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Glutamate regulation of calcium and IP3 oscillating and pulsating dynamics in astrocytes

Recent years have witnessed an increasing interest in neuron-glia communication. This interest stems from the realization that glia participates in cognitive functions and information processing and is involved in many brain disorders and neurodegenerative diseases. An important process in neuron-glia communications is astrocyte encoding of synaptic information transfer: the modulation of intracellular calcium dynamics in astrocytes in response to synaptic activity. Here, we derive and investigate a concise mathematical model for glutamate-induced astrocytic intracellular Ca2+ dynamics that captures the essential biochemical features of the regulatory pathway of inositol 1,4,5-trisphosphate (IP3). Starting from the well-known two-state Li-Rinzel model for calcium-induced-calcium release, we incorporate the regulation of the IP3 production and phosphorylation. Doing so we extended it to a three-state model (referred as the G-ChI model), that could account for Ca2+ oscillations triggered by endogenous IP3 metabolism as well as by IP3 production by external glutamate signals. Compared to previous similar models, our three-state models include a more realistic description of the IP3 production and degradation pathways, lumping together their essential nonlinearities within a concise formulation. Using bifurcation analysis and time simulations, we demonstrate the existence of new putative dynamical features. The cross-couplings between IP3 and Ca2+ pathways endows the system with self-consistent oscillator properties and favor mixed frequency-amplitude encoding modes over pure amplitude modulation ones. These and additional results of our model are in general agreement with available experimental data and may have important implications on the role of astrocytes in the synaptic transfer of information.


I. Introduction
Astrocytes, the main type of glial cells in the brain, do not generate action potentials like neurons do, yet they can transfer information to other cells and encode information in response to external stimuli by employing "excitable"-like rich calcium (Ca 2+ ) dynamics (Volterra and Meldolesi, 2005). Recognition of the potential importance of the intricate inter-and intra-cellular astrocyte dynamics has motivated in recent years, intensive experimental efforts to investigate neuron-glia communication. Consequently, it was discovered that intracellular Ca 2+ levels in astrocytes can be regulated by synaptic activity (Wang et al., 2006;Pasti et al., 1997;Porter and McCarthy, 1996;Parpura et al., 1994;Dani et al., 1992). Responses to low-intensity synaptic stimulation or spontaneous astrocyte activity usually consist of spatially confined Ca 2+ transients (Nett et al., 2002;Pasti et al., 1997;Porter and McCarthy, 1996). On the other hand, high-intensity synaptic activity or stimulation of adjacent sites within the same astrocytic process, are generally associated with Ca 2+ oscillations (Zonta and Carmignoto, 2002) that can bring forth propagation of both intracellular and intercellular waves (Stout et al., 2002;Charles, 1998;Cornell-Bell et al., 1990). Concomitantly, elevation of cytoplasmic Ca 2+ induces the release from astrocytes of several neurotransmitters (or "gliotransmitters"), including glutamate, ATP or adenosine (see Evanko et al., 2004 for a review). These astrocytereleased gliotransmitters feed back onto pre-and post-synaptic terminals. It implies that astrocytes regulate synaptic information transfer (Volman et al., 2007;Araque et al., 1998). Astrocytes can also mediate between neuronal activity and blood circulation , thus extending neuron-astrocyte communications to the level of neuronal metabolism (Bernardinelli et al., 2004).
The physiological meaning of astrocytic Ca 2+ signaling remains currently unclear, and a long-standing question is how it participates in the encoding of synaptic information transfer (De Pittà et al., 2008a;2008b;Volterra and Meldolesi, 2005). Some of the available experimental data suggest a preferential FM (frequency modulation) mode of encoding, namely synaptic activity would be encoded in the frequency of astrocytic Ca 2+ pulsations (Parpura, 2004). Indeed, cytoplasmic Ca 2+ waves in astrocytes often appear as pulse-like propagating waveforms (namely pulses of width much smaller than their wavelength), whose frequency increases when the frequency or the intensity of synaptic stimulation grows (Pasti et al., 1997).
Notwithstanding, the possibility of AM (amplitude modulation) encoding of synaptic activity or even of mixed AFM (amplitude and frequency modulation) encoding has also consistently been inferred (Carmignoto, 2000). For instance, the amplitude of Ca 2+ oscillations in response to external stimuli can be highly variable, depending on the intensity of stimulation (Wang et al., 2006;Finkbeiner, 1993;Cornell-Bell et al., 1990).
Experimental evidence suggests that Ca 2+ dynamics does not simply mirror synaptic activity but is actually much more complex, to a point that astrocytes are suspected of genuine synaptic information processing (Perea and Araque, 2005). The emerging picture is that the properties of Ca 2+ oscillations triggered by neuronal inputs in astrocytes (including their amplitude, frequency and propagation) are likely to be governed by intrinsic properties of both neuronal inputs and astrocytes (Volterra and Meldolesi, 2005;Pasti et al., 1997).
