bootstrap significance test of synchronous spike events—a case study of oscillatory spike trains

STATISTICS IN MEDICINEStatist. Med. 2007; 26:3976–3996Published online 11 July 2007 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/sim.2962

Bootstrap significance test of synchronous spike events—A casestudy of oscillatory spike trains

Hiroyuki Ito∗,†

Faculty of Engineering, Kyoto Sangyo University, Kyoto 603-8555, Japan

SUMMARY

The purpose of this monograph is two folds. Firstly, we introduce challenging spike data to the statisticalanalysis. The data of two neurons recorded from the cat visual pathway show various non-stationarycharacteristics not fitted by the Poisson spike train. Spike firings of both neurons are strongly periodic andtightly synchronized. Our second purpose is a case study of applications of various statistical methodsfor the significance test of the time-varying spike synchrony. We provide various general remarks to thestatistical analysis of the synchronous spike activities. At first, we apply the unitary event analysis. Thesignificance limit for the coincident spike events by the Poisson distribution is compared with the limitgiven by the non-parametric test based on the bootstrap samplings. The bootstrap test performs superiorto the Poisson test in two respects: (1) avoids false positives due to the sudden change of spike density;and (2) takes into account the non-stationary change of the spiking pattern at different sampling windows.When the spike trains are highly periodic, the histogram of the number of accidental coincident spikeevents over the bootstrap samples has a systematically larger variance than the Poisson distribution. Wefind that a large variance originates from the correlation between the successive coincident spike events inthe structured spike trains. The significance of the time-varying synchrony is tested by another statisticalmethod by Ventura et al., which is based on the adaptive smoothing method and the bootstrap significancetest. Copyright q 2007 John Wiley & Sons, Ltd.

KEY WORDS: bootstrap; bootstrapping; spike; synchrony; correlation; unitary event

1. INTRODUCTION

Since any neuron in the brain cannot function in isolation, the basic unit in information processingshould be the cell assembly embedded in the neuronal network [1]. However, traditional view on

∗Correspondence to: Hiroyuki Ito, Faculty of Engineering, Kyoto Sangyo University, Kamigamo, Kita-ku, Kyoto603-8555, Japan.

†E-mail: [email protected]

Contract/grant sponsor: Ministry of Education, Culture, Sports, Science and Technology of Japan; contract/grantnumber: 17021036

Received 8 May 2006Copyright q 2007 John Wiley & Sons, Ltd. Accepted 16 April 2007

BOOTSTRAP SIGNIFICANCE TEST OF SYNCHRONOUS SPIKE EVENTS 3977

information coding assumed that the relevant variable was the change in the mean firing rate ofa single neuron to sensory stimuli or behavioral events. For the last two decades, we have beenfacing a paradigm shift in the concept of relational coding. That is, a part of the information iscoded only by the relationship among activities of multiple neurons. This relationship might beeither a coherent co-modulation of mean firing rates (e.g. cell assembly [1] and population coding[2]) or a precise temporal correlation of spike events (e.g. synfire chain [3], correlative theory ofbrain function [4], dynamical cell assembly [5, 6], and synchronous oscillations [7]). To test thiscoding hypothesis experimentally, we need to record activities of multiple neurons simultaneouslyand then test the causal relationship between the change in spatio-temporal neuronal activities andthe sensory stimuli or behavioral events. Recently, neuroscientists have realized a strong need foradvanced statistical methods that facilitate the analysis of multi-neuronal spike data, which areoften multivariate, non-stationary, correlated, and in limited amount [8, 9].

1.1. Synchronous oscillatory spike activities

Temporally correlated (synchronous) spike activities between the two neurons have been studiedby the cross-correlation histogram (CCH) [10]. Initially, the CCH was applied to investigate theanatomy of synaptic connections [11, 12]. In the last two decades, however, there have been manyreports suggesting that the spike correlation is not a fixed characteristic between the two neuronsbut a dynamic and functional entity. It can be modulated qualitatively as well as quantitativelydepending on the change in the functional context of the stimulus or the behavioral task [7, 13, 14].Although many areas in the brain are known to show synchronous spike activities, the visual cortexhas been intensively investigated [15]. Interestingly, synchrony often appears as the phase lockingbetween the oscillatory spike firings among multiple neurons. Synchronous activity is thought toact as a dynamical mechanism for the perceptual grouping of the local features distributed overmultiple neurons [4, 7, 15]. Stimulus-induced synchronization was observed also in the thalamus(lateral geniculate nucleus, LGN) sending afferent inputs to the cortex [16–18].

1.2. Need for a novel analysis method of non-stationary synchronous spike events

In the previous studies [7, 11–14, 16], synchronous firings were analyzed by the CCH. In com-putation of the CCH, the number of correlated spike event is summed up over the trial duration.Therefore, we lose any information on the non-stationary properties of the synchrony. Even whenthe CCH has a significant peak, we wonder whether the synchrony would be stationary through-out the trial duration. Recently, we applied the unitary event analysis [19–21] to synchronousoscillations in the LGN [18, 22–24]. We found that, even during the presentation of a station-ary light spot stimulus, the synchrony was not stationary in time. The number of synchronousspike events showed a non-stationary modulation that could not be predicted by the change infiring rates.

Some statistical methods have been proposed to visualize and analyze non-stationary syn-chrony, e.g. Joint peri-stimulus time histogram (Joint-PSTH) [25], sliding window CCH [17, 26],unitary event analysis [19–21], and the bootstrap test with adaptive smoothing method [27].Contrary to the popularity of the CCH, however, the study of non-stationary synchronous spikeevents has still been very limited. This may be partly because the formulations of these analysismethods are not intuitively simple as the CCH. Especially, the significance test of synchronyshould be formulated by taking account of the modulation of the spike firing rates in the trialduration.

Copyright q 2007 John Wiley & Sons, Ltd. Statist. Med. 2007; 26:3976–3996DOI: 10.1002/sim

3978 H. ITO

In this paper, we report a case study of the significance test of synchronous spike activitiesin the cat LGN. Different statistical methods are applied to the same data set and their resultsare compared to demonstrate how each method practically works for the actual data. The mainbody of the analysis is spent for the unitary event analysis. In this method, the significance test isperformed against the null hypothesis of independent firings of the two neurons. The number ofaccidentally synchronous spike events is assumed to obey the Poisson distribution. Since actualneuronal spike patterns are not always Poisson like [28], validity of the significance test has beendiscussed for non-Poisson spike trains [20] and a non-parametric test based on bootstrap samplingwas proposed [29, 30]. In these studies, validity of the method was examined for the simulateddata by a renewal point process with no correlation between successive spike events. The maindifferences of our research from those previous studies are in: (1) testing the validity of the methodfor actual experimental data; and (2) testing the validity for oscillatory (periodic) spike trains withstrong correlation between successive spike events.

1.3. Outline of this monograph

Following the explanation of the experimental methods in Section 2, we introduce the data of ourcase study by showing the results of basic analyses of the spike trains. Correlation histogramsshow a tight synchronization between the oscillatory spike firings of the two neurons. The slidingwindow CCH suggests that the synchronous oscillation is non-stationary within the trial duration.The method of the unitary event analysis is explained in Section 3. According to its originalformulation, the significance of synchrony is tested against the null hypothesis of independentfirings obeying the Poisson distribution. We demonstrate how this method works for the significancetest of time-varying synchrony. We also discuss the problem of artifact (false positives) due tothe dense spike burst at the stimulus onset. In Section 4, the validity of the Poisson-based test isexamined by re-analyzing the same data by non-parametric test based on the bootstrap samplings.The statistics of the null hypothesis is estimated empirically from the data itself. We find thatthe two tests provide the null hypothesis of very different statistics. Although the mean numberof accidentally coincident events (CEs) is estimated similarly in either test, the distribution ofthe bootstrap samples is systematically broader than the Poisson distribution. The origin of thisdeviation is examined by inspecting individual bootstrap samples in Section 5. We find that thebroader distribution originates from a strong correlation between successive spike events in thestructured (periodic) spike trains. In Section 6, this conclusion is further confirmed by the simulationstudies of bootstrap samples based on a simple statistical model (the random phase model). Thesame data are examined by the other statistical tool based on the adaptive smoothing method[27] in Section 7. Although the two methods are based on rather different formulations, both areeffective in the significance test of non-stationary synchrony. Finally, we discuss our results inSection 8. We conclude that, when the spike trains are highly oscillatory (periodic), the significancetest based on the Poisson distribution gives inadequately low significance limit and has a dangerof producing false positives. Preliminary results have been published in abstract form [31].

In this monograph, we focus our attention on statistical methods for the significance test oftime-varying synchronous spike events. Rather than applying the same method to many spike data,we apply different analysis methods to single data to compare their results. However, our resultcan apply generally to any spike trains showing oscillatory firings. The global statistics of thecoincident spike events over all the recorded unit pairs in LGN and their physiological significanceare discussed elsewhere [18, 22–24].



