Stochastic neural network model for spontaneous bursting in hippocampal slices

PHYSICAL REVIEW E 66, 051908 ~2002!

Stochastic neural network model for spontaneous bursting in hippocampal slices

B. Biswal1,2 and C. Dasgupta3,4

1ICA1, University of Stuttgart, Pfaffenwaldring 27, D-70569 Stuttgart, Germany2Department of Physics, Sri Venkateswara College, University of Delhi, New Delhi, India

3Department of Physics, Indian Institute of Science, Bangalore 560 012, India4Condensed Matter Theory Unit, JNCASR, Bangalore 560 064, India

~Received 19 June 2002; published 15 November 2002!

A biologically plausible, stochastic, neural network model that exhibits spontaneous transitions between alow-activity ~normal! state and a high-activity~epileptic! state is studied by computer simulation. Brief excur-sions of the network to the high-activity state lead to spontaneous population bursting similar to the behaviorobserved in hippocampal slices bathed in a high-potassium medium. Although the variability of interburstintervals in this model is due to stochasticity, first return maps of successive interburst intervals show trajec-tories that resemble the behavior expected near unstable periodic orbits~UPOs! of systems exhibiting deter-ministic chaos. Simulations of the effects of the application of chaos control, periodic pacing, and anticontrolto the network model yield results that are qualitatively similar to those obtained in experiments on hippo-campal slices. Estimation of the statistical significance of UPOs through surrogate data analysis also leads toresults that resemble those of similar analysis of data obtained from slice experiments and human epilepticactivity. These results suggest that spontaneous population bursting in hippocampal slices may be a manifes-tation of stochastic bistable dynamics, rather than of deterministic chaos. Our results also question the reli-ability of some of the recently proposed, UPO-based, statistical methods for detecting determinism and chaosin experimental time-series data.

DOI: 10.1103/PhysRevE.66.051908 PACS number~s!: 87.18.Sn, 05.45.Gg, 05.45.Tp, 87.19.La

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I. INTRODUCTION

With the development of many new techniques of nonlear time-series analysis@1#, a number of recent studies havattempted to understand brain dynamics@2# in the context ofdeterministic chaos@3#. Experimental studies on normabrain activity @4# and epilepsy@5# have looked for signatureof chaos. New hypotheses@6,7# have been proposed for posible roles of chaos in the functioning of the brain. A covincing demonstration of the occurrence of determinischaos in brain dynamics is of fundamental importanceunderstanding the collective behavior of neuronal netwoin the brain. It may also lead to the possibility of short-teprediction and control, with potentially important medicapplications.

Due to the large number of neurons involved in any scific brain function, the underlying dynamics is inherenhigh dimensional. This makes reliable detection of lodimensional chaos in brain signals a formidable taDatasets from physiological systems are often inadequasmall, nonstationary, and contaminated with noise, possevere problems for a statistically reliable time-series ansis. Though a number of recent studies have reporteddetection of low-dimensional chaos through time-seranalysis of brain signals, experts often remain skeptabout such claims@8#.

The results of anin vitro experiment of Schiffet al. @9# onspontaneous population bursting in rat hippocampal slbathed in a medium with high potassium concentration hattracted much attention in this context. In this work, tpresence of unstable periodic orbits~UPOs! in the dynamicsof the neuronal network is inferred from an analysis of tfirst return map of successive interburst intervals~IBIs!.

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Since UPOs are known to be embedded in typical chaattractors, the presence of UPO-like trajectories suggestsoccurrence of deterministic chaos. Then a variant of a chcontrol technique proposed by Ott, Grebogi and Yorke@10# isused to attempt to make the IBIs more uniform. Applicatiof this technique is found to make the bursts more periodThese observations are interpreted as evidence for deteristic chaos in the neuronal population bursting behavior. Rsults of periodic pacing and the application of an ‘‘anticotrol’’ method designed to make the IBIs more irregular aalso reported. Since the spontaneous bursting observethis in vitro experiment exhibits many similarities with inteictal spikes in human epilepsy, it is suggested that humepileptic activity may also be chaotic and therefore amenato methods of chaos control and anticontrol. The possibiof application of these methods in the treatment of humepilepsy has attracted wide attention@11#.

Some of the methodology and conclusions of the braslice experiment have been questioned by Christini and Clins @12# who have shown that the method of chaos contused by Schiffet al. works equally well in some simplesto-chasticsystems. These authors consider the effects of ative noise on the behavior of the FitzHugh-Nagumo eqtions which describe a single spiking neuron. The preseof the noise causes the interspike interval~ISI! to be irregu-lar. The first return map of successive ISIs shows evidefor the existence of~apparent! unstable fixed points, and thapplication of the chaos control method makes the ISIs mregular. The qualitative behavior found in this simple systis similar to that found in the experiment of Schiffet al.Thiswork casts doubt on the interpretation of the results ofbrain-slice experiment as evidence for the occurrence ofterministic chaos.

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B. BISWAL AND C. DASGUPTA PHYSICAL REVIEW E66, 051908 ~2002!

More recently, several surrogate methods@13,14# to assessthe statistical significance of UPOs detected from time-seanalysis have been proposed. Applications of some of thmethods to the time-series data of the brain-slice experimclaim @15,16# to have reestablished the original conclusionSchiff et al. on the existence of deterministic chaos in tbursting dynamics of hippocampal slices. Results similathose in Refs.@9# and @16# have been reported@17,18# re-cently for hippocampal slices studied under three differexperimental conditions@high potassium, zero magnesiumand in the presence ofg-amino butyric acid (GABAA)-receptor antagonists#. A similar conclusion about the occurence of chaos in epileptic brain activity has also bereached from a surrogate analysis@19# of time-series datarecorded from a human subject.

In this paper, we address the issue of whether or notspontaneous population bursts found in brain-slice preptions and analogous interictal spikes in the human epilebrain are manifestations of deterministic chaos. Our winvolves extensive simulations of an existing neural netwmodel for focal epilepsy@20#. This network exhibits a low-activity ~normal! attractor and a high-activity~epileptic! one.Brief excursions of the network to the high-activity statethe absence of any external stimulus appear as spontanbursts analogous to those observed in the brain-slice exment. In our model, however, the irregularity of the IBIsstrictly a consequence of stochasticity arising from the rdomness of the sequence in which the neurons are updThe network model and its bursting dynamics are descriin Sec. II.

Although the variability of the IBIs in our network modeis a consequence of stochasticity, we find trajectoriesmay be interpreted as evidence for the presence of~apparent!UPOs@unstable fixed points~UFPs! in the first return map#.This allows us to apply the methods of chaos control, podic pacing, and anticontrol considered by Schiffet al. to ourstochastic neural network model. We have also studiedeffects of demand pacing on the network dynamics. Resof our simulations of the effects of these control methodsdescribed in detail in Sec. III. These results are found toquite similar to those obtained in brain-slice experiments

Section IV contains the results of surrogate analysisthe statistical significance of the UPO-like trajectories fouin the IBI datasets. Three different methods are considereour analysis. These methods include the ones used recto claim evidence for the presence of statistically significUPOs in time-series data obtained from brain-slice expments@16,17# and from human epileptic activity@19#. Appli-cations of these methods to the IBI time series obtained fsimulations of our stochastic neural network model yieldsults that are similar to some of those reported in Re@16,17,19#.

Thus, our stochastic neural network model with bistadynamics reproduces most of the results obtained in brslice experiments. This result casts doubt on the interprtion of the experimental data as evidence for determinichaos, and suggests that epileptic brain activity may bmanifestation of stochastic bistability. Our work also quetions the reliability of some of the UPO-based statisti

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methods used in recent studies@16,17,19# to infer the pres-ence of determinism and chaos in experimental time-sedata. A summary of our main results and conclusions is psented in Sec. V. Some of the results reported here wsummarized in a recent Letter@21#.

II. THE NEURAL NETWORK MODELAND ITS DYNAMICS

In this section, we define the neural network model usin our simulations and describe its bursting dynamics. Tmodel was first proposed by Mehta, Dasgupta, and Ullal@20#for describing the process of ‘‘kindling’’@22# of focal epi-lepsy. Kindling is a phenomenon in which a part of the brais externally stimulated, either through electric pulsesthrough chemicals, to produce spontaneous epileptic actiIt is believed that kindling provides a good animal model fthe development of an understanding of epilepsy in the mmalian brain. In the present work, we have used kindnetworks to model spontaneous population bursting in hpocampal slices and interictal spikes in human epilepsy.

A. The model

The network consists ofN McCulloch-Pitts~binary! neu-rons$Si%, i 51,2, . . . ,N, each of which is excitatory and catake the two values 0 and 1, representing low and high firrates, respectively. The inhibitory neurons are collectivmodeled by a background inhibition, which is assumed toproportional to the number of active excitatory neurons. Omodel with binary neurons is clearly inappropriate for dscribing phenomena that depend crucially on the detabiophysical properties of individual neurons. However, sumodels are expected@23# to be adequate for qualitative descriptions of the collective dynamical behavior of biologicnetworks.

The ‘‘local field’’ hi(t) at thei th neuron at timet is givenby

hi~ t !5(j 51

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@$Ji j Sj~ t !2wSj~ t !%

1l$Ki j Sj~ t2t!2wSj~ t2t!%#, ~1!

wherew is the relative strength of inhibition,l is the relativestrength of the delayed signal, andt is the time delay associated with the delayed signal. The four terms from leftEq. ~1! represent, respectively, fast global excitation, faglobal inhibition, slow global excitation, and slow global inhibition. These four ingredients are believed@24# to be es-sential for a realistic modeling of neural oscillations in thippocampus. The ‘‘time’’t is discrete, with each unit~re-ferred to as one ‘‘pass’’! corresponding to the updating ofNneurons. The neurons are updated one at a time in arandomsequence, according to the ruleSi(t11)51 if hi(t)>0 andSi(t11)50 if hi(t),0.

