Abstract. In this paper, we investigate the use of partialcorrelation analysis for the identification of functionalneural connectivity from simultaneously recorded neuralspike trains. Partial correlation analysis allows one todistinguish between direct and indirect connectivities byremoving the portion of the relationship between twoneural spike trains that can be attributed to linearrelationships with recorded spike trains from otherneurons. As an alternative to the common frequencydomain approach based on the partial spectral coher-ence we propose a new statistic in the time domain. Thenew scaled partial covariance density provides addi-tional information on the direction and the type,excitatory or inhibitory, of the connectivities. In simu-lation studies, we investigated the power and limitationsof the new statistic. The simulations show that thedetectability of various connectivity patterns depends onvarious parameters such as connectivity strength andbackground activity. In particular, the detectabilitydecreases with the number of neurons included in theanalysis and increases with the recording time. Further,we show that the method can also be used to detectmultiple direct connectivities between two neurons.Finally, the methods of this paper are illustrated by anapplication to neurophysiological data from spinaldorsal horn neurons.

1 Introduction

Major progress has been made in understanding thefunctions of the nervous system on molecular, cellular,and systemic levels. In contrast, the organization andfunction of local neuronal networks are largely un-known. With the exception of the retina, the goal ofrelating network connectivity to function has not been

achieved in any area of the vertebrate nervous system.Thus, a major challenge of future neuroscience will be toapproach the complex interplay of functionally con-nected ensembles of neurons.

Functionally relevant neuronal connections can bedefined and identified by changes in discharge proba-bility in a postsynaptic neuron by activity in excitatoryor inhibitory presynaptic neurons. While direct, i.e.,monosynaptic, connections are devoid of interneurons,polysynaptic pathways involve one or more intercalatedneurons. A number of experimental methods are nowavailable to simultaneously record action potential dis-charges of large numbers of neurons. These includemultiple single-neuron recordings with an array of mi-croelectrodes (Kruger 1983) or tetrodes (Gray et al.1995), confocal calcium imaging (Fetcho and O’Malley1995), multisite recordings from cultured neurons withplanar electrode arrays (Maeda et al. 1995), or multiplesingle-neuron recordings with voltage sensitive dyes(Grinvald et al. 1988).

Any reliable description of neuronal network func-tion needs to identify excitatory vs. inhibitory andmonosynaptic vs. polysynaptic connections. The neu-rophysiological identification of synaptic connections istypically based on the recorded times of discharges ofthe neurons under study. The association between suchneural spike trains is commonly measured by the cross-correlation histogram (Perkel et al. 1967), which is thehistogram of the times of occurrence of all output spikesrelative to the times of occurrence of all input spikes.The peaks and troughs in such histograms indicate thechange of probability for the occurrence of an outputspike due to the presence of an input spike, thusrevealing the type and the time delay of the synapticconnection. However, when analyzing the structure of alarger neural ensemble we cannot infer from the cross-correlation histogram to what extent these changes aredue to a direct connection between the two neurons, toindirect connections via other recorded neurons, or tocommon inputs.

A frequency domain approach based on the methodof linear partialization has been discussed by Brillinger

Correspondence to: M. Eichler(e-mail: eichler@statlab.uni-heidelberg.de,Tel.: +49-6221-548970, Fax: +49-6221-545331)

Biol. Cybern. 89, 289–302 (2003)DOI 10.1007/s00422-003-0400-3� Springer-Verlag 2003

Partial correlation analysis for the identification of synaptic connections

Michael Eichler1, Rainer Dahlhaus1, Jurgen Sandkuhler2

1 Institut fur Angewandte Mathematik, Universitat Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany2 Brain Research Institute, Vienna University Medical School, Spitalgasse 4, 1090 Vienna, Austria

Received: 2 August 2002 /Accepted: 28 January 2003 / Published online: 14 October 2003

et al. (1976), Rosenberg et al. (1989), and Dahlhaus(2000) for neurophysiological data. Although themethod allows one to distinguish direct from indirectconnections, no distinction is made between excitatoryand inhibitory connections.

In this paper, we present a partialization analysis inthe time domain. The method is based on a scaled ver-sion of the partial covariance density, which can beestimated by frequency domain methods. The scaledpartial covariance density combines the advantages ofthe crosscorrelation histograms and the partializationanalysis in the frequency domain. In particular, it can beinterpreted in the same way as the crosscorrelation his-tograms with peaks and troughs indicating excitatoryand inhibitory connections, while on the other hand itallows for the discrimination of direct and indirectconnections and common inputs. The theoretical back-ground is presented in Sect. 2. In Sect. 3, the resultsfrom a simulation study are given. Furthermore, themethod is applied to a data set consisting of ten spiketrains from spinal dorsal horn neurons.

2 Methods

2.1 Multivariate point processes

The method discussed in this section is entirely based onthe relation between the sequences of times of occur-rence of discharges recorded from different neurons.Therefore, these neural spike trains can be representedby stochastic point processes. A point process in generalrefers to an ordered sequence of isolated events fsjgj2Zoccurring randomly in time. Such a point processassociated with the spike train of neuron a can berepresented by the counting process NaðtÞ, where NaðtÞ isthe total number of action potentials of neuron a fromtime 0 up to time t. For the details about point processeswe refer to the monographs of Cox and Isham (1990)and Daley and Vere-Jones (1988).

Subsequently, it will be assumed that the multivari-ate point process N is stationary, that is, its propertiesare independent of the time at which the process isobserved. We further assume that increments of theprocess N well separated in time are independent and,finally, that the points of the process do not occursimultaneously.

