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Recent years have witnessed an increasing interest in neuron–glia communication. This interest stems from the realization that glia participate in cognitive functions and information processing and are 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 (Ca2+) 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-variable (intracellular Ca2+ and inactive IP3 receptors) Li–Rinzel model for calcium-induced calcium release, we incorporate the regulation of IP3 production and phosphorylation. Doing so, we extend it to a three-variable model (which we refer to as the ChI model) that could account for Ca2+ oscillations with endogenous IP3 metabolism. This ChI model is then further extended into the G-ChI model to include regulation of IP3 production by external glutamate signals. Compared with previous similar models, our three-variable models include a more realistic description of 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 endow the system with self-consistent oscillatory 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 for the role of astrocytes in the synaptic transfer of information.
Electronic supplementary material The online version of this article (doi:10.1007/s10867-009-9155-y) contains supplementary material, which is available to authorized users.
Astrocytes, the main type of glial cells in the brain, do not generate action potentials as neurons do, yet they can transfer information to other cells and encode information in response to external stimuli by employing “excitable”-like rich calcium (Ca2+) dynamics . Recognition of the potential importance of the intricate inter- and intracellular astrocyte dynamics has motivated, in recent years, intensive experimental efforts to investigate neuron–glia communication. Consequently, it was discovered that intracellular Ca2+ levels in astrocytes can be regulated by synaptic activity [2–6]. Responses to low-intensity synaptic stimulation or spontaneous astrocyte activity usually consist of spatially confined Ca2+ transients [3, 4, 7]. On the other hand, high-intensity synaptic activity or stimulation of adjacent sites within the same astrocytic process are generally associated with Ca2+ oscillations  that can bring forth propagation of both intracellular and intercellular waves [9–11]. Concomitantly, elevation of cytoplasmic Ca2+ induces the release from astrocytes of several neurotransmitters (or “gliotransmitters”), including glutamate, ATP, or adenosine (see Evanko et al.  for a review). These astrocyte-released gliotransmitters feed back onto pre- and postsynaptic terminals. This implies that astrocytes regulate synaptic information transfer [13–15]. Astrocytes can also mediate between neuronal activity and blood circulation , thus extending neuron–astrocyte communications to the level of neuronal metabolism .
The physiological meaning of astrocytic Ca2+ signaling remains currently unclear, and a long-standing question is how it participates in the encoding of synaptic information transfer [1, 18, 19]. Some of the available experimental data suggest a preferential frequency modulation (FM) mode of encoding, namely synaptic activity would be encoded in the frequency of astrocytic Ca2+ pulsations . Indeed, cytoplasmic Ca2+ 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 .
Notwithstanding, the possibility of amplitude modulation (AM) encoding of synaptic activity or even of mixed amplitude and frequency modulation (AFM) encoding has also consistently been inferred . For instance, the amplitude of Ca2+ oscillations in response to external stimuli can be highly variable, depending on the intensity of stimulation [2, 11, 22]. Experimental evidence suggests that Ca2+ 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 . The emerging picture is that the properties of Ca2+ 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 [1, 3].
From the modeling point of view, simplified or two-variable models for intracellular Ca2+ signaling can, in principle, be used to account for the diversity of the observed Ca2+ dynamics when the biophysical parameters are varied. We recently presented evidence that one of these two-variable models proposed by Li and Rinzel  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 among AM of Ca2+ oscillations, FM of Ca2+ pulsations, or combined AM and FM (AFM) Ca2+ pulsations [18, 19]. We emphasize that one of the cardinal simplifications of the Li–Rinzel model is neglecting the regulation of inositol 1,4,5-trisphosphate (IP3) dynamics, that is its production and degradation. Since IP3 production is regulated by synaptic activity (via extracellular glutamate signaling), IP3 dynamics has to be included for proper modeling of synapse–astrocyte communication. Only such modeling can provide a realistic account of astrocytic Ca2+ 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 Ca2+ dynamics that can have important implications for the role of astrocytes in synaptic information transfer. Our model incorporates current biological knowledge related to the signaling pathways leading from extracellular glutamate to intracellular Ca2+, via IP3 regulation and IP3-dependent Ca2+-induced Ca2+ release (CICR). First, we extend the Li–Rinzel model to incorporate the regulation of IP3. This yields a three-variable model, called hereafter the “ChI” model, for its state variables that are the intracellular Ca2+ level C, the fraction of inactive IP3 receptors h, and the available IP3 concentration I. Similar three-variable models have already been introduced in previous works [25–30] (see Falcke  for a review), yet our modeling includes a more realistic description of IP3 dynamics, in particular with regard to the complex regulatory pathways of IP3 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 IP3 production and include this contribution as an additional production term into the IP3 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: AM, pulsation FM, and mixed AFM. We also present results of time simulations of the model, illustrating the richness of intracellular Ca2+ 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 Ca2+ dynamics in other cell types whose coordinated activity is based on intra- and intercellular Ca2+ signaling, such as heart cells, pancreas cells, and liver cells.