From the modeling point of view, simplified or two-variable models for intracellular Ca 2+ signaling can in principle be used to account for the diversity of the observed Ca 2+ dynamics when the biophysical parameters are varied. We recently presented evidence that one of these two-variable models proposed by Li and Rinzel (Li and Rinzel, 1994) actually predicts that the same cell could encode information about external stimuli by employing different encoding modes. In this model, changes of biophysical parameters of the cell can switch amongst amplitude modulation (AM) of Ca 2+ oscillations, frequency modulation (FM) of Ca 2+ pulsations or combined AM and FM (AFM) Ca 2+ pulsations (De Pittà et al., 2008a;2008b). We emphasize that one of the cardinal simplifications of the Li-Rinzel model is neglecting the regulation of inositol 1,4,5-trisphosphate (IP 3 ) dynamics, that is its production and degradation. Since IP 3 production is regulated by synaptic activity (via extracellular glutamate signaling), IP 3 dynamics has to be included for proper modeling of synapse-astrocyte communication. Only such modeling can provide a realistic account of astrocytic Ca 2+ variations induced by nearby synaptic inputs.
Here, we introduce and investigate a concise model for glutamate-induced intracellular astrocytic dynamics. Using this model we show new putative features of Ca 2+ dynamics that can have important implications on the role of astrocytes in synapse information transfer. Our model incorporates current biological knowledge related to the signaling pathways leading from extracellular glutamate to intracellular Ca 2+ , via IP 3 regulation and IP 3 -dependent Ca 2+ -induced Ca 2+ release (CICR). First we extend the Li-Rinzel model to incorporate the regulation of IP 3 . This yields a three-variable model, called hereafter the "ChI" model, for its state variables that are the intracellular Ca 2+ level C, the fraction of inactive IP 3 receptors h, and the available IP 3 concentration I. Similar three-variable models have already been introduced in previous works (Kazantsev, 2009;Politi et al., 2006;Höfer et al., 2002;Sneyd et al., 1995;Dupont and Goldbeter, 1993;Meyer and Stryer;see Falcke, 2004 for a review), yet our modeling includes a more realistic description of IP 3 dynamics, in particular with regard to the complex regulatory pathways of IP 3 formation and degradation. Furthermore, while we reduce these complex regulatory pathways to a concise mathematical description, we make sure to keep their essential nonlinearities. We then model the contribution of glutamate signals to IP 3 production and include this contribution as an additional production term into the IP 3 equation of the ChI model. We refer to this case as the "G-ChI" model.
We utilize bifurcation theory to study the coexistence of various encoding modes of synaptic activity by astrocytes: amplitude modulation (AM), pulsation frequency modulation (FM) and mixed amplitude and frequency modulations (AFM). We also present results of time simulations of the model illustrating the richness of intracellular Ca 2+ dynamics (hence, of the encoding modes) in response to complex time-dependent glutamate signals.
We note that although the model presented here is derived for the specific case of astrocytes, our approach can be readily adopted to model Ca 2+ dynamics in other cell types whose coordinated activity is based on intra-and inter-cellular Ca 2+ signaling, such as heart cells, pancreas cells and liver cells.

II. Derivation of the three-variable ChI model of intracellular Ca 2+ dynamics
In this section, we describe the concise ChI model for intracellular Ca 2+ dynamics in astrocytes with realistic IP 3 regulation. Given the relative intricacy of this signaling pathway (see Figure 1), each basic building block of the model is described separately in the following sections.

II-1. CICR core and the two-variable Li-Rinzel model
Intracellular Ca 2+ levels in astrocytes (as in most other cell types) can be modulated by several mechanisms. These include Ca 2+ influx from the extracellular space or controlled release from intracellular Ca 2+ stores such as the endoplasmic reticulum (ER) and mitochondria (Berridge et al., 2000). In astrocytes, though, IP 3 -dependent calciuminduced calcium release (CICR) from the ER is considered the primary mechanism responsible of intracellular Ca 2+ dynamics (Agulhon et al., 2008).
Calcium-induced Ca 2+ release (see Figure 1c) is essentially controlled by the interplay of two specific transports: efflux from the ER to the cytoplasm that is mediated by Ca 2+dependent opening of the IP 3 receptor (IP 3 R) channels, and influx into the ER which is due to the action of (Sarco-) Endoplasmic-Reticulum Ca 2+ -ATPase (SERCA) pumps. In basal conditions however (when CICR is negligible), intracellular Ca 2+ levels are set by the respective contributions of a passive Ca 2+ leak from the ER, SERCA uptake and plasma membrane Ca 2+ transport (Li et al., 1994;De Young and Keizer, 1992).