A fixed number ~q! of random low-activity patterns~memories! $j i

m%,i 51,2, . . . ,N; m51,2, . . . ,q, are storedin the synaptic matricesJi j andKi j in the following way:

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STOCHASTIC NEURAL NETWORK MODEL FOR . . . PHYSICAL REVIEW E 66, 051908 ~2002!

Ji j 51 if (m51

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1 , and Jii 5Kii 50. To simulate low averageactivity of the network in the absence of any external stimlus, the net activity of each memory,( i 51

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value n!N. The n ‘‘active’’ neurons in each memory archosen randomly. The ‘‘fast’’ synaptic connectionsJi j act tostabilize the system in one of the memory states, while‘‘slow’’ connections Ki j tend to induce transitions frommemory state to the next one in the sequence after;tpasses. The global inhibition represented by thew terms pre-vents simultaneous activation of more than one memorWith appropriate choice of the values of the parameters~spe-cifically, for l.1 andw,1), the network exhibits a low-activity limit cycle in which all theq memories are visitedsequentially@20#.

The simulations carried out in this study are for the flowing parameter values:N5200, q520, n510, w50.6,l52, t52 passes. Proper functioning of the network in tabsence of any external input~the resting state! is verified bystarting the network in the memory statej i

1 at timet50, andthen monitoring the time evolution of the overlaps,mm(t)5( i 51

N j imSi(t), m51,2, . . . ,q, of the network with the

memory states. The network evolution is found to followlimit cycle in which it traverses sequentially the 20 memorin about 60 passes. The resulting low-amplitude, noisy oslations of the net activity,Sup(t)5( i 51

N Si(t), are shown inFig. 1~a!.

B. Spontaneous population bursting

To simulate the excess excitability of brain slices bathin a high-potassium medium that induces spontaneous bing, the network is ‘‘chemically kindled’’@20# through aHebbian learning mechanism defined in the following wIf, over n1 consecutive passes,Si51 andSj51 more oftenthann2 times, then the fast synaptic strengthJi j between thetwo neurons is permanently set to 1. In our simulation,n1510 andn256. The networks are chemically kindled bswitching on the Hebbian learning for 50 initial passes unreduced inhibition (w is reduced from 0.6 to 0.24 during thperiod!. The Hebbian learning process generates manyexcitatory synaptic connections, leading to increased contivity between different memories. This, in turn, leads to tformation of a new high-activity~epileptic! attractor of thedynamics corresponding to excess correlated firing ofneurons. Different networks~i.e., networks with different re-

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alizations of the random memories$j im%) exhibit varying de-

grees of kindling due to differences in the initial connectivpattern.

For random sequential updating of the neurons,kindled network makes occasional, short-lived, spontanetransitions to the high-activity attractor@20#. This is similarto the spontaneous population bursting observed in the brslice experiment. These bursts of high correlated activitypear as ‘‘spikes’’ in the net activity of the network. The IBare measured using the threshold crossing method@12#. Likea low-pass filter in experiments, the net activity,Sup(t), isfirst smoothened toSsm(t)51/tsm( t85t2tsm11

t Sup(t8), with

tsm540 passes. WheneverSsm(t) crosses an appropriatelchosen upper threshold in the positive direction, a bursrecorded at the corresponding timet. Before the occurrenceof the next burst,Ssm(t) must decrease below a lower thresold corresponding to the low-activity state of the network.avoid rare occurrences of persistent high activity,refractori-nessis introduced in the network in the following manner.Ssm(t) remains higher than the upper threshold value ctinuously for 20 passes, the neuron configurations~the cur-rent one as well as the previous ones accounting fordelayed signal! are reset to one of the low-activity memorpatterns chosen randomly.

The net activity of a kindled network with refractorinesis shown in Fig. 1~b!, and the smoothened net activityplotted in Fig. 1~c!. Positive crossings denoted by dacircles correspond to the occurrence of bursts aT1 ,T2 , . . . , are theIBIs. Individual IBIs are integers measured in units of passes.

We have studied 10 networks with different realizationsthe random memories. A detailed analysis was carried ou

FIG. 1. ~a! Network dynamics before kindling, showing noislow-amplitude oscillations of the net activitySup around an averagevalue nearn510; ~b! dynamics of the kindled network, showinspontaneous bursting at irregular time intervals;~c! smoothenednetwork activity with IBIs (T1 ,T2 etc.! computed from positivethreshold crossings denoted by dark circles. These results weretained using random sequential updating.

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six of these networks. Each network shows a wide distrition of IBIs and the value of the average IBI varies substtially from one network to another, reflecting different dgrees of kindling. Similar sample-to-sample variations habeen found@25# in hippocampal slices very similar to thosused in the experiment of Ref.@9#. The variability of IBIsshown in the raw-data plots in Ref.@25# is qualitatively simi-lar to that found in our simulations.

C. Stochasticity of the bursting dynamics

The irregular bursting shown in Fig. 1~b! is found only forrandom sequential updating~RSU!, i.e., when the neuronare updated one at a time in a random sequence. Operaally, one of theN neurons is selected randomly for updatinits local field is calculated, and its state variableSi is set to 0or 1 depending on the sign of the local field. This procesrepeatedN times to complete one pass. To ascertain theture of the dynamics, we also checked the time evolutionthe network without the randomness in the updating scheand the refractoriness. In these simulations, refractorinwas implemented by always resetting the network to a fimemory and the network was updated following two diffeent deterministic schemes:fixed sequential updating~FSU!andparallel updating~PU!.

In the FSU, a specific sequence for updating theN neu-rons is chosen randomly, and then the neurons arealwaysupdated in that sequence. For most choices of the upsequence, the network remains confined to the low-actiattractor andSup exhibits perfectly periodic, low-amplitudeoscillations. This is shown in Fig. 2~b!. However, for certain‘‘special’’ choices of the update sequence, the network

FIG. 2. ~a! Network dynamics under random sequential upding, showing excursions to the high-activity attractor at irregutime intervals, and noisy low-amplitude oscillations of the nettivity Sup(t) between these excursions;~b! fixed sequence updatin~switched on att51000) that traps the network in the low-activitattractor, leading to low-amplitude periodic behavior ofSup(t); ~c!fixed sequence updating~switched on att51000) that producesperiodic transitions to the high-activity attractor.

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found to make periodic transitions to the high-activity attrator, as shown in Fig. 2~c!. In the PU scheme, all the neuronare updated simultaneously. The dynamics under PU is qutatively similar to the behavior found for FSU. Dependingthe network state at the time of switching on the PU, tnetwork either remains confined to the low-activity attracor makes periodic transitions to the high-activity attracwith somewhat higher degree of correlated activity than tfound for FSU. These observations establish that the unlying stochasticity in the updating scheme is an essenrequirement for the network to exhibit spontaneous burstsimilar to that observed in brain slices~large variability inthe IBIs!.

Among the three different updating procedures discusabove, the RSU scheme also appears to be most ‘‘physifor describing the behavior of brain slices. A realistic dscription of the dynamics of a network of biological neuromay be obtained@26# in terms of coupled differential equations that govern the time evolution of membrane potentand ionic currents. In this description, a neuron fires~i.e.,generates an action potential! if its net membrane potentiaexceeds a threshold. Once a neuron fires, it cannot fire aunless a short refractory period has elapsed. In the discrtime dynamics of our simple model, the unit of time~onepass! approximately corresponds to this refractory peri@23#: each neuron can, on an average, fire once duringperiod. However, if no external ‘‘clocklike’’ mechanism ipresent, then there is no reason to expect that the membpotentials of different neurons in a network would reachthreshold value at exactly the same time. Since externaputs that may synchronize or order in time the firing ofthe neurons are not expected to be present in such ‘‘in vitro’’brain slices, the asynchronous dynamics of the RSU schshould be appropriate for describing such systems.therefore, used the RSU scheme in all our simulations.results described above suggest that the time series ofobtained from these simulations are stochastic.

We have carried out several tests to check whether orthe IBI time series are truly stochastic. We looked for corlations in the IBI time series by calculating the autocorretion function

C~n!51

m2n (i 51

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~Ti2T!~Ti 1n2T!/s2, ~4!

where m is the total number of IBIs in the dataset,T51/m( i 51

m Ti is the average IBI, ands251/m( i 51m (Ti

2T)2. As shown in Fig. 3, the IBIs are completely uncorrlated in time in all the networks. A similar absence of corrlation has been found in the experiment of Ref.@25# on hip-pocampal slices.

The probability distribution of the IBIs follows a Poissoform, P(Tn)}exp(2aTn), in all the networks. Sample resultfor network 1 are presented in Fig. 4, in which both tsimulation data and the fit to a Poisson form are shown. Tsmoothening procedure we have used for low-pass filteof the data does not allow the occurrence of IBIs smathan about 2tsm580 passes. For this reason, the Poisson

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tribution is cut off near the origin. The value of the paramea shows large sample-to-sample variation in the rangetween 831024 and 331023. The form of the distribution ofthe IBIs recorded in brain-slice experiments@9,17,25# hasnot been reported in the literature. The data in the first retmaps of the IBI time series shown in Refs.@17,18# indicate adistribution that exhibits a sharp maximum close to the lend and decreases slowly to zero at the high end. Thisimilar to a Poisson distribution truncated at the low end

Our results suggest that the transition of the network frthe low-activity state to the high-activity one is a stochasprocess that has a small probabilityp512exp(2a) of oc-curring at each step. For this value ofp, the probability ofhaving two successive transitions separated by a time inval of Tn units would be proportional to (12p)Tn

FIG. 3. Autocorrelation functionC(n) of the IBIs from six net-works.