2. BASIC STATISTICAL CHARACTERISTICS OF TEST DATA

2.1. Experimental methods and basic data analyses

Methods of experimental procedures and data analyses up to the correlation histogram are explainedelsewhere [32]. Briefly, the data were recorded from the LGN of anesthetized and paralyzedcats. Anesthesia was maintained using Isoflurane in a mixture of nitrous oxide and oxygen. Twoelectrodes (tetrodes) were inserted separately into the stainless guide tubing of the micro-drivethat allowed independent manipulation of two tetrodes separated by 500 �m. Neurons (units) werestimulated over their receptive fields by stationary light spots on a dark background. We typicallyran M = 20–40 trials of presentation of the identical stimulus. Multiunit activities recorded by eachtetrode were sorted to multiple single-unit activities by the custom-made spike-sorting software,which was based on the feature clustering and principal component analysis [33]. For each recordedunit, we computed PSTH (bin width 50ms), which represents the modulation of the trial-averagedfiring rate along the trial duration. To study rhythmic neuronal firing, we computed auto-correlationhistogram (ACH) averaged over trials with time lags of ±128ms (bin width 1ms). The powerspectrum of each ACH was calculated and the frequency of the spectrum peak was extracted inthe range of 30–160Hz. For the statistics of response synchronization between the two units, wecomputed trial-averaged CCH for all neuron pairs recorded simultaneously (time lags of ±128ms,bin width 1ms). All experimental procedures were in accordance with institutional and NIHguidelines.

2.2. Synchronous oscillatory spike activities

One unit pair was selected for this case study.‡ The two units were recorded by different electrodesseparated by 500 �m. Figure 1 shows the raster plots of spike trains of each unit over M = 20 trialsof the light spot stimulation. Each row shows a spike train of a single trial and the spike trainsof 20 trials are plotted from top to bottom. We find very little response variability over differenttrials. Both units show a dense cluster of spikes (spike burst) in their transient responses to suddenappearances of the light spots. Basic statistics of the spike trains are summarized in Figure 2. Asshown in the PSTH (Figure 2(a)), both units increase their firing rates during the interval whentwo light spots were presented at the two receptive fields simultaneously (ON response). Increasesin the firing rates are evoked also by sudden disappearances of the light spots (OFF response).Remember that the presentation of the light spots was stationary in time and did not introduce anycharacteristic temporal scale other than the total stimulus duration.

ACH of the spike train of each unit (Figure 2(b), Unit 0; Figure 2(d), Unit 2) and its power spectra(Figure 2(c), Unit 0; Figure 2(e), Unit 2) show oscillatory (periodic) spike firings with a frequencyof 96Hz only during ON response.§ CCH of the spike trains of the two units (Figure 2(f)) shows

‡The raw spike data of this case study are available at http://www.cc.kyoto-su.ac.jp/∼hiro/stat in med.html§Some units followed the vertical refresh of the video monitor (80Hz) on which spot stimulus was presented. Weidentified those artificially oscillatory activities through the following procedures. Firstly, we stimulated the units bya DC light source (ophthalmoscope) to test the artifact of the monitor refresh. Secondly, we checked the existenceof residual oscillatory components in the trial-shifted ACH. Since the onset of the data acquisition was synchronizedwith the phase of the monitor refresh at a resolution of 300 �s in every trial, artificial oscillatory structure was noteliminated even after the trial shifting. We confirmed that the oscillatory firings in the data of this case study werenot such a monitor artifact.


3980 H. ITO

Trials

Trials

Time (s)

Uni

t 0U

nit 2

0 1 2

Figure 1. Raster plot of the two LGN neurons (Unit 0 and Unit 2) used for this case study. Spike trainsof each unit are plotted for successive trials (M = 20). A horizontal bar along the time axis represents theinterval of stimulus presentation of the light spots. Both units show highly dense clusters of spikes (spike

burst) in their transient responses to a sudden appearance of light spots.

synchronization between the oscillatory firings at the two cells. The correlation peak at −1msdelay suggests that Unit 2 tends to fire 1ms before the firing of Unit 0.

2.3. Non-stationary change of synchronous oscillatory spike firings

As explained previously, any non-stationary characteristic in the trial duration was averaged outin the correlation histograms. The simplest way to visualize the non-stationary spiking statisticsis computing the sliding window correlation histogram [17, 26]. The temporal evolution of theoscillatory firings and their synchrony are shown, respectively, in the sliding window ACH (Plate1(a), Unit 0; Plate 1(b), Unit 2) and the sliding window CCH (Plate 1(c)). We compute thosecorrelation histograms only by using the spikes within a finite sampling window (size of 200ms).The sampling window is shifted by a 10ms step along the trial duration. The histogram is nor-malized by the spike count (for ACH) in the sampling window or the geometric mean of thespike counts of the two units (for CCH) and the amplitude at each delay value is represented ina color scale. We find that both the oscillatory spike patterns and their synchrony are not


Unit 0

Time (s)

0.2

0.30

0.00

0 1 2

Del

ay (

ms)

0

30

-30Unit 2

Time (s)

0.2

0 1 2

Del

ay (

ms)

0

30

-30

Unit 0 − Unit 2

Time (s)

0.2

0 1 2

Del

ay (

ms)

0

30

-30

0.40

0.00

0.40

0.00

(a) (b)

(c)

Plate 1. Non-stationary change of spiking statistics within the trial duration. Sliding window ACHsof the two units ((a) Unit 0; (b) Unit 2) and sliding window CCH between the two units (c). Eachcorrelation histogram is computed only by using the spikes within a finite sampling window (smallhorizontal bar, 200ms). The histogram is normalized by the spike count in the sampling windowand the amplitude at each delay value is represented in a color scale. The changes in the correlationhistograms are visualized by shifting the sampling window by 10ms step along the time axis. Athick horizontal bar represents the interval of stimulus presentation. The two vertical dashed lines

represent t = 450 and 660ms in the trial duration, respectively.

Copyright q 2007 John Wiley & Sons, Ltd. Statist. Med. 2007; 26(21)DOI: 10.1002/sim

Time (s)

0.1

Uni

t 0U

nit 2

10

Num

ber

of C

Es

raw-CEpredicted-CEPoisson sig. limitmean of trialshuffled samples

0 1 2

Plate 2. Unitary event analysis of synchronous oscillatory spike activities based on the Poisson significancetest. Top: raster plots of the two units (M = 20 trials); bottom: raw CE number (red line), predicted CEnumber (green line), and 99 per cent significance limit by the Poisson distribution (blue line) [bin widthb= 1ms]. Thick and thin horizontal bars, respectively, represent the interval of stimulus presentation andthe size of the sampling window (101ms). A red scale bar along the ordinate represents 10 counts. Whenthe number of the raw CEs exceeds the significance limit, all the CEs in the corresponding samplingwindow are assigned as the unitary events and highlighted by red circles in the raster plots. The meanCE number averaged over all possible trial-shuffled pairs (magenta line) agrees well with the predicted

CE number throughout the trial duration, except for the stimulus onset and offset.



207

-128 1280

Unit 0 Auto

Unit 2 Powerf = 96Hz

Unit 0- Unit 2-1

Cou

nt

Cou

nt

Pow

er

Pow

er

Unit 0 powerf = 96Hz

Unit 2 Auto

Frequency (Hz)Time (s)

2

0Unit 0

Unit 2Cou

ntC

ount

10 2

60

5

64 128 1920

Delay (ms)

Delay (ms)

-128 128

287

0

Delay (ms) Delay (ms)

0 128 192 -16 0 16

179

64

(a) (b) (c)

(d) (e) (f)

Figure 2. Summary of basic statistics of the spike trains in the raster plots (Figure 1). (a) PSTHs of thetwo neurons (bin width 50ms). In the stimulus duration (horizontal bar), two light spots were presentedat the two receptive fields (inset) simultaneously. Auto-correlation histogram (ACH, bin width 1ms) ofUnit 0 (b) and its power spectrum (c) during ON response show oscillatory spike firings with a frequencyof 96Hz. Oscillatory firings with the same frequency are observed also at Unit 2, ACH (d) and the powerspectrum (e). (f) Cross-correlation histogram (CCH, bin width 1ms) of the two neurons, which is a timehistogram of spike events of Unit 2 triggered by every spike firing of Unit 0. The peak in the CCH

suggests that Unit 2 tends to fire 1ms before the firing of Unit 0.

stationary throughout the trial duration. The sudden decrease of the spike synchrony starting aroundt = 550ms is associated with the change on firing patterns of the two units from oscillatory to non-oscillatory. However, there is no rapid change in their spike numbers around that epoch in the PSTHs(Figure 2(a)).

3. UNITARY EVENT ANALYSIS BASED ON THE POISSON DISTRIBUTION

Although the sliding window CCH is the simplest method to visualize non-stationarity in the spikesynchrony, we need a more advanced method for a significance test of the synchrony at eachtemporal domain. We apply the unitary event analysis to the unit pair of the case study.

3.1. Method of unitary event analysis

The original formulation of the unitary event analysis is given by the following procedures [19–21].(1) Divide the trial duration into multiple bins with a small interval b (1ms in this case study).

In each bin of each trial (totally M trials), place a value 0 when there is no spike event, and1 otherwise (Figure 3(a)).


