Modeling the functional network of primary intercellular Ca2+ wave propagation in astrocytes and its application to study drug effects

Modeling the functional network of primary intercellular Ca2þ wavepropagation in astrocytes and its application to study drug effects

Marcelo Pires a,b, Frank Raischel a,c, Sandra H. Vaz d,e, Andreia Cruz-Silva d,e,Ana M. Sebastião d,e, Pedro G. Lind a,f

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a Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, Campo Grande 1649-003 Lisboa, Portugalb Departamento de Física, Universidade Federal do Amapá, Jardim Marco Zero, 68903-419 Macapá/AP, Brazilc Centro de Geofísica, Instituto Dom Luiz, Universidade de Lisboa, 1749-016 Lisboa, Portugald Instituto de Farmacologia e Neurociências, Faculdade de Medicina, Universidade de Lisboa, 1649-028 Lisboa, Portugale Unidade de Neurociências, Instituto de Medicina Molecular, Universidade de Lisboa, 1649-028 Lisboa, Portugalf Institute für Physik and ForWind, Carl-von-Ossietzky Universität Oldenburg, DE-26111 Oldenburg, GermanyQ2

H I G H L I G H T S

� Novel approach complementary to the standard drug tests.� Extraction of the functional connectivity network from measurements of Ca2þ .� Tests drugs effects in the functional web of tissues where Ca2þ signals propagate.

a r t i c l e i n f o

Article history:Received 15 November 2013Received in revised form17 March 2014Accepted 17 April 2014

Keywords:Signal propagationCellular tissuesComplex networksDrug tests

a b s t r a c t

We introduce a simple procedure of multivariate signal analysis to uncover the functional connectivityamong cells composing a living tissue and describe how to apply it for extracting insight on the effect ofdrugs in the tissue. The procedure is based on the covariance matrix of time resolved activity signals. Bydetermining the time-lag that maximizes covariance, one derives the weight of the correspondingconnection between cells. Introducing simple constraints, it is possible to conclude whether pairs of cellsare functionally connected and in which direction. After testing the method against synthetic data weapply it to study intercellular propagation of Ca2þ waves in astrocytes following an external stimulus,with the aim of uncovering the functional cellular connectivity network. Our method proves to beparticularly suited for this type of networking signal propagation where signals are pulse-like and haveshort time-delays, and is shown to be superior to standard methods, namely a multivariate Grangeralgorithm. Finally, based on the statistical analysis of the connection weight distribution, we proposesimple measures for assessing the impact of drugs on the functional connectivity between cells.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Astrocytes, which were long thought to perform only auxiliaryfunctions in the brain, are known to exhibit complex patterns of Ca2þ

waves propagating in their cellular network (Newman, 2001). Thebasic biological mechanisms that underly the functional links betweenastrocytes, leading to consecutive elevations in calcium signal(Haydon, 2001; Devinsky et al., 2013). The predominant mechanismis mediated by ATP, which activates metabotropic P2Y receptors in theastrocytic membrane, leading to the formation of inositol-3-phosphate(IP3), which then signals to release calcium from the intracellularstores. Calcium elevation in an astrocyte leads to further release of ATPto the extracellular media, which quickly acts in receptors in the

membrane of neighboring cells, leading to calcium elevations in thosecells, which leads to a continuous cascade of Ca2þ signal propagation.Transfer of IP3 across gap-junctions (connexins) may also contributefor the calcium elevation (Hoefer et al., 2002), though in a minordegree (Haydon, 2001; Hassinger et al., 1996; Bennett et al., 2005).Other recent studies have shown that intracellular Ca2þ oscillationsare basically a form of correlated noise (Perc et al., 2008), which raisesthe question of stochasticity and reproducibility of the signal propaga-tion and the corresponding network. While questions concerning thedetails of the propagating mechanisms remain an important matter ofdiscussion (Falcke, 2004), we are here interested in extracting insightfrom the inter-connectivity between cells by keeping track of Ca2þ

signals.

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Journal of Theoretical Biology

http://dx.doi.org/10.1016/j.jtbi.2014.04.0240022-5193/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: Pires, M., et al., Modeling the functional network of primary intercellular Ca2þ wave propagation in astrocytesand its application to study drug effects. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.04.024i

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Uncovering the functional connectivity of astrocytes in these net-works provides a better understanding of the functionality of theastrocytic network itself, but also allows us to assess drug effects, notonly at the single cell level, but also upon the spreading of the signalthroughout the cellular tissue (Falcke, 2004). Such activity can be wellcharacterized by measuring the concentration of calcium ions (Ca2þ)using noninvasive techniques of calcium imaging. One of the mostpopular of such techniques uses fluorescent dye indicators, which bindselectively to free Ca2þ ions, undergoing a conformational change andconsequently a variation in its fluorescence excitation and/or emissionproperties when bounded to Ca2þ . These variations can be used toevaluate changes in intracellular Ca2þ concentration. Here concentra-tion is measured by the ratio R between maximum amplitudes at 340and 380 nm, as illustrated in Fig. 1.

The evolution of Ca2þ concentration in each cell dependstypically on the diffusion of the signaling molecules through theintercellular environment and on the direct connection from oneastrocyte to its neighbors. While the former mechanism is slow,the latter is fast and dominates for measurement series with highsample rates. This study focuses on the latter case. Therefore, theseries of measurements of Ca2þ concentrations at each cellularlocation reflects the flow of Ca2þ , and consequently the propaga-tion of this ion from each cell to the neighboring ones. It should benoted that the brightening front generated at each cell does notnecessarily propagate radially to its neighbors, since the physicalconnections between neighboring cells are heterogeneously dis-tributed, which introduces spatial inhomogeneities in the signalpropagation (Hoefer et al., 2002; Giaume and Venance, 1998).

One possible way to model Ca2þ signal propagation is througha bottom-up approach, where the kinetic constants of intracellularand intercellular signal pathways, the spatial distributions ofconnections and receptors in the tissue, and the relative impor-tance of signal propagation mechanisms need to be known andincorporated in a detailed mechanical simulation. However, thisapproach is quite cumbersome and case dependent.

We argue, however, that the detailed physical structure ofinterconnected cells is not necessary to characterize the responseof the interconnected tissue to a signal. For that one only needs touncover the so-called functional connectivity between cells, whichdescribes the synchronization patterns between the cells. Whilefunctional connectivity is at most loosely related to causal con-nectivity, it reflects the way the tissue as an interconnectedstructure of cells responds to external stimuli, and how thisfunctional network is changed when applying drugs.

