sinusoidally AM tone bursts (Fig. 3e) with a carrier frequency of 500 Hz, the units both were excited by a band-limited range of AM rates around 100 Hz. This suggests that the unit pair exhibits stimulus dependencies of both stimulus-induced cross-correlations as revealed by the shift predictor as well as neurally mediated cross-correlations because of changes in form and width of peaks in the neural synchrony as compared with the shift predictor.
Fig. 3

Unit pair with a stimulus-dependent neural correlation. (a) and (b) dot displays of single-unit activity to tone pips with carrier frequencies in the range 100–3200 Hz. The stimulus inhibits the unit shown in (a), whereas it activates the unit shown in (a). (cf) simultaneous and nonsimultaneous (shift predictor; shaded) cross-correlation histograms for a variety of stimuli. Bin width is 4 ms. (From Epping and Eggermont 1987)

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.

The example shown (Fig. 4) was based on spontaneous multielectrode recordings from primary auditory cortex in an anesthetized cat where the EEG showed spindle waves in the frequency range of 8–10 Hz. In anesthetized animals (sleep), spindle oscillations are frequently observed. The oscillations likely reflect common input as the correlograms typically appear symmetric around zero lag, so their effect should disappear in the corrected correlogram using a deconvolution procedure. These periodic oscillations are reflected in the firing times of the spike trains and in the oscillatory character of the cross-correlogram (red curves). This effect was corrected by deconvolving the cross-correlation function by the geometric mean of the autocorrelation functions of the two spike trains (Eggermont and Smith 1996). The corrected result is shown as the blue curves. One notices also the strong contribution (>50%) of these spindle-related correlations on the peak value of the cross-correlogram.
Fig. 4

The deconvolution-based correction procedure eliminates the oscillatory aspects resulting from anesthesia-induced spindle oscillations in auditory cortex. Here we show all pair correlations between the spike activity between two eight-electrode arrays; one array with electrodes numbered 1–8 and the other with electrodes numbered 9–16. The red curves show the raw cross-correlograms for spontaneous activity in cat primary auditory cortex. The blue curves represent the corrected cross-correlograms. All units showed an oscillatory autocorrelation at the same (spindle) frequency, and the effect thereof is eliminated using the deconvolution procedure. All correlograms are scaled to the peak of the raw cross-correlation coefficient value (the red curve), which is shown in each panel. (From Eggermont (unpublished))

The deconvolution procedure is carried out in the frequency domain by taking the Fourier transform of Cxy(τ), resulting in the cross spectrum Sxy(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:
$$ {\gamma}_{xy}(f)=\frac{S_{xy}\left(,f\right)}{\sqrt{S_{xx}\left(,f\right){S}_{yy}\left(,f\right)}} $$
(15)

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.

In auditory cortex evidence for direct excitation or inhibition has been scant, unless the neuron pairs were recorded on the same electrode, and nearly all correlograms are of the common input type (Eggermont 1992, 2000). In visual cortex several interaction types have been shown for recordings with dual electrodes from cells at different depth in the same cortical column (Toyama et al. 1981). However, more variety in correlogram shapes was found in the auditory midbrain (Epping and Eggermont 1987). In the examples shown in Fig. 5, we show (A) stimulus-induced common excitatory effects without neural correlation for two units recorded on the same electrode and (B) stimulus-induced common suppressive effects for a pair recorded on electrodes separated by 130 μm. In both cases the shift predictor is equal to the neural synchrony. Weak stimulus-induced activation combined with a direct excitatory effect from one neuron upon the other recorded on the same electrode (C) is inferred because the shift predictor is hardly visible and the peak of the correlogram is displaced by 6 ms from the origin and highly asymmetrical. This is interpreted as a unidirectional excitatory influence of unit 1 on unit 2. In these three examples, the stimulus consisted of random frequency tone pips presented once per second. Part D shows stimulus-induced (Poisson-distributed clicks) common input for a pair recorded with electrodes separated by 290 μm putatively via an unobserved neuron (D), because it has a pronounced shift predictor. For two spontaneous recordings (E, F), reason for the flat shift predictors, unidirectional excitatory input was observed for a single-electrode pair (E) and common input for a dual-electrode pair (separation 160 μm; F). The highly asymmetrical peak of the neural synchrony displaced from the origin in part E indicates that unit 1 exerts a unidirectional excitatory influence on unit 2. However, there may be a complication due to the dead time of the spike-sorting procedure, and it may well be a common input neural correlation as well. This is a known drawback of using sorted units recorded on the same electrode.
Fig. 5

