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Journal of Neuroscience Methods 179 (2009) 142–149

Contents lists available at ScienceDirect

Journal of Neuroscience Methods

journa l homepage: www.e lsev ier .com/ locate / jneumeth

ssessment of bursting activity and interspike intervals variability:case study for methodological comparison

ristelle Robina,1, Nicolas Mauriceb,2, Bertrand Degosb, Jean-Michel Deniaub,acques Martineriec, Laurent Pezardd,∗

École Supérieure de Physique et de Chimie Industrielles de la ville de Paris (ESPCI), FranceDynamique et Physiopathologie des Réseaux Neuronaux, INSERM U667 - Collège de France - Université P. & M. Curie, FranceNeurosciences Cognitives et Imagerie Cérébrale, LENA - CNRS UPR 640 - Hôpital de La Salpêtrière - Université P. & M. Curie, FranceNeurosciences Théoriques et Systèmes Complexes, LNIA - CNRS UMR 6149 - Aix-Marseille Université, France

r t i c l e i n f o

rticle history:eceived 18 June 2008eceived in revised form 19 January 2009ccepted 19 January 2009

a b s t r a c t

The detection and characterization of bursting activity remains a topic where no consensual definitionhas been reached so far. We compare here three different approaches of spike trains variability: statisticalcharacterization (average frequency, coefficient of variation), burst detection (Poisson and rank surprise)and multi-scale analysis (detrended fluctuations analysis). Using both real and simulated data, we show

eywords:pike trains variabilityurstingong-range correlations

that Poisson surprise provides information closely related to the coefficient of variation and that ranksurprise detects significant bursts which are associated with long-range correlations. Since these long-range correlations are only adequately characterized with multi-scale analysis, this study emphasizes thecomplementarity of these approaches for the complete characterization of spike trains.

etrended fluctuation analysisoisson surprise methodank surprise method

. Introduction

The characterization of spike trains variability is a centralroblem in the study of brain functions (e.g., Nawrot et al., 2008).

ndeed changes in the firing pattern of a neuron often reveallterations in physiological dynamics either normal or patholog-cal. Among these firing pattern modifications, the appearance ofursting activity is certainly one of the most obvious one. Although

t is often simple to visually recognize a burst as a temporaryncrease in the firing rate from the background activity, the conceptf bursting lacks of a clear definition. Numerous methods havehus been proposed to detect bursting activities (see Kaneokend Vitek, 1996; Gourévitch and Eggermont, 2007 and referencesherein) each based on a specific definition of what a “burst” is.

he rationale of the present study lies in the two main drawbacksf the majority of burst detection techniques:

∗ Corresponding author at: LNIA-CNRS UMR 6149,Pôle 3C - Case B, Université derovence, Centre St-Charles, 3 Place Victor Hugo, F-13331 Marseille Cedex 3, France.el.: +33 491 106 574.

E-mail address: laurent.pezard@univ-provence.fr (L. Pezard).1 Now at: Thales Research and Technology, Palaiseau, France.2 Now at: Institut de Biologie du Développement de Marseille-Luminy, IBDML -

NRS UMR 6216 - Aix-Marseille Université, France.

165-0270/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.jneumeth.2009.01.020

© 2008 Elsevier B.V. All rights reserved.

• First, methods for burst detection are as various as burst defini-tions and often need some ad hoc parameters adjustments. Thissituation (1) leads to results dependent upon the choice of themethod and of its parameters and (2) makes difficult, if not impos-sible, to compare results obtained in different studies that useddifferent methods.

• Second, burst detection is a local description of spikes trains. Timeperiods of bursting activity are localized as deviations from someglobal statistics (e.g., mean interspike interval). Nevertheless, thislocal analysis is often summed up in global indices such as per-centage of spikes in bursts, averaged burst length or number ofbursts. This procedure misses the possible multi-scale organiza-tion between local and global time scales which has been shownto be important in several biological processes (Havlin et al., 1995,1999; Stanley et al., 1999).

The aim of this study was to compare and re-evaluate existingmethods used to characterize spike trains in order to get the bestassessment of bursting activity and interspike intervals variabil-ity. We thus compared local, global and multi-scale descriptions ofspike trains obtained with the following approaches:

Statistical description: It computes the mean firing rate (f ) andthe coefficient of variation (CV) and estimates the parameters of theputative stochastic process underlying the neuronal discharge.