FIG. 4. Probability distribution of IBIs from network simulatio~circles! and Poisson simulation~triangles! in the absence of controin network 1. The dashed line is the best fit to an exponen~Poisson! form.

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5exp(2aTn), which would match the Poisson distribution othe IBIs found in our simulations. To check this, we hacarried out Monte Carlo simulations of a Poisson procwith probability p of occurrence at each time step. In thsimulation, a random numberr distributed uniformly be-tween 0 and 1 is generated at each time step, and the orence of a ‘‘burst’’ is recorded ifr<p. Sincer is distributeduniformly between 0 and 1, the probability ofr being lessthan or equal top is p itself. So, this Monte Carlo procedurensures that the bursts occur with probabilityp. Once a burstoccurs, the next one cannot occur during the next 2tsm timesteps. We use this procedure to generate the same numb‘‘IBIs’’ as in the original time series and compute their ditribution function. As shown in Fig. 4, the distribution functions obtained from the network and Poisson simulationsessentially identical. Similar close agreements betweenresults of the network and Poisson simulations of the behior under different control procedures~see Sec. III F for de-tails! provide strong support to the conclusion that the Itime series obtained in our simulations are stochastic inture.

A defining feature of deterministic chaos is extremely sesitive dependence on initial conditions, manifested in anponential divergence of trajectories starting from neapoints. We have carried out a simple calculation to lookthis behavior in our IBI time series. This involves a calcution of the quantity^ ln@d(k)#& where d(k)5uTi 1k2Tj 1ku,and the average•••& is over pairsi and j with d(0)5uTi2Tj u,n, n being a small number. A similar calculation fothe chaotic logistic map shows an exponential growth^d(k)& with increasingk, and a crude estimate of the valuethe Lyapunov exponent can be obtained by fitting the depdence of^ ln@d(k)#& on k to a straight line. For our IBI timeseries, on the other hand, we find that^ ln@d(k)#& is indepen-dentof bothk andn, indicating that the IBIs are independerandom variables. Similar results are obtained for a twdimensional delay embedding,

d~0!5uTi2Tj u1uTi 112Tj 11u,n,~5!

d~k!5uTi 1k2Tj 1ku1uTi 1k112Tj 1k11u.

Here, we find that ln@d(k)#& is independent ofn, and inde-pendent ofk for all k.1. These observations confirm thstochastic nature of theTi time series and indicate that thuse of more sophisticated methods to calculate Lyapuexponents from our time-series data would not be meanful. Further evidence for the stochastic nature of the IBI timseries is obtained from the surrogate analysis describeSec. IV.

The behavior ofd(k) found for our IBI time series issimilar to the observation in Ref.@17# that a small cluster ofpoints in the first return map of the experimentally obtainIBI time series expands rapidly and covers nearly the whallowed region within two iterations. Due to this rapid epansion, conventional methods of estimating the value oflargest Lyapunov exponent from time-series data couldbe used to analyze the experimental data. Instead, atechnique called ‘‘expansion rate analysis’’@17# was used.l

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III. SIMULATIONS OF CONTROL METHODS

As discussed in the preceding section, the IBI time seobtained from simulations of the stochastic neural netwexhibit general characteristics that are quite similar to thof the IBI time series obtained in brain-slice experime@9,25#. It is, therefore, interesting to enquire whether somethe control results reported in Ref.@9# can also be reproducein simulations of the network model. We have carried oextensive simulations of the behavior of six networks unthe application of various control methods. The resultsthese simulations are described in this section.

A. Apparent unstable periodic orbits

Following the analysis of IBIs from spontaneously buring brain slices@9#, we searched for UPO-like trajectoriesfirst return maps of IBIs from six different networks. UPOof period one appear as UFPs in the first return map defias a plot ofTn vs Tn21. Unlike typical return maps fromdeterministic chaotic dynamics, the points in the return mconstructed from our IBI time series fill up the entire spareflecting the underlying stochastic dynamics of the netwoThe Poisson distribution of the IBIs leads to a clusteringpoints near the origin and the axes. A return map construfrom a 1320-IBI time series collected over 53105 passes fornetwork 1 is shown in Fig. 5. The lack of any geometricstructure in the return maps of our IBI time series is simito the behavior found in brain-slice experiments@9,17,18#.

Although the return maps do not have the structurepected for deterministic dynamics, we found many UPO-ltrajectories which have multiple occurrences and satisfythe criteria adopted by Schiffet al. @9#. Many such purely

FIG. 5. Recurrent UPO-like trajectories in network 1 chosencontrol. Three sequences starting at different times~denoted by 1)are shown along with the calculated stable and unstable manifdenoted by arrows.

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accidental occurrences of UPO-like trajectories were fouin all the six networks studied. The apparent UPOs are idtified from trajectories in the return map that first approacpoint on the identity line (Tn5Tn21) along a line~stablemanifold! and then diverge away along a different line~un-stable manifold!. Specifically, we searched for trajectoriethat satisfy the following criteria. If the sequence of pointsmarked as 1,2,3. . . , then we have the following.

~1! The point 1 is away from the identity line, and thpoint 2 lies close to the identity line.

~2! At least two subsequent points diverge away fropoint 2 along a straight line with a negative slope with manitude greater than unity. The intersection of this line wthe identity line is taken to be the location of the UFP.

~3! The points on the unstable manifold alternate ontwo sides of the identity line~flip-saddle criterion!.

~4! At least two such sequences of points, starting at dferent times, exhibit similar trajectories near the same U~recurrence criterion!.

~5! The best-fit straight lines to the first points of all therecurrent sequences intersect the identity line near the UThis line, whose slope is usually close to zero~the magnitudeof the slope must be less than unity!, defines the stable manifold.

With short sequences~3–5 points!, it is not possible tocheck with certainty that the sequence of points definingunstable manifold diverge exponentially away from the UFThis condition is approximately satisfied by most of the squences obtained with the criteria listed above. The mafolds are determined through least-square fits to the cosponding points in all the recurrent sequences. If the stamanifold does not pass through the UFP, a straight line pallel to this line and passing through the UFP is taken asstable manifold. An example of recurrent UPO-like trajecries found in network 1 is shown in Fig. 5 along with thcomputed stable and unstable manifolds. Another examfrom the same network may be found in Ref.@21#.

After the determination of the UFPs and the correspoing stable and unstable manifolds, simulations of variocontrol methods~chaos control, periodic pacing, demanpacing and anticontrol! around one selected UFP for eachthe six networks were carried out. The chosen UFPs andslopes of the manifolds used for these control applicatiare summarized in Table I. These control methods invothe generation of bursts at appropriately chosen times byapplication of an external stimulation. An external stimution was modeled in our simulations by a reduction of tinhibition strengthw from 0.6 to 0.24 for five passes.

For each of the control methods, we carried out simutions in which 5000 bursts are generated. A statistical sumary of the results of these control simulations is providedTable II. This table gives the values of the average IBIT, thestandard deviations of the IBIs, and the percentage of IBIabove (Tn.T* 1tsm), below (Tn,T* 2tsm), and near(T* 2tsm<Tn<T* 1tsm) the selected UFPT* . The percent-age ~sgs! of IBIs generated by applications of the contrstimulation is also listed for each of the control methods.

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B. Chaos control

The applied chaos-control method@9,12# is a variant ofthe original technique proposed by Ott, Grebogi, and Yo@10#. In this method, also referred to as the proportional pturbative feedback~PPF! method @27#, artificial bursts areproduced through timely stimulations that attempt to plathe state point on the stable manifold of the UPO selected

TABLE I. The UFP (T* ), the slope of the stable manifold (ms),and the slope of the unstable manifold (mus) chosen for control insix networks. The percentages of IBIs above, below, and nearstable manifold~SM! during chaos control are given in columns5, and 6, respectively.

No. T* ms mus Above Below 6tsm

SM SM of SM

1 345 20.041 21.0623 0.1 55.0 44.92 1121 20.127 21.18 1.6 55.8 42.63 823 0.097 21.54 0.6 56.6 42.84 447 0.342 21.43 7.0 30.2 62.85 1099 20.257 21.468 2.1 71.1 26.86 937 20.476 21.683 0.0 65.5 34.5

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control. Whenever a burst occurs, the current IBITn is com-puted and the time at which the next burst should occuorder to place the state point on the stable manifold inreturn map is calculated from the equation of the stamanifold. This time istn111ms(Tn2T* )1T* , wheretn11

is the time of occurrence of the last burst (Tn5tn112tn) andms is the slope of the stable manifold. An external stimulusapplied at this time if no natural burst occurs before its aplication. If a natural burst occurs prior to the applicationthe stimulus, then the time of application of the stimulusrecalculated using the value of the new IBI. In Fig. 6, twsequences of IBIs with such control from networks 1 andare plotted. The control clearly increases the number of Iwith values close toT* . These control results are qualitatively similar to those reported by Schiffet al. for brainslices.

However, a close inspection of the control results shothat the control mechanism ‘‘works’’ mostly by preventinthe occurrence of natural IBIs above the stable manifold. Texternal stimulus that attempts to place the state point onstable manifold almost always produces a burst. This mathe occurrence of bursts withTn11.ms(Tn2T* )1T* ex-tremely rare. The occurrence of spontaneous IBIs below

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TABLE II. Comparison of IBI statistics from six networks during no control~NC!, chaos control~CC!,

periodic pacing~PP!, and demand pacing~DP!. The average IBIT, the number of passes to generate 50IBIs, and the standard deviations of the IBIs are given in columns 3, 4, and 5, respectively. Columns 6and 8 list the percentage of IBIs generated above, below, and near the UFP, respectively. The last~sgs! lists the percentage of IBIs generated by external stimuli.

