^{2+}Wave Propagation in Astrocytes

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Conceived and designed the experiments: MG MDP VV HB EBJ. Performed the experiments: MG MDP HB. Analyzed the data: MG MDP VV HB EBJ. Wrote the paper: MDP VV HB EBJ.

The authors have declared that no competing interests exist.

A new paradigm has recently emerged in brain science whereby communications between glial cells and neuron-glia interactions should be considered together with neurons and their networks to understand higher brain functions. In particular, astrocytes, the main type of glial cells in the cortex, have been shown to communicate with neurons and with each other. They are thought to form a gap-junction-coupled syncytium supporting cell-cell communication via propagating Ca^{2+} waves. An identified mode of propagation is based on cytoplasm-to-cytoplasm transport of inositol trisphosphate (IP_{3}) through gap junctions that locally trigger Ca^{2+} pulses via IP_{3}-dependent Ca^{2+}-induced Ca^{2+} release. It is, however, currently unknown whether this intracellular route is able to support the propagation of long-distance regenerative Ca^{2+} waves or is restricted to short-distance signaling. Furthermore, the influence of the intracellular signaling dynamics on intercellular propagation remains to be understood. In this work, we propose a model of the gap-junctional route for intercellular Ca^{2+} wave propagation in astrocytes. Our model yields two major predictions. First, we show that long-distance regenerative signaling requires nonlinear coupling in the gap junctions. Second, we show that even with nonlinear gap junctions, long-distance regenerative signaling is favored when the internal Ca^{2+} dynamics implements frequency modulation-encoding oscillations with pulsating dynamics, while amplitude modulation-encoding dynamics tends to restrict the propagation range. As a result, spatially heterogeneous molecular properties and/or weak couplings are shown to give rise to rich spatiotemporal dynamics that support complex propagation behaviors. These results shed new light on the mechanisms implicated in the propagation of Ca^{2+} waves across astrocytes and the precise conditions under which glial cells may participate in information processing in the brain.

In recent years, the focus of Cellular Neuroscience has progressively stopped only being on neurons but started to include glial cells as well. Indeed, astrocytes, the main type of glial cells in the cortex, dynamically modulate neuron excitability and control the flow of information across synapses. Moreover, astrocytes have been shown to communicate with each other over long distances using calcium waves. These waves spread from cell to cell via molecular gates called gap junctions, which connect neighboring astrocytes. In this work, we used a computer model to question what biophysical mechanisms could support long-distance propagation of Ca^{2+} wave signaling. The model shows that the coupling function of the gap junction must be non-linear and include a threshold. This prediction is largely unexpected, as gap junctions are classically considered to implement linear functions. Recent experimental observations, however, suggest their operation could actually be more complex, in agreement with our prediction. The model also shows that the distance traveled by waves depends on characteristics of the internal astrocyte dynamics. In particular, long-distance propagation is facilitated when internal calcium oscillations are in their frequency-modulation encoding mode and are pulsating. Hence, this work provides testable experimental predictions to decipher long-distance communication between astrocytes.

The 20^{th} century witnessed crystallization of the neuronal doctrine, viewing neuron as the fundamental building block responsible for higher brain functions. Yet, neurons are not the only cells in the brain. In fact, almost 50% of the cells in the human brain are glial cells

In particular, astrocytes, the main type of glial cells in the cortex, have attracted much attention because they have been shown to communicate with neurons and with each other. Indeed, astrocytes can integrate neuronal inputs and modulate the synaptic activity between two neurons ^{2+} elevations in the astrocyte cytoplasm

Two main types of neuronal activity-dependent Ca^{2+} responses are observed in astrocytes ^{2+} increases that are restricted to the very extremity of their distal processes ^{2+} elevations propagating along these processes as regenerative Ca^{2+} waves, eventually reaching the cell soma. The latter kind of event can even propagate to neighboring astrocytes, thus forming intercellular Ca^{2+} waves ^{2+} waves have been extensively observed in astrocyte cultures