Potential functional connectivity underlying cross-correlation functions. Simultaneous and nonsimultaneous (shift predictor; shaded) cross-correlation histograms are superimposed. (a) Neural synchrony, incremental correlogram. (b) Neural synchrony, decremental correlogram. (c) Unidirectional excitatory input. (d) Neural shared input. (e) Unidirectional excitatory input. (f) Neural shared input. Stimulus-driven activity in (ad); spontaneous activity in (e) and (f). Note different time scales. Simple connectivity schemes that can account for the observed phenomena are shown as insets. S, stimulus; 1, neuron 1; 2, neuron 2; u, unobserved neuron. Open and closed triangles indicate excitatory and inhibitory connections, respectively. (From Epping and Eggermont 1987)

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

Calculating a shift predictor or any other type of stimulus predictor is a way to extract the effects of stimulation on the effective neural connectivity despite the drawbacks that we discussed. However, as we have previously argued (Tomita and Eggermont 2005), it is unlikely that the nervous system performs a correction for stimulus-induced correlation as estimated by the various predictors. It is the actual spike coincidences, i.e., the neural synchrony, that are affecting the potential for firing in a target neuron and not the stimulus-corrected ones. Thus, raw correlations may effectively estimate those coincident firings between neurons that could play a role in neural population coding of sound. The effect of removing common periodicities in spike firing and common bursting activity from the cross-correlation can be justified because these contributions typically do not occur, or occur much less, in awake animals and are often the result of using anesthesia. Nevertheless, in our data for 15-min steady-state stimuli, the effects of a stimulus correction based on the shift predictor are minimal (Fig. 6), so using the stimulus-correction procedure has minimal effect on the interpretation of the data. However, for transient and optimal stimuli, the stimulus correction can be extensive and even remove all correlation (Eggermont 1994), so in this case one has to consider what the correlation estimate needs to support. If one is concerned with estimating connectivity or deciding which wiring scheme is most likely, then both stimulus corrections and corrections for common, network-induced, activity need to be performed. If one is concerned with how the brain performs its task in vivo, then these corrections may obscure the very aspects one wants to understand.
Fig. 6

This set of three cross-correlograms illustrates the stimulus-correction procedure applied to spike trains recorded from primary auditory cortex in the cat during a 900 s long random frequency stimulus where the second half was equal to the first. Stimuli were tone pips presented randomly in frequency and time with an overall rate of 28/s across 7 octaves (112 frequencies, so an average rate of 0.25 frequencies/s). The recording comprised 2 arrays of 16 electrodes each. Here we show the pair correlation between two neighboring electrodes separated by 0.5 mm. The uncorrected cross-correlation coefficient function is shown in solid line; the peak cross-correlation is approximately 0.082. The frequency tuning of the two recording sites was nearly identical, yet the stimulus correlation obtained as a shift predictor (dashed line) shows only modest stimulus locking. The stimulus-corrected cross-correlation coefficient function is shown in a connected dot line. (With kind permission from Springer Science + Business Media: Analysis of parallel spike trains, Chap. 5, Pair-correlation in the time and frequency domain, 2010, page 77–102, J. J. Eggermont, Fig. 5.3)

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Physiology and PharmacologyUniversity of CalgaryCalgaryCanada
  2. 2.Department of PsychologyUniversity of CalgaryCalgaryCanada

Section editors and affiliations

  • Sonja Grün
    • 1
    • 2
  1. 1.Lab for Statistical Neuroscience, Institute of Neuroscience and Medicine (INM–6) and Institute for Advanced Simulation (IAS–6)Jülich Research Centre and JARAJülichGermany
  2. 2.Theoretical Systems NeurobiologyRWTH Aachen UniversityAachenGermany