Burst detection methods: They are so numerous that we chooseto compare only two of them: Poisson surprise (PS) maximisation

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K. Robin et al. / Journal of Neuro

Legendy and Salcman, 1985) and rank surprise (RS) maximisationGourévitch and Eggermont, 2007). They both provide the tempo-al occurrence of bursts and global descriptive indices about theeuronal discharge such as percentage of spikes in bursts (p) andveraged burst length (l). Nevertheless, the first one implements aarametric approach whereas the second one is non-parametric.

Multi-scale analysis: It characterizes the statistical scalingepicted by the data. We focussed here on detrended fluctuationnalysis (DFA) (Peng et al., 1994, 1995) which characterizes long-ange correlations through the computation of a scaling exponent�).

For this study we used, as a benchmark, a set of neuronal data forhich changes in the spike trains characteristics have been previ-

usly reported (Degos et al., 2005). The conclusions provided by these of the three different approaches on these physiological data arehen compared with those provided by the analysis of simulatedata which reproduce as closely as possible the statistical prop-rties of the physiological data. These simulated data thus allows to delineate the validity of the conclusions that can be drawnrom each method and it is shown that those obtained with statis-ical description and burst detection methods remain incomplete.urther directions for the characterization of spike trains are thenroposed.

. Material and methods

.1. Data sets

A set of data drawn from a previous study (Degos et al., 2005)as used here to provide a simple case study of several analysisethods. They were then compared to a set of simulated data which

eproduce as closely as possible their statistical properties.

.1.1. Physiological dataPhysiological data were obtained in vivo by performing single-

nit extracellular recordings in the substantia nigra pars reticulataf rats anesthetized with chloral hydrate. The complete experi-ental procedure is described in the previous report (Degos et

l., 2005). Briefly, the same neurons were successively recordedn control condition and following a systemic injection of neu-oleptics consisting in simultaneous injections of both SCH-233900.5 mg kg−1, s.c., an antagonist of the D1-class dopaminergic recep-ors) and Raclopride (2 mg kg−1, i.p., an antagonist of the D2-classopaminergic receptors). This blocked the dopaminergic trans-ission, resulting in a clear change of the discharge pattern that

resented pauses interrupted by close spikes discharges.Time windows were selected to provide data segments of at

east 1000 interspike intervals (ISI) with no change in the dis-harge statistics. Our study deals with the data obtained fromN = 8ells recorded in two experimental conditions: before neurolepticsnjection and after the neuroleptics injection. Since our purpose isurely methodological and that we are not directly interested here

n the very effect of neuroleptics on neuronal discharge, those twoonditions will be respectively denoted Physiological 1 and Physio-ogical 2.

The average duration of time windows was 63.50 s (±16.83,in.: 29.58, max.: 89.98; Physiological 1: 61.55 ± 19.52, Physiolog-

cal 2: 65.47 ± 14.72, Wilcoxon test: V = 7, p = 1 n.s.) and theirverage number of interspike interval was 1868.13 (±777.40, min.:036, max.: 3789; Physiological 1: 1745.25 ± 750.97, Physiological

: 1991 ± 834.60, Wilcoxon test: V = 9, p = 0.25 n.s.).

.1.2. Simulated data setsIn the second step of our study, a set of simulated data was gener-

ted to reproduce as closely as possible the statistical characteristics

e Methods 179 (2009) 142–149 143

of the physiological data. In order to reproduce the Gamma-like dis-tribution of the interspike intervals distribution, integrate-and-fireneurons with random threshold distributed according to a Gammadistribution of order k and shape parameter � = Vth/k (Vth: meanthreshold value) were used (Gabbiani and Koch, 1998). ThresholdVth follows the probability density function of the Gamma distribu-tion:

fk(Vth) = Vk−1th

�(k)�ke−Vth/� (1)

with �(k) = (k − 1)! and so do the interspike intervals. As kincreases, one obtains increasingly regular spike trains (the casek = 1 corresponds to Poisson spike train and k → ∞ to the per-fect integrate-and-fire neuron). These data were simulated usingour Python translation of a MatLab® routine (fastgammatrain.m)3

contributed by K. Sen.

2.2. Numerical methods

All the following numerical methods were implemented usingthe Python language (http://www.python.org/) and its Numpylibrary (http://numpy.scipy.org/) except for Poisson Surprise imple-mented in R language. All the statistical analysis were performedusing the R language for statistical computing (R Development CoreTeam, 2006).4

Let tn be the time of spike occurrence, interspike intervals seriesare defined as X = {xn} such that xn = tn+1 − tn and 1 ≤ n ≤ N (i.e.N: number of interspike intervals).