Therefore, for our purposes, we focus here on the functionalconnectivity. Similar approaches to assess the functional connec-tivity in biological systems have already been presented, e.g. inislets of Langerhans from mouse pancreas tissue slices (Stozer etal., 2013).

From the mathematical perspective, we quantify the connec-tivity and the causality of information flow between the cells as aweighted graph. A graph is a collection of nodes interconnectingthrough edges according to some specific rules (Diestel, 2005).

Our aim here is mapping the complex interactions and temporalinformation flow patterns to a much simpler representation. Thegraph is completely characterized by a single matrix, the adjacencymatrix, whose entries are the weights of the edges between nodes.In case of cellular tissues, nodes represent the cells, and edgesbetween them the functional connections between a respective pairof cells.

To reconstruct this graph we consider the paths connectingneighboring astrocytes in a confluent culture of astrocytes, used asa tissue model, analyzing the time series of Ca2þ concentrationmeasurements observed at each astrocyte separately (see Fig. 2).From these experimentally measured synchronization patterns,we employ the correlation measures between the Ca2þ timeseries.

Standard algorithms exist for this problem based on the Grangercausality (Granger, 1969; Blinowska et al., 2004; Dhamala et al.,2008). However, although they can be adapted to suit multivariatenon-stationary data series (Granger and Hatanaka, 1964; Granger andNewbold, 1977), these methodologies assume the existence of arandom process with a certain level of stationarity and then applyspectral analysis, decomposing the data series into a certain numberof uncorrelated components, each one corresponding to a givenspectral frequency.

For our particular case where one has sets of externallystimulated signals that are typically pulse-like and with shorttime-delays between them, a different approach must be consid-ered, as will become apparent below.

Here, we show that these particular signals observed in signalpropagation of astrocytic tissues can best be recovered by oursimple and accurate procedure, which can also be applied forassessing the efficiency and the irreversibility of drug infusion onthe structure of the tissue, by studying signal propagation.

This report is organized as follows. We start Section 2 bydescribing the general properties of the signal and then give adetailed account of our procedure. In particular, we argue thatusing the normalized covariance between astrocytes for specifictime-lags enables one to quantify the connectivity of each pair ofastrocytes. We then show that, for synthetic networks, ourprocedure is more efficient and accurate than standard algorithmssuch as the Granger causality. In Section 3 we describe theexperimental setup and the data extracted from samples of living

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Fig. 1. Set of 20 astrocytes observed through calcium imaging procedures. Brightregions indicate the location of cytoplasm and organelles, where the concentrationof Ca2þ is higher than in the dark regions indicating the intercellular medium,where diffusion processes take place. Concentration is measured by the ratio Rbetween the radiation emitted at 510 nm when cells are excited at 340 nm overemission upon excitation at 380 nm.

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astrocyte tissues. In Section 4, we extract the connectivity net-works for each sample and for each stimulus, using our procedure,and afterward analyze the moments of the weight distributionfound for each tissue sample to discuss the effect of two drugs onthe connectivity structure and the signal propagation, and thevariability of the results. Finally, Section 5 concludes the papergiving also a brief description of how this method can be appliedto drug tests. Our auxiliary model for creating synthetic signal datais described in Appendix A. The Granger method and algorithm isbriefly described in Appendix B.

2. Extracting cellular functional connectivity

2.1. Properties of the experimental signal propagation

A brief description of the common numerical features of theCa2þ cascade follows. A detailed description of the experimentalsetup can be found in Section 3.

In all experiments, cell 1 is externally stimulated 10 times, with10 μM ATP (focally applied for 200 ms), the eleventh peak beinginduced by a supramaximal concentration of ATP (100 μM) to testthe maximal activity of the cell. The Ca2þ signal cascade isobserved by measuring the intracellular Ca2þ levels, cf.Figs. 1 and 2, through the radiation amplitude ratio R.

The measured temporal signals, cf. Fig. 2, are a series of pulses,accompanied by an increasing trend, the latter not being ofinterest here. The signals are of similar shape, which is initiallyGaussian, before decaying with a slower-than exponential tail. Theamplitudes, however, are widely varying. Generally speaking, itcan be observed in this example that the stimulated cell 1 – andalso cell 4 – has more or less constant signal amplitude ratios,whereas the cells farther away from cell 1 exhibit lower ampli-tudes and less regular peak heights. The signals are all delayedwith respect to cell 1, although this crucial property is not visibleat the resolution level provided by Fig. 1. This delay, specificallythe delay in correlations related to it, is the cornerstone of ourmethod of reconstructing the signal network.

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Fig. 2. Intercellular Ca2þ signaling cascade. When one has a set of time series of the radiation ratio R measured at each cell of a tissue sample, how can one infer the flow ofCa2þ through the tissue? The central plot shows a case of eight signals extracted from the indicated positions, one per cell, forming the culture of confluent astrocytes that isused as a tissue model. In the ten surrounding signal plots, each plot shows the respective response of an individual cell to ten different ATP stimuli (see text) applied to cell1. The final, eleventh peak shows the maximum cellular response, recorded by applying a supramaximal ATP stimulus (100 μM) to cell 1, and was not used in theanalysis below.

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2.2. Modeling functional connectivity strength from signalcorrelation

Following the experimental results, we propose a model toobtain the network of functional connectivity, i.e. the existence,direction and strength of network links between the networknodes, which are the cells, from the measured signal correlations.The result is then a network comparable to Fig. 3a.

We consider a number M of cells from which the Ca2þ can bemeasured composing the time series Xi(t), i¼ 1;…;M and tlabeling time-steps. To derive the connectivity between a pair ofcells, say i and j, we consider primarily how strong the corre-sponding signals, Xi(t) and Xj(t), are correlated. Since we aredealing with signal propagation, one must also consider a time-delay τ separating the two measures Xi(t) and Xjðt�τÞ. In parti-cular, we assume that a proper choice of the value of τ maycompletely reproduce at cell j the shape of the signal occurringpreviously in cell i, i.e. XiðtÞ ¼ Xjðt�τÞ for all times t ¼ nΔt, moduloan attenuation factor. Here, Δt is the inverse of the sample ratetaken for extracting the series of measurements composing thesignal at steps n¼ 1;…;N, with N indicating the total number ofmeasurements.