The connectivities between neurons are commonlystudied by pairwise second-order statistics. In the timedomain, the conditional intensity function mabðuÞ de-scribes the probability of occurrence of a spike of type aat time u > 0 given that a spike of type b has occurred attime 0

mabðuÞ ¼ ProbfdNaðuÞ ¼ 1jNbð0Þ ¼ 1g=du

where dNaðtÞ ¼ Naðt þ dtÞ � NaðtÞ. Alternatively, onemight consider the cross-covariance density at lag ubetween processes Na and Nb:

qabðuÞ ¼ covfdNaðtÞ; dNbðt þ uÞg=ðdt duÞ ð1Þ

which is related to the conditional intensity functionby qabðuÞ ¼ mabðuÞpb � papb, where pa ¼ ProbfdNaðtÞ ¼1g=dt is the mean intensity of process Na. Since theincrements dNaðtÞ of a point process are discrete with val-ues of only 0 or 1, it follows that varfdNaðtÞg ¼ padt and,consequently, the right-hand side of Eq. 1 does notconverge for dt; du! 0 if a ¼ b and u ¼ 0. In thissituation, the autocovariance density qaa is defined suchthat it is continuous also at u ¼ 0. Because of thisbehavior of the variance the correlation

corrfdNaðtÞ; dNbðt þ uÞg ¼ffiffiffiffiffiffiffiffiffiffi

dt dup qabðuÞ

ffiffiffiffiffiffiffiffiffi

papbp

for u 6¼ 0 or a 6¼ b converges to 0 as dt; du! 0. Theexpression, however, suggests that the following scaledcovariance density (SCD)

sabðuÞ ¼qabðuÞffiffiffiffiffiffiffiffiffi

papbp

be used as a measure for the linear dependence of thetwo processes Na and Nb. This statistic is not boundedlike an ordinary correlation but still has a number ofuseful properties. For example, in the case of Hawkes’self-exciting processes (Hawkes 1971a,b), which havelinear relationships, the scaled covariance densities stayconstant if the mean intensities of all componentprocesses are increased by a common factor. We notethat van den Boogaard et al. (1986) used a differentscaling for the covariance density that, however, hasbeen motivated only by giving particularly simpleexpressions for the second-order dependencies in thediscussed neuron model.

In the frequency domain, the corresponding second-order parameters are the cross spectrum between pro-cesses Na and Nb:

fabðkÞ ¼1

2p

Z

1

�1

qabðuÞ expð�ikuÞ du

and for a ¼ b the autospectrum faaðkÞ

faaðkÞ ¼pa

2pþ 1

2p

Z

1

�1

qaaðuÞ expð�ikuÞdu

The association between two processes Na and Nb is

measured by the spectral coherence jRabðkÞj2 where

RabðkÞ ¼fabðkÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

faaðkÞfbbðkÞp

The spectral coherence is bounded and takes valuesbetween 0 and 1. Let dðT Þa denote the finite Fouriertransform of process Na:

dðT Þa ðkÞ ¼Z

T

0

hðt=T Þ expð�iktÞ dNaðtÞ

290

where h is a data taper (cf. Brillinger 1981). Then thespectral coherence satisfies

jRabðkÞj2 ¼ limT!1

�

�corrfdðT Þa ðkÞ; dðT Þb ðkÞg

�

�

2

If the spectral coherence vanishes for all frequencies, thetwo spike trains are linearly independent, whereas avalue of one indicates a perfect linear relationshipbetween the two processes.

If the process has been observed on the interval ½0; T �,the estimates for the second-order parameters in the timedomain are typically based on the crosscorrelation his-togram:

mmabðuÞ ¼#�

ðj; kÞju� h < rj � sk < uþ h; rj 6¼ skg2hNbðT Þ

where ‘‘#’’ stands for ‘‘the number of ’’ and frjg andfskg denote the observed spike times for neuron a and b,respectively, up to time T . In the frequency domain, thefinite Fourier transform dðT Þa can be computed efficientlyat the Fourier frequencies kj ¼ 2pj=T by a fast Fourieralgorithm based on a discrete approximation of theprocess Na (Rigas 1992). The spectral density matrixf ðkÞ ¼

�

fabðkÞ�

a;b¼1;...;d can then be estimated compo-nentwise by

f ðT Þab ðkÞ ¼1

2pH2T

X

j2ZwðT Þðk� kjÞdðT Þa ðkjÞdðT Þb ð�kjÞ

where Hk ¼R 1

0 hðtÞkdt, wðT ÞðkÞ ¼ MT wðMT kÞ for somesequence MT !1, and w is some kernel function withbounded support (Brillinger 1981). A more detaileddiscussion of frequency methods may be found inBrillinger et al. (1976) and Rosenberg et al. (1989).

2.2 Partial correlation analysis

When analyzing the structure of a larger neural net, thequestion arises as to what extent the association betweentwo processes Na and Nb is due to a direct connectionbetween the two neurons or whether the observedcorrelation can be attributed to an indirect connectioninvolving other neurons, say, c1; . . . ; cr, or is only due toa common input from these neurons. This question canbe addressed by second-order statistics using the methodof partialization. The basic idea is to measure theassociation between the two processes Na and Nb afterthe linear effects of the multivariate processNC ¼ ðNc1 ; . . . ;NcrÞ

0 have been subtracted. It is clearthat in this approach the meaning of direct and indirectconnections depends on the measured network and thatthe inclusion of any additional neurons may lead to theidentification of a formerly direct connection as beingindirect.