When synaptic activity is large enough, synaptically-released glutamate may spill over the synaptic cleft and bind to the extracellular part of astrocytic metabotropic glutamate receptors (mGluRs) (Porter and McCarthy, 1996). Binding of glutamate to mGluRs increases cytosolic IP 3 concentration and promotes the opening of few IP 3 R channels (Berridge, 1993). As a consequence, intracellular Ca 2+ slightly increases. Since the opening probability of IP 3 R channels nonlinearly increases with Ca 2+ concentration (Bezprozvanny, 1991), such initial Ca 2+ surge increases the opening probability of neighboring channels. In turn this leads to a further increase of cytoplasmic Ca 2+ . These elements therefore provide a self-amplifying release mechanism (hence the denomination of CICR). The autocatalytic action of Ca 2+ release however reverses at high cytoplasmic Ca 2+ concentrations, when inactivation of IP 3 R channels takes place leading to CICR termination (Iino, 1990). In parallel, SERCA pumps, of which activity increases with cytoplasmic Ca 2+ (Lytton et al., 1992), quickly sequester exuberant cytoplasmic Ca 2+ by pumping it back into the ER lumen. The intracellular Ca 2+ concentration consequently recovers towards basal values which suppress IP 3 R channels inactivation. Hence, if glutamatergic stimulation is prolonged, intracellular IP 3 remains high enough to repeat the cycle and oscillations are observed (Keizer et al., 1995).
The SERCA pump rate can be taken as instantaneous function of cytoplasmic [Ca 2+ ] (denoted hereafter by C) by assuming a Hill rate expression with exponent 2 (see Appendix A): where v ER is the maximal rate of Ca 2+ uptake by the pump and K ER is the SERCA Ca 2+ affinity, that is the Ca 2+ concentration at which the pump operates at half of its maximal capacity (Carafoli, 2002).
The nonspecific Ca 2+ leak current is assumed to be proportional to the Ca 2+ gradient across the ER membrane by r L , the maximal rate of Ca 2+ leakage from the ER: where C ER is the Ca 2+ concentration inside the ER stores (De Young and Keizer, 1992).
IP 3 R channels can be thought of as ensembles of four independent subunits with three binding sites each: one for IP 3 and two for Ca 2+ . The latter include an activation site and a separate site for inactivation (De Young and Keizer, 1992). IP 3 -binding sensitizes the receptor towards activation by Ca 2+ but only if both IP 3 and activating Ca 2+ are bound to a fixed set of three out of four subunits, the channel is open.
Assuming that the kinetic rates of the binding reactions are ordered such as IP 3 -binding>> Ca 2+ -activation>> Ca 2+ -inactivation, Li and Rinzel proposed the following equation for the Ca 2+ current through the IP 3 R channels (Li and Rinzel, 1994): with the channel open probability that is given by and h account for the three gating reactions, respectively IP 3 -binding, activating Ca 2+ -binding and Ca 2+ -dependent inactivation of the receptor. The power of 3 was directly suggested by experimental data (De Young and Keizer, 1992;Bezprozvanny et al., 1991). Finally, I stands for the intracellular IP 3 concentration and r C is the maximum channel permeability.
Since Ca 2+ fluxes across the plasma membrane have been proven not necessary for the onset of CICR (Li et al., 1994;Foskett et al., 1991;Rooney et al., 1991), they can be neglected so that the cell-averaged total free Ca 2+ concentration (C 0 ) is conserved. Hence the ER Ca 2+ concentration (C ER ) can be rewritten in terms of equivalent cell parameters where c 1 is the ratio between the ER and the cytosol volumes. It follows that J chan and J leak can entirely be expressed as functions of cell parameters, namely: Adding together the above terms (equations 1,4), the cytoplasmic Ca 2+ balance is given by: This equation is coupled with an equation for h that accounts for the kinetics of IP 3 Rs (Li and Rinzel, 1994): where: Equations (5) and (6) form the so-called Li-Rinzel (L-R) model of CICR and constitute the core mechanism of our model for astrocyte Ca 2+ signaling. We discuss below some of its properties.