Typically, the signal at one cell is not completely reproduced inanother cell, since the signal propagation involves variousmechanisms, such as gap junction transport and diffusion. Todetermine how strong the signals are correlated, we take theircovariance with a time-delay τ:

CðXi;Xj; τÞ ¼∑N

t ¼ τðXiðtÞ�XiÞðXjðt�τÞ�XjÞðN�1Þsisj

ð1Þ

where Xi and si stand for the average and standard deviation ofsignal Xi(t) respectively. This measure has the propertiesCðXi;Xi;0Þ ¼ 1 and limτ-1CðXi;Xj; τÞ ¼ 0.

The value of the covariance between the signals at twoneighboring cells may vary depending on the type of stimulusapplied at the source-cell. Also, when repeatedly applying thesame stimulus to the same array of cells, signal propagation maybe altered by aging or learning effects. One would also expect thatthe effect of drugs is reflected by the covariance, such that, whenthe covariance increases in absolute value, it could indicate anexcitatory effect of the stimulus substance. Or, when it decreases,it could reveal an inhibitory effect.

We stress once more that, while it remains only a hypothesisthat the strength of the connection between two cells can beascertained from the covariance their functional connectivity isindeed reflected in the covariance: the larger the covariancebetween two signals is in absolute value, the stronger theconnection between the functional behavior of the correspondingcells should be. Q3

Fig. 4 shows the covariance between two astrocytes in thesample shown in Fig. 1. Depending on the value of the delay τ, thecovariance between two series can be large (close to 1 or �1) orsmall (close to 0). The successive maxima shown in Fig. 4 corre-spond to distinct stimuli.

However, the covariance alone does not suffice for showinghow strongly connected two astrocytes are. Indeed, the covariancefor different delays corresponds to different interaction modes,rates and paths joining two cells. Here, we focus in the “first”interaction mode, corresponding to the fastest path propagatingthe signal from the unique source-cell to each of the other cells inthe tissue. We assume that detecting for instance a reversible or anirreversible effect of a drug in the rapidity of these propagatingpaths is sufficient to determine the reversible or the irreversibleeffect in the tissue.

To determine the fastest paths, the time-delay in Eq. (1) mustalso be considered, specifically the smallest one that maximizesthe covariance: the delay needed for characterizing the connectionbetween astrocytes is the lowest delay for which a local maximumof the covariance is observed. This value will be referred to as τmax,cf. Fig. 4, and the corresponding covariance is henceforth calledCmax ¼ CðXi;Xj; τmaxÞ.

The time-delay τmax in fact measures the typical time for thesignal to propagate from astrocyte i to astrocyte j. Notice that wedo not consider the relative distances of the cells because, asmentioned in the introduction above, the medium in which thesignals propagates is highly heterogeneous and therefore prefer-ential and non-preferential pathways of signal propagation exist

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Q4 Fig. 3. (a) A sample directed graph of functional connectivity as it can be extractedby our method, here for M¼8 interconnected (artificial) astrocytes. Nodes repre-sent the cells (blue bullets) and edges represent their connections (yellow arrows).(b) The connections can be evaluated from the time series at each node using twodifferent procedures: Granger causality (Granger, 1969; Blinowska et al., 2004;Dhamala et al., 2008) (black arrows) or our method (red arrows). Comparing theresults with the synthetic network of astrocytes in (a), our method can be taken asbetter for this particular purpose (see text and Fig. 7). (c) Particular cases of signalcorrelations, namely Pearson coefficient equal to one, may represent different butequivalent signal propagation topologies. (For interpretation of the references tocolor in this figure caption, the reader is referred to the web version of this paper.)

0 50 100 150 200 250 300 350 400τ

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

C(X1,X

2,τ)

T

τmax

Fig. 4. The covariance in Eq. (1) between the two cells 1 and 2, shown in Fig. 3, as afunction of time-delay τ. Here, T is the time-window between successive stimuli.Typically, during each T period, the covariance decreases from a maximum value toa minimum. The value of τmax giving the delay for which the covariance ismaximized is the lowest time-lag corresponding to a relative maximum.






and the geometrical distance is not a suitable criterion for signalpropagation velocity (Giaume and Venance, 1998).

In the particular situation that a cell propagates its signal to twoneighbors with neither delay nor dumping, the respective covariancesare exactly one, see Fig. 3c. In this case both situations drawnwith blueand green arrows are equivalent and indistinguishable. Either one, andonly one, of them is known to connect the set of cells.

If the delay τmax that maximizes the covariance between twoastrocytes is small, it means that they should be closely connectedand the corresponding connection should be strong. If the time-delay is large compared to other pairs, the corresponding strengthshould be small compared to those other pairs. Therefore, theweight of the connection is reasonably assumed to be proportionalto the inverse of the delay τmax.

Combining all the considerations above we define the strengthwij of the connection between astrocytes i and j as

wij ¼jCmaxðXi;Xj; τmaxÞj

τmaxþ1; ð2Þ

where one unit in the denominator is added for convenience to avoidsingular behavior. Notice that the weight wij, being the quotientbetween correlation of two signals and time, can be interpreted as acorrelation flux, which in this case measures the causality – and notthe strength – of the flow of information between cells.

Still, the weight value alone cannot reveal the structure ofsingle signals through the tissue, as we explain next.1

2.3. Constraints for signal propagation

For any pair i and j, the strength wij is typically a non-zerovalue. Therefore, by solely considering the values of wij computedas in Eq. (2), one cannot immediately infer the structure shown inFig. 3a, since typically there is a non-zero covariance for any pair ofcells. A final step is still necessary to filter out redundantconnections.

The filter is based on two simple constraints for the connec-tions. First, there is one single source-cell characterized by havingone or more outgoing connections but no incoming connection.In all experiments shown, this is the first cell, i¼1. Second, all cells,different from the source-cell, have only one incoming connection,but can have several outgoing connections. These two constraintsare the sufficient and necessary ones for cell-to-cell signal propa-gation, which is the situation we are considering here.

From these two constraints the task reduces to extract theincoming functional connection of each cell that establishes anoptimal (fastest) path from the source-cell to it. We say that a pathPOði; jÞ between two cells, i and j, is the optimum path betweenthese cells if it has the minimum (total) time-delay cost from allpossible paths Pði; jÞ. In case one has more than one pathcorresponding to the minimum time-delay, one should choosethe path maximizing the number of connections, in order tominimize the time between each two adjacent cells in the path.Fig. 5, together with Table 1, illustrates how the optimum pathbetween each pair of cells is computed.