2.2.1. Statistical indicesWe define the average frequency f as the averaged instantaneous

frequency:

f = 1N

N∑n=1

1xn

(2)

and coefficient of variation CV as the standard deviation � of theinterspike interval distribution normalized by the mean interspikeinterval x:

CV = �

x(3)

2.2.2. Burst detectionTwo methods for burst detection were used in this study. They

both detect a burst when the likelihood of a set of successive inter-spike intervals is low, i.e. when the “surprise” is high. This “surprise”is computed as S = − log P where P is the probability to observe sucha set of intervals. These methods differ in the definition of P.

In Poisson surprise method (PS) (Legendy and Salcman, 1985),P is the probability that, in a Poisson spike train of the same averagespikes rate r as the studied spike train, a given time interval of lengthT contains q or more spikes and it is given by

P = e−rT

∞∑k=q

(rT)k

k!(4)

In the rank surprise method (RS) (Gourévitch and Eggermont,

3 Available at http://glab.bcm.tmc.edu/signal processing techniques/signal proc.html.

4 Most of the computation were realized on a computer with 64Bits architecturerunning Linux Debian with adequate packages.

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44 K. Robin et al. / Journal of Neuro

ariates Tq between 1 and N is lower than the sum of the ranks Rf the q interspike intervals in a suspicious burst, i.e. P = Pr(Tq ≤n + · · · + Rn+q−1) obtained by

= Pr(Tq ≤ u) = 1Nq

∑0≤k≤(u−q)/N

(−1)k(u − kN)!k!(q − k)!(u − kN − q)!

(5)

ig. 1. A sample of results for a typical cell for Physiological 1 (left) and Physiological 2 (righurprise (PS) burst dectection. (c) Rank surprise (RS) burst detection. (d) Detrended fluctraph.

e Methods 179 (2009) 142–149

For each method, an iterative construction of a burst allows tomaximize the “surprise” S attached to a suspicious burst. Namely,interspike intervals that fulfill a selection criterion are detected.

In the case of Poisson surprise, this criterion corresponds to threespikes spaced by less than the average interspike interval. In thecase of rank surprise, the largest interspike interval included inbursts was 75% of the interspike intervals distribution. Then con-tiguous spikes are added to the suspicious burst so that S is

t) conditions. (a) Distribution of interspike intervals (ISI in milliseconds). (b) Poissonuations analysis (DFA). The corresponding values for the indices are given on each

science Methods 179 (2009) 142–149 145

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Table 1Statistical description (mean±S.D.) of experimental data and results of Wilcoxonsigned-rank test with paired data. *Significant difference (p < 0.05), n.s. non sig-nificant. f : mean firing rate, CV: coefficient of variation, pPS and lPS: respectivelypercentage of spikes in bursts and averaged burst length for Poisson surprise method,pRS and lRS: respectively percentage of spikes in bursts and averaged burst length forthe rank surprise method, �: scaling exponent for detrended fluctuations analysis.

Physiological 1 Physiological 2 Wilcoxon test

f 35.08 ± 13.49 42.78 ± 18.46 V = 1, p = 0.02∗

CV 0.43 ± 0.12 0.60 ± 0.24 V = 3, p = 0.05∗

pPS 5.26 ± 7.85 22.52 ± 18.65 V = 0, p = 0.01∗

lPS 2.65 ± 2.41 8.37 ± 4.54 V = 0, p = 0.01∗∗

K. Robin et al. / Journal of Neuro

aximized and above a minimum level of significance (taken heres ˛ = 0.01) for the rank surprise method.

Standard global indices were extracted from the burst detectionbtained with each method. We used the percentage of spikes inursts (p) and the average burst length (l), i.e. mean number ofpikes in a burst. Four indices were thus computed (pPS , pRS , lPS andRS) for each cellular discharge.