II-2. AM, FM and AFM encoding modes in the Li-Rinzel model
Calcium acts as a second messenger and transmits information from the extracellular side of the plasma membrane to targets within the cell (Berridge and Bootman, 1997;Jaffe, 1993;Berridge, 1990). In the case of Ca 2+ signaling in astrocytes however, the information usually arrives as a non-oscillatory stimulus at the plasma membrane and is translated into intracellular Ca 2+ oscillations. For instance, glutamate concentration at the extracellular side of astrocyte membrane determines the degree of activation of mGluRs and therefore can be directly linked to intracellular IP 3 concentration (Verkhratsky and Kettenmann, 1996). It follows that in the L-R model, the level of IP 3 can be thought as being directly controlled by glutamate signals impinging on the cell from its external environment. In turn, the level of IP 3 determines the dynamics of intracellular Ca 2+ . In physiological conditions glutamate-induced astrocyte Ca 2+ signaling is synaptically evoked (Wang et al., 2006;Pasti et al., 1997;Porter and McCarthy, 1996). One can therefore think of the Ca 2+ signal as being an encoding of information about the level of synaptically-released glutamate and ultimately, of synaptic activity. Notably, this information encoding can use amplitude modulation (AM), frequency modulation (FM) or both modulations (AFM) of Ca 2+ oscillations and pulsations.
We have recently shown that these encoding modes may actually depend on inherent cellular properties (De Pittà et al., 2008a;2008b). In particular the stronger the SERCA uptake with respect to Ca 2+ efflux from the ER, the more pulsating and FM-like the encoding. A fast uptake by SERCAs in fact firmly counteracts CICR so that higher Ca 2+ levels are required for the onset of this latter. When this happens though, the effects of CICR are large and the increase of intracellular Ca 2+ is fast and remarkable. Accordingly, inactivation of IP 3 R channels is also faster and basal Ca 2+ levels are recovered rapidly. In these conditions, the IP 3 level modulates the onset of CICR (through m  ) thus setting the frequency of pulsation (FM encoding). On the contrary the AM case is observed with weaker Ca 2+ uptakes by SERCAs. Weaker SERCA rates in fact allow for smoother oscillations whose amplitude is mainly dependent on the interplay between CICR onset and Ca 2+ -dependent inactivation. Hence the amplitude of oscillations in these latter conditions depends on IP 3 whereas their frequency does not, as it is essentially fixed by IP 3 R channel recovery from Ca 2+ -dependent inactivation (De Pittà et al., 2008b). Finally, it is important to note that the L-R model assumes that IP 3 does not vary with time nor depends on the other variables (that is, its concentration I, in equations (5) and (6), is a parameter of the model). Yet examination of the underlying signaling pathways ( Figure 1) immediately hints that IP 3 concentration indeed depends on both intracellular Ca 2+ and extracellular glutamate, so that IP 3 should be an additional variable in the model. Our aim in the present article is to devise a model that incorporates these dependencies.
The activation properties of these two isoenzymes are different and so likely are their roles. PLC is primarily controlled by cell surface receptors, hence its activity is linked to the level of external stimulation (i.e. the extracellular glutamate) and as such, it pertains to the glutamate-dependent IP 3 metabolism and will be addressed in the next section.
On the contrary PLC is essentially activated by increased intracellular Ca 2+ levels ( Figure 1d) (Rhee and Bae, 1997). Structural and mutational studies of complexes of PLC with Ca 2+ and IP 3 revealed complex interactions of Ca 2+ with several negatively charged residues within its catalytic domain (Rhee, 2001;Essen et al., 1997;Essen et al., 1996;), a hint of cooperative binding of Ca 2+ to this enzyme. In agreement with these experimental findings, the PLC activation rate can be written as (Höfer et al., 2002;Pawelczyk and Matcki, 1997): where the maximal rate of activation depends on the level of intracellular IP 3 .
Experimental observations shows that high (>1 M) IP 3 concentrations inhibit PLC activity by competing with PIP 2 binding to the enzyme (Allen et al., 1997). Accordingly, assuming competitive binding (Stryer, 1999), the maximal PLC-dependent IP 3 production rate can be modeled as follows: where   is the inhibition constant of PLC activity. Figure 3 shows the behavior of this term when Ca 2+ and corresponding IP 3 levels obtained from the bifurcation diagrams of the L-R model in Figure 2 are substituted into equation (7). We have set K PLC to a value that is close to the Ca 2+ concentration of the lower bifurcation point. This allows us to translate the large-amplitude Ca 2+ oscillations into oscillations of v  that could preserve the main AM/FM properties.

b) IP 3 degradation
Two major IP 3 degradation pathways have been described so far ( Figure 1b). The first one is through dephosphorylation of IP 3 by inositol polyphosphate 5-phosphatase (IP-5P).
The other one occurs through phosphorylation of IP 3 by the IP 3 3-kinase (IP 3 -3K) and is Ca 2+ dependent (Zhang et al., 1993).