In this section, we describe the concise ChI model for intracellular Ca2+ dynamics in astrocytes with realistic IP3 regulation. Given the relative intricacy of this signaling pathway (see Fig. Fig.1),1), each basic building block of the model is described separately in the next sections.
Intracellular Ca2+ levels in astrocytes (as in most other cell types) can be modulated by several mechanisms. These include Ca2+ influx from the extracellular space or controlled release from intracellular Ca2+ stores such as the endoplasmic reticulum (ER) and mitochondria . In astrocytes, though, IP3-dependent CICR from the ER is considered the primary mechanism responsible of intracellular Ca2+ dynamics .
Calcium-induced Ca2+ release (see Fig. Fig.1c)1c) is essentially controlled by the interplay of two specific transports: efflux from the ER to the cytoplasm that is mediated by Ca2+-dependent opening of the IP3 receptor (IP3R) channels and influx into the ER which is due to the action of sarco-endoplasmic reticulum Ca2+-ATPase (SERCA) pumps. In basal conditions, however (when CICR is negligible), intracellular Ca2+ levels are set by the respective contributions of a passive Ca2+ leak from the ER, SERCA uptake, and plasma membrane Ca2+ transport [35, 36].
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) . Binding of glutamate to mGluRs increases cytosolic IP3 concentration and promotes the opening of a few IP3R channels . As a consequence, intracellular Ca2+ slightly increases. Since the opening probability of IP3R channels nonlinearly increases with Ca2+ concentration , such an initial Ca2+ surge increases the opening probability of neighboring channels. In turn, this leads to a further increase of cytoplasmic Ca2+. These elements therefore provide a self-amplifying release mechanism (hence the denomination of CICR). The autocatalytic action of Ca2+ release, however, reverses at high cytoplasmic Ca2+ concentrations, when inactivation of IP3R channels takes place, leading to CICR termination . In parallel, SERCA pumps, whose activity increases with cytoplasmic Ca2+ , quickly sequester excess cytoplasmic Ca2+ by pumping it back into the ER lumen. The intracellular Ca2+ concentration consequently recovers toward basal values which suppress IP3R channel inactivation. Hence, if glutamatergic stimulation is prolonged, intracellular IP3 remains high enough to repeat the cycle, and oscillations are observed .
The SERCA pump rate can be taken as an instantaneous function of cytoplasmic [Ca2+] (denoted hereafter by C) by assuming a Hill rate expression with exponent 2 (see Appendix 1):
where vER is the maximal rate of Ca2+ uptake by the pump and KER is the SERCA Ca2+ affinity, that is the Ca2+ concentration at which the pump operates at half of its maximal capacity .
The nonspecific Ca2+ leak current is assumed to be proportional to the Ca2+ gradient across the ER membrane by rL, the maximal rate of Ca2+ leakage from the ER:
where CER is the Ca2+ concentration inside the ER stores .
IP3R channels can be thought of as ensembles of four independent subunits with three binding sites each: one for IP3 and two for Ca2+. The latter sites include an activation site and a separate site for inactivation . IP3-binding sensitizes the receptor toward activation by Ca2+ but only if both IP3 and activating Ca2+, 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 IP3-binding >> Ca2+-activation >> Ca2+-inactivation, Li and Rinzel proposed the following equation for the Ca2+ current through the IP3R channels :
with the channel open probability given by , where m∞=Hill( I,d1 ), , and h account for the three gating reactions, respectively, IP3-binding, activating Ca2+-binding, and Ca2+-dependent inactivation of the receptor. The power of 3 was directly suggested by experimental data [36, 38]. Finally, I stands for the intracellular IP3 concentration and rC is the maximum channel permeability.
Since Ca2+ fluxes across the plasma membrane have been proven not necessary for the onset of CICR [35, 43, 44], they can be neglected, so that the cell-averaged total free Ca2+ concentration (C0) is conserved. Hence, the ER Ca2+ concentration (CER) can be rewritten in terms of equivalent cell parameters as CER=(C0−C)/c1 where c1 is the ratio between the ER and the cytosol volumes. It follows that Jchan and Jleak can entirely be expressed as functions of cell parameters, namely:
This equation is coupled with an equation for h that accounts for the kinetics of IP3Rs :
Calcium acts as a second messenger and transmits information from the extracellular side of the plasma membrane to targets within the cell [33, 45, 46]. In the case of Ca2+ signaling in astrocytes, however, the information usually arrives as a nonoscillatory stimulus at the plasma membrane and is translated into intracellular Ca2+ oscillations. For instance, glutamate concentration at the extracellular side of the astrocyte membrane determines the degree of activation of mGluRs and therefore can be directly linked to intracellular IP3 concentration . It follows that in the L–R model, the level of IP3 can be thought as being directly controlled by glutamate signals impinging on the cell from its external environment. In turn, the level of IP3 determines the dynamics of intracellular Ca2+. In physiological conditions glutamate-induced astrocyte Ca2+ signaling is synaptically evoked [2–4]. One can therefore think of the Ca2+ 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 AM, FM, or both modulations (AFM) of Ca2+ oscillations and pulsations.