No. Type Duration T s Above Below Around sgs

3105 T* T* T*

1 NC 18.442 369 297 34.3 53.8 11.9CC 12.575 251 100 0.1 54.4 45.5 38.5PP 10.915 218 102 2.3 71.0 26.7 47.7DP 12.322 246 99 0.1 56.1 43.8 37.4

2 NC 66.027 1320 1236 42.3 54.8 2.9CC 41.304 826 492 19.9 55.2 24.9 39.0PP 32.075 641 397 1.7 74.5 23.8 32.7DP 40.482 810 469 1.6 55.3 43.1 40.1

3 NC 44.16 883 816 38.2 57.2 4.6CC 27.561 551 290 0.6 73.1 26.3 40.0PP 23.071 461 272 1.1 76.7 22.2 45.4DP 27.964 559 283 0.5 59.6 39.9 36.7

4 NC 48.938 979 930 62.3 32.1 5.6CC 21.977 440 392 12.4 48.6 39.0 57.0PP 17.876 358 186 7.3 48.9 43.8 50.1DP 22.326 447 397 7.2 31.5 61.3 55.4

5 NC 47.191 944 898 28.9 68.3 2.8CC 35.945 719 468 19.4 71.0 9.6 24.6PP 27.853 558 376 2.5 83.6 13.9 39.7DP 34.966 699 442 2.8 67.7 29.4 26.9

6 NC 49.383 988 953 36.4 60.2 3.4CC 33.041 661 369 23.2 68.1 8.7 32.8PP 25.477 510 303 0.4 78.5 21.1 49.2DP 31.113 622 314 0.0 60.0 40.0 38.1

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FIG. 6. Comparison of control methods. Results of chaos control~CC!, periodic pacing~PP!,demand pacing~DP!, and anticontrol~AC! runsthat produce 300 IBIs each in~a! network 1 and~b! network 4 are shown. No control is applieduring the intermediate periods of 150 IBIs. IBIgenerated by the external stimuli are denoted bdifferent symbol~stars! to distinguish them fromthe spontaneous IBIs.

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stable manifold are not significantly affected by the applition of the control. Consequently, the control proceduworks better~i.e., the fraction of IBIs nearT* increases! ifT* is small and the slope of the stable manifold is closezero. This is illustrated in Fig. 6 where the ‘‘quality’’ ocontrol in network 4, for which bothT* and ms are largerthan those for network 1~see Table I!, is clearly found to beworse than that in network 1: the fraction of IBIs nearT* issmaller, and the fraction of IBIs greater thanT* is higher innetwork 4~see Table II for the values of these fractions!. Theresults of control are also better if the average IBI in tundisturbed network is large: this decreases the numbespontaneous IBIs lying below the stable manifold. All thecharacteristics of the control results may be found instatistical summary given in Table II. These results are qutatively similar to those reported in Ref.@12# for a stochasticsingle-neuron model.

In a truly deterministic chaotic system, intermittent appcations of the external stimulus should lead to effective ctrol. Once the state point is correctly placed close tostable manifold by an external stimulus, a few subsequpoints should continue to remain near the stable maniwithout any external intervention. The trajectory would thdeviate away, needing external intervention after some tto put the state point back in the vicinity of the stable mafold. In our simulations, intermittent applications of the cotrol stimulus did not produce effective control. This is cosistent with the purely stochastic nature of the burstdynamics of our model. Intermittent control applicatiowere also unsuccessful in the brain-slice experiment@9#, al-though this may also be attributed to inaccuracies in thetermination of the slope of the stable manifold from noi~and possibly nonstationary! physiological data.

If the state point remains close to the stable manifold aapproaches the UFP for some time after it has been plathere by a control stimulus~as it should in a system exhibiting deterministic chaos!, then a majority of the IBIs foundnear the fixed point in the presence of control should ocnaturally rather than through applications of the exter

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stimulus. In our simulations where the underlying dynamis stochastic, almost all the IBIs clustering around the fixpoint value and giving the impression of successful conare generated by the control stimulus. This can be seeFig. 6 where the stimulus-generated IBIs and the naturoccurring ones are plotted with different symbols. As empsized in Ref.@28#, such an analysis is absolutely necessfor an unambiguous interpretation of the apparent succeschaos control found in the brain-slice experiment. Sincechaos control procedure attempts to put the system pointhe stable manifold, a relevant quantity to consider isfraction of IBIs found within6tsm of the stable manifoldduring the application of chaos control. Values of this quatity for the six networks are given in Table I. These valuare close to~slightly higher than! the fractions of stimulus-generated IBIs given in Table II. This confirms that nearlypoints found near the stable manifold during chaos conare generated by the control stimulus.

As indicated in Table I, a small fraction of IBIs are founto lie above the stable manifold even when the control isThis is due to rare failures of the external stimulus to genate a burst. Increasing the strength of the stimulus~i.e., in-creasing the amount by whichw is decreased during the application of the stimulus! improves the quality of control bydecreasing the number of IBIs lying above the stable mafold. This is analogous to the improvement in control fouin the brain-slice experiment when two pulses instead ofwere used to generate a burst at the appropriate time.

Slutzky et al. @18# have recently reported the resultsapplications of chaos control methods to spontaneoubursting hippocampal slices similar to those studied in R@9#. One of the control methods used by them is similarours, with the difference that in their experiment, the contstimulus to generate a burst was applied only when thetance of the state point from the selected UFP exceedesuitably chosen ‘‘control radius’’Rc . Large values ofRccorrespond to intermittent applications of the control stimlus, and this method reduces to the one used by us inlimit of very small values ofRc . Slutzky et al. studied the

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effects of changing the value ofRc on the efficacy of control.They also kept track of the fraction of bursts generatedthe control stimulus. They found that the quality of contrmeasured by the variance of the IBIs generated during ctrol, increases asRc is decreased. However, decreasingRc

also increased the fraction of stimulus-generated IBIs, incating that the improvement of control was mostly due togeneration of many IBIs nearT* by the external stimulusThis is similar to the results of our simulations. The valuesT* and ms for the UPOs selected for control were not rported in Ref.@18#. It appears from the plots shown in thpaper that good control was achieved only whenT* is closeto the lower limit of the IBI distribution and the value ofumsuis close to zero. As mentioned above, a very similar behawas observed in our simulations of PPF control. Slutzet al. have also reported the results of applications ofadaptive tracking method@29# that continuously refines thestimates ofT* andms during the application of control, ana new protocol called ‘‘state point forcing’’ that helps in dtermining the validity of fixed point estimates. Simulatioof these methods for our network model are planned fornear future.

C. Periodic pacing

In periodic pacing, the network is stimulated at fixed itervals equal toT* irrespective of the occurrence of naturspikes. As a result, many artificial bursts at intervals equaT* are generated, and naturally occurring bursts correspoing to IBIs larger thanT* are eliminated. Periodic pacinalso produces many new IBIs smaller thanT* —these aredue to spontaneous bursts occurring between two succeapplications of the control stimulus.

Two sequences of IBIs with periodic pacing from neworks 1 and 4 are plotted In Fig. 6. From the densitypoints nearT* , it is evident that in network 1 periodic pacing is less effective than chaos control in increasing theriodicity of the bursts. The reverse is true in network 4. Tquantitative differences between the results obtained fortwo networks may be found in Table II. These differencesmainly due to differences in the slopes of the stable mafolds. As discussed in the preceding section, chaos conworks well in network 1 because bothT* and umsu are rela-tively small. Sinceumsu is close to zero in this case, trying tplace the state point on the stable manifold during chcontrol is basically the same as periodic pacing, exceptthe external stimulation is not applied if a spontaneous boccurs before its application. This eliminates the large nuber of small IBIs found in periodic pacing. In networkchaos control produces a large number of IBIs near the stmanifold, but only a small fraction of them lie nearT* be-causeumsu is relatively large. So, if the fraction of IBIs neaT* is taken to be the measure of the effectiveness ofcontrol procedure, then periodic pacing works better thchaos control in this network.

It is clear from Fig. 6 that a large fraction of the IBIfound nearT* during periodic pacing are generated by texternal stimulus. In contrast to the behavior during chcontrol, a large number of IBIs lower thanT* are also gen-

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erated by the external stimulus. This is due to spontanebursts between two successive applications of the stimulusmall number of IBIs with values larger thanT* are alsofound. As noted above, these are due to occasional failurethe stimulus in producing the required burst.

These features are similar to the results of the brain-sexperiment@9#. However, some of the successful periodpacing results in the brain-slice experiment, where the nuber of IBIs belowT* are also apparently reduced, cannotreproduced in our network simulation. Periodic pacing in tnetworks always increases the number of IBIs belowT* ~seeTable II!. In fact, the amount of time required to generatefixed number of IBIsdecreases~the value of the average IBbecomes smaller! whenever any form of control is applied tour networks. Applications of the external stimulus to genate bursts at appropriate times during control increasesnumber of bursts, but the naturally occurring bursts in ostochastic model are not affected by the control. As a resthe total number of bursts in a fixed period always increaduring control. This is clear from the data shown in TableA convincing demonstration that the average IBI can bein-creasedby the application of one of these control methodsbrain-slice experiments would provide strong support toclaim @9,16# of deterministic behavior in such systems.