In order to assess this hypothesis though, several aspects of Ca^{2+} signaling in astrocytes remain to be elucidated. Experimental data suggest that a stimulus impinging on an astrocyte is preferentially encoded in the modulation of the frequency (FM) of astrocytic Ca^{2+} oscillations ^{2+} oscillations can be highly variable, depending on cell-specific properties such as Ca^{2+} content of the intracellular stores, or the spatial distribution, density and activity of (sarco-)endoplasmic reticulum Ca^{2+}-ATPase (SERCA) pumps

Much effort has also been devoted to understand the mechanisms responsible for initiation and propagation of intercellular Ca^{2+} waves. From a single-cell point of view, intracellular Ca^{2+} dynamics in astrocytes is mainly due to Ca^{2+}-induced Ca^{2+} release (CICR) from the endoplasmic reticulum (ER) stores and its regulation by inositol trisphosphate (IP_{3}) _{3} molecules directly from the cytosol of an astrocyte to that of an adjacent one through gap junction intercellular hemichannels _{3} levels therein

Albeit experimental protocols monitor wave propagation as variations of intracellular Ca^{2+}, the molecule that is transmitted through gap junctions to neighboring astrocytes is not Ca^{2+}, but IP_{3} _{3} in a given cell increases, some of it can be transported through a gap junction to a neighbor astrocyte. This IP_{3} surge in the neighbor cell can in turn trigger CICR, thus regenerating the original Ca^{2+} signal. Yet, the transported IP_{3} is required to reach a minimal threshold concentration to trigger CICR in the neighboring cell. If this threshold is not reached, propagation ceases _{3} levels _{3} transport, could induce local IP_{3} concentrations large enough to trigger CICR ^{2+} wave propagation. Production of IP_{3} by Ca^{2+}-dependent PLCδ has been suggested as a plausible candidate regeneration mechanism ^{2+} waves simulated with this mechanism are hardly reconcilable with experimental observations, hinting a critical role for gap junction IP_{3} permeability

In the present study, we investigated the intercellular propagation of Ca^{2+} waves through the gap-junctional route by a computer model of one-dimensional astrocyte network. To account for intracellular Ca^{2+} dynamics, we adopted the concise realistic description of IP_{3}-coupled Ca^{2+} dynamics in astrocytes previously introduced in Ref. ^{2+} dynamics on the wave propagation distance. By means of bifurcation analysis and numerical solutions, we show that nonlinear coupling between astrocytes can indeed favor IP_{3} partial regeneration thus promoting large-distance intercellular Ca^{2+} wave propagation. Our study also shows that long-distance wave propagation critically depends on the nature of intracellular Ca^{2+} encoding (i.e. whether Ca^{2+} signals are FM or AM) and the spatial arrangement of the cells. Furthermore, our results suggest that, in the presence of weak coupling, nonlinear gap junctions could also explain the complex intracellular oscillation dynamics observed during intercellular Ca^{2+} wave propagation in astrocyte networks

We describe calcium dynamics in astrocytes by an extended version of the Li-Rinzel model ^{2+} regulation by IP_{3}-dependent CICR as well as IP_{3} dynamics resulting from PLCδ-mediated synthesis and degradation by IP_{3} 3-kinase (3K) and inositol polyphosphate (IP) 5-phosphatase (5P). The temporal evolution of astrocytic intracellular calcium in our model is described by three coupled nonlinear equations:_{3}_{3}R channels on the ER membrane, and the cell-averaged concentration of IP_{3} second messenger, respectively. Each one of these variables is coupled to others via the set of equations that describe contributions of different biochemical pathways, as described in details in Supplementary Information (equations S1–S4) alongside the complete mathematical analysis of the model features.