The rate of both IP-5P dephosphorylation (v 5P ) and IP 3 -3K phosphorylation (v 3K ) of IP 3 can be considered as of Michaelis-Menten type (Dupont and Erneux, 1997;Togashi et al., 1997;Irvine et al., 1986). Therefore: Since K 5 >10 M (Sims et Allbritton, 1998;Verjans et al., 1992;), and physiological levels of IP 3 are in general below this value, IP-5P is likely not to be saturated by IP 3 . It follows that the rate of IP 3 degradation by IP-5P can be linearly approximated: where P r 5 is the linear rate of IP 3 degradation by IP-5P and can be defined by parameters in equation (9) as In basal conditions, phosphorylation of IP 3 by IP 3 -3K is very slow. The activity of IP 3 -3K is substantially stimulated by Ca 2+ /calmodulin (CaM) via CaMKII-catalyzed phosphorylation ( Figure 1b) (Communi et al., 1997). However, other experimental reports have suggested that Ca 2+ -dependent PKC phosphorylation of IP 3 -3K could have inhibitory effects (Sim et al., 1990). Notwithstanding, evidences for this latter possibility are contradictory (Communi et al., 1995). Hence, for the sake of simplicity, we have chosen in the present model to consider the simplified case were only CaMKII-catalyzed phosphorylation of IP 3 -3K is present ( Figure 1f).
Phosphorylation of IP 3 -3K by active CaMKII (CaMKII*) only occurs at a single threonine residue (Communi et al., 1999;Communi et al., 1997), therefore we can . Activation of CaMKII is Ca 2+ /CaM-dependent and occurs in a complex fashion because of the unique structure of this kinase, which is composed of ~12 subunits with three to four phosphorylation sites each (Kolodziej et al., 2000). Briefly, Ca 2+ elevation leads to the formation of a Ca 2+ -CaM complex (CaM + ) that may induce phosphorylation of some of the sites of each CaMKII subunit. CaMKII quickly and fully activates when two of these sites (at proximal subunits) are phosphorylated (Hanson et al, 1994).  , theoretical investigations showed that these two regimes are essentially brought forth by the Ca 2+ -dependent Hill term in the expression of v 3K , irrespectively of the assumption of Michaelis-Menten kinetics for IP 3 -dependence of IP 3 -3K ( Figure 4).
Accordingly, a linear approximation for v 3K such as: , could also be considered instead of equation (12), in agreement with previous investigations found in literature (Politi et al., 2006;Sims et Allbritton, 1998).
Indeed, the behaviors of v 3K in equations (12) and (13) for IP 3 and Ca 2+ concentrations obtained from the corresponding Li-Rinzel bifurcation diagrams are qualitatively similar ( Figure 5). Moreover, the overall bifurcation diagrams are largely conserved (results not shown). The main quantitative difference is that the linear approximation yields stronger degradation rates. In particular, the IP 3 -3K rate can be up to 2-fold higher in equation (13) than in equation (12). This is particularly marked when high [Ca 2+ ] are reached, such as in FM conditions (Figures 5c-d). Notwithstanding, the Michaelis-reach such micromolar concentrations in vivo (Mishra and Bhalla, 2002). Therefore, in the following, we will keep the Michaelis-Menten formulation for v 3K (equation 12).
Finally, experimental measurements show that for [Ca 2+ ]>1 M and low IP 3 levels, the IP 3 -3K activity exceeds that of IP-5P by almost 20-fold. In the model, this means that if ), then v 3K  20v 5P . Accordingly, we set the maximal degradation rates in the following such that

c) Model analysis
In summary, our model of Ca 2+ dynamics with endogenous IP 3 metabolism is based on the two L-R equations (equations 5-6), but the IP 3 concentration (I) is now provided by a third coupled differential equation (summing the terms given by equations 7,10,12): Equation (14) together with equations (5) and (6) define our three-variable "ChI" model, in the name of the letters used to denote its state variables.
Consistency of the ChI model with respect to the L-R core model was sought by comparing two curves for pseudo-steady states. First, we set 0  I  and 0  C in equation (14), and solved for I as a function of C in the resulting equation. In parallel, we in equation (5)

III. Modelling glutamate regulation of IP 3 production: the G-ChI model
The contribution of glutamate signals to IP 3 production can be taken into account as an additional production term in the IP 3 equation of the above three-variable ChI model. The resulting new model is referred to as the "G-ChI" model.
Although it is likely that the type of mGluRs expressed by astrocytes depends on the brain region and the stage of development (Gallo and Ghiani, 2000), it seems reasonable to assume that such differences are negligible in terms of the associated secondmessenger pathways (Abe et al., 1992;Masu et al., 1991).