We have recently shown that these encoding modes may actually depend on inherent cellular properties [18, 19]. In particular, the stronger the SERCA uptake with respect to Ca2+ 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 Ca2+ levels are required for the onset of this latter one. When this happens, however the effects of CICR are large and the increase of intracellular Ca2+ is fast and remarkable. Accordingly, inactivation of IP3R channels is also faster and basal Ca2+ levels are recovered rapidly. In these conditions, the IP3 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 Ca2+ uptake by SERCAs. Weaker SERCA rates in fact allow for smoother oscillations whose amplitude is mainly dependent on the interplay between CICR onset and Ca2+-dependent inactivation. Hence, the amplitude of oscillations in these latter conditions depends on IP3, whereas their frequency does not, as it is essentially fixed by IP3R channel recovery from Ca2+-dependent inactivation .
From a dynamical systems perspective, AM and FM encoding are associated with well-distinct bifurcation diagrams. Amplitude modulations of Ca2+ oscillations are typically found when the system exhibits Hopf bifurcations only. In particular, when oscillations are born through a supercritical Hopf bifurcation at low IP3 concentration, then AM encoding exists (Fig. (Fig.2a–c).2a–c). Alternatively, if the Hopf bifurcation is subcritical, AFM might be found . On the contrary, in FM (Fig. (Fig.2d–f),2d–f), the presence of a saddle-node homoclinic bifurcation accounts for pulsatile oscillations which arise at arbitrarily small frequency but with amplitude essentially independent of the IP3 value .
Finally, it is important to note that the L–R model assumes that IP3 does not vary with time nor depends on the other variables (that is, its concentration I, in (5) and (6), is a parameter of the model). Yet examination of the underlying signaling pathways (Fig. (Fig.1)1) immediately hints that IP3 concentration indeed depends on both intracellular Ca2+ and extracellular glutamate, so that IP3 should be an additional variable in the model. Our aim in the present article is to devise a model that incorporates these dependencies.
In astrocytes, IP3 together with diacylglycerol (DAG) is produced by hydrolysis of phosphatidylinositol 4,5-bisphosphate (PIP2) by two phosphoinositide-specific phospholipase C (PLC) isoenzymes, PLCβ and PLCδ . The activation properties of these two isoenzymes are different and so, it is likely, are their roles. PLCβ 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 IP3 metabolism and will be addressed in the next section.
On the contrary, PLCδ is essentially activated by increased intracellular Ca2+ levels (Fig. (Fig.1d)1d) . Structural and mutational studies of complexes of PLCδ with Ca2+ and IP3 revealed complex interactions of Ca2+ with several negatively charged residues within its catalytic domain [50–52], a hint of cooperative binding of Ca2+ to this enzyme. In agreement with these experimental findings, the PLCδ activation rate can be written as [27, 53]:
where the maximal rate of activation depends on the level of intracellular IP3. Experimental observations show that high (>1 μM) IP3 concentrations inhibit PLCδ activity by competing with PIP2 binding to the enzyme . Accordingly, assuming competitive binding , the maximal PLCδ-dependent IP3 production rate can be modeled as follows:
where is the inhibition constant of PLCδ activity.
Figure Figure33 shows the behavior of this term when Ca2+ and corresponding IP3 levels obtained from the bifurcation diagrams of the L–R model in Fig. Fig.22 are substituted into (7). We have set to a value that is close to the Ca2+ concentration of the lower bifurcation point. This allows us to translate the large-amplitude Ca2+ oscillations into oscillations of that could preserve the main AM/FM properties.
Two major IP3 degradation pathways have been described so far (Fig. (Fig.1b).1b). The first one is through dephosphorylation of IP3 by inositol polyphosphate 5-phosphatase (IP-5P). The other one occurs through phosphorylation of IP3 by the IP3 3-kinase (IP3-3K) and is Ca2+ dependent .
Since M [57, 66] and physiological levels of IP3 are in general below this value, IP-5P is likely not to be saturated by IP3. It follows that the rate of IP3 degradation by IP-5P can be linearly approximated:
where is the linear rate of IP3 degradation by IP-5P and can be defined by parameters in (9) as .