.2.3. Multi-scale analysis of fluctuationsMulti-scale analysis of fluctuations has been performed using

he Detrended Fluctuations Analysis (DFA) method (Peng et al.,994, 1995). It is based on the root mean square fluctuation F(�),btained at different timescales �, for the integrated time series(k) = ∑k

n=1(xn − x), defined as

(�) =

√√√√ 1N

N∑k=1

(y(k) − y�(k))2 (6)

here y�(k) is the local trend for time scale �.For process with long-range power-law correlations, i.e. with

uto-correlation function C(t) = t−˛, F(�) behaves as F(�) = �� with= 1 − ˛/2. For a sequence of interspike intervals with no cor-

elation, the scaling exponent � = 0.5, whereas � = 1 representsself-similar or scale-invariant fractal sequence. For 0 < � < 0.5,

he sequence is an anti-persistent correlated process, whereas for.5 < � < 1, the sequence depicts a persistent long-range cor-elations. The degree of deviation of the scaling exponent from.5 to 1 represents the strength of long-range correlation in theequence.

This multi-scale method has been implemented on the basis ofhe C routine available in the Physionet toolkit (Goldberger et al.,000). The estimate of the scaling exponent follows a proceduredapted for small data sets (Govindan et al., 2007). In this case, theetrending uses an order-two polynomial and the scaling exponent

s estimated as the averaged value of a set of (100) scaling exponentsbtained with Fast Fourier Transform surrogate data. In fact, theFT surrogates keep the correlation function of the data and thusllows one to reduce the variability in the estimation of the scalingxponent.

The presence of long-range correlations in the original sequenceas tested using a set of (100) random shuffle surrogate data.

he shuffled sequence preserves the distribution of the origi-al sequence, but lacks correlations. Long-range correlations wereonsidered as statistically significant if the estimate of the scaling-xponent of the original sequence was higher than that of thehuffled sequences (probability p < 0.01).

. Results

.1. Physiological data

Fig. 1 gives an example of the results obtained in the anal-sis of the spike train variability for a typical cell. The changesbserved between conditions Physiological 1 and Physiological 2 inhis cell are similar to those observed on average for our data set: anncrease of bursting activity and the appearance of long-range cor-elations in Physiological 2 when compared to Physiological 1. Frommethodological point of view, one can notice that the two meth-ds used for burst detection do not provide the same results, e.g.,he percentages of spikes in bursts are noticeably different between

hese two methods. This observation strengthens our introductiveemarks.

Characteristics of the spikes trains are summarized in theable 1. The statistical results obtained for the whole set of N = 8ells are then presented and discussed.

pRS 7.78 ± 3.66 18.92 ± 5.51 V = 0, p = 0.01lRS 7.03 ± 2.53 10.48 ± 4.49 V = 6, n.s.� 0.54 ± 0.08 0.68 ± 0.10 V = 0, p = 0.01∗

3.1.1. Comparison of the indicesThe whole set of measurements obtained from our physiological

data and the linear correlation between these measurements aredepicted and quantified in Fig. 2.

The average frequency does not depict any significant correla-tion with any other index.

The coefficient of variation (CV) is significantly correlated withthe percentages of spikes in bursts, both pPS and pRS; the highestcorrelation (r = 0.979) being observed with pPS .

The measurements obtained from both burst detection methodsare significantly correlated except for the percentage of spikes inbursts detected using PS and the length of the burst detected withRS, i.e. for the pair pPS : lRS .

The scaling exponent (�) is significantly correlated with 3burst indices, namely: lPS , pRS and lRS . This result emphasizes that,although the scaling exponent does not have an obvious physiolog-ical interpretation yet, such as a burst description, it neverthelessconveys valuable physiological information.

3.1.2. Differences between Physiological 1 and Physiological 2conditions

The comparisons between the physiological conditions for allthe scalar indices studied here are given in Table 1. All these indicesare able to differentiate between Physiological 1 and Physiologi-cal 2 except the burst length obtained with RS (lRS). Thus from thepragmatic viewpoint of characterizing a difference between exper-imental conditions, all these indices are comparable.

Nevertheless, the statistical test of long-range correlations usingshuffle surrogate data provides a further information: in Physio-logical 1 condition, 7 cells (out of 8) do not depict any long-rangecorrelation (i.e. their scaling exponent was not significantly dif-ferent from that of the shuffled surrogates) and in Physiological 2condition, 7 cells depict long-range correlations (i.e. their scalingexponent was significantly higher than that of the shuffled surro-gates). We can thus conclude that when recorded in Physiological2 condition the discharge of neurons of the substantia nigra parsreticulata significantly presented long-range correlations that werenot present when recorded in Physiological 1 condition (Fisher test:p = 0.01).