The G protein associated with astrocyte mGluRs is a heterotrimer constituted by three subunits: ,  and . Glutamate binding to mGluR triggers receptor-catalyzed exchange of GTP from the G subunits to the G subunit. The GTP-loaded G subunit then dissociates from the G protein in the membrane plane and binds to a co-localized PLC (Figures 1a,e). Upon binding to G, the activity of PLC substantially increases, thus promoting PIP 2 hydrolysis and IP 3 production. Activation of PLC can therefore, at first approximation, be directly linked to the number of bound mGluRs, hence to the level of external stimulation. It follows that glutamate-dependent IP 3 production can be written in the following generic form: where v  is the maximal PLCrate, that depends on the surface density of mGluRs, and R(,C) is the fraction of activated (bound) mGluRs. Experimental evidence shows that PLC activity (i.e. v  in equation 15) is also dependent on intracellular Ca 2+ (Rhee and Bae, 1997). Notwithstanding, such dependence seems to occur for [Ca 2+ ]>10 M, hence out of our physiological range (Allen et al., 1997). Therefore, Ca 2+ dependence of PLC maximal rate will not be considered here.
R(,C) can be expressed in terms of extracellular glutamate concentration () at the astrocytic plasma membrane, assuming a Hill binding reaction scheme, with an exponent ranging between 0.5 and 1 (Suzuki et al, 2004). In the current study, we choose 0.7 yielding: In equation (16), R(,C) is expressed as a Hill function with a midpoint that depends on glutamate and intracellular Ca 2+ concentrations. This choice was motivated by the termination mechanism of PLC signaling, that occurs essentially through two reaction pathways (Rebecchi and Pentyala, 2000): (i) reconstitution of the inactive G-protein heterotrimer due to the intrinsic GTPase activity of activated G subunits and (ii) PKC phosphorylation of the receptor, or of the G protein, or of PLC or some combination thereof. We lump both effects into a single term, K  (,C), such as the effective Hill midpoint of R(,C) increases as PLC termination takes over, namely: Here K R is the Hill midpoint of glutamate binding with its receptor whereas K p measures the increment of the apparent affinity of the receptor due to PLC terminating signals.
Hill( 0.7 ,K R ) accounts for the intrinsic GTPase-dependent PLC activity termination, as this effect is linked to the fraction of activated G subunits and therefore can be put in direct proportionality with the fraction of bound receptors.
instead accounts for PKC-related phosphorylation-dependent termination of PLC activity. Experimental data suggest that the target of PKC in this case is either the G protein or PLC itself (Ryu et al., 1990). Generally speaking, phosphorylation by PKC may modulate the efficiency of ligand-binding by the receptor, the coupling of occupied receptor to the G protein, or the coupling of the activated G protein to PLC (Fisher, 1995). All these effects indeed are lumped into equation (17), as explained below.
PKC is activated in a complex fashion (Figure 1e). Indeed its activation by mere intracellular Ca 2+ is minimal (Nishizuka, 1995), while full activation is obtained by binding of the coactivator DAG. In agreement with this description, PKC activation can be approximated by a generic Hill reaction scheme whereas Ca 2+ dependent PKC phosphorylation can be assumed Michaelis-Menten (Ryu et al., 1990) (17).
To complete the model, it can be shown by numerical analysis of K  (,C) (equation 17) that the term related to the GTPase-dependent PLC termination pathway, i.e.

IV. Dynamical behaviors and coding modes of the G-ChI model
The dynamical features of the G-ChI model for different extracellular concentrations of glutamate can be appreciated by inspection of bifurcation diagrams in Figures 10 and 11.
We note that the choice of  v , the maximal rate of glutamate-dependent IP 3 production which is linked to the density of receptors on the extracellular side of the astrocyte membrane, can substantially influence the bifurcation structure of the model and the extent of the oscillatory range. Indeed, as  v decreases the oscillatory range expands towards infinite glutamate concentrations but the amplitude of oscillations concomitantly decreases (at least with regard to the IP 3 concentration).
The extension of the oscillatory range is due to the shift towards infinity of the subcritical were derived using different sets of parameters that pertain respectively to AM and FM encoding in the ChI model as well as in the L-R core model (see Table 1  Notably, in such conditions, oscillations resemble pulsating dynamics. In other words, rather than pure AM encoding, as we could expect by a set of parameters that provides

AM in the ChI model (Figures 1a-c in Online Supplementary Material), it seems that in
the G-ChI model, Ca 2+ oscillations become AFM encoding. Notably, IP 3 dynamics appears to be always AFM encoding both in the AM (Figures 10b,e) and in the FMderived bifurcation diagrams (Figures 11b,f,i).
Conversely, mere FM encoding is essentially preserved for Ca 2+ dynamics derived from FM encoding sets of parameters in the ChI model, although a significant increase of the range of amplitudes of pulsations can be pointed out (compare Figure 11d with Figure 3d in Online Supplementary Material). These observations indicate that the G-ChI model accounts either for FM or AFM encoding Ca 2+ oscillations, which are though always coupled with AFM encoding IP 3 oscillations. In addition they provide further support to the above-stated notion that IP 3 metabolism could consistently modulate the frequency of Ca 2+ pulsating dynamics more than their amplitude (see Section II-3c).