Brillinger et al. (1976), Rosenberg at al. (1989), andDahlhaus et al. (1997) have discussed the partializa-tion analysis in the frequency domain and recom-mended the use of the partial spectral coherencejRabjCðkÞj2 where

RabjCðkÞ ¼fabjCðkÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

faajCðkÞfbbjCðkÞp

with the partial spectral densities

fabjCðkÞ ¼ fabðkÞ � faCðkÞfCCðkÞ�1fCbðkÞ ð2Þ

Let NNajCðtÞ be the best predictor of NaðtÞ based on theprocess NC, which is given by

NNajCðtÞ ¼ lþZ

t

0

Z

1

�1

/ðs� uÞdNCðuÞ ds

where / has Fourier transform //ðkÞ ¼ faCðkÞfCCðkÞ�1and l ¼ pa � //ð0ÞpC . We can then define the partialresidual process

eajCðtÞ ¼ NaðtÞ � NNajCðtÞ

The association between Na and Nb after removing thelinear effects of NC can now be described by thecorresponding second-order parameters of the partialresidual processes eajC and ebjC . For example, the partialspectral coherence can be seen to be the correlationbetween the Fourier transforms dðT ÞajC and dðT ÞbjC of thepartial residual processes eajC and ebjC :

jRabjCðkÞj2 ¼ limT!1

�

�corrfdðT ÞajCðkÞ; dðT ÞbjC ðkÞg

�

�

2

In the time domain, we can similarly define the partialcovariance density of Na and Nb by

qabjCðuÞ ¼ covfdeajCðt þ uÞ; debjCðtÞg

which is related to the partial cross-spectral density by

qabjCðuÞ ¼Z

1

�1

fabjCðkÞ expð�ikuÞ dk

Similarly, the partial autocovariance density qaajC is theFourier transform of faajCðkÞ � pa=2p, where the secondterm corresponds to the variation of the increments ofthe partial residual process eajC. Since the best predictorNNajC is continuous in time, the variation of deajCðtÞ isdominated by the variation of the predicted process Na,whereas the variation due to the best predictor isnegligible, that is

varfdeajCðtÞg ¼ varfdNaðtÞg þ terms of order dt2

¼ padt þ terms of order dt2

This leads to the following definition of the scaled partialcovariance density (SPCD)

sabjCðuÞ ¼qabjCðuÞffiffiffiffiffiffiffiffiffi

papbp

Like the conditional intensity, the scaled partial covari-ance density gives information not only on the strengthof a connection between two neurons but also on thetype and the direction of the connection. For an

291

excitatory connection from neuron b to neuron a thetime delay u0 due to the propagation of the actionpotential along the axon and the synaptic delay results ina peak of sabjCðuÞ at u ¼ u0. Similarly, for an inhibitoryconnection we obtain a trough at u ¼ u0.

The partial spectra and partial spectral coherencescan be estimated by replacing the spectral densities fijðkÞin Eq. 2 by their estimates ff ðT Þij ðkÞ. For a multivariatepoint process N with components Na, a ¼ 1; . . . ; d, allpartial spectral coherences jRabjV nfa;bgðkÞj2 with V ¼f1; . . . ; dg can be computed efficiently by inversion ofthe spectral matrix (Dahlhaus 2000). Let gðkÞ ¼ðgabðkÞÞa;b¼1;...;d denote the inverse spectral matrix. Thenwe have

faajV nfagðkÞ ¼1

gaaðkÞ

RabjV nfabgðkÞ ¼ �gabðkÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gaaðkÞgbbðkÞp

faajV nfabgðkÞ ¼faajV nfagðkÞ

1� jRabjV nfabgðkÞj2

fabjV nfabgðkÞ ¼RabjV nfabgðkÞ

1� jRabjV nfabgðkÞj2

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

faajV nfagðkÞfbbjV nfbgðkÞq

The first two equations have been proved by Dahlhaus(2000), while the other two equations follow from theseand the inverse variance lemma (e.g. Whittaker 1990,Prop. 5.7.3). Replacing the spectral matrix by itsestimate we obtain estimates for all partial second-orderstatistics in the frequency domain. Further, we canestimate the partial covariance density qabjCðuÞ for a 6¼ bby the Fourier transform of the partial spectral densityestimate:

qqðT ÞabjCðuÞ ¼Z

p=bT

�p=bT

ff ðT ÞabjCðkÞ expð�iukÞ dk ð3Þ

where bT converges to zero as T !1. The properties ofthe estimate can be further improved by the introductionof a convergence factor cT ðkÞ, which is monotonicallydecreasing in k and increasing in T . Finally, the scaledpartial covariance densities can be estimated by substi-tuting the empirical means ppðT Þa ¼ NaðT Þ=T for the meanintensities pa.

2.3 Partial correlation graphs

Partial second-order statistics can be used for thedefinition of undirected graphs that visualize the corre-lation structure of a multivariate point process. Suchpartial correlation graphs have been used by Dahlhauset al. (1997) for the identification of synaptic connec-tions from neural spike train data. A detailed discussionof partial correlation graphs and their properties in thecontext of multivariate time series can be found in

Dahlhaus (2000). The same results also hold for pointprocesses.

Suppose that we observe a multivariate point processN ¼ ðN1; . . . ;NdÞ0. We identify the components Na withthe vertices of a graph, i.e., we have the set of verticesV ¼ f1; . . . ; dg. An edge between vertices a and b isdefined to be missing if the corresponding processes Naand Nb are uncorrelated at all lags after the linear effectsof all other components have been subtracted. With thedefinitions of the last section this is the case if and only ifthe partial residual processes eajV nfa;bg and ebjV nfa;bg areuncorrelated at all lags and consequently the scaledpartial covariance density sabjV nfa;bgðuÞ is zero for all u.