Starting from the source-cell, we traverse the network using aburning breadth-first algorithm (Sedgewick, 1983): start at a rootnode and inspect all its neighbors; for each of those neighbors,inspect their neighbor nodes which were still not visited; and soon. Then, for the source-cell we compute the optimum path fromit to each one of its neighbors, removing all redundant incomingconnections of each neighboring cell. We iteratively repeat this

procedure for each one of these neighbors, and therefore for allcells. Fig. 6 summarizes the full algorithm.

2.4. Verification of the network reconstruction algorithmwith a simple model of Ca2þ signal propagation

In order to numerically verify the reliability of our reconstruc-tion algorithm, described in Sections 2.2 and 2.3, we will considerin this section a synthetic network of cells, as sketched in Fig. 3a,joined by directed connections (yellow arrows). We simulate theinformation flow through these artificial networks by a simpleauxiliary model of information flow described in detail inAppendix A. At cell 1 a Gaussian stimulus X1ðtÞ is introduced.Here, X1 corresponds to the experimental ratio R of both radiationamplitudes (see Fig. 1) measured at cell 1.

By prescribing a time-delay to each connection, which controlsthe necessary delay for the signal to propagate to the neighboringcells, we extract the series of values composing the signal at eachof the other cells. Using the algorithm described in the previoussubsections, we were able to accurately uncover the connectivitystructure sketched in Fig. 3a with yellow arrows solely by analyz-ing the separated signals. In Fig. 3b we indicate the result of ourreconstruction with red arrows and compare it with resultsobtained from the standard Granger algorithm for multivariatedata (Blinowska et al., 2004) (black arrows). Comparison withFig. 3a shows the accuracy of our reconstruction, whereas theGranger model identifies spurious connections not present inthe network. Next we carefully compared our algorithm with thestandard Granger causality, analyzing a set of 100 artificial net-works of 11 synthetic cells (10 connections). For details on thegeneration of synthetic data we again refer the reader to AppendixA, and for details on Granger causality procedures see Appendix B.

Fig. 7 shows the frequency of artificial networks that correctlydetect a given percentage of connections (efficiency). As can beclearly seen, in all cases at least 50% of the connections wereproperly extracted, and typically the percentage of connectionscorrectly identified lies between 80% and 90%. In comparison withthe standard Granger algorithm (red histogram), one concludesthat for these signal-propagating networks the above procedureshows a high efficiency.

Here efficiency e was computed directly from the number Nc ofconnections that were correctly predicted and the number ofconnections Ne equivalent to the original networks, as sketchedin Fig. 3c, yielding e¼ ðNcþNeÞ=NT , NT being the total number ofconnections correctly predicted. For our algorithm the totalnumber of connections equals the number of cells minus one,NT ¼ nc�1. For the Granger algorithm NT is variable and liesbetween ne�1 and neðne�1Þ, since it is insensitive to the con-straints for cell-to-cell propagation introduced above.

It should be noted that there are two fundamental differencesbetween Granger's method and ours. First, our procedure considers aconnectivity matrix weighted by the time-lag between signals. Second,it introduces two constraints necessary for uncovering the so-calledprimary graph in the particular case of externally stimulated tissues ofinterconnected cells. From these tissues one extracts multivariatesignals that result from one single source signal – the externalstimulus – which propagates throughout a spatially extended system.By uncovering the primary graph, our procedure will not guaranteethat reversal connections do not exist. Still, focusing on the primarygraph, we see that it retrieves the first order effects of the stimulus inthe interconnected functional structure among cells propagating thesignal, particularly in the case when a drug is used. We show belowthat these effects are complementary to the usual effects uncoveredthrough standard drug tests (see Fig. 8). With our procedure, non-distinguishable connections are mutually exclusive, avoiding redun-dant connections.

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1 The full implemented algorithm can be shared for research purposes. Forthat, contact the authors.






3. Experimental setup and data extraction

As mentioned above in Section 1, activation of specific mem-brane receptors localized in the astrocytic plasma membranetriggers a rapid and brief rise of intracellular Ca2þ concentrationin this cell, which promotes the release of gliotransmitters thatwill lead to the increase of intracellular Ca2þ concentration on aneighboring astrocyte. Thus, stimulation of a single astrocyte inculture, with ATP (adenosine-5’-triphosphate), induces intracellu-lar Ca2þ elevation in the stimulated cell, which is then followed byCa2þ increases in neighboring astrocytes.

The transmission of intercellular Ca2þ signals between astro-cytes is achieved through two distinct pathways: (i) release ofgliotransmitters that will bind receptors located on neighboringastrocytes, and/or (ii) Ca2þ itself, or a Ca2þ liberating secondmessenger (as IP3) permeate gap junction channels and then acton similar intracellular targets in neighboring coupled cells.

The calcium signal, as the ones recorded in the top plots ofFig. 8, corresponds to a calcium signal by one cell, i.e. the variationof the fluorescence ratio, proportional to calcium concentrationwithin the cell versus time. In a monolayer culture of astrocytes,such as the ones studied here, the calcium signal propagates fromone astrocyte to another creating a propagating calcium wave. Thepropagation velocity of Ca2þ waves reaches 28:2 μm=s (Newman,2001) and it is also known that intracellular velocity ranges from9.4 to 61:2 μm=s (Cornell-Bell et al., 1990). Thus, in order to haveintracellular Ca2þ waves (Kang and Othmer, 2009) that could beused to uncover the connections between the astrocytes weperformed calcium imaging using primary cultures of corticalastrocytes. For the results shown in Fig. 8 we have a statisticalsignificance of Po0:05 (Students t-test) for the hypothesis thatthe (Gaussian) distribution is the same when the signal amplitudeduring drug perfusion, upon ATP stimulation, is compared withthe responses immediately before drug perfusion (control).

Primary cultures of cortical astrocytes from Wistar rats (0–2days old) were prepared as reported previously (Vaz et al., 2011)

and in accordance with Portuguese laws and the European UnionDirective 86/609/EEC on the protection of Animals used forExperimental and other scientific purposes.