(^{2+}-induced Ca^{2+} release (CICR) from the endoplasmic reticulum (ER) is the main mechanism responsible for intracellular Ca^{2+} dynamics in astrocytes. (^{2+} dynamics and IP_{3} metabolism in the astrocyte. (_{3} production is brought forth by hydrolysis of PIP_{2} by PLCδ (the activity of which is regulated by Ca^{2+}). (_{3} mainly occurs through IP_{3} 3-kinase- (3K-) catalyzed phosphorylation and inositol polyphosphate 5-phosphatase (IP-5P)-mediated dephosphorylation. For simplicity, Ca^{2+}-dependent PKC-mediated phosphorylation of IP_{3}-3K _{4} to IP-5P are not considered in this study. The legend of different arrows is given below (

In a single-cell context, this model reproduces most of the available experimental data related to calcium oscillations in astrocytes. In particular, it faithfully reproduces the experimentally reported changes of oscillation frequency and wave shape caused by SERCA pump activity modulations

Experimental evidence shows that chemical signaling between astrocytes usually takes the form of propagating Ca^{2+} pulses that are elicited following the gap-junctional transfer of IP_{3} second messenger molecules _{3} activates the CICR pathway, giving rise to the observed rapid transient elevations in cytosolic free calcium. We considered three scenarios to describe the exchange of IP_{3} between a pair of adjacent astrocytes: (1) linear, (2) threshold-linear (composed of a linear term operating after a threshold) and (3) non-linear (here described as sigmoid) coupling (see _{3} only when the IP_{3} gradient between the two adjacent cells overcomes a threshold value.

Shown is the relative flux, i.e. the value the IP_{3} flux divided by the coupling force _{3} gradient (Δ_{3}_{3}^{thr} = 0.3 µM, _{3}^{scale} = 0.05 µM.

Our investigation of nonlinear coupling case was motivated by the experimental observations suggesting that gap junction permeability in itself can be actively modulated by various factors, among them different second messengers. Indeed, there is growing evidence that gap junctions may have greater selectivity and more active gating properties than previously recognized _{3} degradation ^{2+} _{3} and Ca^{2+} dynamics ^{2+} wave propagation in astrocytes.

The linear model simply results from Fick's law of diffusion. The flux _{3} molecules (where

Threshold-linear coupling only partially keeps the linear characteristics of the “classical” gap junction adding a threshold on IP_{3} gradient below which the flux _{3} gradient value, above which the IP_{3} flux is constant.

Sigmoid coupling is defined as:_{3}^{thr} is the predetermined threshold value and _{3}^{scale} is the width of the transition zone in the sigmoid function (see

We consider chains of ^{i}^{i}_{3}^{i}^{2+} concentration, the ratio of open IP_{3}Rs and the intracellular IP_{3} concentration in this astrocyte. The dynamics of these internal variables is given by the _{3}

Reflective (zero-flux) boundaries assume that IP_{3} exiting cell 1 or _{3} fluxes. This is the case of absorbing boundary conditions in which IP_{3} can flow from cell 2 to cell 1, but the reverse flux (from cell 1 to 2) is always null (and similarly for cells

To induce wave propagation in the astrocyte chain, one cell (referred to as the “driving” cell) is stimulated by a supplementary exogenous IP_{3} input. This external stimulus is supplied through a (virtual) “dummy” cell, coupled to the driving cell by one of the coupling functions described above. In this sense, the dummy cell acts as an IP_{3} reservoir in which the level of IP_{3} is kept fixed to a constant value _{3}^{bias}.

Let _{3} dynamics in the

Most simulations done in this work were driven by a constant value of _{3}^{bias}. In the last section though, a square positive wave stimulus was applied to the model.

Initial conditions for all cells were set in agreement with experimental values reported in astrocytes for Ca^{2+} and IP_{3} at basal conditions

The chain model consists of 3^{th}-order Runge-Kutta scheme with a time step of 10 ms as this value showed to be the best compromise between integration time and robustness of the results. The computational model was implemented in Matlab (2009a, The MathWorks, Natick, MA) and C. Bifurcation analysis was done using XPPAUT (