In practice, the partial correlation graph must beestimated from the empirical scaled partial covariancedensities

ssðT ÞabjV nfa;bgðuÞ ¼qqðT ÞabjV nfa;bgðuÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ppðT Þa ppðT Þb

q

which in the case of a missing edge are only approxi-mately zero. Therefore, statistical tests must beemployed to decide whether an edge is missing. Thiscan be done by constructing a threshold for theempirical scaled partial covariance density. If thisthreshold is exceeded for some lag u, we decide thatthere is an edge between vertices a and b.

An approximate threshold can be obtained from theasymptotic distribution of ssðT ÞabjV nfa;bgðuÞ. In the appendix,

we show that under the hypothesis of sabjV nfa;bgðuÞ ¼ 0and under the further assumption that bT ! 0 andTbT !1 as T !1 the empirical scaled partialcovariance density is asymptotically normal with meanzero and variance

limT!1

TbT var�

ssðT ÞabjV nfa;bgðuÞ�

¼ H4=H22

We note that the asymptotic variance does not dependon the process N , which is a consequence of the chosenscaling. As a measure of the detectability of aninteraction between two processes we use the detectabil-ity index D (Aertsen and Gerstein 1985), which is definedby D ¼ d=r, where d is the absolute amplitude of themaximum peak or trough of ssðT ÞabjV nfa;bgðuÞ and r2 is theabove asymptotic variance.

For a neural network the partial correlation graphderived from the spike trains does not directly reflect thephysiological connectivity structure of the network. Theconnectivity of the network can be described by adirected graph in which a directed edge from a to b ispresent if and only if there is a neural pathway fromneuron a to neuron b that does not involve any of theremaining neurons in V . In this case, we say that a is aparent of b and b is a child of a. The edges in the partialcorrelation that indicate linear relationships between thespike trains now do not necessarily coincide with theedges in the directed graph since for converging con-nections the input processes become correlated afterpartialization. Thus, in the directed graph all parents ofa joint child must be connected by an edge in order to

292

get the partial correlation graph. This so-called morali-zation of the directed graph is well known in the theoryof graphical models (Whittaker 1990).

Dahlhaus et al. (1997) have shown that partial cor-relation graphs nevertheless can be used for the identi-fication of the synaptic connectivities by application of arecursive procedure. This procedure is based on theobservation that for a directed graph without cycles allterminal vertices, i.e., vertices without children, and theiradjacent edges can be correctly identified from the scaledpartial covariance density. After these vertices have beenremoved from the vertex set, the partial correlationgraph for the remaining vertices is estimated and theterminal vertices in the new graph are determined. Byrecursion of this scheme all directed edges can eventuallybe identified.

2.4 Simulation

A network of simulated neurons has been used toinvestigate the properties of the proposed partialcorrelation analysis. The spike trains of the modelneurons are described by a stochastic point process Ngenerated from a nonlinear integrate-and-fire modelwith refractory period. An extensive discussion of themodel, in terms of stochastic equations, is given inJohannesma and van den Boogaard (1985).

Let saðtÞ denote the last time neuron a has fired beforetime t. Then the intracellular potential of neuron a isgiven by

uaðtÞ ¼X

d

b¼1

Z

t

saðtÞ

wabðt � uÞ dNbðuÞ ð4Þ

The weight functions wabðuÞ determine the size andshape of the postsynaptic potential for the connectionfrom neuron b to neuron a. In the simulations, anexponential shape has been chosen for the weightfunctions

wabðuÞ ¼ wab exp�

� ðu� uabÞ=rab�

for u > uab and zero elsewhere. The amplitude factor wabmeasures the strength of the connection and alsoindicates whether the link is excitatory or inhibitory.Furthermore, uab is the time delay of the connection andrab determines the duration of the postsynaptic poten-tial. In the case a ¼ b, the integral in Eq. 4 becomeswaa�

t � saðtÞ�

and models the refractoriness of theneuron.

The probability that an output spike of type a isgenerated can be written as

ProbfdNaðtÞ ¼ 1juaðtÞ ¼ ug ¼ gaðuÞ dt

where the pulse generator function gaðuÞ is of exponen-tial shape:

gaðuÞ ¼ la expðuÞ

The constant la determines the spontaneous firing rate.

2.5 Local neuronal connectivity in spinal cord

The proposed method has been used to analyze neuronalspike trains from ten simultaneously recorded neurons.Methods have been described in detail elsewhere (Sand-kuhler and Eblen-Zajjur 1994). In brief, multiple singleneuron recordings were made with tungsten microelec-trodes (impedance 4–5 MX) in deep layers of lumbarspinal cord of adult, pentobarbital anaesthetized Spra-gue-Dawley rats. Spinal cord was transected rostral tothe recording sites to remove any descending inhibitionand facilitation from supraspinal sites. Recordings weredigitized at 32 kHz by an A/D converter card (Data-Translation DT2821). Discrimination of action poten-tials from individual neurons was made with theprincipal component method based on the shape of thewaveform using the Discovery software package version3.1 (BrainWave Sytems). Background activity in theabsence of intentional stimulation was recorded for 10–30 min to collect 2000–5000 action potential dischargesof each neuron under study.

3 Results

3.1 Comparison of SPCD and SCD

In order to illustrate the effect of partialization on theassociation of neural spike trains, we have simulatedvarious networks of three neurons using the integrate-and-fire-model described in Sect. 2.4. The scaled partialcovariance densities s21j3ðuÞ and the scaled covariancedensities s21ðuÞ estimated from the simulated data aregiven in Fig. 1.