D. Demand pacing

In demand pacing, the network is prevented from proding any IBI higher than the UFP by timely applications of thexternal stimulus. Given the occurrence of a burst at timetn ,the external stimulus is applied at timetn1T* if no sponta-neous burst occurs before its application. This proceduressentially the same as the PPF chaos-control methodscribed above if the slope of the stable manifold is closezero. This similarity between PPF control and demand ping has been noted in a recent experimental study@18#. Forstable manifolds of larger slope, however, demand pacinmore effective than chaos control in increasing the periodity of the bursts. Demand pacing always works better thperiodic pacing by eliminating some of the short IBIs genated in periodic pacing through the occurrence of spontaous bursts between two successive applications of the enal stimulus. In all the networks, the number of IBIs near tfixed point in demand pacing is close to the sum of tnumber of stimulus-generated bursts and the numberspontaneous bursts expected near the fixed point in thedisturbed network during the period of application of contr

The results of applications of demand pacing to netwo1 and 4 are plotted in Fig. 6. The characteristics of dempacing mentioned above are evident from this figure, aalso from the quantitative results shown in Table II. Theresults are consistent with the underlying stochastic dynics of the networks. Specifically, the increased effectivenof demand pacing over chaos control whenumsu is large isexpected only if the dynamics is stochastic. In truly chaodynamics, a chaos-control mechanism that makes use osaddle structure near a candidate UPOs is expected tqualitatively different from, and more effective than demapacing which amounts to just cutting off the IBIs above t

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UFP. Therefore, if the bursting in brain slices were trudeterministic, then, irrespective of the slope of the stamanifold, chaos control should always be more effectthan demand pacing in increasing the periodicity ofbursts. An experiment in which the effectivenesses of thtwo methods are compared would be very helpful in undstanding the nature of the underlying dynamics and in eslishing unambiguously the occurrence of chaos.

E. Anticontrol

Anticontrol methods@30–33# use the underlying deterministic dynamics to increase or preserve its complexSchiff et al. @9# tried to achieve this by applying PPF-typcontrol for an appropriately chosen anticontrol line~‘‘repel-lor’’ line ! that passes through the selected UFP. They mthe ad hocchoice of the repellor line to be the mirror imagof the unstable manifold about the identity line. It wclaimed that this choice effectively diverted the state poaway from the UFP and increased the variability of IBIs.the brain-slice experiment, anticontrol applications had liited success: only a small fraction of the attempts appeato be effective in increasing the variability of the IBIs. Quatatively similar results were obtained by Christini and Clins @12# in their simulations of a stochastic single-neurmodel.

A proper assessment of the success of anticontrol requa quantitative measure of its effectiveness in increasingvariability of the IBIs. No such measure was defined in Re@9# and @12#. We have defined two dimensionless quantitthat may be used for this purpose. The first one is the r(r 1) of the standard deviations of the IBIs recorded duringanticontrol run and during a run of the same duration withany control. Values ofr 1 greater than unity would indicatsuccess of anticontrol. A second quantity we have consideis the ratio (r 2) of the standard deviation and the mean of tIBIs recorded during anticontrol. This quantity measuresscaled variability of the IBI distribution. Values ofr 2 ob-tained from simulations of our networks in the absenceany control lie between 0.8 and 0.95, the smaller valuesing obtained in networks with relatively small values of tmean IBI ~see Table II!. Deviations of these values fromunity, the value expected for a Poisson distribution, areto the small-T cutoff in the IBI distribution. Substantiallyhigher values ofr 2 during the application of anticontrowould indicate its success.

Repellor lines obtained from the prescription of Schet al. @9# have negative slopes with magnitude smaller thunity. As a result, the control stimulus attempts to generalarge IBI after the occurrence of a small one, and vice veAlso, stimulus-generated IBIs cannot exceed the value giby the point at which the repellor line intersects they axis.As noted above, this prescription for selecting the slopethe repellor line does not have any clear justification. A dferent choice was made by Christini and Collins@12# whotook the repellor line to be the mirror image of the unstamanifold about a line passing through the UFP and perpdicular to thex axis. Since the slope of the unstable manifois always negative with magnitude larger than unity, t

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slope of such a repellor line is always positive and largthan unity. Such lines impose no upper bound on stimulgenerated IBIs. A small IBI leads to a subsequent smallinduced by the control stimulus, and a large IBI leads tolarge one unless a natural burst occurs prior to the appltion of the control stimulus.

In the absence of any clear-cut prescription for choosthe slopes of the repellor line, we have systematically stuied the effectiveness of anticontrol for many different valuof s. We found that if the number of IBIs recorded during aapplication of anticontrol is sufficiently large (.103) to givereliable statistics, then the standard deviation of the IBIs ding anticontrol is smaller than that for the undisturbed nwork (r 1,1) for all choices ofs. The reason for this behavior is the same as that mentioned above for the result thaaverage IBI obtained from our simulations in the presenceany form of control is smaller than that for the undisturbnetwork. In our stochastic networks, applications of the ctrol stimulus generate additional bursts, but cannot prevthe occurrence of spontaneous bursts. Therefore, the numof relatively small IBIs can be increased, but the upper limof the IBI-distribution cannot be extended beyond the vafound in the same network without any control. However,simulations of anticontrol over relatively small periods duing which only a few hundred bursts are generated, ‘‘scessful’’ anticontrol (r 1.1) was observed in a few runs witlarge, negative values ofs. This is purely due to statisticafluctuations: values ofr 1 smaller than unity were obtainewhen the simulations were continued for longer perioSimilar to the experimental results, anticontrol failed in mocases even during these small-duration runs. Applicationanticontrol in brain slices for longer durations would be usful to determine whether the reported successes of antictrol were due to the deterministic nature of the dynamicsdue to inadequate statistics.

While this anticontrol method cannot increase the stdard deviation of the IBIs in our stochastic networks, it cincrease the relative variabilityr 2 of the IBI distribution forappropriate choices of the slopes of the repellor line. Wefound that values ofs close to unity are most effective inproducing large values ofr 2. Repellor lines withs slightlyabove unity do not impose any upper or lower bound onstimulus-generated IBIs, and are definitely off the manifolAnticontrol with such values ofs produces many additionaartificial IBIs of small magnitude. This decreases both tmean value and the standard deviation of the IBIs. Tformer is decreased more in comparison with the latter,that the value ofr 2 is increased. Our simulations withs closeto unity yield values ofr 2 in the range 1.2–1.6. The resulof two such applications of anticontrol over relatively smdurations are shown in Fig. 6. These plots are visibly simto those shown in Ref.@12#. We have also found that increasing the strength of the control stimulus increases the efftiveness of anticontrol, essentially by ensuring that the stimlus produces the desired burst. This is similar to the resreported in Ref.@9# for ‘‘double-pulse’’ control methods.

F. Poisson simulations

We have carried out Poisson simulations of all the conprocedures described above. As mentioned in Sec. II C, th

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simulations are based on the assumption that the transitiothe network to the high-activity state~i.e. the occurrence of aburst! is a stochastic process that has a probabilityp of oc-curring at each time step. The values ofp for the six net-works studied were computed from the results of netwsimulations. The additional ingredient required for simuling control methods is the process of generating burstsapplying an external stimulus. This was modeled by anotindependent stochastic process that has a high probabilitp8of occurrence at the time of application of the external stimlus. Values ofp8(0.9<p8<1.0) were obtained from networsimulations where we kept track of the fraction of applistimuli that produced bursts.

All the measurements made in the network simulatio~the measured quantities are listed in Tables I and II! wererepeated in our Poisson simulations. A comparison of thesets of results for each network showed good agreemenall cases. Such comparisons for the probability distributof the IBIs during chaos control, periodic pacing and demapacing in network 1 are shown in Figs. 7, 8, and 9, resptively. These plots illustrate the close agreement betweenresults of network and Poisson simulations. Similar resfor network 6 may be found in Ref.@21#. These results confirm the stochastic nature of the IBI time series generateour network model.

FIG. 7. Probability distribution of IBIs from network simulatio~dotted line! and Poisson simulation~dark triangle! during chaoscontrol in network 1.

FIG. 8. Probability distribution of IBIs from network simulatio~dotted line! and Poisson simulation~dark triangle! during periodicpacing in network 1.

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IV. STATISTICAL SIGNIFICANCE OF UPOS

As discussed in Secs. II and III, simulations of our schastic neural network model qualitatively reproduce allfeatures found in the brain-slice experiment of Ref.@9#. Thisobservation raises the following important question: doresults reported in Ref.@9# establish the occurrence of deteministic chaos in spontaneously bursting hippocampal slicor alternatively, do these results reflect apparent determinfeatures that appear purely as a matter of chance in achastic time series. This issue was first raised by Chrisand Collins@12# who showed that simulations of a stochassingle-neuron model reproduce most of the results repoin Ref. @9#.

More recently, a number of new methods have been pposed for locating UPOs in experimental time-series dand for assessing their statistical significance through sugate analysis. Soet al. @14# have proposed a method bason adynamical transformationthat utilizes the local dynamics to enhance the probability measure about distinct UPin state space. In subsequent work@15,16#, this method hasbeen applied to analyze IBI datasets obtained from the brslice experiment. These studies claim to have found UPOa statistically significant number of windowed dataseSimilar results have been found in a recent study@17# ofhippocampal bursting under three different experimenconditions. In an analysis using the same dynamical transmation, but a different criterion for assessing the statistisignificance of candidate UPOs, Van Quyenet al. @19# havereported detection of statistically significant UPOs in humepileptic activity. Pierson and Moss@13# have proposed atopological recurrencecriterion for locating UPOs in noisytime-series data. Using this method in combination with srogate analysis, Pei and Moss@34# have detected statisticallsignificant UPOs in the dynamics of the crayfish caudal ptoreceptor. Using the same method, de la Pridaet al. @35#have found statistically significant UPOs in ISI data obtainfrom immature hippocampal networks.

To test the reliability of these statistical methods, andcheck whether or not simulations of our model can reprodsome of the results obtained from their applications toperimental data, we have used these methods to analyzstochastic IBI time series obtained from our network sim

FIG. 9. Probability distribution of IBIs from network simulatio~dotted line! and Poisson simulation~dark triangle! during demandpacing in network 1.