For calcium measurements, microglia contamination was mini-mized by following a standard shaking procedure (McCarthy andde Vellis, 1980). After 6 days in culture (DIC 6), cells in the T-75culture flasks were shaken for 4–5 h at 37 1C, the supernatant wasremoved and DMEM supplemented medium was added. At DIC 7,the T-75 culture flasks were shaken again for 2–3 h at 37 1C, thesupernatant containing mostly microglia was removed and thencells were washed once with PBS. After removing microgliacontamination, astrocytes to be used in calcium imaging experi-ments were plated (7�104 cells/ml) in γ-irradiated glass bottommicrowell dishes. Before plating, cells were gently detached bytrypsinization (1% trypsin–EDTA) for 2 min, the process beingstopped by the addition of 4.5 g/l glucose DMEM medium contain-ing 10% fetal bovine serum with 0.01% antibiotic/antimycotic.

Astrocytes were loaded with the Ca2þ-sensitive fluorescent dyefura-2 acetoxymethyl ester (fura-2 AM; 5M) at 22 1C for 45 min.After loading, the cells were washed three times in externalphysiological solution (composition in mM: NaCl 125, KCl 3,NaH2PO4 1.25, CaCl2 2, MgSO4 2, D(þ)-glucose 10 and HEPES 10;pH 7.4 adjusted with NaOH) (Rose et al., 2003).

Dishes were mounted on an inverted microscope with epi-fluorescent optics (Axiovert 135TV, Zeiss) equipped with a xenonlamp and band-pass filters of 340 and 380 nm wavelengths.Throughout the experiments, the cells were continuously super-fused at 1.5 ml/min with physiological solution with the aid of aperistaltic pump. Calcium signals were induced by ATP, appliedfocally, for 200 ms, through a FemtoJet microinjector (Eppendorf,Hamburg, Germany), coupled to an ATP (10 μM) filled micropip-ette placed under visual guidance over a single astroglial cell.

In this study we address the following two drugs: 4-[2-[[6-Amino-9-(N-ethyl-β-D-Ribofuranuronamidosyl)-9H-purin-2-yl]amino]ethyl]benzene]propanoic acid hydrochloride, henceforthnamed as “Drug A”, and N6-Cyclopentyl-9-β-D-Ribofuranosyl-9H-purin-6-amine, henceforth named as “Drug B”. Both drugs wereadded to the external solution under perfusion. Changeover ofsolutions was performed by changing the inlet tube of the peristalticpump from one flask to another; changeover of solutions with equalcomposition did not lead to appreciable changes of the responses. Ineach experiment and for each cell, responses to the stimulus(pressure applied ATP) were first obtained in the absence of thedrug (control, Pre-C), then in the presence of the drug, after change-over of solutions, and lastly after returning to the drug free condi-tions (washout, Post-C). Image pairs obtained every 250 ms by

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Fig. 5. Illustration of the computation of optimum paths. (a) At each edge the corresponding delay τmax is indicated. In this example the optimum path between cells 1 and8 is Pð1;8Þ ¼ f1;2;3;5;8g, because this path combines the least time cost (0.7) and greatest number of edges (4). See Table 1. (b) Consequently, the connections f1;8g, f3;8gand f7;8g are redundant and therefore are filtered out by the procedure.

Table 1Time cost and number of edges of all paths between cells 1 and 8 in the examplesketched in Fig. 5.

Path Time cost Number of edges

f1;8g 0.9 1f1;2;3;8g 0.7 4f1;2;3;5;8g 0.7 5f1;2;3;5;6;7;8g 1.2 7






exciting the preparations at 340 and 380 nm were taken to obtainratio images. Excitation wavelengths were changed through a highspeed wavelength switcher, Lambda DG-4 (Sutter Instrument,Novato, CA, USA), and the emission wavelength was set to 510 nm.

Image data were recorded with a cooled CCD camera (PhotometricsCoolSNAP fx) and processed and analyzed using the softwareMetaFluor (Universal Imaging, West Chester, PA, USA).

Regions of interest were defined manually over the cell profile.Typically one chooses the cytoplasmic region for measuring theratio R, given preference to the brightest regions in each astrocyteappearing in the photo images during one stimulus.

It is important to know whether the effect of a certain drug isinhibitory or excitatory, as well as whether it is reversible orirreversible. To that end we compute the ratio R (induced by ATPapplication) at three different time steps: before introducing thedrug (left), in the presence of the drug (middle), and after washingout the effect of the drug (right). At each one of these momentsone measures the magnitude of R, having values RH1 , RD and RH2 .If RDoRH1 the drug has an inhibitory effect, while in the oppositecase, RD4RH1 the drug proves to be excitatory. In our case one seesthat the effect of drug A is excitatory (Fig. 8, top left) while theeffect of drug B is inhibitory (Fig. 8, top right).

To ascertain the reversibility of drug effects one takes inaddition the magnitude RH2 . If RD�RH1 � RD�RH2 the effects arereversible. If not they should not be reversible, yielding typicallyRD � RH2 . In the case illustrated in Fig. 8, drug A is reversible whiledrug B is irreversible.

Notice that the responses in left plot of Fig. 8 are from one cell.Responses from the right plot are from another cell and anotherculture. The stimulation parameters in the condition illustrated in thetop-left plot of Fig. 8 were empirically adjusted (by changing therelative position of the stimulating electrode) to induce a weakresponse under control conditions since the protocol was designed totest the influence of a drug known to have excitatory actions. Drug Ais a well known and selective agonist of excitatory adenosine A2Areceptors, known to be present in astrocytes (Vaz et al., 2011). The

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Fig. 6. Infography of the algorithm for extracting the connectivity network of astrocytic tissues (see Section 2).

Fig. 7. Testing the algorithm for extracting the connectivity network in livingtissues of astrocytes. For the procedure described in Section 2 (blue histogram) allnetworks show a correctness larger than 50% and most of them reach 80–90%. Suchcorrectness is significantly larger than the one obtained with standard Grangercausality (red histogram). The plot shows the percentage of connections correctlyextracted from 100 artificial networks, each with 11 cells and 10 connections. (Forinterpretation of the references to color in this figure caption, the reader is referredto the web version of this paper.)






stimulation parameters in the condition illustrated in top-right plotof Fig. 8 were empirically adjusted (by changing the relative positionof the stimulating electrode) to induce a stronger response undercontrol conditions since the protocol was designed to test theinfluence of a drug known to have inhibitory actions. Drug B is awell known and selective agonist of inhibitory adenosine A1 recep-tors, also known to be present in astrocytes (Cristóvão-Ferreira et al.,2013). Importantly, within each panel, the responses shown are allfrom the same cell, under exactly the same stimulation conditions.Once set they were not changed up to the end of the recordingperiod. The difference being the absence or the presence of the drugin the perfusion solution.