In the first two networks (a) and (b), neurons 1 and 2are directly connected. In this case, partialization has noeffect on the linear relationship of N1 and N2 and thediagrams for the SPCD and the SCD are virtually thesame. The results are different if the two neurons areconnected indirectly via a third neuron as in networks (c)and (d). For these networks the peak (trough) in theSCD indicating an excitatory (inhibitory) connectionvanishes after the linear effects of the intermediate pro-cess have been removed. The same effect can be observedfor networks (e) and (f) with diverging connectivitiesafter partialization on the common input. Altogether theSPCD correctly distinguishes direct connections fromindirect connections and common inputs.

The next two examples (g) and (h) show that fornetworks with converging connections the behavior ofthe SCD and the SPCD is contrary to the case ofdiverging connections. In this case, the SCD between theinput processes does not show any significant deviationfrom zero, whereas after partialization on the outputprocess the input processes become correlated. This so-called marrying-parents effect (Dahlhaus et al. 1997)leads to a central trough in the SPCD if the two inputsare of the same type, while for inputs of opposite typesthe SPCD has a central peak.

In practice, the activity of only a small part of theneural network can be measured simultaneously. In

293

general, we cannot infer by partial correlation analysisto what extent an association between two spike trains isdue to a direct connection between the correspondingneurons or only the result of an incomplete monitoringof the network since it is not possible to remove thelinear effects of neurons whose activity has not beenrecorded. Therefore, the partial correlation analysis onlyreveals the connectivity relative to the set of monitoredneurons. However, if the SPCD has a central peak ortrough, the SCD can be used to distinguish between the

two possible causes: if the peak or trough vanishes in theSCD, it has been due to a marrying-parents effect. Onthe other hand, if the peak is also visible in the SCD, thissignifies a common input from outside the measurednetwork, as depicted in (i) and (j).

3.2 Detectability of SPCD

The next two sets of simulations have been performed toevaluate the limitations of the partial correlationanalysis for the discrimination of the connectivitypatterns discussed in the previous section. In the firstset of simulations, we studied the sensitivity of the SPCDand the SCD on the strength of connectivity and theneural activity. The simulation with recording timeT ¼ 60 s is based on 100 replications for each configu-ration. The results for three neuron networks withdirect, indirect, and converging connectivities are sum-marized in Fig. 2.

For direct connections (first two rows) the detect-ability curves have approximately the same shape for theSPCD and the SCD, although in the case of inhibitionthe detectability is slightly deteriorated by partialization.If the connectivity strength is increased, both types ofconnectivities become better detectable but the maxi-mum of detectability shifts to lower intensities. This shiftcan be explained by superposition of postsynapticpotentials (see, e.g., Melssen and Epping 1987). Com-paring the detectability of excitatory and inhibitoryconnectivities, the former were always better detectablethan the latter ones. This observation is in accordancewith previous findings for the crosscorrelation histogramby Aertsen and Gerstein (1985) and Melssen and Epping(1987).

The third row shows that in the case of excitation, theSPCD can discriminate direct and indirect connectivitiesover a large range of intensities. While for weak con-nectivities (jwabj ¼ 1) the association due to indirectconnectivities can be completely eliminated by partial-ization, significant peaks appear in the SPCD forintensities la � 64 Hz and la � 128 Hz, respectively, ifthe strength of connectivity is increased. For theseparameters the impulse transmission becomes moredeterministic, which prevents the discrimination of di-rect and indirect connections. In fact, in a purelydeterministic system, the firing pattern of an indirectconnection from a to c via b would be the same as that ofa common input of a on b and c, thus making an indirectand a direct connection between a and c indistinguish-able. Still, partialization causes a substantial reductionof the detectability and thus reveals that the timingrelation between the two spike trains is dominated by anindirect connectivity. In the case of inhibition, direct andindirect connections are correctly discriminated by theSPCD irrespective of the firing rate and the connectivitystrength. However, it should be noted that Melssen andEpping (1987) observed situations in which inhibitoryconnectivities have a much improved detectability,which may lead to a similar behavior as in the case ofexcitation.

Fig. 1. SCD (left) and SPCD (right) for three different neuronnetworks. The horizontal dashed lines signify pointwise 95% thresh-olds for the hypothesis that the scaled (partial) covariance density iszero

294

In the case of converging connectivities (last threerows), the shape of the detectability curves of theSPCD is similar to the shape for direct connectivities,but the detectability index is significantly smaller at allintensities and for all connectivity strengths. In par-ticular, if both converging connections are inhibitory,the resulting trough in the SPCD is hardly discerniblefor any intensity. The different detectability for directand converging connectivities is important as it allows

the complete identification of the converging structurein the network whenever a marrying-parents effectleads to a significant peak or trough in the SPCD.Since the SCD does not show any significant deviationfrom zero irrespective of intensity or connectivitystrength, the converging connectivity can finally bedistinguished from a common input from outside themonitored network by an appropriate reduction of thenetwork.