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lations. The results of these studies are described in thistion.

A. Method of So et al.

We first consider period-1 orbits that correspond to fixpoints in the first return map. In the method of Soet al., suchfixed points are detected by performing the following dnamical transformation@14# on the IBI time series$Tn%:

Tn5Tn112sn~k!Tn

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where

sn~k!5Tn122Tn11

Tn112Tn1k~Tn112Tn!. ~7!

Under this transformation, most of the points in the lineregion around a fixed pointT* are mapped to the vicinity oT* . Therefore, a histogram of the probability distributior(T) of the transformed time series$Tn% shows a sharp peaat each fixed point. Spurious peaks, which depend stronon the parameterk, are eliminated by averaging the distribtion over a large number of random values ofk obtained ask5kR whereR is a random number distributed uniformly i@-1,1#, and k is a parameter chosen to ensure that theterms on the right-hand side of Eq.~7! have the same ordeof magnitude. The statistical significance of candidate UPcorresponding to peaks ofr(T) is assessed from a comparson with the distribution obtained for a large number of cafully constructed, randomized versions~surrogates! of theoriginal dataset.

In the analysis@15,16# of the brain-slice data using thimethod, it was argued that in biological systems the pareters governing the underlying dynamics are not stationartime. This argument contradicts to some extent the interptation of the control results reported in Ref.@9#, where it wasclaimed that candidate UPOs detected from a search duri‘‘learning’’ period can be used to control the dynamicssubsequent time intervals of substantial duration. Neverless, assuming the system parameters to vary slowly in tthe IBI dataset obtained from the experiment was brokeninto small overlapping@15# or nonoverlapping@16# windowsof size m, which is chosen to be small enough to maintastationarity and large enough to provide adequate statisThe transformation of Eq.~6! was then applied to the data ieach window to obtain the distributionr(T).

Using many realizations of Gaussian-scaled phashuffled surrogates@36# ~random shuffling that approximately preserve the power spectrum! of the original data ineach window, the statistical probability that the observpeaks in the transformedr(T) could be found in randomsurrogates was estimated. For each realization of the sugate data, the same procedure was applied to calculatedistributionrsur(T). Then, from this collection of$rsur(T)%,the ensemble-averaged distributionrsur(T) was obtained.

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w~ T!5rsur~ T!2 rsur~ T!. ~8!

For each surrogate, the maximum deviationW5max@w(T)# was measured, and the fractionJ(W8) of sur-rogates with maximum deviationW.W8 was found using alarge number of surrogates. Then, the statistical significaof a candidate UPO atT* was estimated from the value oJ(W* ) whereW* 5r(T* )2 rsur(T* ). Specifically, the sta-tistical confidence level for the detection of an UPO westimated to be@12J(W* )#3100%.

We have carried out the same analysis using windowdatasets obtained from our network simulations. We scathe IBI data to match the experimental data used inanalysis of Ref.@15#. In that work, 1834 ISIs collected ove25 min were analyzed for the detection of UPOs. In netwo1, the same number of IBIs are generated in 685 409 pasWe, therefore, scaled our time unit by setting one pass eto 0.0022 s. In our analysis, we used 100 surrogates, tk55.0, and averaged the distribution over 500 values ofrandom numberR. Representative results of this analysis ashown in Fig. 10. We found that this method successfurejects the UPO-like trajectories found in the first retumaps of the IBI time series obtained from our network simlationsonly if the dataset used in the analysis is sufficienlarge (m>2048). Our results for a dataset withm54096,

FIG. 10. Statistical significance of UPOs~method of Soet al.,see Sec. IV A! in windowed datasets. The plots show histograms

the probability distributionr(T) ~solid line! of the transformed se-ries for the original dataset, and the average probability distribu

r(T) ~dotted line! of the transformed series for the surrogates. Rsults for a 512-IBI dataset and a 4096-IBI dataset are shownpanels~a! and ~b!, respectively. Inset: Plots ofJ(W), the fractionof surrogates with maximum deviation exceedingW ~see text!. The

maximum deviationWmax betweenr(T) and r(T) occurs atT

50.31 s(Wmax54.97) for ~a!, and atT50.51 s (Wmax58.11) for~b!. Arrows at Wmax indicate the statistical significance leveJ(Wmax) ~see Sec. IV A!.

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10~b!. The maximum value of the deviationW5r(T)2 rsur(T) is found to beWmax58.11, which occurs atT50.51. However, the fraction of surrogates with maximudeviation exceeding this value is fairly large:J(Wmax).0.39, as shown by the arrow in the inset of Fig. 10~b!. Thismeans that this candidate UPO should not be consideredtistically significant at the confidence level of 95% usedthe analysis of Ref.@16#. In a similar analysis carried out fo35 datasets withm>2048, we found a statistically significant UPO in only one dataset withm52048. Thus, this sur-rogate analysis recognizes our IBI time series to be stoctic if a sufficiently large dataset is used in the analysis.

However, the same analysis leads to false detection‘‘statistically significant’’ ~at the 95% confidence leve!UPOs in a substantial fraction of windowed datasets ifm issmall. The results for a window withm5512, obtained fromsimulations of network 1 and showing the presence of a ‘‘stistically significant’’ UPO, are plotted in Fig. 10~a!. In thiscase, the probability of finding the peak (Wmax54.97 atT50.31) in the surrogates is very low, around 1%, as incated by the arrow in the inset of the figure, leading tofalse identification of a statistically significant UPO at 95confidence level. A similar plot for am5128 dataset may befound in Ref.@21#.

We have carried out a systematic analysis for differvalues ofm and observed that the criteria adopted in Re@15# and@16# fail to recognize our IBI datasets as stochasin a significant fraction of trials ifm is small. Results of thisanalysis, carried out for IBI datasets collected from ten nworks, are summarized in Table III. The total numbernonoverlapping windows used in the analysis for each wdow sizem is listed in the third column of the table. Thsecond column lists the percentage of networks with at leone window with a statistically significant UPO at 95% cofidence level, and the fourth column lists the percentagef ofwindows~combined from all networks! with statistically sig-nificant UPOs. These results are quite similar to thoseported in Ref.@16# for extracellular brain-slice data at relatively low potassium levels.

A similar analysis for a two-dimensional delay embeddiof the IBI time series obtained from the brain-slice expe

TABLE III. Statistical significance of UPOs in windowedatasets from ten networks, according to the method of Soet al.~see Sec. IV A!. The number of IBIs in each nonoverlapping window and the percentage of networks showing statistically significUPOs are given in columns 1 and 2, respectively. The total numof windows used in the analysis and the percentage of windshowing statistically significant UPOs are given in columns 3 a4, respectively.

Window size % significant No. of windows % significannetworks windows

32 80 544 8.8264 100 272 10.29128 50 136 8.82

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ment was performed in Ref.@15#. We have carried out thesame analysis using simulation data obtained for networIn our analysis, we scaled the IBI data to match the tiscale of the experiment, used parameter values identicathose in Ref.@15# (k55.0, 500 values ofR, and 100 surro-gates!, and considered a large number of overlapping anonoverlapping windows of different size. The results of thanalysis are very similar to those described above. Forample, we found ‘‘statistically significant’’ UPOs in 75 out o1024 nonoverlapping 32-IBI windows. The values off werefound to vary between 1.6% and 9.4% as the window sizmwas varied between 32 and 1024, and no UPO was foundm>2048. In Ref.@15#, the results of the surrogate analysfor the detection UPOs in overlapping 128-IBI windowwere presented in the form of a gray-scale plot from whichis difficult to extract quantitative information such as thvalue off. For this reason, we could not compare our resudirectly with those of Ref.@15#.

The method of Ref.@14# can be extended@15# to the de-tection of orbits of period higher than 1. Using this methoSo et al. @16# have detected statistically significant UPOshigher period in intracellular data obtained from brain slicSince intracellular recordings reflect a combination of colletive and single-neuron dynamics, our network with binaneurons is not appropriate for the modeling of such daNevertheless, we have carried out a simple analysis fordetection of period-2 orbits in the simulation data for nwork 1. Since a period-2 orbit corresponds to a fixed pointhe second iterate of the dynamics, we considered the tseriesT1 ,T3 ,T5 , . . . ,Tm21 (T2 ,T4 ,T6 , . . . ,Tm) consistingof only the odd-numbered~even-numbered! terms in am-IBIwindow (m even! of the original time series, in our analysfor the detection of period-2 orbits. The method describabove for the detection of statistically significant fixed poinwas then applied separately to the odd- and even-numbdatasets. If a period-2 orbit is present in them-IBI window ofthe original dataset, then this calculation should show tstatistically significant fixed points in both the odd- aneven-numbered datasets, and the locations of these fipoints should be nearly the same in the two datasets.have confirmed that this method succeeds in detecperiod-2 UPOs in the Henon map.

Since the IBIs in our stochastic time series are indepdent random variables, the time series obtained by retainonly even- or odd-numbered terms are statistically identto the original one. We, therefore, expect that a perioanalysis for the even- and odd-numbered datasets wyield results that are very similar to those obtained in oanalysis for period-1 orbits in the original dataset. In ooriginal period-1 analysis, we found very few windows wimore than one statistically significant UPOs. Further, sinthe even- and odd-numbered datasets are statistically ipendent, it is highly improbable that both of them wouexhibit UPOs at approximately the same locations. So,probability that the even- and odd-numbered datasets woeach exhibit two statistically significant UPOs at appromately the same locations is expected to be very smallour stochastic time series. The results of our analysisconsistent with this expectation. In our analysis of a 32 7

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IBI time series from network 1 with window sizes varyinfrom 128 to 8192, we did not find any window in which thcriteria mentioned above for the detection of period-2 orbwere satisfied. As expected, the fractionf of windows show-ing statistically significant period-1 orbits was approximatethe same for even- and odd-numbered datasets, andnumber was close to that obtained in our original perioanalysis. However, very few windows showed two periodorbits, and the results obtained for the even-numberedfrom a particular window did not match the results for todd-numbered data from the same window. These resconfirm the stochastic nature of our IBI time series. It woube interesting to carry out a similar analysis for the braslice data.