Before proceeding to study the propagation of calcium waves in spatially extended networks of astrocytes, it was necessary to understand the dynamical response of a single model cell in response to IP_{3} stimulation. To this end, we performed a detailed bifurcation analysis of our model astrocytes. A wealth of dynamical regimes was discovered, allowing model astrocytes to encode information about IP_{3} stimulus in amplitude-modulated (AM), frequency-modulated (FM) or mixed (AFM) modes, depending on parameter values (see _{3}^{bias} values, which can turn into complex oscillations for larger _{3}^{bias} values. Because stable oscillation regimes could coexist in the bifurcation diagrams with stable fixed points, it could not be predicted from these diagrams whether an IP_{3} input to the cell would trigger pulse-like oscillations or not, i.e. whether it would switch the system from the fixed point to the oscillatory regime. Thus, we resorted to extensive numerical simulations to investigate under what conditions one could observe propagation of Ca^{2+} waves along the astrocyte chain.

In agreement with previous studies (see _{3}-triggered CICR indeed allows intercellular Ca^{2+} wave propagation in our modeling framework, as shown in Supplementary Information _{3} dynamics ^{2+} wave propagation, under different coupling modes (linear vs. nonlinear) and different encoding regimes (FM vs. AFM).

Propagation patterns both for the linear and nonlinear cases are presented in _{3}^{bias} = 1.0 µM) was used and always applied to the first cell of the chain. Model analysis (see _{3} would trigger periodic Ca^{2+} pulses at least in the stimulated cell and possibly in the other ones as indeed confirmed by simulations (see Astrocyte 1) both in the case of linear and non-linear coupling. For the linear coupling case, we observed propagation failure at 6^{th}–7^{th} cell from the driving one (^{2+} pulses can propagate for the whole length of the chain (

The astrocyte chain was composed of 12 FM-encoding cells with reflective boundary conditions. Stimulation triggered by _{3}^{bias} = 1.0 µM from

Analysis of the IP_{3} pattern for the nonlinear coupling function (_{3} and Ca^{2+} pulses. The IP_{3} pulses are followed in time by the Ca^{2+} ones, suggesting that pulsed Ca^{2+} propagation is mediated by the propagation of IP_{3} across the cells. By contrast, in the case of linear coupling, the correlation between the propagating Ca^{2+} pulses and the intracellular IP_{3} signals is not so apparent (_{3} seems to diffuse smoothly from the stimulated cell without any effective propagation pattern.

The observed difference in the propagation distance between linear and nonlinear gap-junction couplings can be understood from this analysis. Indeed, in the case of linear gap-junction coupled cells, the IP_{3} arriving in cell _{3} displays the almost diffusive pattern of _{3} concentration becomes too small to trigger CICR. This stops Ca^{2+} wave propagation. Conversely, with nonlinear gap junctions, IP_{3} can accumulate in cell _{3} concentration evolves to the locally regenerative spatiotemporal pattern illustrated in ^{2+}

The distances (measured in units of number of cells) travelled by the propagating Ca^{2+} waves as a function of the stimulation amplitude for an astrocyte chain of _{3}^{bias}, but never exceeds one third of the chain length. On the contrary, with nonlinear sigmoid coupling, Ca^{2+} oscillations propagate along the whole chain as soon as the oscillatory regime is engaged (that is for _{3}^{bias} >0.72 µM, see

The stimulus is applied to the first cell and the traveled distance is expressed in number of cells. With moderate coupling strength (^{2+} propagation is observed in the case of FM (_{3}^{bias} = 2.0 µM) is shown in (_{5P}-ν_{δ}

_{3} concentrations. Hence, these results indicate that the significant parameter for long-distance wave propagation through nonlinear gap junctions is the presence of an IP_{3} concentration threshold below which the junction is closed (this property is shared by the two nonlinear models), rather than the saturation of the transport at high IP_{3} concentrations (found only in the sigmoid model).