Fig. 2. Detectability of synapticconnections: detectability index(mean � SD) of SPCD (solidlines) and SCD (dashed lines) atmaximal peak or trough fordirect connections, indirect con-nections, and converging con-nections. Horizontal axis:spontaneous activity of theneurons

295

In the second set of simulations we investigated theeffect of the number of recorded neurons on thedetectability of direct connectivities by partialized sta-tistics. For this we considered networks that consisted oftwo directly connected neurons and a varying number oftotally disconnected neurons. Figure 3 depicts thedetectability of excitatory and inhibitory connectivitiesas a function of the number of neurons included in thestudy for three different recording times (T ¼ 50 s,100 s,and 200 s). For both types of connections the detect-ability curve exhibits a strong dependence on the num-ber of neurons and the recording time. The detectabilityis diminished by an increase of the size of the networksup to some breakpoint at which connectivities becometotally undetectable. For sampling length T ¼ 50 s thisbreakpoint occurs at a network size of 11 and 10 neu-rons for excitatory and inhibitory connectivities,respectively. If the recording time is increased, thedetectability improves and the breakpoint shifts tohigher numbers of neurons. This deterioration of the

detectability in large networks is caused by the finitelength of the spike train recordings. This allows for thefeatures in the realization to be reproduced arbitrarilywell by a linear combination of a sufficiently largenumber of, e.g., Poisson processes. Therefore, if thenumber of neurons increases, partialization eventuallyleads to the total elimination of any information on adirect connectivity between two neurons.

If the number of connectivities in the measured net-work increases, the detectability can be further dimin-ished by partialization even for very long recordings ofthe spike trains. This effect is due to partialization onsucceeding neurons and thus related to the marrying-parents effect described in the previous section. Toillustrate the effect, we simulated a network of fourconnected neurons (Fig. 4) with sample periodT ¼ 500 s. The SCD and the SPCD for the direct con-nection from neuron 1 to neuron 2 are displayed inFig. 4. The area between the curves signifies the decreasein detectability if the SPCD is used instead of the SCD.Despite the long recording, we observe a reduction inheight of about 25% over the full width of the peak.This reduction can be explained as follows: the spiketrains of neurons 3 and 4 improve the prediction of thefiring times of neuron 2 since both neurons are excitedby neuron 2. Therefore, partialization on neurons 3 and4 decreases the probability for a discharge of neuron 2 ifthis is followed by a discharge in neuron 3 or 4. Moreprecisely, the partial residual process e2j3;4ðtÞ has asmaller mean intensity than the spike train N2ðtÞ. Thesame holds if we consider instead the conditional meanintensity of e2j3;4ðtÞ given that neuron 1 has fired, whichis equivalent to the SPCD. Just as for the marrying-parents effect, the reduction is approximately symmetricand therefore causes a small trough, which precedes theasymmetric peak on its steep side.

3.3 SPCD for multiple connections

We examined the ability of partialization analysis in thetime domain to detect multiple direct connections

Fig. 3. Detectability of synaptic connections: detectability index(mean � SD) of SPCD for different recording times [T ¼ 50 s(dotted), T ¼ 100 s (solid), T ¼ 200 s (dashed)] estimated fromdifferent number of neurons for excitatory connection (a) andinhibitory connection (b)

Fig. 4. SPCD (bold) and SCD (shaded) for the direct connectionbetween model neurons 1 and 2 in a four-neuron network forT ¼ 500 s

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between two neurons. For this we analyzed data from asimulated network of six neurons with an excitatoryfeedback loop (Fig. 5a). Such circuits have been used asa model for the short time memory. The same data havebeen analyzed by Dahlhaus et al. (1997) with frequencydomain methods in the case where only neurons 1, 2, 4,and 6 are monitored. In this reduced network, neurons 2and 4 are joined by two direct connections of oppositedirections.

Figure 5c and d presents the results of the frequencydomain analysis. The partial spectral coherencejR4;2j1;6ðkÞj2 (Fig. 5c) correctly detects that neurons 2 and4 are directly connected. The slope of the correspondingpartial phase spectrum (Fig. 5d) now indicates that im-pulses are transmitted in the direction of neuron 4,which corresponds to the main pathway in the network.Neither the partial coherence nor the partial phaseprovide any information about the presence of anexcitatory feedback connection.

The corresponding partialized time domain statistics2;4j1;6ðuÞ is depicted in Fig. 5b. We observe two distinctpeaks at lag u ¼ 36 ms and at lag u ¼ �36 ms, whichcorrectly indicates that impulses between the two neu-rons are transmitted in both directions. Both connec-tions are excitatory and have approximately the sametime delay. Thus the SPCD leads to a complete identi-fication of the connectivity structure. Comparing theheight of the peaks we further find that the signaltransmission in the direction of neuron 4 is slightlystronger. Therefore, the averaged time delay of the linearrelationship between the two spike trains is positive,which explains the negative slope of the partial phasespectrum.

3.4 Temporal variation of neuronal interactions

The SCD and SPCD are average measurements for theeffective connectivity over the entire length of the

recorded data. Their validity therefore depends on thestationarity of the recorded spike trains. However, inmany experimental situations, the network showsdynamic activity due to, for example, behavior orsensory stimulation. In such situations, the firing ratesand correlations between neurons may vary dramaticallyeven in sign (e.g., Aertsen and Gerstein 1985; Vaadiaet al. 1991). Such modifications of neuronal interactionscan be exposed by segmenting the data into shortersections and computing the estimates from the data ineach section. Plotting the results as a function of time wecan assess the temporal variation of the neuronalinteractions.

For illustration purposes we have simulated a net-work of three neurons in which the spontaneous firingrates la and the amplitude factors wab varied randomlyover time. For each component the firing rate and theamplitude factor were obtained from a Markov processwith two states: high or low for the firing rate and po-sitive or negative for the amplitude factor. The connec-tivity of the network is depicted in Fig. 6; the totalsimulation period was 500 s.