3.2. Simulated data

Since standard indices (f , CV, pPS and lPS) are based on theassumption that neuronal discharge can be described as a stochas-tic process, we simulated a set of (N = 8) “pseudo-cells” with

stochastic interspike intervals (Gamma distribution). These simu-lated data were then analyzed with the same procedures as thephysiological data. This study allowed us to question the validityof this assumption and to demonstrate the complementarity of themethods.

146 K. Robin et al. / Journal of Neuroscience Methods 179 (2009) 142–149

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ig. 2. Pairs of indices (lower left) and correlation coefficients (Pearson’s r, upper riorrelation (p < 0.05).

To obtain data with coefficient of variation similar to those ofhe physiological data, Gamma distribution of order 6 (k = 6) wassed to reproduce a pseudo-Physiological 1 condition and Gammaistribution of order 3 (k = 3) was used for a pseudo-Physiological 2ondition (Gabbiani and Koch, 1998). These data are respectivelyenoted Simulated 1 and Simulated 2.

For each physiological series, one simulated series was obtainedith the same number of interspike intervals and the closest aver-

ge frequency (f ) and coefficient of variation (CV). These series ofnterspike intervals were thus analyzed using the same algorithmss those used to characterize the physiological data. Fig. 3 depictshe results obtained for the simulated cell which corresponds tohat of Fig. 1 for the physiological data. The results obtained for the

et of 16 simulated series are given in Table 2 and depicted in Fig. 4.

The results obtained for mean frequency (f ), coefficient of vari-tion (CV) and percentage of spikes in bursts for Poisson surprisepPS) are similar to those obtained with the physiological data. This

able 2tatistical description (mean±S.D.) of simulated data and results of Wilcoxon signed-ank test with paired data. *Significant difference (p < 0.05), n.s.: non significant.he indices are abbreviated as in Table 1.

Simulated 1 Simulated 2 Wilcoxon test

¯ 45.13 ± 12.86 64.51 ± 25.48 V = 0, p = 0.01∗

V 0.46 ± 0.15 0.61 ± 0.21 V = 0, p = 0.01∗

PS 8.36 ± 10.29 19.36 ± 14.56 V = 0, p = 0.01∗

PS 2.64 ± 0.64 3.08 ± 0.84 V = 2, p = 0.02∗

RS 9.36 ± 1.62 10.07 ± 1.36 V = 14, n.s.RS 5.90 ± 1.01 5.86 ± 0.67 V = 19, n.s.

0.51 ± 0.02 0.51 ± 0.02 V = 21, n.s.

r the whole set of indices for the physiological data in both conditions. *Significant

result is in accordance with the hypothesis underlying these indices.For the other indices (lPS , pRS , lRS and � see Table 2 and Fig. 4), onlythe results obtained for the Simulated 1 condition are similar tothose of the physiological data. For the Simulated 2 condition, theseindices are no longer similar to those of the physiological data andremain constant in comparison with the results obtained for theSimulated 1 condition. As expected, no long-range correlation wasobserved for these simulated data in both conditions. Thus the char-acteristics of bursts associated with long-range correlations cannotbe adequately reproduced by the interspike intervals obtained witha Gamma generator.

4. Discussion

On the basis of a case study of physiological and simulated data,we have compared the results obtained with methods used to char-acterize the variability of interspike intervals: statistical description(mean firing rate, f and coefficient of variation, CV), burst detectionmethods (Poisson and rank surprise maximisations) and multi-scale analysis of fluctuations (DFA, scaling exponent: �).

Our first observation concerns the percentage of spikes in burstsdetected using Poisson surprise method (pPS). This index depicts

a high correlation with the coefficient of variation (CV) for bothphysiological and simulated data. In fact, both indices measurethe deviation of the interspike intervals distribution from a Pois-son distribution5. This result shows that these indices are almost

5 Recall that CV = 1 for Poisson distribution and CV = 0 for regular spike trains andthat PS uses an algorithm based on the “surprise” conveyed by a set of successiveinterspike intervals when one expects a Poisson distribution.

K. Robin et al. / Journal of Neuroscience Methods 179 (2009) 142–149 147

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ig. 3. A sample of results for the simulated cell used to mimicked the cell of Fig. 1ntervals (ISI in milliseconds). (b) Poisson surprise (PS) burst detection. (c) Rank suralues for the indices are given on each graph.

edundant. Although the interpretations of these indices differ, CVppears as a better global statistical characteristics of spike trainshan pPS since it is computationally less demanding and does not

uppose any parameter adjustment.