3982 H. ITO

1*

*1

1

1

Coincident Event (CE)

* can be 1 or 0

-10 0 10

Delay (ms)

Cou

nt

Unit A-Unit B

Unit A

b

1

1

0

1

1

0

0

1

0

0

10110

01010

Unit B

Unit A

Unit B

Tw = 101×b

Unit A

Unit B

(a)

(c)

(b)

Figure 3. Methods of unitary event analysis are schematically depicted. (a) spike trains of the two units(Unit A and Unit B) are transformed to a matrix of event lists. The trial duration is subdivided intomultiple bins with a small interval b (1ms). In each bin of each trial, we place a value 0 when thereis no spike event, and 1 otherwise. (b) CCH of the two units. The time delay at the peak is detected ina precision of bin width b. This example has a peak at +1ms. (c) constellation of the coincident event(CE) corresponding to the correlation peak: Unit B fires 1ms after the firing of Unit A. Bins indicated byasterisks can take either 0 or 1 value. The number of CEs within a small sampling window (Tw = 101× b)is counted up over all the trials to obtain the raw CE number. Sliding the sampling window in steps of

b, a temporal modulation of the raw CE number is visualized.

(2) Select a specific time delay d in a precision of bin width b for the target joint spikeevent between the two units. In general, we choose a time delay at the peak of the CCH(Figure 3(b)). A more objective way for selection of both the bin width b and the delay dis discussed elsewhere [18].

(3) In the pair of spike trains in each trial, detect the joint spike events having exactly thistime delay (CEs) throughout the trial duration. In the example shown in Figure 3(b), sincethe CCH has a peak at 1ms, the corresponding CE has a constellation: Unit B fires 1msafter the firing of Unit A (Figure 3(c)). By definition, the states of other bins indicated byasterisks in Figure 3(c) can be either 0 or 1.

(4) At each bin position, place a small sampling window of duration Tw (101× b=101ms, extending 50 bins for both sides around the central bin). Count the number of theCEs within the sampling window and sum it up over M trials to obtain the raw CEnumber.

(5) Sliding the sampling window in steps of b, obtain the temporal modulation of the raw CEnumber in the trial duration.

The total spike number of each unit is also counted in the sampling window and summed up overall the trials. Dividing it by both the window size Tw and the number of trials M , we obtain thefiring rate (probability) of each unit in the sampling window.



The raw CE number should be tested for its statistical significance at each sampling window.In the null hypothesis of independent firings of the two units, the probability of an accidental CE,PCE, is given by the product of the firing rates of the two units in the sampling window. Whenthe number of CEs within the sampling window is small, we may assume that each CE occursindependently. This is the most critical assumption in this method. Then the number of accidentalCEs in the sampling window, n, obeys the binomial distribution

�(n)=(Tw − d

n

)PnCE(1 − PCE)Tw−d−n, (n = 0, . . . , Tw − d)

where d is the number of bins separating the two spikes in the CE (d = 1 in the example inFigure 3(c)). The number of CEs in the sampling window is bounded by Tw−d due to a geometricalconstraint. For the cumulative number of CEs over M trials, Tw−d in the above formula is replacedby (Tw − d)M . Except for cases of very low firing rate, the binomial distribution can be wellapproximated by the Poisson distribution

�(n) = �n

n! e−�

where the mean of the distribution (the predicted CE number) is given by � = PCE(Tw −d)M . Theraw CE number in the sampling window is judged as significant (unitary event) when it exceedsthe significance limit of the Poisson distribution (99 per cent limit in this study, that is, P<0.01).

3.2. Unitary event analysis of synchronous oscillatory activities—Poisson-based test

Since the CCH of the unit pair of the case study has a peak at −1ms delay (Figure 2(f)), wesearch for occurrences of the CEs with a constellation: Unit 2 fires 1ms before the firing of Unit 0.The results of the unitary event analysis are summarized in Plate 2. The two top panels in Plate2 are the raster plots of the two units. Below the raster plots, a red line represents the raw CEnumber counted over all the trials within the sliding sampling window. When the raw CE numberexceeds the 99 per cent significance limit of the Poisson distribution (blue line), all the CEs inthe corresponding sampling window are classified as unitary events, that is, a highly significantsynchronous event. Spike pairs forming the unitary events are enclosed by red circles in the rasterplots.

As expected from the result of the sliding CCH, the raw CE number shows a transient increaseduring the ON response. Although the firing rates of both units remain still high in the latephase of the ON response, the raw CE number shows a sudden decrease to the chance level (thepredicted CE number, green line in Plate 2). Interestingly, the firings of the two units in the OFFresponse are totally independent. The mean CE number averaged over all possible trial-shuffledpairs [M × (M − 1) = 380 samples] (magenta line) agrees well with the predicted CE numberthroughout the trial duration, except for the stimulus onset and offset. This agreement suggests acharacteristic that occurrences of the CEs are locked to the stimulus onset only loosely in eachtrial. Remember that the stimulus onset and the recording start were synchronized in every trial.If the occurrences of the CEs are be tightly locked to the stimulus onset, the CE number cannotbe reduced even by the trial shuffling. In later section, we will show that the phase of oscillatoryfirings of each unit itself is not locked to the stimulus onset [16, 18]. Although the precise timings


3984 H. ITO

of the occurrence of the CEs are not identical over different trials, there is a common interval inwhich the CEs occur more frequently.

3.3. False positives due to sudden change of spike density

The unitary events appearing immediately after the stimulus onset and offset are artifacts (falsepositives) due to sudden changes the spike density. Since the unitary event analysis assumes astationarity of spiking statistics within the sampling window, any violation of this assumption maylead to inadequate results. While the predicted CE number is a rate variable, the raw CE numberis a count variable. In case of inhomogeneous spike distribution within the sampling window, thedifferent characteristics of the two variables lead to the artifacts [21]. Consider a case of extremelyinhomogeneous spike distribution such that there is no spike in the left half of the sampling window(zero spike rate) and the spike rate in the right half is � for both units. Further assume that thefirings of the two units are independent. Averaging the rate within the sampling window, the meanspike rate of either unit is given by �/2. Then the predicted CE number is estimated as (�/2)2TwM .On the other hand, the CE number is 0 in the left half of the sampling window and �2(Tw/2)Min the right half. Since the raw CE number is the total count of the CEs in the sampling window,it is given not by the average but by the sum of the two numbers, �2(Tw/2)M . In this way, theraw CE number becomes twice as large as the predicted CE number even without any intrinsicsynchrony.

The different characteristics of the two variables become apparent also in the case of suddenchange in the spike density. As seen in the raster plots in Figure 1, the spike trains of both unitsshow a highly localized spike burst at the stimulus onset. Sliding the sampling window alongthe time axis, both the spike rates and the predicted CE number (green line in Plate 2) increasegradually as the window starts to include the spike burst. On the other hand, the raw CE number,which is a count variable, jumps to a larger value and keeps a similar value as long as the samplingwindow contains the spike burst (red line in Plate 2). Note that the mean CE number of the trial-shuffled samples takes a value similar to the raw CE number and also exceeds the significancelimit in this region. Since the occurrences of the spike burst are very tightly time locked to thestimulus onset, the trial shuffling cannot destroy this type of artificial synchrony.

These artifacts may be attenuated by adopting the sampling window of a smaller size. However,the size should be reasonably small for such a highly localized spike burst and it might be too smallfor a reasonable statistics in other regions of moderate firing rate. Even applying the samplingwindow of a half size (50ms), the artificial unitary events still exist at the ON transient and theCE number in the other region starts to show more statistical fluctuation and jiggling (figure notshown). When the spike trains show highly inhomogeneous spike distributions within the trialduration, the sampling window should not be fixed to the same size but should be variable to adaptto different spike densities.

4. UNITARY EVENT ANALYSIS BASED ON BOOTSTRAP SAMPLINGS

Since the spike trains of our case study show oscillatory firings and are not fitted by the Poissonprocess, we need to investigate the validity of the significance test based on the Poisson distribution.We execute non-parametric significance test by bootstrap sampling [27, 29, 30, 34] and examinehow far the Poisson-based test departs from the non-parametric test.



4.1. Method of bootstrap samplings

In the non-parametric approach, the distribution of the null hypothesis is predicted by the statisticsof the bootstrap samples generated from the original data itself.

(1) Generate a bootstrap sample by randomly shuffling the combination of M trial pairs of thespike trains of the two units, so that no spike train of Unit A makes a pair with that ofUnit B in the same trial.

(2) For each trial shuffled pair, count the CE number within the sampling window and sum itup over M pairs to get one sample of the CE number.

(3) Repeat steps 1–2 N times to get N samples of the CE number at each sampling window.Since the number of possible shuffled combinations becomes very large, we apply uniformMonte Carlo samplings.

(4) At each sampling window, 99 per cent significance limit was given by the 0.99 quantity ofthe N values.

Since any bootstrap sample is generated by the shuffling of the same set of spike trains as theoriginal data, both the PSTHs and the predicted CE number remain unchanged. Trial shufflingdestroys a fine temporal correlation between the spike events at the two units, which exists only inthe simultaneously recorded trial. This type of non-parametric test was known as the permutationtest in literature [34].

4.2. Comparison between bootstrap test and Poisson-based test

The results of the unitary event analysis based on the bootstrap test are summarized in Plate 3. Thesignificance limit by the bootstrap test (black line) is shown along with the limit by the Poissondistribution (blue line). The mean CE number averaged over N = 1000 bootstrap samples (magentaline) agrees with the predicted CE number (green line) except for the regions with sudden changesin spike density. Naturally, the mean CE number of the bootstrap samples is almost identical tothat of all possible trial-shuffled pairs in Plate 2. The significance test by bootstrap sampling issuperior to the test by the Poisson distribution in two respects. Firstly, the false positives at thesampling windows including sudden change in spike density are avoided. Since the significancelimit is estimated on the basis of the statistics of the CE numbers, it includes the characteristicof the count variable that is not predicted by the firing rates. Secondly, a non-parametric test caninclude the statistical characteristics of the spike trains not fitted by the Poisson distribution. Asseen in Plate 3, the variance of the bootstrap samples, that is the 99 per cent significance limit,shows systematic deviations from the Poisson distribution at some regions, although the means ofthe two distributions (green line and magenta line) are in good agreement. Generally, when thebootstrap significance limit is above (below) the limit by the Poisson distribution, firing patternsare more (less) variable than the Poisson process. It is interesting to see that the firing patternchanged its statistical characteristics in a non-stationary manner even in the response duration tothe stationary light spots.