On the contrary, the amplitude and shape of IP 3 oscillations appear to be dramatically Simulations of physiologically equivalent glutamate stimulation and associated astrocyte Ca 2+ -IP 3 patterns are shown in Figure 13. Real MEA-recording data were considered as inputs of a single glutamatergic synapse (modeled as in Tsodyks et al., 1997) and a fraction of the released glutamate was assumed to impinge the astrocyte described by our model. We may notice that from the stimulus up to Ca 2+ dynamics the smoothness of the patterns seems to increase. Indeed, the highly jagged glutamate stimulus turns into a less indented IP 3 signal which is coupled with even smoother Ca 2+ oscillations. Depending on the inherent cellular properties (Figure 13,

V. Discussion
Calcium dynamics in astrocytes can be driven by extra-cellular signals (such as glutamate neurotransmitter) through regulation of the intracellular IP 3 levels. Therefore, a prerequisite towards unraveling the response of astrocytes to such signals is a thorough understanding of the complex IP 3 -related metabolic pathways that regulate intracellular  (Rebecchi and Pentyala, 2000).
We linked the relative expression of these two isoenzymes to the expression of PKC and to the strength of PLC regulation by PKC. Indeed, Ca 2+ -dependent PKC activation can phosphorylate the receptor or PLC or a combination thereof, leading to termination of IP 3 production (Ryu et al., 1990). In astrocytes, this mechanism has been suggested to limit the duration of Ca 2+ oscillations thus defining their frequencies (Codazzi et al., 2001). In agreement with this idea, a stronger PKC-dependent inhibition of PLC shrank the oscillatory range in our model astrocyte and led to the progressive loss of long-period oscillations.

In our model, the PKC-dependent inhibition of PLCis counteracted if PLC expression
is high enough to support high IP 3 production levels and the resulting release of Ca 2+ from the intracellular stores. This observation raises the possibility of phase-locked Ca 2+ oscillations under conditions of intense stimulation. Phase-locked Ca 2+ oscillations were also found in other models of agonist-dependent intracellular Ca 2+ dynamics (Chay et al., 1995a;Chay et al., 1995b;Cuthbertson and Chay, 1991) and are often associated with pathological conditions (Uhlhaas and Singer, 2006;Shrier et al., 1987). In our model, persistent pulsating Ca 2+ dynamics that are essentially independent of the level of stimulation are observed for weak maximal rates of IP 3 production by PLC (Figures 10d-e,11g-h). In astrocytes, such persistent oscillations could also be interpreted as a fingerprint of pathological conditions (and Meldolesi, 2005;Balászi et al., 2003). In fact, a decay of PLC activity is likely to occur for instance, if the density of effective metabotropic receptors in the astrocytic plasma membrane decreases, such as in the case of epileptic patients with Ammon's Horn sclerosis (Seifert et al., 2004).
We note that although focusing on stimulus-triggered Ca 2+ oscillations, our study also hints a possible link between modulation of frequency and amplitude of Ca 2+ pulsations and spontaneous Ca 2+ dynamics. Recently it has been shown that the inter-pulse interval of the spontaneous Ca 2+ oscillations is inherently stochastic (Skupin and Falcke, 2008;Skupin et al., 2008). In particular, experimental observations are compatible with model studies of a local stochastic nucleation mechanism that is amplified by the spatial coupling among IP 3 R clusters through Ca 2+ or IP 3 diffusion (Skupin et al., 2008;Falcke, 2003). Our analysis may provide meaningful clues to identify what factors and processes within the cell could affect the rate of wave nucleation. More specifically we may predict that putative intracellular IP 3 dynamics could affect the statistics of Ca 2+ inter-pulse intervals not only in terms of spatial coupling amongst IP 3 R clusters by means of intracellular IP 3 gradients, but also by modulation of either the recovery from Ca 2+ inhibition or the progressive sensitization of IP 3 Rs by Ca 2+ (Tang and Othmer, 1995).
The resulting scenario therefore, would still be that of a local stochastic nucleation mechanism amplified by IP 3 R spatial coupling, but the local IP 3 R and SERCA parameters would vary according with to the biochemical regulation system presented in the current work.
A critical question in experiments is the identification of the mechanism that drives IP 3 oscillations and pulsations (Young et al., 2003;Nash et al., 2001;Hirose et al., 1999;Sims and Allbritton, 1998). In our model, self-sustained IP 3 oscillations are brought about by the coupling of IP 3 metabolism with Ca 2+ dynamics. In other words, our model can be considered as a self-consistent astrocytic generator of Ca 2+ dynamics. This might have broad implications for astrocyte encoding of information and neuron-glia communication.