The partialization analysis based on the entire data(Fig. 6b) reveals three excitatory direct connections,while the second peak in the SCD between neurons 1and 3 is correctly attributed to an indirect connection vianeuron 2. Additionally, for neurons 2 and 3 both SCDand SPCD show a peak at the origin, which indicates thepresence of a common input from an unmeasuredsource. To investigate the temporal variation of thedependence structure we have computed the time-vary-ing SPCD and SCD from overlapping segments oflength 20 s and shift 2.5 s (Fig. 6a). Here the changesalong the vertical axis reflect the nonstationarity of thefunctional relationships between the processes. In par-ticular, we find that the direct connection between neu-rons 1 and 3 is not purely excitatory but becomesinhibitory over the last 150 s of the simulations. Fromthe plot we can identify time intervals over which the

Fig. 5. Identification of multiple connections ofopposite directions. a Simulated network withexcitatory feedback. b Estimated SPCDss4;2j1;6ðuÞ. c Estimated partial coherence

jRR4;2j1;6ðkÞj2. d Estimated partial phase //4;2j1;6ðkÞwith 95% confidence interval

297

processes seem to be approximately stationary. For twosuch intervals (150–250 s and 350–500 s) we have com-puted SPCD and SCD to analyze the effective connec-tivity of the network at these times. As can be seen inFig. 6b, the analysis of these shorter segments leads to acorrect identification of the network. In particular, we

note that the results reject the presence of a commoninput for neurons 2 and 3 as the peak at the origin of theSPCD has vanished.

3.5 Partial correlation analysisof measured spike trains

Finally, we analyzed spike train data from the lumbarspinal dorsal horn of a pentobarbital-anaesthetized ratduring noxious stimulation. The firing times of tenneurons were recorded simultaneously by a singleelectrode with an observation time of 100 s. The datahave been measured and analyzed by Sandkuhler andEblen-Zajjur (1994), who studied discharge patterns ofspinal dorsal horn neurons under various conditions.

Figure 7b displays the estimated SCD and SPCD forthe ten neurons. The SCD shows for 32 (71%) pairs ofneurons a peak at or near the origin. In 18 (40%) cases,these peaks are accompanied by smaller bilateral peaks.From the estimated autospectra (Fig. 7a) we find thatthese smaller peaks are associated with rhythmic dis-charges at 5 Hz in one or both neurons, which corre-sponds to the observed time lag of 200 ms to the mainpeak.

A much clearer picture of the connectivity structure isprovided by the SPCD. The majority of the peakspresent in the SCD have disappeared after partializa-tion. Only for nine pairs of neurons do we observe ahighly significant peak at u ¼ 12 ms preceded by asmaller trough. These troughs may be due to eitherpartialization on successive neurons as described inSect. 3.2 or the periodic firing of the neurons, which ingeneral leads for a direct connection to a similar com-bination of peak and trough. In the latter case, thetrough appears also in the SCD as, for example, forneurons 8 and 10, but the difference in the depth of thetrough in the SCD and in the SPCD may still be due topartialization. Furthermore, only two pairs exhibit smallbilateral peaks besides the sharp main peak. These canbe ascribed to the strong periodicity of discharges ofneuron 1, which is included in both pairs. Additionally,we find a small central trough for the SPCD betweenneurons 1 and 3.

To identify the connectivities for this set of the neu-rons, we apply the recursive identification procedure byDahlhaus et al. (1997). In the first step, we estimate thepartial correlation graph (Fig. 8a) using the approxi-mate threshold for the SPCD derived in Sect. 2.3. Todetermine the terminal vertices, i.e., vertices that have noadjacent edges pointing to other edges, in the underlyingdirected graph, we identify preliminary directions for theedges from the lags of the peaks and troughs. The ninepeaks are interpreted as direct excitatory connectionsbetween the neurons according to the remarks above,whereas the direction of the edge associated with thecentral trough remains open. Figure 8b shows thedetermined terminal vertices and the identified finaldirections for the adjacent edges. In the next step, theterminal vertices are removed from the vertex set and thepartial correlation graph is estimated from the SPCD for

Fig. 6. a Estimated time-varying SPCD and SCD for simulated three-neuron network. b Estimated SPCD (solid) and SCD (dotted) forsimulated three-neuron network for different sampling periods: 0–500s (left), 150–250 s (middle), and 350–500 s (right)

298

the thus reduced set of neurons. In the SPCD betweenneurons 1 and 3 (Fig. 9), the trough has now disap-peared. Therefore, the edge between 1 and 3 in the ori-ginal partial correlation graph has been due to theconverging of the connections from neurons 1 and 3 toneuron 4. Since for all other pairs the SPCD remainsbasically the same, we obtain the partial correlationgraph in Fig. 8c. Repeating the recursive scheme threemore times the final directions of all edges are deter-mined and we obtain the connectivity graph in Fig. 8das the final estimate of the connectivities between theneurons.

Comparing the connectivity graph with the SCD wefind that neuron 9 is totally disconnected from theother neurons, although it is weakly correlated withneurons 1 and 2. The autospectra (Fig. 7a) reveal thatall three neurons exhibit a strong periodicity at 5 Hz.Therefore, the time lags between successive dischargestend to be the same for these neurons, which then leadsto a correlation that is not associated with a connec-tivity.

We note that this graph displays only the relativeconnectivity structure, and it cannot be excluded thatsome or all of the direct connections are only due toindirect connectivities or common inputs involvingunmeasured neurons. Nevertheless, the results allowone to falsify certain hypotheses that are in conflictwith the estimated graph. For example, the periodicbehavior of many of the measured neurons mightsuggest one common input for those neurons. In thatcase, the corresponding subset of vertices in the con-nectivity graph must be complete, that is, any twovertices in the subset are joined by an edge. This isclearly not the case for the estimated connectivitygraph in Fig. 8d.