4.3. Distribution of bootstrap samples

At two different sampling windows in the trial duration, we compare the distribution of the CEnumbers over N = 1000 bootstrap samples with the Poisson distribution quantitatively. In the firstsampling window at t = 450ms (left dashed line in Plates 1 and 3), the spiking pattern is highly


3986 H. ITO

oscillatory. The histogram of the bootstrap samples (Plate 4(a)) shows a significantly broaderdistribution than the Poisson distribution with a mean of the predicted CE number (red line).However, the two distributions have almost the same mean value. The deviation is not due toa finite sample size, because the histogram over N = 10 000 samples (Plate 4(b)) also shows asimilar deviation. The 99 per cent significance limit by the bootstrap distribution (blue dashed line)is significantly larger than the 99 per cent limit of the Poisson distribution (red dashed line). If wewould apply the significance limit by the Poisson distribution, the ratio of the bootstrap samplesexceeding this limit (false positives) to the total sample size is P = 0.036, which is much morethan P = 0.01. On the other hand, in the second sampling window at t = 660ms (right dashed linein Plates 1 and 3), the spiking pattern is non-oscillatory. The histograms of the bootstrap samples(Plate 4(c), N = 1000 samples; Plate 4(d), N = 10 000 samples) are well fitted by the Poissondistribution and the two distributions provide similar significance limits.

4.4. Significance of deviation from the Poisson distribution

The magnitude of false positives, that is, the ratio of the bootstrap samples exceeding the Poissonlimit to the total sample size, itself is a statistical variable. We examine the significance of thisdeviation over many sets of bootstrap samples. One bootstrap sample is given by one shuffled combi-nation of M = 20 trial pairs of spike trains. The distribution of the CE numbers in Plate 4(a), whichwas computed by one set of N = 1000 bootstrap samples, gives one value for the magnitude of falsepositives. We generate 1000 different sets of bootstrap samples to get 1000 values for this statisticalvariable. Figure 4(a) shows the distribution of the rate of bootstrap samples exceeding 91 CE counts(P<0.0104 of the Poisson distribution in Plate 4(a)). The distribution is well approximated by anormal distribution with mean 0.031 ± 0.006. The same analysis is carried out by increasing thesize of one set of bootstrap samples to N = 10 000 (see Plate 4(b)). The distribution over 1000sets in Figure 4(b) becomes narrower with mean 0.031± 0.004. Therefore, the magnitude of falsepositives is about three times larger than the estimate of the Poisson distribution.

When the spike trains are not oscillatory in the sampling window at t = 660ms, the distributionof the bootstrap samples is close to the Poisson distribution (Plate 4(c) and (d)). The significance of

Cou

nt

80

0

40

P

0.01

mean: 0.031 STD: 0.006

0.050.03

N=1000, 1000 set

0

P

0.01 0.050.03

Cou

nt

120

60

N=10000, 1000 set

mean: 0.031 STD: 0.004

(a) (b)

Figure 4. Histograms of the magnitude of false positives—that is, the rate of bootstrap samples exceedingthe P = 0.01 limit of the Poisson distribution to the total sample size, N ((a) N = 1000 and (b) N = 10 000).We generated 1000 sets of the bootstrap samples in both histograms. Distribution of the bootstrap samples

has a significant departure from the Poisson distribution.


Time (s)

0.1

Uni

t 0U

nit 2

10

Num

ber

of C

Es

raw-CEpredicted-CEPoisson sig. limit

mean of bootstrapsamplesbootstrap sig. limit

0 1 2

Plate 3. Unitary event analysis of synchronous oscillatory spike activities based on the bootstrap significancetest. Top: raster plots of the two units (20 trials); bottom: raw CE number (red line), predicted CEnumber (green line), and 99 per cent significance limit by the Poisson distribution (blue line), 99 percent significance limit by the bootstrap samples (black line) and mean CE number of N = 1000 bootstrapsamples (magenta line). Scale bars and red circles in the raster plots are used in the same way as in Plate 2.

The two vertical dashed lines represent t = 450 and 660ms in the trial duration, respectively.


Number of CEs

Prob

abili

ty

Prob

abili

ty0. 0.

0.1 0.1

t=450ms, N=1000 t=450ms, N=10000

t=660ms, N=1000 t=660ms, N=10000

20 40 60 80 100

Number of CEs

20 40 60 80 100

Number of CEs

Prob

abili

ty

0.

0.1

20 40 60 80 100

Number of CEs

Prob

abili

ty

0.

0.1

20 40 60 80 100

(a) (b)

(c) (d)

Plate 4. Deviation of the bootstrap samples from the Poisson distribution. Histograms of the CE numbersover bootstrap samples ((a) N = 1000 samples and (b) N = 10 000 samples) and the Poisson distributionwith a mean of the predicted CE number (red line). In the sampling window at t = 450ms, the spikingpattern is highly oscillatory. The 99 per cent significance limit by the bootstrap distribution (blue dashedline) is significantly larger than the limit by the Poisson distribution (red dashed line). Histograms ofthe bootstrap samples ((c) N = 1000 samples and (d) N = 10 000 samples) and the Poisson distribution(red line) in the sampling window at t = 660ms, where the spiking pattern is not oscillatory. The two

distributions provide similar 99 per cent significance limit.


1-

17

2-

210

3-

34

4-

48

5-

55

6-

67

7-

79

8-

85

9-

98

10-1

07

11-1

16

12-1

26

13-1

38

14-1

410

15-1

57

16-1

66

17-1

75

18-1

89

19-1

99

20-2

06

Tim

e (m

s)

t=45

0ms,

TE

ST1-

5 0

2- 1

53-

1810

4-13

55-

20 4

6- 8

67-

6 2

8- 3

79-

10 7

10-

9 4

11-1

4 4

12-1

7 7

13-

7 8

14-1

1 3

15-1

9 5

16-

4 8

17-

2 2

18-1

2 1

19-1

5 6

20-1

610

t=45

0ms,

MA

X1-

5 0

2- 7

33-

11 2

4- 3

05-

16 3

6-10

27-

17 0

8-20

69-

2 0

10-

4 5

11-1

2 1

12-1

5 2

13-1

9 2

14-

8 3

15-1

8 0

16-

6 1

17-1

3 0

18-

9 0

19-

1 1

20-1

4 2

t=45

0ms,

MIN

t=45

0ms,

TE

STt=

450m

s, M

AX

t=45

0ms,

MIN

05

10

Count 08

Num

ber

of C

Es

Num

ber

of C

Es

Num

ber

of C

Es

400

500

Tim

e (m

s)40

050

0T

ime

(ms)

400

500

05

10

Count

08

05

10

Count

08

(a)

(b)

(c)

(e)

(d)

(f)

Plate5.Rasterplotsof

spiketrains

inthesamplingwindow(sizeof

101ms)att=

450msforthreedifferentb

ootstrap

samples:

(a)testdata;(b)

bootstrapsamplegiving

themaxim

umCEnumber;and(c)bootstrapsamplegiving

theminim

umCEnumber.

The

upperspike

trainineach

rowrepresentsthefirings

ofUnit0

forsuccessivetrials(M

=20).The

lowerspiketrainrepresents

firings

ofUnit2

intheshuffledtrialspecifiedby

thesecond

numberin

theleftmargin.The

jointspike

eventsform

ingthetarget

CEsarehighlig

hted

inred.

Foreach

trialpair,the

numberof

CEswith

inthesamplingwindow

isshow

nin

therightmargin.

Histogram

sof

theCEnumbers

over

20trialpairsin

each

sample:

(d)test

data;(e)bootstrapsamplegiving

themaxim

umCEnumber;and(f)bootstrapsamplegiving

theminim

umCEnumber.


1- 1 1 2- 2 6 3- 3 1 4- 4 2 5- 5 0 6- 6 3 7- 7 2 8- 8 3 9- 9 2

10-10 311-11 512-12 613-13 314-14 415-15 116-16 217-17 318-18 519-19 320-20 4

Time (ms)

t=660ms, TEST

Number of CEs

0 5 100

10t=660ms, MAX t=660ms, MIN

t=660ms, TEST

Cou

nt

Number of CEs

0 50

10

10

Cou

nt

610 710

0

10

Cou

nt

(a)

Number of CEs

0 5 10

(b)

(d)(c)

Plate 6. (a) Raster plot of the test data in the sampling window (size of 101ms) att = 660ms, where the spiking pattern is not oscillatory. Histograms of the CE number over20 trial pairs; (b) test data; (c) bootstrap sample giving the maximum CE number; and

(d) bootstrap sample giving the minimum CE number.



this agreement is also tested by 1000 sets of bootstrap samples of size N = 10 000. The distributionof the rate of bootstrap samples exceeding 73 CE counts (P<0.0108 of the Poisson distributionin Figure 4(d)) is again well approximated by a normal distribution with mean 0.008 ± 0.003(figure not shown). This result suggests that in this sampling window the bootstrap distribution issignificantly close to the Poisson distribution and even has a smaller variance.