In basal conditions, phosphorylation of IP3 by IP3-3K is very slow. The activity of IP3-3K is substantially stimulated by Ca2+/calmodulin (CaM) via CaMKII-catalyzed phosphorylation (Fig. (Fig.1b)1b) . However, other experimental reports have suggested that Ca2+-dependent PKC phosphorylation of IP3-3K could have inhibitory effects . Notwithstanding, evidences for this latter possibility are contradictory . Hence, for the sake of simplicity, we have chosen in the present model to consider the simplified case where only CaMKII-catalyzed phosphorylation of IP3-3K is present (Fig. (Fig.11f).
Phosphorylation of IP3-3K by active CaMKII (CaMKII*) only occurs at a single threonine residue [67, 70], therefore we can assume that [CaMKII*]. Activation of CaMKII is Ca2+/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 . Briefly, Ca2+ elevation leads to the formation of a Ca2+–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 . In spite of the occurrence of multiple CaM+ binding to the inactive kinase, experimental investigations showed that KII activation by CaM+ can be approximated by a Hill equation with unitary coefficient . Hence, if we surmise the following kinetic reaction scheme for CaMKII phosphorylation:
it can be demonstrated that [CaMKII*] Hill(C4, KD) (see Appendix 2).
Accordingly, and the equation for IP3-3K-dependent IP3 degradation reads:
Experimental observations show the existence of three regimes of IP3 metabolism . At low [Ca2+] and [IP3] (<400 nM and <1 μM, respectively), IP-5P and IP3-3K degrade roughly the same amounts of IP3. Then, at high [Ca2+] (≥400 nM) but low [IP3] (≤8 μM), IP3 is predominantly metabolized by IP3-3K. Eventually, for [IP3] greater than 8 μM, when IP3-3K activity saturates, IP-5P becomes the dominant metabolic enzyme, independently of [Ca2+].
In our modeling, the third regime—corresponding to [IP3] > 8 μM—exceeds the range of validity for the linear approximation of IP-5P degradation (10) and therefore cannot be taken into account. However, it can be shown that the first two regimes are sufficient to reproduce Ca2+ oscillations and pulsations, thus restricting the core features of IP3 metabolism to the maximal rates of IP3 degradation by IP3-3K and IP-5P and to the Ca2+ dependence of IP3-3K. In particular, by opportune choice of parameters such as , theoretical investigations showed that these two regimes are essentially generated by the Ca2+-dependent Hill term in the expression of v3K irrespectively of the assumption of Michaelis–Menten kinetics for IP3 dependence of IP3-3K (Fig. (Fig.4).4). Accordingly, a linear approximation for v3K such as:
Indeed, the behaviors of v3K in (12) and (13) for IP3 and Ca2+ concentrations obtained from the corresponding Li–Rinzel bifurcation diagrams are qualitatively similar (Fig. (Fig.5).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 IP3-3K rate can be up to twofold higher in (13) than in (12). This is particularly marked when high [Ca2+] is reached, such as in FM conditions (Fig. (Fig.5c–d).5c–d). Notwithstanding, the Michaelis–Menten constant of IP3-3K for its substrate is experimentally reported to be M [56, 59] and it is likely that intracellular IP3 levels can reach such micromolar concentrations in vivo . Therefore, in the following, we will keep the Michaelis–Menten formulation for v3K (12).
Finally, experimental measurements show that for [Ca2+] > 1 μM and low IP3 levels, the IP3-3K activity exceeds that of IP-5P by almost 20-fold. In the model, this means that if I << K3 (i.e., , then v3K≈20v5P. Accordingly, we set the maximal degradation rates in the following such that .
In summary, our model of Ca2+ dynamics with endogenous IP3 metabolism is based on the two L–R equations ((5) and (6)), but the IP3 concentration (I) is now provided by a third coupled differential equation (summing the terms given by (7), (10), (12)):
Consistency of the ChI model with respect to the L–R core model was sought by comparing two curves for pseudosteady states. First, we set and C → 0 in (14) and solved for I as a function of C in the resulting equation. In parallel, we set in (5) and solved for I as a function of C in the resulting equation as well. The two resulting I–C curves should be as similar as possible. Analysis showed that they are indeed relatively similar (Fig. 6) if one chooses , KD≈H2, K3>H2, where H1 and H2 denote Ca2+ and IP3 concentrations at the two Hopf bifurcations in the L–R bifurcation diagrams (Fig. (Fig.2).2). Such choice of parameters together with the others given in Table Table11 ensures the existence of Ca2+ and IP3 oscillations with amplitudes that are in agreement with those reported in the literature (; see Fig. 3 of Online Supplementary Material).
An important feature of our model is that despite the coupling between Ca2+ and IP3, the equation for Ca2+ dynamics (5) does not contain parameters found within the equation for IP3 dynamics (14). This means that the equation of the C-nullcline does not change with respect to the L–R model. Because the shape of this nullcline is crucial for the encoding mode (see Fig. Fig.2a,2a, c), the occurrence of AM, FM, or AFM modes in the ChI model is essentially established by the parameters of the L–R core model.