Because the effects due to the different shapes of coupling curves could be conflated in the above observations, we computed the dependence of wave propagation range on the maximal strength of coupling, for linear vs. nonlinear coupling cases (^{2+} wave propagation distance was significantly larger for the nonlinear case as compared to the linear case, ruling out the possibility that our findings are just a trivial confound. The only exceptions to this claim were noted for low ^{2+} waves (except in the “chaotic” low

_{δ}_{3} production by PLCδ) and _{5P} (max. rate of IP_{3} degradation by IP-5P) vary. We locate with black dots the (_{δ}_{5P}) pairs for which Ca^{2+} waves propagate across the whole cell chain. Clearly, long-range propagation is found for a wide region of this parameter space. As expected, larger IP_{3} synthesis rates must be balanced by larger IP_{3} degradation rate to allow long range propagation, hence the diagonal-like aspect of the black region in the panel.

These first results thus indicate that the propagation distance of Ca^{2+} waves in our model is much smaller with linear gap junctions than with nonlinear ones. This observation remains valid when the number of cells in the chain is much larger (we have simulated up to 120 cells in the chains) or/and when up to the 20 first cells in the chain receive the stimulation simultaneously (not shown). The above results are also robust with respect to the changes in boundary conditions (see ^{2+} wave propagation for a chain of

This confirms that the difference of propagation distance between linear and nonlinear gap junction-coupling is a robust and fundamental property of our model. Hence, the existence of a threshold concentration for cell-to-cell IP_{3} diffusion, similar to the one displayed by nonlinear gap junctions may be a critical factor for long-distance propagation of Ca^{2+} waves across astrocytes. In what follows, we examine the influence of a second physiological characteristic of Ca^{2+} signaling in astrocytes, namely their stimulus encoding mode (FM-encoding or AFM-encoding chains).

As illustrated in ^{2+} waves do not propagate in our model of AFM-type astrocyte chains. In ^{2+} variations are observed only in the driving cell (cell 1 in ^{2+} changes or no Ca^{2+} change at all. Importantly, _{3}^{bias} = 1.5 µM, namely an intensity deeply inside the oscillatory region of the bifurcation diagrams in _{3} diffusion through gap junctions. First, in the case of nonlinear gap junctions, the stimulus strength is well beyond the diffusion threshold (_{3}^{thr}). Secondly, failure is also observed with linear gap junctions, where no coupling threshold can impede cell-to-cell diffusion. Hence the propagation failure likely stems from an intrinsic inability of AFM astrocytes to build up sufficient intracellular IP_{3} levels to trigger CICR in neighboring cells.

The astrocyte chain was composed of 12 AFM astrocytes. Stimulus protocol and other parameters as in

This intrinsic difference in the propagation properties brought about by AFM or FM modes can be explained on the basis of the single-cell bifurcation diagrams (^{2+} oscillations decreases with decreasing IP_{3} stimulations. Hence the IP_{3} generation in AFM is such that a local depression of transmitted IP_{3} will be accentuated in the next cell. Any decline of IP_{3} production in a given cell will thus be transmitted outward and amplified along the chain, until the signal eventually fails. This phenomenon is not observed with FM cells because, by definition of the FM mode, the peak amplitude of the IP_{3} oscillations in cell _{3} stimulus coming from cell _{3} stimulus falls within the oscillatory range of cell _{3} values in cell

Moreover, the range of IP_{3} input that gives rise to oscillations in the AFM encoding regime is much narrower than in the FM case. Thus, a perturbation of the IP_{3} stimulation from cell _{δ}_{5P}) pairs that were tested (results not shown).

Therefore, these results suggest a neat functional difference between AFM and FM oscillations in astrocytes: while FM could support long distance propagation of pulse-like Ca^{2+} waves, AFM is rather expected to give rise to localized Ca^{2+} signalling with diffusion-like spatial patterns for IP_{3}. Hence, any parameter relevant to ^{2+} signaling and able to switch the cell between AFM and FM modes (e.g. the affinity or activity of the SERCA pumps) is predicted to play a key role in the ^{2+} signals in astrocytes.

Because the astrocyte population within the brain is heterogeneous ^{2+} wave propagation across astrocytes with different properties. Here we tackled this issue using composite astrocyte chains, namely chains constituted of both FM and AFM cells, and investigated under what conditions propagation is possible with nonlinear sigmoid gap junctions.