In Ref. @16#, the values off obtained from the analysis ointracellular data are found to be significantly higher ththose calculated from the data obtained from extracellurecordings. This may be due to the following reason. Textracellular data reflect the collective dynamics of the nwork, whereas intracellular recordings can resolve‘‘spikes’’ that occur during the bursting of the neuron frowhich the recordings are made. Therefore, extracellularcordings provide information about the times of occurrenof population bursts, while the time series obtained frointracellular recordings is one of interspike intervals~ISIs!,which reflect a combination of collective and single-neurdynamics. This difference is clear from the characteristime scales@16# of these two kinds of data: the typical IBI ithe extracellular data is of the order of 1 s, whereastypical ISI in the intracellular data is much smaller, of thorder of 0.05 s. Given these facts, the larger values ofobtained in the analysis of the intracellular data may beterpreted as evidence for determinism in the spiking dynaics of individual neurons. This interpretation is supportedthe results of de la Pridaet al. @35#, who carried out a carefusurrogate analysis of intracellular data recorded from immture hippocampal networks. They found that the IBIs in thtime series are purely stochastic, whereas the spikesform the intrinsic structure of individual bursts show evdence for deterministic dynamics. These results suggestdeterminism and stochasticity need not be mutuaexclusive—the same system may show both kinds of dynics at different levels and time scales. The dynamics ofnetworks illustrates the same point: the time evolution ofnetwork in the low- and high-activity states is mostly detministic, whereas transitions between these two statesstochastic.

Since the confidence level for the detection of statisticasignificant UPOs was set at 95% in our calculations, valof f for the stochastic IBI data obtained from simulationsour network model are expected to be near 5%. Our resfor datasets withm>2048 are consistent with this expecttion. The significantly higher values off ~near 10%, as shownin Table III! obtained from our analysis of windowedatasets withm<128 must then be attributed to insufficiesampling. The similarity between our results and thoseported in Ref.@16# for the extracellular brain-slice data sugests that the relatively large values off obtained in Ref.@16#may also be due to insufficient sampling. It would be int

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esting to check this possibility by carrying out a surrogaanalysis of the brain-slice data using windows of larger siA reduction of the values off with increasingm would sug-gest stochastic behavior. However, a similar trend would abe found if the dynamics of brain slices is nonstationary.it would be difficult to distinguish between stochasticity anonstationarity from an analysis of this kind. This appearsbe a drawback of this method of detecting UPOs in expmental time-series data. Another possible source of insucient statistics in the analysis@16# of the brain-slice data isthe relatively small number~between 33 and 71! of win-dows. Our studies show that the difference between theculated and expected values of the fractionf of windowsshowing statistically significant UPOs decreases as the nber of windows used in the analysis is increased.

A recent study@17# has reported large values off ~be-tween 12.5% and 42.1%! obtained from analysis of win-dowed brain-slice data. In this work, IBI data were recordunder three different experimental conditions: high-@K1#,low-@Mg21#, and the presence of GABAA-receptor antago-nists. The method of Soet al. @15# with a two-dimensionaldelay embedding was applied to overlapping 256-IBI wdows. The reported values of the fraction of experimeshowing at least one window with a statistically significaUPO are similar to those obtained in our simulations. Hoever, the values off are considerably larger. The statisticsignificance of the results obtained in the low-@Mg21# andGABAA-antagonist experiments is questionable becaussmall number~16 and 19, respectively! of windows wereused in these experiments. In a dataset of 256 nonoverping 32-IBI windows obtained from simulations of netwo1, we found two separate 19-window and 16-window sements for which the values off are 42.1% and 31.3%, respectively, although the value off obtained for the full 256-window dataset is much smaller~10.5%!. Short segments o;20 windows with similar values off were also found inother datasets. Thus, our model can produce such largeues off if a small number of windows is considered. However, we did not find any segment of 60 or more windowswhich the value off is more than about 15%. So, it is unlikely that our model would reproduce the period-1 resultRef. @17# for high-@K1# ( f 529% for 62 windows! and theresult of Ref.@16# for 10.5 mM K1 ( f 521% for 67 win-dows!. More data to ascertain the statistical significancethese experimental results would be very useful.

B. Method of Van Quyen et al.

Van Quyenet al. @19# used a similar method to analyzISI data obtained from human epileptic activity. We hacarried out the same analysis for our simulation data.repeat their method as closely as possible, the IBI data fnetwork 1 were rescaled so that the range over whichIBIs vary remains approximately the same as that in theperimental data. The range of the experimental data is ab250 ms, whereas that for the IBI data from network 1about 1000 passes. So, one pass in the network dynawas assumed to be equal to 0.25 ms. Other parametersin our analysis, such as bin sizes for histograms, the radiu

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circles on the identity line~see below! and the number ofsurrogates, were maintained at the values used by Van Qet al.

We used a dataset withm51024 IBIs ~this number isclose to 1062, the number of points in the dataset used inanalysis of Ref.@19#! obtained from simulations of networ1 in our analysis. The IBI dataset$Tn%, n51,2, . . . ,1024,used in our analysis and a histogram~bin size5 5ms! of thedistribution of the IBIs are shown in Figs. 11~a! and 11~b!.Our IBI data exhibit a Poisson distribution which is differefrom the nearly Gaussian distribution found in Ref.@19#. Thedensity of points within a small circle of radius 8 ms aroua pointTn on the diagonal line in the return map providescrude measure of the probability of successive IBIs to remnear the sameTn @19#. A plot of this density~probability ofrecurrence! as a function ofTn is shown in Fig. 11~d!. Fivemain peaks in the recurrence probability, atT150.027 s,T250.045 s,T350.06 s,T450.087 s, andT550.125 s, arefound. The locations of these peaks in the return mapindicated by circles in Fig. 11~c!. This inhomogeneous clustering of points in the return map is qualitatively similarthat observed in Ref.@19#. Therefore, the interpretation@19#of this clustering as a significant indication of correlationsuccessive IBIs may not be correct.

Since our IBIs follow a Poisson distribution, we could nuse one of the methods used in Ref.@19# to construct thesurrogate datasets. We considered three types of randomrogates that preserve the distribution of the IBIs. These arfollows.

~1! Gaussian-scaled phase-shuffle [36].Random shufflingof the dataset in a manner that approximately preservespower spectrum. This is the same as the method use

FIG. 11. Probability of recurrence of IBIs, according to thmethod of van Quyenet al. ~see Sec. IV B!. Panel~a!: 1024 IBIs(Tn) from network 1, rescaled to match the time scale of theperiment. Panel~b!: Distribution of the IBIs. Panel~c!: Return mapwith clustering of points~recurrence! along the identity line,marked with circles of radius 8 ms. Panel~d!: the probability den-sity of the points along the diagonal.

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generate surrogates of type II in Ref.@19#.~2! Random shuffle.Simple random shuffling of the IBI

dataset. This maintains the distribution but destroys corrtions.

~3! Poisson simulation.Surrogate datasets generated frosimulations of a Poisson process, as described in Sec. I

The transformation of Eq.~6!, with k55 and 500 valuesof R, was then applied to the original dataset and the sugates, and the probability distributionr(T) for the originaldata and the distributionsrsur(T) for each of the surrogatewere calculated. Following Ref.@19#, we considered 39 dif-ferent realizations of each type of surrogate data. The statical significance of a candidate UPO corresponding topeak ofr(T) at T5T* was estimated by a one-tailed MonCarlo test@19#:

P5@number of cases$rsur~T* !>r~T* !%#11

~number of surrogates!11. ~9!

Van Quyen et al. considered peaks withP50.025, i.e.,rsur(T* ) , r(T* ) for all 39 surrogates, to be statisticallsignificant.

The results of our analysis for surrogates of type 1plotted in Fig. 12. The lower panel shows histogramsr(T) and the average ofrsur(T). The upper panel shows thdegree of statistical confidence in the detection of UPThis is defined as 100(12P)%. This analysis finds thepeaks atT5 to be statistically significant according to thcriterion ~97.5% confidence level! adopted by Van Quyenet al. for all three types of surrogates. It is important to nothat the remaining four peaks are found to be spuriousspite of the high weightage of these peaks in Fig. 11~d! andlarge clustering of points in this region of the return m@Fig. 11~c!#. In the surrogate analysis, peaks atT50.125 s,0.1875 s for surrogate type 1,T50.015 s, 0.125 s, 0.1875 s

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FIG. 12. Statistical significance of candidate UPOs nearpeaks in Fig. 11~d!. The solid line in panel~b! shows the histogram

of r(T) and the dashed line shows the average ofrsur(T) obtainedfrom 39 surrogates. The upper panel~a! shows the degree of statistical confidence in the detection of UPOs near the location ofpeaks. The confidence level is defined as 100(12P)% with P de-termined from a Monte Carlo test@Eq. ~9!#.

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0.21 s for surrogate type 2 andT50.015 s, 0.125 s, 0.21 s fosurrogate type 3 were found to be statistically significan97.5% confidence level. The peaks atT50.1875 s,0.21 s,found to be statistically significant for two of the three typof surrogates, are probably irrelevant, as they are in theof the distribution and no clustering of points on the identline in the return map@Fig. 11~c!# is found around these twopoints.