5. ORIGIN OF LARGE VARIANCE IN BOOTSTRAP SAMPLES

5.1. Inspection of coincident events in bootstrap samples

The origin of a large variance in the distribution of the bootstrap samples is further examined byinspecting the CEs in the spike trains of individual bootstrap samples. Three raster plots in Plate 5show the spike trains of different bootstrap samples in the sampling window of 101ms. The upperspike train in each row represents the firings of Unit 0 for successive trials (M = 20). The lowerspike train represents the firings of Unit 2 in the shuffled trial specified by the second number inthe left margin. Different bootstrap samples correspond to different shuffle combinations of thetrial numbers. For each trial pair, the spike pairs forming the target CEs are highlighted in red andtheir number in the sampling window is shown in the right margin.

In case of the sampling window at t = 450ms, the raster plot of the test data shows many CEsassociated with synchronous oscillatory firings (Plate 5(a)). The number of CEs counted up over alltrials is 142, which is far above the bootstrap significance limit, 98. As shown in the histogram ofthe CE numbers over 20 trial pairs (Plate 5(d)), any trial pair of the test data consistently containsa large number of CEs. On the other hand, Plate 5(b) shows the spike train pairs of the bootstrapsample giving the maximum CE number in the distribution of Plate 4(a) (N = 1000 samples).Since the spike trains of both units show periodic firings with the same frequency, once the spiketrains from different trials make an accidental synchrony at one oscillation cycle, synchrony islikely to occur also at other cycles. That is, the occurrences of successive CEs are not independentin the sampling window. Such a large number of accidental CEs seldom occur in the Poissonspike train. Although many trial pairs have a large CE number, some pairs fail to get accidentalsynchrony (Plate 5(e)). Only the combination of the same trials, which is realized only in the testdata, can provide a large CE number systematically for all the trial pairs. Vice versa, once thespike trains from different trials fail to get an accidental synchrony at one oscillation cycle, theyhave little chance of synchrony also at the other cycles. The bootstrap sample giving the minimumCE number in the distribution of Plate 4(a) has many trial pairs of such a systematic asynchrony(Plate 5(c) and (f)). Many trial pairs have not even any accidental CE in the sampling window,which is very unlikely in case of the Poisson spike train. We conclude that the deviation from thePoisson distribution originates from the increased variance of the bootstrap samples due to thecorrelation between the successive CEs in the structured (oscillatory) spike trains.

In the sampling window at t = 660ms, the raster plot of the test data shows only accidentalCEs (Plate 6(a)). The total number of CEs is close to the predicted CE number and well belowthe bootstrap significance limit (see Plate 3). The histogram of the CE numbers over 20 trialpairs suggests a simple stochastic distribution around the mean (Plate 6(b)). The histograms inPlate 6(c) and (d), respectively, correspond to the bootstrap samples giving the maximum CEnumber and the minimum CE number in the distribution of Plate 4(c). Contrary to a systematicbias in case of oscillatory firings, they may be regarded as just a sampling inhomogeneity. Even


3988 H. ITO

in the sample giving the minimum CE number (Plate 6(d)), any trial pair is likely to have at leasta single accidental CE within the sampling window.

6. SIMULATION OF BOOTSTRAP SAMPLES BY THE RANDOM PHASE MODEL

In the previous section, we found that the correlation between successive CEs leads to a largevariance in the bootstrap samples. Here we try to abstract the basic mechanism by simulating thebootstrap samples with simple statistical model (the random phase model).

6.1. Computing the phase difference between two oscillatory spike trains

In the sampling window at t = 450ms, the spike trains of both units are highly oscillatory, withthe same frequency as seen in the ACHs (Figure 2(b) and (d)). Therefore, we can define a phaseof oscillation for each spike train and compute a phase difference between two spike trains of anytrial-shuffled combination. In a general formulation, the phase difference between the spike trainof Unit A in the mth trial and that of Unit B in the m’th trial can be computed by the followingprocedures (m,m′ = 1, . . . , M, M = 20):

(1) The spike train of Unit i (i =A or B) in the mth trial is transformed to a continuous spikedensity function D(m)

i (t) by replacing a �-function to a Gaussian kernel (� = 0.75ms) ateach spike event [35, 36].

(2) Compute the overlap function F (m,m′)(�) by averaging the product D(m)A (t)D(m′)

B (t + �)within the sampling window for each of the different temporal shifts �. We limit the rangeof � to −5, −4, . . . , 4, 5 (ms) covering the oscillation period (∼ 10ms, 96Hz).

(3) Find the value of � giving the peak of F (m,m′)(�). We assign this value as the phase difference�∗ between the two oscillatory spike trains.

Plots in Plate 7 summarize the results of two different shuffled trial combinations for m and m′.In case of in-phase oscillations (Plate 7(a) and (b)), the two spike trains have a large number ofaccidental CEs with a delay of −1ms (spikes in red) and the overlap function F (m,m′)(�) between

the spike density functions D(m)0 (t) and D(m′)

2 (t) takes a peak at �∗ = −1ms. On the other hand,in case of out-of-phase oscillations (Plate 7(c) and (d)), there is no accidental CE and the peak ofthe overlap function occurs at �∗ = 4ms.

6.2. Characteristic curve between the mean CE number and phase difference

For every possible shuffled combination of the two trials m and m′ [M × (M − 1) = 380 cases,M = 20], we computed the number of CEs and the phase difference �∗ in the sampling window att = 450ms. Since the phase of oscillation in each trial is not locked to the stimulus onset, the phasedifferences �∗ between the trial-shuffled spike trains are uniformly distributed over all possiblevalues (Figure 5, inset). The mean CE number NCE(�∗) was computed by averaging the samplesbelonging to a class of each phase difference �∗. As expected from the discussion in the previoussection, the characteristic curve NCE(�∗) shows a definite tendency: the closer the phase difference�∗ is to −1ms, the larger the number of CEs NCE(�∗) is.


3-18

10

UNIT 0

UNIT 2

UNIT 2

UNIT 0

Time (ms) Phase τ (ms)400 500 -5 0 5

Am

plitu

de

t * = -1

1- 5

0

Am

plitu

de(a) (b)

UNIT 0

UNIT 2

UNIT 2

UNIT 0

Time (ms)400 500

(c) Phase τ (ms)-5 0 5

(d)

Plate 7. Computation of the phase difference between spike trains of the two units in a shuffled trial combi-nation: two examples ((a) and (b) in-phase case; (c) and (d) out-of-phase case). (a) and (c) spike trains of the

two units (Unit 0 and Unit 2) and their spike density functions D(m)0 (t) and D(m′)

2 (t) in the sampling windowat t = 450ms. Pair of numbers m–m′ above the left corner represent the shuffled trial combination of themth trial of Unit 0 and the m′th trial of Unit 2. Every target CE (delay −1ms) is highlighted in red and theircount in the sampling window is shown at the right margin. Each spike train is transformed to a continuousspike density function by replacing a �-function into a Gaussian kernel (�= 0.75ms) at each spike event.(b) and (d) plot of the overlap function F (m,m′)(�) between the two spike density functions as a functionof the temporal shift � over the range (−5,−4, . . . , 4, 5ms). The peak of the overlap function (red circle)

occurs at �∗ = −1ms ((b) in-phase case) and �∗ = 4ms ((d) out-of-phase case), respectively.


Number of CEs

Prob

abili

ty

7030 50 90 110

0.00

0.05t=450ms, M=20, N=10000

Plate 8. Distribution of the CE numbers over N = 10 000 bootstrap samples simulated by the randomphase model. The distribution has a larger variance than the Poisson distribution of the same mean(� = 69.6, red line). The red vertical dashed line represents the significance limit of the Poisson distribution(P = 0.01). The distribution of the simulated samples is well fitted to the theoretical estimate—that is, a

normal distribution with mean � = 69.6 and variance V = 153.4 (green line).

Prob

abili

ty

x0 10 20

0.2

0.0

0.1

Plate 9. Probability distribution of the CE number in the spike trains of a shuffled trial pair,Pmean(x), which is obtained by uniformly averaging the Poisson distributions of different mean val-ues PPoisson{x; � = NCE(�)} over all possible phase �. Pmean(x) has a larger variance (V = 7.67)than the Poisson distribution with the same mean (red line). A blue dashed line represents the mean

of the distributions, �= 3.48.



-5 0 50

5

10

0

50

5−5 0

Cou

nt

Phase τ (ms)

Phase τ (ms)

Mea

n N

umbe

r of

CE

s N

CE(τ

)⎯

Figure 5. Plot of the mean CE number NCE(�) in the sampling window at t = 450ms (size 101ms),which was computed by averaging the samples belonging to a class of each phase difference �. Thevertical bar at each data point represents the standard deviation around the mean. Inset: the distributionof the phase difference � between the spike trains of the two units over all possible trial-shuffled

combinations [M × (M − 1)= 380 samples, M = 20].

6.3. Simulating bootstrap samples by the random phase model

We simulate the bootstrap samples by using a simple statistical model based on the characteristiccurve NCE(�) (the random phase model).