The only possible way that IP3 metabolism could affect the encoding mode is by modulating the dynamics of the channel inactivation variable h. This mechanism is suggested by the projection of the surfaces for , , and (Fig. (Fig.7)7) onto the C–I plane for different values of h and C (Fig. (Fig.8).8). We note indeed that the C-nullcline depends on the value of h but not the I-nullcline. In contrast, both the h-nullcline and the I-nullcline change with C, which suggests that the coupling between Ca2+ and IP3 dynamics essentially occurs through h. We may expect that, since h sets the slow time scale of the oscillations, the effect of IP3 metabolism on Ca2+ dynamics in our model is mainly a modulation of the oscillation frequency. This aspect is further discussed in Sections 4 and 5, following the introduction in the next section of the last term of our model, namely the glutamate-dependent IP3 production.
The contribution of glutamate signals to IP3 production can be taken into account as an additional production term in the IP3 equation of the above three-variable ChI model. The resulting new model is referred to as the “G-ChI” model.
Glutamate-triggered Ca2+ signals in astrocytes are mediated by group I and II mGluRs . Metabotropic GluRs are G-protein coupled receptors associated with the phosphotidylinositol signaling-cascade pathway . Although it is likely that the type of mGluRs expressed by astrocytes depends on the brain region and the stage of development , it seems reasonable to assume that such differences are negligible in terms of the associated second-messenger pathways [77, 78].
The G protein associated with astrocyte mGluRs is a heterotrimer constituted by three subunits: α, β, and γ. Glutamate binding to mGluR triggers receptor-catalyzed exchange of GTP from the Gβγ subunits to the Gα subunit. The GTP-loaded Gα subunit then dissociates from the G protein in the membrane plane and binds to a colocalized PLCβ (Fig. (Fig.1a,1a, e). Upon binding to Gα, the activity of PLCβ substantially increases, thus promoting PIP2 hydrolysis and IP3 production. Activation of PLCβ can therefore, to a first approximation, be directly linked to the number of bound mGluRs, and hence to the level of external stimulation. It follows that glutamate-dependent IP3 production can be written in the following generic form:
where is the maximal PLCβ rate that depends on the surface density of mGluRs and R(γ,C ) is the fraction of activated (bound) mGluRs. Experimental evidence shows that PLCβ activity (i.e., in (15)) is also dependent on intracellular Ca2+ . Notwithstanding, such dependence seems to occur for [Ca2+] > 10 μM, hence out of our physiological range . Therefore, Ca2+ dependence of PLCβ maximal rate will not be considered here.
R(γ,C ) can be expressed in terms of extracellular glutamate concentration (γ) at the astrocytic plasma membrane, assuming a Hill-binding reaction scheme, with an exponent ranging between 0.5 and 1 . In the current study, we choose 0.7, yielding:
In (16), R(γ,C ) is expressed as a Hill function with a midpoint that depends on glutamate and intracellular Ca2+ concentrations. This choice was motivated by the termination mechanism of PLCβ signaling that occurs essentially through two reaction pathways : (a) reconstitution of the inactive G-protein heterotrimer due to the intrinsic GTPase activity of activated Gα subunits and (b) PKC phosphorylation of the receptor, or of the G protein, or of PLCβ, or some combination thereof. We lump both effects into a single term, Kγ(γ,C ), such that the effective Hill midpoint of R(γ,C ) increases as PLCβ termination takes over, namely:
Here, KR is the Hill midpoint of glutamate binding with its receptor whereas Kp measures the increment of the apparent affinity of the receptor due to PLCβ terminating signals.
Hill(γ0.7, KR) accounts for the intrinsic GTPase-dependent PLCβ activity termination, as this effect is linked to the fraction of activated Gα subunits and therefore can be put in direct proportionality with the fraction of bound receptors. Hill instead accounts for PKC-related phosphorylation-dependent termination of PLCβ activity. Experimental data suggest that the target of PKC in this case is either the G protein or PLCβ itself . Generally speaking, phosphorylation by PKC may modulate the efficiency of ligand-binding by the receptors, the coupling of occupied receptors to the G protein, or the coupling of the activated G protein to PLCβ . All these effects indeed are lumped into (17), as explained below.
PKC is activated in a complex fashion (Fig. (Fig.1e).1e). Indeed, its activation by mere intracellular Ca2+ is minimal , 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 Ca2+-dependent PKC phosphorylation can be assumed Michaelis–Menten  so that . Remarkably, [DAG] can itself be related to intracellular Ca2+ concentration  so that [PKC*] can be rewritten as . Finally, KDAG<<Kπ [61, 81, 82] so that we can eventually approximate the product of the two Hill functions by that with the highest midpoint (see Appendix 1 for the derivation of this approximation). That yields: , which accounts for the second Hill function in (17).