In ^{2+} wave in this cell. The intensity of the stimulus was set close to the upper edge of the cell oscillatory range according to the bifurcation diagrams in ^{2+} wave in a chain of alternating FM (black traces) and AFM (gray traces) cells and shows that propagation abruptly terminates at the second AFM cell in the chain (cell 4). Notably, closer inspection of IP_{3} dynamics in the subsequent FM cell (i.e. cell 5 in _{3} concentration intermittently passes across the predicted threshold for oscillations in this cell. However the time spent above the threshold is never large enough to trigger CICR, so that propagation halts. Extending propagation to further cells in the chain thus demands faster endogenous IP_{3} production. This can for instance be obtained with larger values of the maximal rate of PLCδ, _{δ}_{3} degradation rates, _{3K}_{5P}

_{3}^{bias} = 1.2 µM was applied between _{δ}^{(AFM)} = 0.108 µM/s; _{δ}^{(FM)} = 0.832 µM/s; _{5P}^{−1}; _{3}^{thr} = 0.215 µM; (_{δ}^{(AFM)} = 0.15 µM/s. In all these simulations, a 5-minute-long stimulus was applied.

The former possibility is considered in _{δ}_{3} production, all AFM cells in the chain in fact maintain intracellular IP_{3} concentration either beyond or within the oscillatory range. Moreover, since this range essentially overlaps with the lower part of the oscillatory range for FM cells, the IP_{3} transported from one AFM to the next FM cell in the chain can trigger CICR there, thus perpetuating propagation.

Another possible mechanism to facilitate wave propagation across AFM cells in heterogeneous conditions consists in increasing the frequency of the wave pulses in the FM cell preceding the AFM one. This effect is actually naturally obtained when several successive FM cells are placed between two AFM ones. We illustrate this in ^{2+} pulses, and thus the frequency of elementary diffusion events of IP_{3} in the next AFM cell. In turn this increases both the frequency of the IP_{3} oscillations in the AFM cell and their minimal level, thus allowing Ca^{2+} wave propagation in the subsequent FM cells.

The presence of homogenous FM cell domains between AFM cells is therefore likely to enable long traveling distances for propagating Ca^{2+} waves. One may even assume that if the number of successive FM cells in the FM domains is large enough, the Ca^{2+} wave should propagate over the entire network, whatever its size. Although we did not further investigate this possibility, our simulations hint on the contrary that there likely exists an upper bound for the travelling distance because the frequency of propagating Ca^{2+} waves in FM domains is not constant, but tend to decrease after each AFM cell, as can be seen by comparison of the pulse frequency in the Ca^{2+} traces of FM cells in _{3} diffusion through gap junctions, thus terminating propagation (results not shown).

We note that modifications of the IP_{3} threshold for diffusion in the nonlinear gap junctions, _{3}^{thr}, should facilitate transmission of Ca^{2+} pulses from cell to cell in the chain, thus increasing the frequency of propagation (

The Ca^{2+} and IP_{3} dynamics observed so far were all obtained using a rather high value of the coupling strength (^{−1}). In these conditions, the properties of the propagated waves are rather simple: a pulse-like (or not) wave front travels across astrocytes, with conserved shape and either stops after a few cells or invades the whole cell chain. However, our system is a spatially extended dynamical system with large numbers of degrees of freedom. Such systems (e.g. coupled map lattices) are known to manifest complex spatiotemporal behaviors when the coupling strength changes. To get an insight on the possible propagation behavior exhibited by our model with weaker coupling, we considered the dynamics with reduced levels of gap junction permeability (setting ^{−1}).

^{2+} dynamics of 41 coupled FM cells and a square wave periodic stimulation applied to the central cell #21 (see figure caption for details). Visual inspection of the Ca^{2+} traces in each cell (^{2+} waves that can propagate along the whole chain. Importantly, this figure also evidences the occurrence of occasional propagation failures that do not seem to result from a simple spatiotemporal pattern. Actually, observation of the temporal traces of each individual cell reveals the occurrence of pulse-like events showing up with no apparent regularity. Accordingly, the distribution of the time-intervals between two such pulses can be very broad for some cells, with large intervals often almost as probable as small ones (see

The stimulus is an oscillatory input (positive square wave) applied to the central cell of an _{3}^{2+} and IP_{3} time series. Simulation performed on FM-encoding astrocytes with reflective boundary conditions and sigmoid gap junctions. Stimulus protocol: positive square wave of 50-second period and duty cycle of 0.4.