(1) Choose one phase � uniform randomly in the range of −5, −4, . . . , 4, 5 (ms).(2) Generate a random number from the Poisson distribution with a mean NCE(�), PPoisson{x; � =

NCE(�)}, (x = 0, 1, 2, . . .). This is used as one sample CE number.(3) Repeat 1–2 for M trials (M = 20) to get M samples of CE numbers. The sum of these

numbers gives one bootstrap sample of the total CE number.(4) Repeat 1–3 for N = 10 000 times, to get 10 000 bootstrap samples of the total CE number.

The histogram of the simulated bootstrap samples (Plate 8) actually reproduces a distributionhaving a larger variance than the Poisson distribution with the same mean (red line), PPoisson{x; � = 69.6}.

6.4. Theoretical estimate of the distribution of bootstrap samples

Since the phase difference � is sampled uniform randomly, the above procedures 1–3 are equivalentto computing the sum of M random numbers generated from the mean probability distribution,which is obtained by averaging the Poisson distributions of different mean values over all possible �,

Pmean(x)= 1

11

5∑�= −5

PPoisson{x; � = NCE(�)} (x = 0, 1, . . .)

The mean of Pmean(x) is given by � = 111

∑5�=−5 NCE(�) = 3.48. As seen in the graphs of the

two distributions (Plate 9), Pmean(x) has a larger variance (V = 7.67) than PPoisson{x; � = 3.48}


3990 H. ITO

irrespective of the same mean �. Only when the characteristic curve NCE(�) is homogeneous overall � could Pmean(x) become identical to PPoisson{x; �}.

Each simulated bootstrap sample is the sum of M independent variables obeying Pmean(x) (mean� and variance V ). Its distribution is expected to be well approximated by a normal distributionwith mean �M and variance VM when M is large enough (law of large numbers). Actually,in case of M = 20, the histogram of the simulated bootstrap samples in Plate 8 is well fittedby a normal distribution (green line) with mean (3.48M = 69.6) and variance (7.67M = 153.4).With such a large mean, the Poisson distribution PPoisson{x; � = 69.6} (red line) can also beapproximated by a normal distribution (mean 69.6 and variance 69.6). The red dashed line inPlate 8 represents the P = 0.01 limit of the Poisson distribution, which is located at 2.32 times ofthe standard deviation from the center. On the other hand, the standard deviation of the distributionof the bootstrap samples is 1.48 times larger than that of Poisson distribution. Then the rate ofthe false positives exceeding the limit can be estimated as the area of the normal distributionlarger than 2.32/1.48= 1.56 times of the standard deviation from the center. That is, P = 0.0594.This theoretical value overestimates the empirical value obtained by the trial-shuffled bootstrapsamplings of actual spike data, 0.031 ± 0.004. We consider that this deviation can originate fromthree factors that are not taken into account in the random phase model: (1) the actual distributionof the phases is not completely uniform (Figure 5, inset); (2) the distribution of the CE numbersin a class of each phase difference may not obey the Poisson distribution; and (3) the samplingsof M phases are not completely independent in the actual trial-shuffled samples, but dependent sothat the spike train of each trial appears only once in any single bootstrap sample.

We conclude that, although there is some quantitative disagreement, the simulation study bythe random phase model abstracts the basic mechanism leading to a broader distribution of thebootstrap samples. Non-Poisson characteristic originates from the addition of Poisson variableswith inhomogeneous mean values. It is noted that an increase in the number of trials M doesnot lead to a convergence of the two distributions and a decrease in the rate of false positives. Itsimply leads to more convergence of the bootstrap distribution to a normal distribution with mean�M and variance VM and the rate of false positives remains constant.

7. BOOTSTRAP SIGNIFICANCE TEST WITH THE ADAPTIVE SMOOTHING METHOD

The time-varying spike synchrony can be studied by various statistical methods. Ventura et al.[27] showed that a combination of an adaptive smoothing method and the bootstrap significancetest works more effectively than a bin-based approach by Joint-PSTH [25]. We analyze the data ofour case study by their method and compare the result with the result by unitary event analysis.

7.1. Smoothed estimate of time-varying synchrony

In the formulation by Ventura et al. [27], a measure of the correlated spike activity is defined by

��(t) = PAB(t, t + �)

PA(t)PB(t + �)

where PA(t) and PB(t + �), respectively, represent the probability of spike firing of Unit A attime t and the probability of spike firing of Unit B at time t + �. PAB(t, t + �) is the probabilityof the CE with a delay �—that is, the joint spike events: Unit A fires at time t and Unit B firesat t + �. The ratio ��(t) would be equal to 1 for all t and all �, if the firings of the two units


Spik

e R

ate

(Hz)

500

0

250

Time (s)

0 21

Spik

e R

ate

(Hz)

500

0

250

CE

Rat

e (H

z)

100

0

50

150

Unit 0 Unit 2

(a)

(c)

(b)Time (s)0 21

Time (s)0 21

Plate 10. PSTHs of the two units ((a) Unit 0 and (b) Unit 2) and the number of the CEs (c) [20 trials, binwidth 10ms]. The red line in each plot represents the smoothed estimates of each histogram by BARS.

Time (s)0 21

Time (s)0 21

0.

1.

4.

2.

3.

5.

0.

30.

60.

(a) (b)

Plate 11. Measure of synchrony �̂−1(t) (red line) in two different scales ((a) global view and (b) magnifiedview). The two blue dashed lines in each plot represent the 0.975 significance limit and the 0.025significance limit obtained by 1000 bootstrap samples. The mean of �̂−1(t) over all bootstrap samples(green dot-dashed line) stays around 1, the value of independent firings, except for the regions of ON andOFF transients. Excursion G test of �̂−1(t) beyond the significance limit is the area of a hatched region in (b).



were independent. Ventura et al. assumed that all those probabilities should change continuouslyin time. Thus, they computed a smoothed estimate of ��(t), �̂�(t) using the smoothed estimates ofprobabilities P̂A(t), P̂B(t + �), and P̂AB(t, t + �).

In evaluation of �̂�(t), we compute the smoothed estimates of the PSTHs of the two units and thenumber of CEs. The smoothing procedure is carried out by the algorithm based on bayesian adaptiveregression splines (BARS) [37]. Plate 10 shows the PSTHs of the two units (a) Unit 0; (b) Unit (2)averaged over 20 trials (bin width 10ms). The smoothed estimates (red lines) fit those histogramsfairly well, except for the transient spike burst responses at the stimulus onset. Especially, thechange in spike counts of Unit 0 is too sudden and too large to be fitted by a smoothed curve.In evaluation of the smoothed estimate of the CE number, at first, the occurrences of CEs aredetected at each original bin (b= 1ms) in each trial. Then we count up the number of CEs overall the trials within a coarser bin (10ms) to get the histogram (Plate 10(c)). Again, the smoothedestimate (red line) well fits the global modulation profile except for the sharp transient increase.

Since the binary variable (0 or 1) of both the spike event and the CE was originally defined in1ms bin, the probability P̂(t) can be obtained by normalizing the count of the histogram in Hz by1000. Since a shift by delay � =−1ms makes almost no change in the PSTH with a 10ms bin, weuse P̂B(t) in place of P̂B(t + �) in the estimation of �̂�(t). Note that we have correctly estimatedP̂AB(t, t +�) with the delay � =−1ms and with a precision of 1ms. �̂�(t) shows a large departurefrom 1 at the three temporal domains (Plate 11(a), red line): ON transient, ON response, and OFFtransient. In the magnified view (Plate 11(b), red line), �̂�(t) keeps departing from 1 during theON response and stays close to 1 during the OFF response, which is consistent with a departureof the raw CE number from the predicted CE number in the unitary event analysis (see Plate 2).

7.2. Estimation of the significance limit by bootstrap samplings

The significance limit for �̂�(t) in the null hypothesis of independent firings is estimated again bythe bootstrap samplings of trial-shuffled spike trains [27].

(1) Use the same N = 1000 bootstrap samples that were generated and used in the unitary eventanalysis.

(2) For each sample, compute �̂�(t) using the same P̂A(t) and P̂B(t), because in any trial-shuffled sample the PSTHs remain the same as the test data. We get 1000 estimates of�̂�(t).

(3) For each time t , define 95 per cent significance limits by selecting the 0.025 (the 25thsmallest value) and 0.975 (the 25th largest value) quantities of 1000 values.

The significance limits are plotted in Plate 11 (blue dashed lines). The mean of 1000 estimates of�̂�(t) (green dot-dashed line) stays around 1, the value of independence, except for the region ofON and OFF transients.

7.3. Significance test of excess synchrony

The magnitude of excess synchrony is quantified by G test, which is the largest area of any con-tiguous portion of �̂�(t) that exceeds the significance limit (hatched area in Plate 11(b)). Thestatistical significance of G test is evaluated by P-value, estimated again by the bootstrap sam-ples [27]. We compute the magnitude of excess synchrony in �̂�(t) for each bootstrap sample,


3992 H. ITO

Gboot. The bootstrap P-value of G test is estimated by the ratio of the bootstrap samples havingGboot>G test to the total sample size. We obtain P<0.01 for the hatched area in Plate 11(b). How-ever, �̂�(t)’s of all the bootstrap samples having Gboot>G test exceed the significance limit only inthe transient region at either ON or OFF onset. Since �̂�(t) takes a very large value in those regions(Plate 11(a)), even a small departure from the significance limit leads to a large Gboot. If we excludethose transient regions from the significance test, G test is larger than any Gboot, that is, P = 0.Although both G test and Gboot allow the contiguous portion of �̂�(t) below the 0.025 significancelimit, our data seldom show such less synchrony.