We previously demonstrated that modulation by astrocytes of synaptic information transfer could account for some of the peculiar dynamics observed in spontaneous activity of cultured cortical networks (Volman et al., 2007). In particular, a simple neuron-glia circuit composed of an autaptic neuron "talking" with a proximal astrocyte could serve as a self-consistent oscillator when fed by weak external signals. The results presented in the current study suggest an alternative, more robust, way (independent of synaptic architecture) to form glia-based self-consistent oscillators. The relative contribution and significance of either the astrocytic or the IP 3 -based hypotheses to the spontaneous network activity, need to be assessed by future combined experiments and modeling. Meanwhile, the analysis of our present model suggests that in astrocytes, different second messenger molecules are engaged in an intricate dialogue, likely meaning that those non-neural cells might be crucially important to decipher some of the enigmas of neural information processing.
Another significant prediction of our model is that IP 3 dynamics is essentially AFM and Ca 2+ oscillations/pulsations are inherently FM encoding, that is they can be either FM or AFM but not AM (Berridge, 1997;De Pittà et al, 2008a;2008b). In FM, Ca 2+ oscillations resemble pulses. In AFM instead, their shape is smoother and necessarily depends on the stimulus dynamics.
The assumption that IP 3 oscillations are always AFM-encoding could provide an optimal interface between agonist stimuli and intracellular Ca 2+ signals. The stimuli impinge the cell in the form of trains of pulses or burst of pulses and information is carried in the timing of these pulses rather than in their amplitude (Sejnowski and Paulsen, 2006). AFM features in IP 3 signals could perfectly match these stimuli, embedding the essential features of the spectrum of the signal into the spectrum of the IP 3 transduction. Hence, IP 3 signaling with FM features could offer an efficient way to keep the essence of the information of the stimulus. On the other hand, because Ca 2+ signals are triggered primarily by sufficiently ample elevations of IP 3 (Li et al., 1994), the coexistence of AM features within the IP 3 signal seems to be a necessary prerequisite in order to trigger CICR.
The fact that coupling of IP 3 metabolism with CICR does not allow pure AM-encoding is in general agreement with experimental data on intracellular Ca 2+ signaling in several cells (Berridge, 1998;Woods et al., 1986) including astrocytes (Pasti et al., 1997).
Notwithstanding the possibility of AFM encoding Ca 2+ oscillations has recently come up as a reliable alternative mechanism to explain gliotransmitter exocytosis that is dependent on the specific agonist that triggers astrocyte Ca 2+ dynamics (Montana et al., 2006;Carmignoto, 2000).
The above could be relevant to understand the origin of the integrative properties of Ca 2+ signaling in astrocytes (Perea and Araque, 2005a). Our analysis in fact shows that such properties could result from at least two steps of integration, one at the transduction of the agonist signal into IP 3 signal and the other at the cross-coupling between IP 3 and Ca 2+ signals. Indeed AFM-encoding IP 3 dynamics could deploy smoothing of the highly indented agonist stimulus thus hinting possible integrative properties for IP 3 signals ( Figure 13). On the other hand, the associated Ca 2+ patterns look even smoother suggesting a further integration step that likely relies only on the inherent features of CICR.

Appendix A
For the sake of simplicity we have adopted throughout the text the following notation for the generic Hill function:  This result can be extended to the product of N Hill functions, that is: Notably, the product of Hill function is not the only case in which a function composed (see Figure 2 in Online Supplementary Material).

Appendix B
We seek an expression for [CaMKII*] based on the following kinetic reaction scheme: Let us first consider the reaction chain (B2). We can assume that the second step is very rapid with respect to the first one (De Konick and Schulman, 1998;Thiel et al., 1988)    (e) The contribution of PLC to IP 3 production instead depends on agonist binding to Gprotein coupled metabotropic receptors (mGluRs) found on the surface of the cell.     (5), (6) and (14).    Table 1 except for (d-f) where  v =0.05 Ms -1 . Parameters as in Table 1    feature is a remarkable increase of the signal smoothness, when one goes from glutamatestimulus to IP 3 traces (b-c) and from the latter to Ca 2+ traces (a-b). This fact hints different integrative properties for IP 3 and Ca 2+ with respect to the stimulus, which are likely to be cross-coupled (see Discussion).
. This figure shows that the ChI model can still display oscillations in presence of an external non-specific bias of IP 3 production. This is a first suggestion that the corresponding glutamate-dependent G-ChI model may also display oscillations. The parameters are taken from Table 1