Albeit consistently pulse-like, the shape of the propagated Ca^{2+} waves is also quite variable. Closer inspection of the time series for the driving cell (i.e. cell 21) for instance shows that the generated Ca^{2+} pulses vary from a single-peak waveform to multiple peaks per single pulse (_{3} signals is also very large. The lack of obvious regular behavior is particularly striking on movies showing the parallel temporal evolution of the Ca^{2+} and IP_{3} level in each cell, as in

To further illustrate the complexity of the obtained dynamics, we plot in ^{−1} (depending on the time series under consideration).

The apparent complexity of the dynamics is most likely due to some form of spatiotemporal chaos, the nature of which is beyond the scope of the current article and is left to future work. But whatever the response, these simulations evidence that complex Ca^{2+} wave propagation patterns can manifest at low couplings, even with spatially homogeneous cell properties and in the absence of any stochasticity source.

Calcium-mediated signalling is a predominant mode of communication between astrocytes ^{2+} wave propagation is possible when the gap junctions are rendered by nonlinear permeability but only when most of the model astrocytes are tuned to encode the strength of incoming IP_{3} signal into frequency modulated Ca^{2+} oscillations.

There has been a long-standing debate over the nature and characteristics of intercellular Ca^{2+} waves observed in astrocyte networks. The present article concerns about the purely intracellular route, which involves the transfer of IP_{3} molecules directly from cytosol to cytosol through gap junctions

A critical issue for the modeling studies of intercellular Ca^{2+} waves is to explain the observed variability of Ca^{2+} wave travelling distance ^{2+} waves are obtained via IP_{3} regeneration in each cell by Ca^{2+}-activated PLCδ. However, whenever PLCδ maximal activity is lower, regeneration becomes partial and the Ca^{2+} wave propagation distance decreases. Yet this model does not include Ca^{2+}-dependent IP_{3} degradation, which could be critical for the occurrence of IP_{3}-mediated Ca^{2+} oscillations _{3} production, thus hindering IP_{3} regeneration and Ca^{2+} wave propagation. This calls for additional factors to be taken into account to explain intercellular Ca^{2+} wave propagation.

A first prediction of our model is that, regenerative waves are possible in a network composed in its majority of astrocytes that encode information about incoming IP_{3} signals in the frequency of their Ca^{2+} oscillations (FM). Interestingly, the response of astrocytes in vivo to IP_{3} stimulation is known to exhibit high variability, both in frequency and amplitude ^{2+} of the SERCA pumps. To our knowledge, the kinetic properties of SERCA2b have never been measured in astrocytes. However the hypothesis that SERCA2b affinity for Ca^{2+} shows variability ^{2+} from 170 to ∼400 nM ^{2+} wave propagation via direct interaction with SERCA2b thus modulation of Ca^{2+} uptake by this pump

In our model, the strength and the transfer properties of the gap junction coupling are critical permissive factors that allow long-range intercellular signaling between the astrocytes. In particular, nonlinear gap junctions were found to significantly enhance the range of Ca^{2+} wave propagation (as opposed to the classic linear gap junctions that caused fast dissipation). Gap junctions with dynamic resistance are known to exist in cardiac networks _{3} metabolism ^{2+} waves therein

In the present study, we considered a simplified setup of 1D network implemented as a regular chain of coupled cells. Such 1D chains display attractive aspects. In particular, we could proceed to a numerical bifurcation study of these 1D coupled-cell systems (see