Finally, we computed the time-varying SPCD andSCD to check for temporal variation in the effectiveconnectivity of the network. Here we only give the re-sults for neurons 1–5, plotted in Fig. 10, but the resultsfor the complete network are similar. The shapes ofSPCD and SCD appear stable over the entire observa-tion period, and we therefore conclude that the

Fig. 7. a Logarithm of spectral densities for spinal dorsal hornneurons 1 to 10. b Scaled partial covariance densities (below diagonal)and scaled covariance densities (above diagonal) for spinal dorsal horn

neurons 1 to 10. The horizontal dotted lines signify pointwise 95%thresholds for the hypothesis that the scaled (partial) covariancedensity is zero

299

stationarity assumption for the above partializationanalysis is fulfilled.

4 Discussion

We have seen that the scaled partial covariance density isa useful tool for the identification of functional connec-tivities in neural ensembles. Although the presented timedomain techniques are mathematically equivalent to thefrequency domain approach suggested by Brillingeret al. (1996), Rosenberg et al. (1989), and Dahlhauset al. (1997), both methods emphasize distinct featuresof the data. We have shown that the SPCD can be usedfor the identification of the direction and the type ofsynaptic connections.

The results in this paper have shown that the detec-tion of different neural configurations like direct, indi-rect, diverging, and converging connections depends onthe signal-to-noise ratio of the measured spike trains. Inparticular, high ratios lead to a quasideterministic neural

coupling that prevents the discrimination of direct andindirect connectivities. Another limiting factor in theanalysis of neural ensembles is the number of neuronsincluded in the study. Partialization of an increasingnumber of neurons leads to deterioration of the detect-ability and eventually to the breakdown of the identifi-cation method. The effect can be compensated for by anincrease of the sampling interval.

As we have emphasized before, the partializationstatistics proposed in this paper measure only theeffective connectivity between the observed neurons.Therefore, the connectivity graph obtained from a par-tialization analysis can replicate the true physiologicalconnectivity only in a simplified way; direct connections,for example, may involve additional interneurons. Fur-thermore, only connections that were sufficiently activeto be detectable at the time of observation are includedin the connectivity graph.

These problems apply to steady-state experimentalsituations. Additional problems arise if firing rates andinteractions between the neurons vary over time duringbehavior or sensory stimulation. For the evaluation ofthe temporal modifications of the functional relation-ships between the neurons we suggested applying thepartialization method to short overlapping segments ofthe full spike trains and ploting the resulting statisticsas functions over time. Naturally these statistics willhave a lower precision, but they provide a rough pictureof the temporal firing pattern and functionality of thenetwork. In particular, the results may indicate timeregions over which the process seems to be approxi-mately stationary. The data from each of these regionscan then be analyzed by the proposed partializationmethod.

Acknowledgements. The authors wish to thank an anonymousreferee for his helpful comments on an earlier version of this paper.

Appendix

We give a short proof for the asymptotic distribution ofthe empirical SPCD. For the empirical SCD theasymptotic distribution has been derived by Rigas(1991). Let C ¼ V nfa; bg. We start by noting that underthe assumptions of sabjCðuÞ ¼ 0 the SPCD estimatessðT ÞabjCðuÞ has the same limiting distribution as

Fig. 8. a Estimated partial correlation graph for spinal dorsal hornneurons 1 to 10. b Identification of terminal vertices. cReduced partialcorrelation graph for neurons 1, 2, 3, 6, 7, and 8 with identifiedterminal vertices. d Estimated connectivity graph for neurons 1 to 10

Fig. 9. Estimated SPCD ss1;3j2;6;7;8ðuÞ for spinal dorsal horn neurons

300

qqðT ÞabjCðuÞ=ffiffiffiffiffiffiffiffiffi

papbp

Since the partial cross-spectral density fabjC is a nonlin-ear function of the entries of the spectral matrixf ¼ ðfijÞ, the partial covariance density can be expressedas the integral of a nonlinear function of the spectralmatrix. The asymptotic normality of estimates of suchintegrals of nonlinear functions of the spectral matrixhas been proved by Taniguchi et al. (1996). WritingDij ¼ ofabjC=ofij we further obtain from the expressionfor the best predictor NNajCðtÞ by elementary calculationsthatX

ij

DijðkÞfijðkÞ ¼ fabjCðkÞ;X

ijkl

DijðkÞDklðkÞfikðkÞfljðkÞ ¼ fabjCðkÞfbajCðkÞ;X

ijkl

DijðkÞDklðkÞfilðkÞfkjðkÞ ¼ faajCðkÞfbbjCðkÞ;X

ijkl

DijðkÞDklðkÞfijklðk;�k; lÞ ¼ fababjCðk;�k; lÞ

where fijkl and fababjC are the fourth-order cumulantspectra of the multivariate process N and the bivariatepartial residual process eabjC, respectively. From the

results in the cited paper it now follows that the partialcovariance density estimate given by Eq. 3 is asymptot-ically normal with variance

TbT var�

qqðT ÞabjCðuÞ�

¼ 2pH4bT

H22

Z Z

p=bT

�p=bt

fababjCðk;�k; lÞeiuðk�lÞ dk dl

þ 2pH4bT

H22

Z

p=bT

�p=bt

faajCðkÞfbbjCðkÞ dk ð5Þ

Since fababjC is absolutely integrable, the first termconverges to zero as bT ! 0. Further we have

faajCðkÞ ¼pa

2pþZ

1

�1

qaaðuÞ expð�ikuÞ du

where the second term is absolutely integrable. Conse-quently, the second term in Eq. 5 converges to ðH4=H2

2 Þpapb, from which the stated asymptotic variancefollows.

Fig. 10. Estimated time-varying SPCD and SCD for spinal dorsal horn neurons

301

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