As we will see in the next section, by assessing the connectivitynetwork for each set of cells in one culture of astrocytes we will beable to provide additional insight to the effect of one drug in the livingtissue.

4. Assessing drug effects from cellular connectivity

The standard procedure described in Section 3 for evaluating theexcitatory and inhibitory effects of one drug or their reversible orirreversible character will in this section be extended to a broadercontext. Indeed, the approach done in the previous section con-sidered the signal's total amplitude observed in the entire tissuesample. Now, using the method introduced in Section 2 allows us toretrieve the full structure of the connectivity network throughwhich the injected stimulus propagates. In Fig. 9 three tissuesamples are shown with their respective connectivity network forone particular stimulus. Next, we apply this procedure to a succes-sion of ten stimuli in each tissue sample shown in Fig. 9. For eachstimulus, the connectivity network is defined by the weight matrixwij, quantifying the signal propagation between all sender cells i andall receptor cells j.

For instance, while a drug may have an inhibitory effect on theamplitude, reducing the overall signal strength, it may simultaneously

change the signal propagation network in a way that the signal,though weaker, propagates more easily, i.e. it has a facilitatory effect inthe propagating structure. We introduce four additional quantities,each one of them reflecting the facilitatory or inhibitory – andreversible or irreversible – effects induced by a particular drug.As does wij, these additional properties characterize not the intensityof the signal but the stronger or weaker ability of the signal topropagate throughout the tissue, i.e. the causality of the informationflow. Therefore, they can be taken as properties complementary to theamplitude.

The four additional quantities are all computed directly from theweight wij introduced above, using the auxiliary quantitywij ¼ ð1=10Þ∑10

T ¼ 1wijðTÞ (T ¼ 1;…;10), the average of the weightbetween each pair of cells, over the ten experimental phases. Byascertaining how the weights wij change from one stimulus to thenext, we are able to determine the effects of the substance in thetissue, which is reflected by the moments of the weight distribution.

The first moment of the distribution is simply

⟨w⟩ðTÞ ¼ 1L∑ia j

wijðTÞ; ð3Þ

with L indicating the total number of connections. While the set ofvalues wij represents the time-average strength of one singleconnection in time, ⟨w⟩ðTÞ indicates the average weight – or flux– per connection in the tissue for a particular stimulus at time T.

The second moment is important for ascertaining how influ-ential is a particular drug in inducing a variation of the weightbetween two connected cells. It is computed by accounting for thefluctuations around the means wij and averaging them over the Lconnections:

s2ðTÞ ¼ 1L�1

∑ia j

ðwijðTÞ�wijÞ2: ð4Þ

When s¼ 0 it implies that the connectivity network is preciselythe same for all stimuli and therefore the drug has no influence onthe weights. The larger the value of s is, the stronger the influence

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Fig. 8. For determining if a drug is inhibitory or excitatory or to determine if its effects are reversible or not, one needs to observe the R ratio for three different moments:(i) before applying the drug (Pre-C), (ii) during the effect of the drug and (iii) after washing out the drug (Post-C). In the plots above we plot the R amplitude for these threeinstants using two different drugs drug A (left) and drug B (right). While drug A shows an excitatory and reversible effect, drug B is inhibitory and irreversible. In the plotsbelow the averages of Ca2þ concentration for a total of ten experiments are shown. Statistical significance of the drug effect is Po0:05. Drug A: 4-[2-[[6-Amino-9-(N-ethyl-β-D-Ribofuranuronamidosyl)-9H-purin-2-yl]amino]ethyl]benzene]propanoic acid hydrochloride; Drug B: N6-Cyclopentyl-9-β-D-Ribofuranosyl-9H-purin-6-amine. Through-out the experiments, ATP (10 μM) was used as the stimulus to evoke Ca2þ signals. ATP was pressure applied for 200 ms (arrows in (a) and (b)) through a micropipette placedover the cells.






of the drug to induce a variation in each connection. Typically,large fluctuations are more probable when the average weight isalso large. Therefore, to remove this scaling effect, we consider thenormalized second moment, s=⟨w⟩, which quantifies the fluctua-tions with respect to the observed average weight. We call thiscoefficient the sensitivity coefficient.

For evaluating how powerful a drug is in weakening orstrengthening the connectivity between each pair of cells weconsider the third moment of the weight distribution:

μ3ðTÞ ¼ 1L∑ia j

ðwijðTÞ�wijÞ3: ð5Þ

When μ¼ 0 it means that the amount of connections with astrength below average is the same as the amount of connectionswith a strength above it. If μa0 the weight distribution isasymmetric. A positive value indicates that the values of theweights concentrate on the right-side of the distribution, i.e. thereare few weak connections, while a negative value indicates thatthere are few strong connections. Consequently, when a stimulusleads to a connectivity network with a larger (smaller) μ valuethan previously, it has a strengthening (weakening) effect on theconnectivity of the tissue. Similar to the second moment, weconsider μ=s, normalized to the standard deviation s. We call thiscoefficient the strengthening coefficient.

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Fig. 9. Three different astrocytic cultures with the corresponding derived network structure of Ca2þ signal propagation. Circles denote the location of spots where the signalwas measured, namely in the cytoplasm. From the connectivity between cells and its evolution through a succession of ATP stimuli, one is able to evaluate relevant effects ofthe perfused drug on the culture (see text).

Fig. 10. Assessing the robustness and activity of samples of astrocytic tissues throughout a series of ten stimuli: two initial stimuli (Pre-C), four drug A (first row) or drug B(second row) and four washout (Post-C). Drug A: 4-[2-[[6-Amino-9-(N-ethyl-β-D-Ribofuranuronamidosyl)-9H-purin-2-yl]amino]ethyl]benzene]propanoic acid hydrochlor-ide; Drug B: N6-Cyclopentyl-9-β-D-Ribofuranosyl-9H-purin-6-amine. From left to right one sees result for flux ⟨w⟩ (first column) in Eq. (3), sensitivity s=⟨w⟩ (second column)in Eq. (4), strengthening μ=s (third column) in Eq. (5), variability η (fourth column) in Eq. (6). Here we use a total of ten experiments.






A fourth measure is added to these three moments, which wecall variability η. It is a function of the stimulus T that evaluateshow much the weights between each pair of cells varies from onestimulus to the next one:

ηðTÞ ¼∑i;jjwi;jðTþ1Þ�wi;jðTÞj∑i;jjwi;jðTþ1Þþwi;jðTÞj

: ð6Þ

The variability takes values between zero, when all connectionsremain the same from one stimulus to the next one, and one,when all connections switch from zero at T to one at Tþ1 or vice-versa. The larger the variability the broader the overall inducedchange in the connectivity network.