We conclude that the significance test by Ventura et al. works for our data as effectively asthe unitary event analysis. The two methods provide the significance tests at different temporalscales. The significance of synchrony is tested locally at each sampling window in the unitaryevent analysis. On the other hand, Ventura et al. considered the areas of any contiguous portionof �̂�(t) exceeding the significance limit and selected their global maximum over the entire trialduration as a statistic of synchrony. A smoothed statistic can avoid false positives due to statisticalfluctuations in the bin-based test. However, there is a danger that the significance of the statistic ofthe test data might be tested against any artificial synchrony appearing at the transient regions in thebootstrap samples. To avoid this failure, we need to inspect any inhomogeneous spike distributionin the data and limit the significance test to the relevant temporal domain.

8. DISCUSSION

We introduced spike data of the two neurons recorded simultaneously from the cat LGN. Thedata show various non-stationary characteristics not fitted by the Poisson spike train. Spike firingsof both neurons are highly periodic (oscillatory) and they are tightly synchronized in a precisionof 1ms. The sliding window correlation histograms suggest that both the oscillations and thesynchrony appear only transiently in the response duration even to a temporally stationary visualstimulus (light spots). Furthermore, the neurons show very transient (phasic) responses, highlydense spike bursts, at both the onset and the offset of the stimulus.

We reported case studies of applications of two statistical methods for the significance testof the time-varying spike synchrony in these test data. At first, we applied the unitary eventanalysis [19–21]. The significance limit for the coincident spike events estimated by the Poissondistribution was compared with the limit given by the non-parametric test based on the bootstrapsamplings (trial shuffling, permutation test). The bootstrap test performs superior to the Poissontest for our non-stationary data: (1) avoids false positives due to the sudden change in spikedensity; and (2) takes into account the non-stationary change of the spiking pattern at differentsampling windows. When the spike trains are highly periodic, the distribution of the numberof accidental CEs over bootstrap samples has a systematically larger variance than the Poissondistribution. By inspecting individual bootstrap samples, we found that a large variance originatesfrom the correlation between the successive CEs in the structured (oscillatory) spike trains. Thebasic mechanism of this phenomenon was abstracted by the simulation study of a simple statisticalmodel (the random phase model). The non-Poisson characteristic of the bootstrap samples originatesfrom the addition of the Poisson variables with inhomogeneous mean values. We conclude that,in the unitary event analysis of synchronous oscillatory activities, the Poisson-based test may givean inadequately low significance limit leading to false positives.



The significance of the time-varying synchrony in the test data was tested also by another statis-tical method by Ventura et al. [27], which is based on the adaptive smoothing method and thebootstrap significance test. We found that the significance test by Ventura et al. works for our dataas effectively as the unitary event analysis. Use of smoothed variables can avoid false positivesdue to statistical fluctuations in the bin-based test of the unitary event analysis. The two methodsprovide the significance tests at different temporal scales. The significance of synchrony is testedlocally at each sampling window in the unitary event analysis. On the other hand, in the method byVentura et al., the significance is tested for a single statistic quantifying the global departure fromthe independent firings over the entire spike trains. Therefore, in case of highly inhomogeneousspike distributions as our test data, there is a potential failure that the significance of the statisticmight be tested against any artificial synchrony appearing at the transient regions in the bootstrapsamples. Our case studies provided various general remarks to the statistical analysis of non-stationary synchronous spike activities. Since emergences of synchronous oscillations and theirfunctional role have been attracting attention, great care should be taken to test their statisticalsignificance.

Since the main aim of this monograph is to discuss the problems in the statistical analysis ofnon-stationary spike synchrony, we limit our analysis to a case study of a single LGN neuron pair.Global statistics of synchrony over all the recorded unit pairs is reported elsewhere [18, 23]. Theseanalyses concluded that a large portion of the unit pairs also show non-stationary modulation ofthe number of CEs. Also, broader distribution of the bootstrap samples is commonly observed inother unit pairs showing highly synchronous oscillations.

8.1. Robustness of the significance test to non-Poisson and non-stationary spike trains

The original framework of the unitary event analysis assumed three properties of the spike data[19–21]: (1) Poisson statistics in spike firings; (2) stationarity of spiking characteristics (densityand pattern) within the sampling window; and (3) stationarity of spike train across different trials.Since these characteristics are not always satisfied in actual physiological data, we need to examinehow sensitive the significance test is to the violations of those assumptions.

Gruen et al. estimated the percentage of false positives by applying the Poisson-basedtest to independent spike trains simulated by a gamma process [20]. By variation of a sin-gle shape parameter �, a gamma process can vary the spike train structure from bursty (�<1,large variance of inter-spike intervals (ISI), CV>1) to regular (�>1, small variance, CV<1).For � = 1, the spike train is Poissonian (CV= 1). They concluded that the percentage of falsepositives was less than 2 per cent of the total samples (for a significance level P = 0.01)over a wide range of parameters (both the firing rate and the shape parameter �). For regularspike trains with �>1, the distribution of the number of accidental CEs was narrower than thePoisson distribution having the same mean. Thus, the significance limit by the Poisson-based testyielded a conservative estimate [38]. In their simulation study, successive CEs occurred inde-pendently within the sampling window. We have shown that, when each spike train is stronglyoscillatory, successive CEs can be strongly correlated even in trial-shuffled spike trains. The dis-tribution of the bootstrap samples shows a large departure from the Poisson distribution and thepercentage of false positives becomes more than 3 per cent (for a significance level P = 0.01).Recently, Nawrot et al. [39] showed that their physiological data had a weak negative serial ISIscorrelation, which should be zero when the spike trains would be generated by a renewal pointprocess.


3994 H. ITO

Gruen et al. discussed non-stationary spiking statistics in the sampling window [21]. Theymodeled the simplest case in which the two spike trains changed their spike densities in a stepwisemanner at the middle of the sampling window (see discussion in Section 3). Estimating a percentageof false positives by varying the magnitude of the density change, they concluded that the changeshould be very large to lead to false positives. However, as known from our case study and theglobal statistics over other unit pairs [18], neurons in the LGN often show a large and sharp changeof spike density at the stimulus onset and offset. The significance test based on the rate-basedpredictor could not work adequately for such highly inhomogeneous spike trains. The bootstraptest based on the counts of the CEs should work more adequately.

The effect of a violation of stationarity in spike trains across trials was also investigated byGruen et al. [40]. This type of non-stationarity has a high chance to lead to false positives, becausethe trial-averaged spike rate does not reflect the correct statistics of spike trains in individualtrials. They attempted to avoid the false positives by calculating the expected number of CEson a trial-by-trial basis and summing them to get a more adequate predictor. Since most LGNneurons show much less trial variability in their spike counts than the cortical neurons, this typeof non-stationarity is not a problem in our case study.

8.2. Other related works on bootstrap test for synchronous spike events

A bootstrap significance test based on trial shuffling and re-sampling was first applied to the unitaryevent analysis by Pipa et al. [29, 30]. Based on the simulated spike trains of a finite trial number, theyestimated the number of Monte Carlo samples needed for a required precision of the rejection prob-ability. Their approach, however, considered only the total number of CEs within the trial durationand did not include non-stationary change of spiking patterns at different sampling windows. Theydid not report a case study of application of the bootstrap test to physiological data.

As introduced in the previous section, Ventura et al. [27] proposed a novel method for thesignificance test of spike synchrony. They examined the validity of the bootstrap significance testto non-Poisson spike trains. Based on numerical simulations of the gamma process, they concludedthat non-Poisson spiking behavior does not damage the properties of the test as long as a departurefrom Poisson is not very large.

8.3. Functional significance of non-stationary synchrony

We showed that, in a pair of LGN neurons, both the number of CEs and the spiking pattern showednon-stationary modulations even during the presentation of stationary light spots. The modulationof these variables had an intrinsic temporal profile that cannot be predicted by the modulations ofthe firing rates. Global statistics over all the recorded neuron pairs in the LGN showed that suchintrinsic modulations of the synchrony are observed commonly [18]. Although more systematicworks are needed for discussion on its functional significance, non-stationary synchrony mightreflect the intrinsic dynamical reorganization of synchronous cell assembly associated with theinternal neuronal processing [41–43]. Non-stationary changes of spike-timing inter-relationshipamong multiple units—that is, effective connectivity [25]—have been observed in multiple corticalareas (prefrontal area [44], motor area [45], visual cortex [46]).

Recently, stimulus-evoked oscillatory spike activities in a high � band were observed also inthe awake monkey LGN [47]. Also, Koepsell et al. [48] reported that the relay cell in the catLGN received spontaneous oscillatory input of the retinal ganglion cell even without any lightstimulation. They found that, during the light stimulation, each cycle of the oscillatory firings at



the relay cell was precisely time locked to the cycle of the oscillatory sub-threshold retinal input.Because of its spontaneous nature, the phase of the retinal oscillation was independent of the onsetof light stimulation. We consider that this independence is probably the reason why the phaseof the oscillatory activities in our data was not time locked to the stimulus onset. These recentobservations suggest the robustness of the oscillatory synchrony in the early visual system.