To complete the model, it can be shown by numerical analysis of (17) that the term related to the GTPase-dependent PLCβ termination pathway, i.e., Hill(γ0.7, KR), can be neglected to a first approximation (Fig. (Fig.9).9). Hence can be simplified as :
The dynamical features of the G-ChI model for different extracellular concentrations of glutamate can be appreciated by inspection of the bifurcation diagrams in Figs. Figs.1010 and and11.11. We note that the choice of , the maximal rate of glutamate-dependent IP3 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 decreases, the oscillatory range expands toward infinite glutamate concentrations, but the amplitude of oscillations concomitantly decreases (at least with regard to the IP3 concentration).
The extension of the oscillatory range is due to the shift toward infinity of the subcritical Hopf bifurcation at high glutamate concentrations (compare Fig. Fig.11a,11a, d). Notably, for some values of receptor density, there seems to be coexistence of oscillations and asymptotic stability at high concentrations of extracellular glutamate, depending on the state of the cell prior to the onset of stimulation (Fig. (Fig.1111d–f).
As decreases, degradation becomes progressively preponderant so that IP3 peak levels are lower and the IP3R channels’ open probability is also reduced. Consequently, CICR is weaker and the increase of cytosolic Ca2+ is smaller. Then Ca2+-dependent PKC activation is reduced and termination of PLCβ signaling by PKC-dependent phosphorylation is limited. Moreover, if saturation of receptors occurs (i.e., R(γ, C )≈1) and oscillations are observed in this case, it follows that higher extracellular glutamate concentrations cannot further affect the intracellular Ca2+ dynamics.
The value of at which intracellular Ca2+ dynamics locks onto stable oscillations also depends on , the strength of the endogen PLCδ-mediated IP3 production. To some extent, increasing decreases the minimal value above which oscillations appear, provided that CICR is strong enough to activate enough PLCδ to keep IP3 levels above the lower Hopf bifurcation (results not shown).
Coupling between IP3 and Ca2+ dynamics in the G-ChI model might have important implications for the encoding of the stimulus. Bifurcation diagrams in Figs. Figs.1010 and and1111 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 Table11 and Fig. 3 in Online Supplementary Material). Notwithstanding, the applicability of these definitions to the G-ChI model might lead to some ambiguity.
We have previously assumed that AM (FM) encoding exists only if the amplitude (frequency) of oscillations (pulsations) throughout the oscillatory range can at least double with respect to its minimum value . Here, if we consider the AM-derived bifurcation diagrams for Ca2+ and IP3 dynamics (Fig. (Fig.10),10), we note that AM is still found since oscillations occur with arbitrarily small amplitude for the supercritical Hopf point at lower stimulus intensity (Fig. (Fig.10a,10a, b, d, e). But the period of oscillations (Fig. (Fig.10c,10c, f) at the upper extreme of the oscillatory range is almost half that observed at the onset of oscillations at the lower Hopf point. Thus, FM also occurs. 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 (Fig. 1a–c in Online Supplementary Material), it seems that, in the G-ChI model, Ca2+ oscillations become AFM encoding. Notably, IP3 dynamics appears to be always AFM encoding both in the AM (Fig. (Fig.10b,10b, e) and in the FM-derived bifurcation diagrams (Fig. (Fig.11b,11b, f, i).
Conversely, mere FM encoding is essentially preserved for Ca2+ 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 Fig. Fig.11d11d with Fig. 3d in Online Supplementary Material). These observations indicate that the G-ChI model accounts either for FM or AFM encoding Ca2+ oscillations, which are, however always coupled with AFM encoding IP3 oscillations. In addition, they provide further support to the above-stated notion that IP3 metabolism could consistently modulate the frequency of Ca2+ pulsating dynamics more than their amplitude (see Section 2.3.3).
On the contrary, the amplitude and shape of IP3 oscillations appear to be dramatically correlated with those of Ca2+ oscillations, as a consequence of the numerous Ca2+-dependent feedbacks on IP3 metabolism. Smooth Ca2+ oscillations such as those obtained in AM-like conditions (Fig. (Fig.12a,12a, AM) are coupled with small zigzag IP3 oscillations (Fig. (Fig.12b,12b, AM). Under FM conditions instead, pulsating large-amplitude Ca2+ variations (Fig. (Fig.12a,12a, FM) can be lagged by analogous IP3 oscillations (Fig. (Fig.12b,12b, FM), with the difference that whereas Ca2+ pulsations are almost fixed in their amplitude, IP3 ones can substantially vary.