For each tissue sample we considered a succession of typicallyten stimuli, similar to what was done above. The first two stimuliwere applied in the absence of drugs (buffer) in order to uncoverthe tissue connectivity when not subjected to drugs. Then one ofthe two drugs, herein referred to as drug A and drug B, wasapplied and the astrocytes were stimulated four times. Drugs werethen removed from the bath and the astrocytes were stimulatedagain four times (washout period).

Fig. 10 shows the four coefficients, ⟨w⟩, s=⟨w⟩, μ=s and η,measured for each stimulus. As for measuring the R ratio inFig. 8, we define a drug effect as inhibitory if RDoRH1 , or asexhibitory if RD4RH1 , and the classification as reversible orirreversible is based on the comparison of RD�RH1 and RD�RH2 .In Table 2 we present a summary of the main results from Fig. 10,where “0” indicates no statistical significance in a T-test (p40:1).As one can see, both the flux and the strengthening are notaffected by any of the drugs, while the variability is. However,different from the amplitude, variability is shown to be affected bydrug B, being facilitatory and irreversible. Drug A also shows anirreversible and a facilitatory effect on the variability but curiouslywith some delay. This may indicate that the time for drug

application while being properly used when evaluating its effectin total signal amplitude (Fig. 8) may be not long enough for otherfeatures of the tissue influencing the signal propagation, in thiscase the variability. Such increase of the variability could be a signthat the structure supporting some robustness of signal propaga-tion is permanently modified by each drug. The same facilitatoryand irreversible effect on the sensitivity seems to also occur, butonly for drug A. It deserves to be noted that in a completelyrandom model of link creation in each time step T, the variabilitywould be peaked at a value of 1=2. This is not supported by ourdata, which is a strong indicator that the reconstructed networkdoes not have overwhelmingly random character.

For these new properties, we may summarize the analysis inthis section by stating that while drug A mainly affects sensitivitys=⟨w⟩, drug B mainly affects variability η.

Comparing the variability η with the amplitude one concludesthat, at least for these two different properties and for thecorrelation measure, the effects of the drugs are of differentnature. Therefore, one should approach such drug effects in amore extended way to analyze not only its overall influence in onesingle property, such as the total amplitude of the signals, but alsoin a set of properties that combined, characterize the functionalstructure of the network.

Finally, it is important to check whether the functional struc-ture does not change randomly among the several experimentsunder the same conditions. Fig. 11 shows the one-step correlation:

ρðTÞ ¼ 1L∑ia j

wijðTþ1ÞwijðTÞ ð7Þ

for all three periods when testing drug A (left) and drug B (right).If we imagine a simple binomial model in which functionalconnections between cells i; j are randomly created in each step,i.e. wijðtÞ ¼ ranð0;1Þ, both correlations would be equal to 1/4.Correlation values above 1/4 indicate the presence of actualcorrelations and memory about previous correlations in thesystem. When applying drug B correlations approximate theuncorrelated regime ρ¼ 1=4. For both pre- and post-periodshowever, correlation deviates significantly from ρ¼ 1=4, eviden-cing a functional structure underlying the sequence of signals. Fordrug A this deviation is even stronger and it is further strength-ened when applying the drug in a irreversible way (see Table 1).

5. Conclusion

In this paper we introduce a procedure for extracting thefunctional connectivity network in living tissues subjected toexternal pulsed stimuli that propagate through it as a signal. Ourmethod is based on the covariance matrix of the separated signalstaken at different time-lags. By adding proper constraints of a

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Table 2Table of evaluation properties for the drugs, drug A and drug B, in signal amplitudeand the properties assessing the structure of the signal propagation network. Foreach property we indicate whether its effects are facilitatory (þ) or inhibitory (�)and reversible (Rev.) or irreversible (Irrev.). Zero indicates no statistical significancein a T-test among a total of ten experiments. Last row shows the correlation (seetext), which deviates from the purely random case ρ¼ 1=4, except when applyingdrug B (see Fig. 11).

Property Drug A Drug B

Amplitude (Fig. 8) þ (Rev.) � (Irrev.)Flux ⟨w⟩ (Fig. 10, first col) 0 0Sensitivity s=⟨w⟩ (Fig. 10, second col) þ (Irrev.) 0Strengthening μ/s (Fig. 10, third col) 0 0Variability η (Fig. 10, fourth col) 0 þ (Irrev.)Correlation ρ (Fig. 11) þ (Irrev.) þ (Rev.)

Fig. 11. Correlations for drug A (top) and drug B (bottom) as shown in Eq. (7). For the period when applying drug B one observes ρ� 1=4 meaning that the functionalstructure is random. Still, before and after the application of drug B, the correlation deviates from the purely random case. The same occurs for the experiments with drug A(see text).






minimum time-delay between pairs of cells and single-sourcestimulus to each cell we are able to filter out all redundant orartificial connections from the covariance matrix. We test ourprocedure with synthetic data. There, our procedure proves to bebetter suited for assessing the connectivity of living tissues ofastrocytic samples used than other standard measures, namely themultivariate Granger causality algorithm. The better results of ourprocedure indicate that for the particular case of signal propaga-tion networks with one single triggering source, standard methodsmay retrieve biased results.

The weight (strength) of each connection is computed directlyfrom the covariance matrix and the minimum delay-time. Further, weshowed how to obtain additional insight relative to the drug used forstimulating the source-cell only by analyzing the distribution ofconnectivity strengths (weights). From the first, second and thirdmoments of the weight distribution we showed how to characterizethe signal flux and also the sensitivity and strengthening of thenetwork underlying the signal propagation in the tissue. We alsocharacterized the temporal stability of the network by its variabilityand a correlation measure, and found these measures to complementthe information from the signal amplitude ratios.

Following what was introduced in the beginning of our paper, it isimportant to stress here that our method is not able to uncover thephysical connectivity structure between cells and that it is also notaimed for studying particular features of the signal itself, such as themeasurement noise (Perc et al., 2008). While our method works forpulsed signals similar to the ones measured in the samples shown inFig. A1, as explained in Appendix A, and the variability alreadyreflects implicitly the impact of measurement noise in the sample ofsignals, it would be interesting to extend this methodology further tosituations where the measurement noise is significant.