Following Ref.@19#, we further examined the significancof the apparent UPO atT5 by looking for recurrent UPO-liketrajectories around this point in the first return map.shown in Fig. 13, such trajectories are indeed present inIBI dataset. The ‘‘recurrent UPO’’ nearT5 has a stable manifold, y50.1735x10.1015, and an unstable manifoly521.4x10.2978. These are indicated in the figurelines with arrows.

Surprisingly, this analysis fails to reject all UPO-like trjectories in our stochastic IBI dataset as spurious. Ratherresults arequantitativelysimilar to those of Ref.@19#. Thisclearly shows that the methods and criteria used byQuyenet al. are not completely reliable for identifying truUPOs. Therefore, some of the apparent UPOs found by tin human epileptic activity may not be real. The false potive results obtained in this analysis may be due to the smsize of the dataset. When a larger dataset containing 8IBIs from the same network simulation was analyzed, snificantly different results with no prominent peaks in tdistributionr(T) were obtained. The use of a small numbof surrogates may be another reason for the failure ofmethod to identify our dataset as stochastic.

C. Topological recurrence criterion

In this method@13,34#, the number of UPO-like trajectories in the first return map of the original dataset is compuand compared with the values of the same quantity obtafor surrogates of the original dataset. The following criteare used for identifying UPO-like trajectories.

Level 0.There must be three sequential points thatproach the identity line at successively decreasing perpdicular distances, followed by three sequential points t

FIG. 13. Detection of UPO-like trajectories around the ‘‘stattically significant’’ UFP atT550.125 s~see Fig. 12!.

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depart at successively increasing distances. The third poicommon to both sequences.

Level 1.A straight-line fit to the three approaching poinmust have a slopems with 21<ms,0 and a straight-line fitto the three departing points must have a slopemus with2`,mus,21.

Level 2.The distance of the point of intersection of thetwo straight lines from the identity line must be smaller thhalf of the mean of the distances of the five points in tapproaching and departing sequences.

We have analyzed IBI datasets from ten networks for ‘‘ecounters’’ that satisfy the criteria mentioned above. LetN bethe number of encounters detected for a dataset. Usingsame criteria, encounters are searched for in 100 Gausscaled phase-shuffled surrogates of the original dataThen, the statistical measureK5(N2^Ns&)/ss is computed,where^Ns& is the mean andss is the standard deviation othe number of encounters found in the surrogate datasetsGaussian statistics,K>2 represents a statistically significadetection of the presence of UPOs with a confidence legreater than 95%.

This surrogate analysis successfully identifies all thedatasets of varying lengths from the ten networks as stoctic, i.e., statistically significant UPOs are not found. Tvalue of K is always close to zero and sometimes negatfor large datasets. For a few datasets of small lengthm<1024), values ofK greater than 1 are found, but they anot statistically significant at the 95% confidence level. Fsome of the datasets, we also calculated the maximum vaNmax, of the number of encounters found in the surrogatIn all these cases, the number of encountersN in the originaldataset was found to be smaller thanNmax. Typical resultsobtained for network 1 are shown in Table IV. Similar resuwere obtained in an analysis of datasets obtained from Pson simulations.

TABLE IV. Results of surrogate analysis of IBI datasets fronetwork 1, using the topological recurrence criterion~see Sec.IV C!. Columns 1–7 list, respectively, the size of the datasem~number of IBIs!, the number of Gaussian-scaled phase-shufflsurrogates used in the analysis, the numberN of UPO-like trajecto-ries ~encounters! found in the original dataset, the average numb^Ns& of encounters found in the surrogates, the standard deviass of the number of encounters found in the surrogates, the sttical measureK5(N2^Ns&)/ss , and the maximum valueNmax ofthe number of encounters found in the surrogates.

m No. of surrogates N ^Ns& ss K Nmax

64 100 1 0.56 0.82 0.54 3

128 100 2 1.34 1.02 0.65 4

256 100 2 2.41 1.72 20.24 7

512 100 3 4.29 2.07 20.62 10

1024 100 8 8.52 3.18 20.16 16

2048 100 18 16.87 3.97 0.28 25

4096 100 35 34.72 6.44 0.04 53

8192 100 80 72.67 8.73 0.84 96

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We have also carried out a systematic analysis of thecentagef of windows showing statistically significant UPOaccording to this criterion. Using a dataset with 32 768 IBobtained from simulations of network 1, we found that tvalues of f for different window sizes (64<m<2048) liebetween 0 and 6.3. These results are consistent with thepectation that about 5% of the windows of our stochasticdata should show statistically significant UPOs at 95% cfidence level.

We, therefore, conclude that this method correctly idefies the datasets from our network and Poisson simulationstochastic. However, we were surprised to find that tmethod also identifies certain datasets of small length, gerated from the logistic map in the chaotic regime, as schastic. For example, this method showed the presencstatistically significant UPOs in only 5 of 32 datasetslength 1024 generated from the logistic map witha53.99.This observation argues against a complete reliability ofmethod. If the probability that a deterministic system visthe neighborhood of a particular UPO during the time spof the dataset being analyzed is comparable to or smathan the probability of purely accidental appearance of UPlike trajectories in a surrogate of the original dataset, ththis method would wrongly identify the original data as schastic. Since the probability of a deterministic system viing the neighborhood of a UPO can, in general, be qusmall ~the logistic map is an example of this!, this methodwould not work in all cases. However, the error that thmethod would produce in such cases is in a direction thaopposite to that of the other methods of surrogate data ansis we have considered. This method may identify a deministic time series as a stochastic one, but it is unlikelyidentify a stochastic time series as a deterministic one.other methods we have considered lead to the~incorrect!conclusion that our stochastic IBI series has some charaistics of a deterministic one.

V. SUMMARY AND DISCUSSION

The main result of our investigation is the demonstratthat simulations of our neural network model reproduce mof the features found in brain-slice experiments@9,18#, and insurrogate analysis@15–17# of the data obtained from thesexperiments. This result suggests that our model is an appriate one for describing spontaneous population burstinhippocampal slices. We have also shown that the variabof the IBIs in our network model is purely stochastic in ogin. This casts serious doubt on the conclusions reacheRefs. @9,15–18# about signatures of deterministic chaosthe underlying dynamics of brain slices. Rather, our wostrongly suggests that the bursting dynamics is wellscribed by ‘‘bistability’’ with stochastic transition betweenlow-activity ~normal! attractor and a high-activity~epileptic!attractor of the underlying neuronal network. The stochasity of the IBIs in our simulations is a consequence of the uof random sequential updating. We believe that this updascheme is the most ‘‘physical’’ one for describing the behior of brain slices, because any external ‘‘clocklike’’ mechnism that may synchronize or order in time the firing of t

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neurons is not expected to be present in suchin vitro sys-tems. However, this may not be ruled out inin vivo systems,where the region of bursting activity continuously receivinputs from many other regions of the living brain@37#.

There exist other results that support the conclusionthe bursting dynamics of brain slices is stochastic. The sptaneous bursting behavior of rat hippocampal slices, ppared in exactly the same way as the ones studied inexperiment of Schiffet al. @9#, were tested for signatures odeterminism in Ref.@25#. Three different tests were conducted for each of six slices. Out of the 18 tests, only oshowed a statistically significant positive signature of detminism. In intracellular recordings from immature hippoampal networks@35#, the IBIs were found to be purely random, following a Poisson distribution. Slutzkyet al. @17# didnot find any evidence of determinism in their ‘‘expansiorate analysis’’ of IBI data from hippocampal slices. A physologically realistic computer model@38# of synchronized epi-leptiform bursts induced by high potassium concentrationrat hippocampal slices had to include stochasticity inform of spontaneous excitatory postsynaptic potentials meled by an independent Poisson process for each pyramcell. This is similar to the behavior found in our simulationwhere the presence of stochasticity in the updating procenecessary for generating irregular IBIs.

Our work, however, does not rule out the possibility ththe bursting behavior in brain slices is a manifestationdeterministic chaos. The demonstration that our stochamodel reproduces the experimentally observed behavior dnot imply that other, possibly deterministic, models wounot be equally~or more! successful in describing the expermental results. Our systematic analysis of the stochasticdata obtained from simulations of our network model poiout several signatures of stochastic dynamics in the conbehavior and in the results of surrogate analysis for thetistical significance of UPOs. These signatures have bdiscussed in detail in Secs. III and IV. It would be interestito look for these signatures in brain-slice experiments. Sinvestigations would be very helpful in establishing the trnature of the collective dynamics of brain slices.

In our simulations, applications of external stimuli durincontrol procedures do not alter the parameters~such as syn-aptic strengths, the relative strength of inhibition and timdelays! that govern the network dynamics. This may nottrue in networks of biological neurons. There are reasonbelieve that repeated external stimulation causes permachanges in the connectivity pattern of neuronal netwothrough synaptic plasticity. The phenomenon of kindli@22#, for which the model studied here was originally deveoped@20#, is a well-known example of changes in netwoproperties caused by repeated external stimulation. Freqstimulations may also alter the dynamics of biological nworks by making the neurons fatigued or refractory. Edence for stimulation-induced changes in network propermay be found in Ref.@18#. Any interpretation of the resultsof brain-slice experiments must take into account the pobility of such changes. These changes may also accounsome of the differences between the results of our netw

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simulation and those of brain-slice experiments.Results of our surrogate analysis of IBI datasets for de

mining the statistical significance of UPO-like trajectorifound in these purely stochastic time series cast serdoubts on the reliability of the statistical methods usedRefs. @15,16,19#, especially for datasets of relatively smalength. Our work suggests that far more stringent criteria

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analysis are required for unambiguously establishing sigtures of deterministic chaos in experimental time-series d

ACKNOWLEDGMENTS

B.B. thanks S. J. Schiff for useful comments and for sugesting the demand pacing analysis, and D. Kaplan for usdiscussions.

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Stochastic neural network model for spontaneous bursting in hippocampal slices