Simulations of physiologically equivalent glutamate stimulation and associated astrocyte Ca2+–IP3 patterns are shown in Fig. Fig.13.13. Real multi-array electrode-recording data were considered as inputs of a single glutamatergic synapse (modeled as in Tsodyks and Markram ) and a fraction of the released glutamate was assumed to impinge on the astrocyte described by our model. We may notice that, from the stimulus up to Ca2+ dynamics, the smoothness of the patterns seems to increase. Indeed, the highly jagged glutamate stimulus turns into a less indented IP3 signal which is coupled with even smoother Ca2+ oscillations. Depending on the inherent cellular properties (Fig. (Fig.13,13, for example, considers two cases associated with different SERCA Ca2+ affinities), the difference of smoothness between IP3 and Ca2+ can be dramatic, more likely in the case of FM encoding Ca2+ pulsations (compare Fig. Fig.13a–b,13a–b, AM and FM).
Calcium dynamics in astrocytes can be driven by extracellular signals (such as the neurotransmitter glutamate) through regulation of the intracellular IP3 levels. Therefore, a prerequisite for unraveling the response of astrocytes to such signals is a thorough understanding of the complex IP3-related metabolic pathways that regulate intracellular Ca2+ dynamics. Here, we have devised and studied a model for agonist-dependent intracellular Ca2+ dynamics that captures the essential biochemical features of the complex regulatory pathways involved in glutamate-induced IP3 and Ca2+ oscillations and pulsations. Our model is simple, yet it retains the essential features of the underlying physiological processes that constitute the intricate IP3 metabolic network.
More specifically, the equation for IP3 dynamics is a central component of our model because of the large number of metabolic reactions that it accounts for and because coupling with intracellular Ca2+ dynamics is resolved through complex feedback mechanisms. Production of IP3 depends on the agonist/receptor-dependent PLCβ activation as well as on the endogenous agonist-independent contribution of PLCδ because both isoenzymes are found in astrocytes .
We linked the relative expression of these two isoenzymes to the expression of PKC and to the strength of PLCβ regulation by PKC. Indeed, Ca2+-dependent PKC activation can phosphorylate the receptor or PLCβ or a combination thereof, leading to termination of IP3 production . In astrocytes, this mechanism has been suggested to limit the duration of Ca2+ oscillations, thus defining their frequencies . In agreement with this idea, a stronger PKC-dependent inhibition of PLCβ 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 PLCβ is counteracted if PLCδ expression is high enough to support high IP3 production levels and the resulting release of Ca2+ from the intracellular stores. This observation raises the possibility of phase-locked Ca2+ oscillations under conditions of intense stimulation. Phase-locked Ca2+ oscillations were also found in other models of agonist-dependent intracellular Ca2+ dynamics [84–86] and are often associated with pathological conditions [87, 88]. In our model, persistent pulsating Ca2+ dynamics that are essentially independent of the level of stimulation are observed for weak maximal rates of IP3 production by PLCβ (Figs. (Figs.10d–e,10d–e, 11g–h). In astrocytes, such persistent oscillations could also be interpreted as a fingerprint of pathological conditions [1, 89]. In fact, a decay of PLCβ 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 .
We note that, although focusing on stimulus-triggered Ca2+ oscillations, our study also hints, at a possible link between modulation of frequency and amplitude of Ca2+ pulsations and spontaneous Ca2+ dynamics. Recently, it has been shown that the interpulse interval of the spontaneous Ca2+ oscillations is inherently stochastic [91, 92]. In particular, experimental observations are compatible with model studies of a local stochastic nucleation mechanism that is amplified by the spatial coupling among IP3R clusters through Ca2+ or IP3 diffusion [92, 93]. 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 IP3 dynamics could affect the statistics of Ca2+ interpulse intervals not only in terms of spatial coupling among IP3R clusters by means of intracellular IP3 gradients but also by modulation of either the recovery from Ca2+ inhibition or the progressive sensitization of IP3Rs by Ca2+ . The resulting scenario therefore would still be that of a local stochastic nucleation mechanism amplified by IP3R spatial coupling, but the local IP3R and SERCA parameters would vary according to the biochemical regulation system presented in the current work.
A critical question in experiments is the identification of the mechanism that drives IP3 oscillations and pulsations [57, 95–97]. In our model, self-sustained IP3 oscillations are brought about by the coupling of IP3 metabolism with Ca2+ dynamics. In other words, our model can be considered as a self-consistent astrocytic generator of Ca2+ 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 . 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 IP3-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 for deciphering some of the enigmas of neural information processing.
Another significant prediction of our model is that IP3 dynamics is essentially AFM, and Ca2+ oscillations/pulsations are inherently FM encoding, that is, they can be either FM or AFM but not AM [18, 19, 98]. In FM, Ca2+ oscillations resemble pulses. In contrast, in AFM, their shape is smoother and necessarily depends on the stimulus dynamics.