To further validate our method, it would be interesting to test itwith an empirically known drug whose effect in the propagatingstructure is known. To our knowledge, there is no such drug test,since they typically focus on the overall (sum) signal in the tissueand not in the features of its propagation throughout the tissue.

All in all, this study proposes a procedure complementary to thestandard approach where only the overall amplitude is tested before,during and after the application of a given drug. Although it is not ableto extract the physical interconnections between cells, our simpleprocedure provides a way for quantifying the functional connectivityin the tissue and to ascertain how it changes due to the application ofdrugs. The framework introduced here as well as the reported findingsshould now be used for a systematic pharmacological study. Othermore sophisticated approaches, namely a spectral analysis (Brouwerand Haemers, 2011) applied to these signal-propagation networks,could be useful. In this case larger tissues, having a larger number ofastrocytes, are needed.

Finally, while the method introduced in this paper is based onconstraints that give results for primary functional connections,some other interactions with a real physiological meaning couldbe further considered. Namely, there can be autocatalytic phe-nomena by self-stimulation, some signals could be reversible ormultiple stimulation of one cell may occur in other situations.Forthcoming studies could further improve the ability for char-acterizing drug effects in living tissues by considering these moregeneral effects, i.e. several time-lags, or local maxima of the matrixsolution, so that a hierarchy of interactions could be determined.

Acknowledgments

The authors thank Matthew Blow for his suggestions on the text.M.P. thanks Ciência sem Fronteiras, Brasil, Ref. MCT/CNPq/CsF/202611/2012-4. Financial support from Fundação para a Ciência e a Tecnologia,Portugal is acknowledged by F.R. (SFRH/BPD/65427/2009), S.H.V.

(SFRH/BPD/81627/2011), P.G.L. (Ciência 2007) and also partial supportunder PEst-OE/FIS/UI0618/2011 and the binational program betweenFCT and DAAD, DRI/DAAD/1208/2013 by P.G.L. and F.R.

Appendix A. Generation of synthetic data

In order to evaluate our method for reconstruction the con-nectivity network, we propose a simple signal propagation modelwith properties similar to those found in the experiment.

The model for signal propagation treats a set of nodes i,i¼ 1;…;M with respective simulated signal time series X̂ iðtÞ thatare linked through connection with strength (weight) wij, either0 or 1.

The signal propagation is governed by the following rules:

(i) The first node, i¼1, is driven by an external Gaussian signalX̂1ðtÞpexpð�ðt�t0Þ2=2s2

0Þ, where t0 is the starting time ands0 is the standard deviation. Here we choose t0 ¼ 30 ands¼ 3.

(ii) Each node ia1 has only one incoming connection j (ja i), i.e.only one nonvanishing wji.

(iii) There are no simple loops, i.e. wii ¼ 0, for all i.(iv) At each time step t the signal departing from cell i is

distributed through its outgoing connections transportingthe signal to its neighbors j. The distribution is implicitlydefined as

X̂ iðtÞ ¼ α∑jwijX̂ jðtþτijÞ; ðA:1Þ

with α¼ 1=ð1þ∑jwijÞ and the delay τij ¼ 1 if wij ¼ 1 or τij ¼ 0otherwise.

This means that the cells receive the incoming signal which isdelayed by an amount in this case given by the strength of theincoming connections themselves. Such synthetic signals, while nothaving exactly the same shape as the empirical signals, do reproduceattenuation and stimulus delay in a similar way (see Fig. A1).

Appendix B. Granger causality

Granger causality procedures aim to test whether one timeseries is able to forecast another one. The main idea can easily beexplained by considering, for simplicity, two time series X1 and X2.One says that X1 Granger-cause X2 if the series of values of X1

provide information about future values of X2. More precisely, weselect a proper autoregression of X2 which has say m previousvalues:

Xð1Þ2 ðtÞ ¼ a0þ ∑

m

i ¼ 1aiX2ðt� iÞþϵð1Þt ; ðB:1Þ

and one computes the estimate errors ϵt.This autoregression is then compared with another one where

the values of X1 are considered:

Xð2Þ2 ðtÞ ¼ a0þ ∑

m

i ¼ 1aiX2ðt� iÞþ ∑

m

i ¼ 1biX1ðt� iÞþϵð2Þt : ðB:2Þ

If the latter estimate error ϵð2Þt is smaller than the former one ϵð1Þt ,X1 is assumed to Granger-cause X2.

In our paper we considered an improved multivariate versionof Granger causality, where the M time-series are taken simulta-neously and the error estimates of autoregressions are now for allpairs of variables and then compared as a matrix. Following theprocedure outlined in Blinowska et al. (2004), we fit to our vectorof measured time series XðtÞ ¼ ½X1ðtÞ;X2ðtÞ;…;XMðtÞ�T a MVAR

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(multivariate auto-regressive) model as

XðtÞ ¼ ∑p

k ¼ 1AðkÞXðt�kÞþEðtÞ ðB:3Þ

with a delay dependent coefficient matrix AðkÞ, a given maximumdelay p, and an error matrix E, which are assumed to be Gaussianwhite noise sources. Transforming the quantities to the domain offrequencies f, one can compute the so-called transfer matrix Hðf Þfrom the relation

A�1ðf ÞEðf Þ ¼Hðf ÞEðf Þ; ðB:4Þfromwhich one can compute the directed transfer function (DTF) γas

γ2ijðf Þ ¼jHijðf Þj2

∑Mm ¼ 1jHimðf Þj2

: ðB:5Þ

In a network of nodes i, j, the DTF is a measure of signalcausality for the propagation of the signal from node j to node i. Inour case, we use a single delay p¼1, effectively eliminating thefrequency dependency of the DTF, which reduces to a matrix ofconnection strengths. It should be noted, however, that the multi-variate Granger approach assumes stationarity of the underlyingtime series in order for the MVAR mechanism to be applicable.Stationarity does not hold in our case, as the measured signals arenon-stochastic soliton-like peaks with a compact support. This isthe fundamental reason for Granger causality to fail when appliedto stimuli signals in living tissues.

Appendix C. Supplementary material

Supplementary data associated with this paper can be found inthe online version at http://dx.doi.org/10.1016/j.jtbi.2014.04.024.

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Fig. A1. Comparison between the empirical signals measured on astrocytic tissues and synthetic signals generated as described in Section Appendix A.







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Modeling the functional network of primary intercellular Ca2+ wave propagation in astrocytes and its application to study drug effects