Recent studies suggest that the astrocytes within the cortex form heterogeneous populations ^{2+} wave propagation in composite 1D networks, consisting of both FM- and AFM-encoding cells. Our simulations predict that the propagation dynamics and distance of intercellular Ca^{2+} waves critically depends both on the encoding property of the cells and on their spatial arrangement. Interestingly, the cell bodies of neighboring astrocytes within the brain are believed to distribute in space in a nonrandom orderly fashion called “contact spacing” ^{2+} wave propagation in astrocyte networks. If, as suggested by our model, the spatial arrangement of the astrocytes, coupled to the heterogeneity of their response, conditions Ca^{2+} wave propagation, then contact spacing may play a critical role in intercellular wave propagations in the brain and the related computational properties of astrocyte networks.

It is now widely accepted that astrocytes and neurons are interwoven into complex networks and are engaged in an intricate dialogue, exchanging information on molecular level ^{2+} waves spreading through the astrocyte network. The connectivity of this astrocyte network is in turn defined by the patterns of electrical activity in neuronal network

Bifurcation analysis of an uncoupled (i.e. isolated) ^{2+} oscillations (f) at almost constant amplitude (d). Legend: (a, b): ^{2+} (c)

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Bifurcation analysis of the astrocyte chain model for _{3}^{bias} values larger than ≈0.8 µM, the stable oscillations become far more complex than in the isolated case. This is due to a very rapid cascade of period-doubling bifurcations, which yields extremely complex limit cycles (with numerous folds) that could not be precisely rendered in the figure (see also Section III.1.b). Moreover, for _{3}^{bias} >1.1 µM, the amplitude of these limit cycles shrinks and numerical investigations evidenced the coexistence of multiple complex stable orbits. Legend:

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IP_{3}-triggered CICR-mediated propagation of a pulsed Ca^{2+} wave within a chain of five FM _{3} stimulation of constant intensity (_{3}^{bias} = 0.8 µM) is applied to cell A1 from t = 10 s to t = 30 s. This increases IP_{3} concentration, thus triggering CICR from the ER and the generation of a Ca^{2+} pulse. (b) By means of communication through gap junctions, suprathreshold IP_{3} from A1 can diffuse to A2, triggering CICR there. The process is essentially regenerative so that a Ca^{2+} pulse almost identical to the original one can be observed in the arrival cells. (c) As soon as the IP_{3} influx to one cell from its neighbors is not sufficient to trigger CICR, the propagation stops. This is indeed the case of cells A4 and A5. Cells were coupled by sigmoid gap junctions and experienced reflective boundary conditions.

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Propagation patterns with non-linear sigmoid-like gap junctions in an astrocyte chain of 12 FM-encoding cells with periodic (a,b) or absorbing (c,d) boundary conditions. Stimulation triggered by _{3}^{bias} = 1.0 µM from

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Calcium traces for wave propagation in composite astrocyte chains constituted of both FM _{3}^{thr} = 0.215 µM or (b) _{3}^{thr} = 0.3 µM. The diffusion threshold is critical to determine the efficiency of transmission of Ca^{2+} waves along astrocyte chains. Other parameters as in

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Interpulse interval distributions for the simulations shown in

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Parameter list and used values. The parameters of the Li-Rinzel core of the ChI model were taken according to previous studies [De Pittà ^{2+} and _{3}

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Supplementary information text.

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Illustration of the chaotic-like behavior at low coupling. This movie shows the evolution with time of the calcium (upper panel, blue bars) and IP_{3} (lower panel, green bars) concentration in each cell (x-axis) of a ^{1}). Stimulation triggered by _{3}^{bias} = 1.0 µM from

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The authors wish to thank Vladimir Parpura, Giorgio Carmignoto and Ilyia Bezprozvanny and Herbert Levine for insightful conversations.

^{2+}dependent glutamate release.

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_{3}receptormediated [Ca

^{2+}]

_{i}oscillations derived from a detailed kinetic model: A HodgkinHuxley like formalism.

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^{2+}feedback on single cell inositol 1,4,5trisphosphate oscillations mediated by Gproteincoupled receptors.

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^{2+}pumps and their modulation by phospholamban.

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