The assumption that IP3 oscillations are always AFM encoding could provide an optimal interface between agonist stimuli and intracellular Ca2+ signals. The stimuli impinge on the cell in the form of trains of pulses or bursts of pulses and information is carried in the timing of these pulses rather than in their amplitude . AFM features in IP3 signals could perfectly match these stimuli, embedding the essential features of the spectrum of the signal into the spectrum of the IP3 transduction. Hence, IP3 signaling with FM features could offer an efficient way to keep the essence of the information of the stimulus. On the other hand, because Ca2+ signals are triggered primarily by sufficiently ample elevations of IP3 , the coexistence of AM features within the IP3 signal seems to be a necessary prerequisite in order to trigger CICR.
The fact that coupling of IP3 metabolism with CICR does not allow pure AM encoding is in general agreement with experimental data on intracellular Ca2+ signaling in several cells [100, 101] including astrocytes . Notwithstanding, the possibility of AFM-encoding Ca2+ oscillations has recently come up as a reliable alternative mechanism to explain gliotransmitter exocytosis, which is dependent on a specific agonist that triggers astrocyte Ca2+ dynamics [21, 102].
The above could be relevant to understanding the origin of the integrative properties of Ca2+ signaling in astrocytes . 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 IP3 signal and the other at the cross-coupling between the IP3 and Ca2+ signals. Indeed, AFM-encoding IP3 dynamics could deploy smoothing of the highly indented agonist stimulus, thus hinting at possible integrative properties for IP3 signals (Fig. (Fig.13).13). On the other hand, the associated Ca2+ patterns look even smoother, suggesting a further integration step that likely relies only on the inherent features of CICR.
Below is the image is a link to a high resolution version
The product of two Hill functions (a-b) with sufficiently distant midpoints is equivalent to the Hill function with the largest midpoint (c). Namely: Hill(x, K1) · Hill(x, K2) ≈ Hill(x, K2) where K1<<K2. Midpoints are marked by vertical dashed lines; K1: red; K2: blue. (GIF 22.5KB)
Hill functions of Hill functions (a-b) can also be approximated by Hill functions. (c-d) Hill (Hill(x, K2), K1) = (1+K1)−1. Hill(x, K1K2(1+K1)−1). In this case the midpoint of the resulting Hill function depends on the specific values of the midpoints of the original Hill functions considered in the composition of the Hill-of-Hill function. (e-h) Hill(x, K1 · Hill (x, K2)) = (x + K2)/(x + K1 + K2) = Hill (x, (K1 + K2)) + f(x), where f(x) = K2/(x + K1 + K2). Notably, f(x0) = K2/(K1 + K2) whereas f(x∞) ≈ 0, so that the resulting Hill curve is essentially comprised within the interval [K2/(K1 + K2),1). (GIF 50.6KB)
Bifurcation diagrams for a modified ChI model and prototypical sets of (a-c) AM-encoding and (d-f) FM-encoding L-R parameters. The bifurcation diagrams were computed after introduction into the ChI model of the rate of glutamate-dependent IP3 production, vglu, as a free bifurcation parameter, namely . This figure shows that the ChI model can still display oscillations in presence of an external non-specific bias of IP3 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. (GIF 44.6KB)
The authors wish to thank Vladimir Parpura, Giorgio Carmignoto, and Ilyia Bezprozvanny for insightful conversations. V. V. acknowledges the support of the U.S. National Science Foundation I2CAM International Materials Institute Award, Grant DMR-0645461. This research was supported by the Tauber Family Foundation, by the Maguy-Glass Chair in Physics of Complex Systems at Tel Aviv University, by the NSF-sponsored Center for Theoretical Biological Physics (CTBP), grants PHY-0216576 and 0225630, and by the University of California at San Diego.
For the sake of simplicity, we have adopted throughout the text the following notation for the generic Hill function:
where n is the Hill coefficient and K is the midpoint of the Hill function, namely the value of x at which Hill.
It can be shown that the product of two Hill functions can be approximated by the Hill function with the greatest midpoint, when the two midpoints are distant enough from each others, that is:
if and only if K1 << K2 (Fig. 1, Online Supplementary Material). Indeed, under such conditions, Hill only when x >> K1, hence
This result can be extended to the product of N Hill functions, that is:
provided that K1<<K2 <<...<<KN.
Notably, the product of Hill function is not the only case in which a functions composed by Hill functions can be approximated by a mere Hill function: other examples are given by functions of the type or (see Fig. 2 in Online Supplementary Material).
We seek an expression for [CaMKII*] based on the following kinetic reaction scheme:
Let us first consider the reaction chain (22). We can assume that the second step is very rapid with respect to the first one [58, 104] so that generation of CaMKII* is in equilibrium with CaMKII consumption, namely:
Then, under the hypothesis of quasisteady state for CaMKII, we can write:
where . Defining as the total kinase II concentration and assuming it constant, we can rewrite (25) as follows:
An erratum to this article can be found at http://dx.doi.org/10.1007/s10867-009-9182-8