Our second observation concerns the set of remaining indiceslPS , pRS , lRS and �) that display high correlations for the physio-ogical data but not for the simulated ones. These indices are alsohose that were adequately reproduced by the numerical simula-

e Simulated 1 (left) and Simulated 2 (right) conditions. (a) Distribution of interspike(RS) burst detection. (d) Detrended fluctuations analysis (DFA). The corresponding

tion of the Gamma interspike intervals distribution in Simulated 1only. In Physiological 2 condition, where the data depict significantlong-range correlations, the numerical simulations were not able

to reproduce adequately the results obtained for these indices withthe physiological data. One can thus suggest that these indices char-acterize, each in its own way, the correlations present in the spiketrains. Since the burst length detected using Poisson surprise (lPS)appears in this set of indices, it can be concluded that pPS and lPS

148 K. Robin et al. / Journal of Neuroscience Methods 179 (2009) 142–149

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ig. 4. Pairs of indices (lower left) and correlation coefficients (Pearson’s r, upperorrelation (p < 0.05).

oes not measure the same characteristics of spike train and thathe former might be less reliable than the latter. In the case of theank surprise method, both indices (pRS and lRS) are correlated withhe scaling exponent (�) which quantifies the long-range correla-ions. The characteristics of significant bursts detected by the RS

ethod are thus closely related with the presence of long-rangeorrelations.

These results show that Poisson surprise method is too closelyelated to the hypothesis of Poisson distribution underlying thepike train and question its adequacy for spike trains characteriza-ion. A method that do not suppose any a priori hypothesis on thenterspike intervals distribution should thus be preferred. In thisespect, the rank surprise method is a valuable candidate since, as aon-parametric technique, it does not suppose any a priori distribu-ion of the interspike intervals. Nevertheless, although rank surprise

ethod is able to detect significant bursts, this method only pro-ides a description of the spike train and does not allow one tonfer the properties of the statistical process which underlies theseursts. Our study has shown that the characteristics of the burstetected using RS are highly correlated to the presence of long-ange correlations. Since among the methods used here, detrendeductuation analysis only can detect and quantify the presence of

ong-range correlations, those methods (RS and DFA) are likely to beomplementary tools for a complete characterization of neuronalischarge.

It has been shown using simulated data that our physiologicalata can be adequately reproduced by a Gamma generator onlyhen long-range correlations are absent. Then, although bursts can

e obtained from renewal process, i.e. neuronal discharge with-ut memory, long-range correlations can only be produced by a

for the whole set of indices for the simulated data in both conditions. *Significant

non-renewal process. Such a process is evidenced using detrendedfluctuations analysis but is ignored when any of the other methodsis used.

In the case studied here, one can conclude that the underlyingprocesses that are responsible for the changes in the neuronal dis-charge in Physiological 2 do not correspond to the change of theparameter of a Gamma generator. In fact, the long-range corre-lations observed in that condition are not taken into account bysuch a stochastic process. The presence of long-range correlationsin single-unit activity has already been described (Bhattacharyaet al., 2005; Blesic et al., 2005; Guo et al., 2007; Yatang et al.,2008) and several mechanisms have been proposed for their gen-esis: either the statistics of inputs correlations (Jackson, 2004) orintrinsic dynamics of neuronal excitability (Brandman and Nelson,2002). Our data do not allow to choose between these two processesbut suggest that the presence of long-range correlation should becarefully searched in neuronal data using a multi-scale approach.Indeed, other methods that deal with multi-scale characterizationof time series, such as multi-scale entropy (Costa et al., 2002, 2005)or scaling properties of the Fano factor (e.g., Teich, 1989) couldprovide other complementary information.

5. Conclusion

As a general methodological principle of data analysis, it is

often preferable to use methods which do not suppose any a priorihypothesis and parameter adjustments. On this basis, rank sur-prise method, detrended fluctuations analysis and coefficients ofvariations provide valuable tools to characterize the variability ofinterspike intervals of spike trains. Since coefficient of variation

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rovide information that does not take into account the temporalorrelations of the data, it only provides first order characteris-ics that have to be completed with correlations quantifications.lthough rank surprise method detects bursts whose characteris-

ics are closely related to temporal correlations this method cannote used to infer the statistical process that underlies the data. It canhus be sketched that RS should be used to localize in time the occur-ence of bursts whereas DFA should be preferred for the statisticalharacterization of an entire period of neuronal activity.

cknowledgments

We wish to thank the editor and the reviewer for their com-ents, which helped us to improve this article.

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