ACKNOWLEDGEMENTS

I thank C. M. Gray, P. E. Maldonado, S. Gruen, and M. Nawrot for useful comments. I also thankthe anonymous referees for their suggestions of further analyses, which led to a deeper understand-ing of the phenomena. This research was supported by Grant-in-Aid for Scientific Research on PriorityAreas—Integrative Brain Research—from the Ministry of Education, Culture, Sports, Science andTechnology of Japan (17021036).

REFERENCES

1. Hebb DO. Organization of Behavior. A Neuropsychological Theory. Wiley: New York, 1949.2. Georgopoulos AP, Schwartz AB, Kettner RE. Neural population coding of movement direction. Science 1986;

233:1416–1419.3. Abeles M. Corticonics—Neural Circuits of the Cerebral Cortex. Cambridge University Press: Cambridge, 1991.4. Von der Malsburg C. The correlation theory of brain function. Internal Report 81-2, Max-Planck Institute for

Biophysical Chemistry, Goettingen, 1981.5. Gerstein GL, Bedenbaugh P, Aertsen AMHJ. Neuronal assemblies. IEEE Transactions on Biomedical Engineering

1989; 36:4–14.6. Fujii H, Ito H, Aihara K, Ichinose N, Tsukada M. Dynamical cell assembly hypothesis—theoretical possibility

of spatio-temporal coding in the cortex. Neural Networks 1996; 9(8):1303–1350.7. Gray CM, Koenig P, Engel A, Singer W. Oscillatory responses in cat visual cortex exhibit inter-columnar

synchronization which reflects global stimulus properties. Nature 1989; 338:334–337.8. Kass RE, Ventura V, Brown EN. Statistical issues in the analysis of neuronal data. Journal of Neurophysiology

2005; 94:8–25.9. Brown EN, Kass RE, Mitra PP. Multiple neural spike train data analysis: state-of-the-art and future challenges.

Nature Neuroscience 2004; 7:456–461.10. Perkel DH, Gerstein GL, Moore GP. Neuronal spike trains and stochastic point processes II—simultaneous spike

trains. Biophysics Journal 1967; 7:419–440.11. Toyama K, Kimura M, Tanaka K. Cross-correlation analysis of interneuronal connectivity in cat visual cortex.

Journal of Neurophysiology 1981; 46:191–201.12. Toyama K, Kimura M, Tanaka K. Organization of cat visual cortex as investigated by cross-correlation technique.

Journal of Neurophysiology 1981; 46:202–214.13. Ahissar M, Ahissar E, Bergman H, Vaadia E. Encoding of sound-source location and movement: activity of

single neurons and interactions between adjacent neurons in the money auditory cortex. Journal of Neuroscience1992; 67:203–215.

14. Sakurai Y. Dependence of functional synaptic connections of hippocampal and neocortical neurons on types ofmemory. Neuroscience Letters 1993; 158:181–184.

15. Singer W. Putative functions of temporal correlations in neocortical processing. In Large-scale Neuronal Theoriesof the Brain, Koch C, Davis JL (eds). MIT Press: Cambridge, MA, 1994; 201–237.

16. Ito H, Gray CM, Viana Di Prisco G. Can oscillatory activity in the LGN account for the occurrence of synchronousoscillations in the visual cortex? Social Neuroscience Abstracts 1994, no. 61.7.

17. Castelo-Branco M, Neuenschwander S, Singer W. Synchronization of visual responses between the cortex, lateralgeniculate nucleus, and retina in the anesthetized cat. Journal of Neuroscience 1998; 18:6395–6410.

18. Ito H, Maldonado PE, Gray CM. Global statistics of synchronous spike activities in the lateral geniculatenucleus—non-stationary unitary events. 2007, in preparation.

19. Gruen S. Unitary Joint-events in Multiple-neuron Spiking Activity. Detection, Significance, and Interpretation.Verlag Harri Deutsch: Frankfurt, 1996.


3996 H. ITO

20. Gruen S, Diesmann M, Aertsen A. Unitary events in multiple single-neuron spiking activity: I. Detection andsignificance. Neural Computation 2001; 14:43–80.

21. Gruen S, Diesmann M, Aertsen A. Unitary events in multiple single-neuron spiking activity: II. Nonstationarydata. Neural Computation 2001; 14:81–119.

22. Hirata A, Maldonado PE, Gray CM, Ito H. Unitary event analysis of synchronous activities in cat LGN. SocialNeuroscience Abstracts 2002, no. 352.14.

23. Hirata A, Maldonado P, Gray C, Ito H. Unitary event analysis of synchronous activities in Cat LGN. In TheNeural Basis of Early Vision, Kaneko A (ed.). Springer: Berlin, Tokyo, 2003; 190–193.

24. Ito H, Maldonado PE, Gray CM. How stable are synchronous oscillations in the lateral geniculate nucleus?Social Neuroscience Abstracts 2005, no. 856.2.

25. Aertsen AM, Gerstein GL, Habib GL, Palm G. Dynamics of neuronal firing correlation: modulation of ‘effectiveconnectivity’. Journal of Neurophysiology 1989; 61:900–917.

26. Neuenschwander S, Singer W. Long-range synchronization of oscillatory light responses in the cat retina andlateral geniculate nucleus. Nature 1996; 397:434–436.

27. Ventura V, Cai C, Kass RE. Statistical assessment of time-varying dependency between two neurons. Journal ofNeurophysiology 2005; 94:2940–2947.

28. Reich DS, Mechler F, Purpura KP, Victor JD. Interspike intervals, receptive fields, and information encoding inprimary visual cortex. Journal of Neuroscience 2000; 20:1964–1974.

29. Pipa G, Gruen S. Non-parametric significance estimation of joint-spike events by shuffling and resampling.Neurocomputing 2003; 52–54:31–37.

30. Pipa G, Diesmann M, Gruen S. Significance of joint-spike events based on trial-shuffling by efficient combinatorialmethods. Complexity 2003; 8(4):79–86.

31. Ito H. Bootstrap significance test of synchronous oscillatory spike events. Social Neuroscience Abstracts 2006,no. 491.1.

32. Gray CM, Viana Di Prisco G. Stimulus-dependent neuronal oscillations and local synchronization in steriatecortex of the alert cat. Journal of Neuroscience 1997; 17:3239–3253.

33. Gray CM, Maldonado PE, Wilson MA, McNaughton BL. Tetrodes markedly improve the reliability and yield ofmultiple single unit isolation from multiunit recordings in cat striate cortex. Journal of Neuroscience Methods1995; 63:43–54.

34. Efron B, Tbshirani RJ. An Introduction of the Bootstrap. Chapman & Hall: New York, 1993.35. Nawrot M, Aertsen A, Rotter S. Single-trial estimation of neuronal firing rates: from single-neuron spike trains

to population activity. Journal of Neuroscience Methods 1999; 94:81–92.36. Nawrot PM, Aertsen A, Rotter S. Elimination of response latency variability in neuronal spike trains. Biological

Cybernetics 2003; 88:321–334.37. DiMatteo I, Genovese CR, Kass RE. Bayesian curve fitting with free-knot splines. Biometrika 2001; 88:1055–1073.38. Pipa G. Entwicklung und Untersuchung einer nicht parametrischen Methode zur Schaetzung der signifikanz

zeitlich koordinierter Spike-Aktivitaet. Diploma Thesis, Frankfurt am Main, 2001.39. Nawrot PM, Boucsein C, Rodriguez-Molina V, Aertsen A, Gruen S, Rotter S. Serial interval statistics of

spontaneous activity in cortical neurons in vivo and in vitro. Neurocomputing 2007; 70:1717–1722.40. Gruen S, Riehle A, Diesmann M. Effect of cross-trial nonstationarity on joint-spike events. Biological Cybernetics

2003; 88:335–351.41. Gray CM, Engel AK, Koenig P, Singer W. Synchronization of oscillatory neuronal responses in cat striate cortex:

temporal properties. Visual Neuroscience 1992; 8:337–347.42. Harris KD, Csicsvari J, Hirase H, Dragoi G, Buzaki G. Organization of cell assemblies in the hippocampus.

Nature 2003; 424:552–556.43. Harris DK. Neural signatures of cell assembly organization. Nature Reviews Neuroscience 2005; 6:339–407.44. Vaadia E, Haalman I, Abeles M, Bergman H, Prut Y, Slovin H, Aertsen A. Dynamics of neuronal interactions

in monkey cortex in relation to behavioural events. Nature 1995; 373:515–518.45. Riehle A, Gruen S, Diesmann M, Aertsen A. Spike synchronization and rate modulation differentially involved

in motor cortical function. Science 1997; 278:1950–1953.46. Aertsen AMHJ, Gerstein GL. Dynamic aspects of neuronal cooperativity: fast stimulus-locked modulations of

effective connectivity. In Neuronal Cooperativity. Krueger J (ed.). Springer: Berlin, 1991; 52–67.47. Royal D, Pouget P, Ruiz O, Schall J, Casagrande V. Receptive field mapping with local field potentials and

single unit activity in macaque lateral geniculate nucleus. Social Neuroscience Abstracts 2006, no. 241.6.48. Koepsell K, Wang X, Wei Y, Wang Q, Vaingankar V, Hirsch JA, Sommer FT. Dynamical encoding of natural

stimuli in the lateral geniculate nucleus of the thalamus. Social Neuroscience Abstracts 2006, no. 241.9.


bootstrap significance test of synchronous spike events—a case study of oscillatory spike trains

Documents