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

Abstract

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 (Ca^2 +) dynamics in astrocytes in response to synaptic activity. Here, we derive and investigate a concise mathematical model for glutamate-induced astrocytic intracellular Ca^2 + 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 Ca^2 + 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 Ca^2 + 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 Ca^2 + 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.

1 Introduction

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 (Ca^2 +) dynamics [1]. 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 Ca^2 + 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 Ca^2 + 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 Ca^2 + oscillations [8] that can bring forth propagation of both intracellular and intercellular waves [9–11]. 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. [12] 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 [16], thus extending neuron–astrocyte communications to the level of neuronal metabolism [17].

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 [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 Ca^2 + pulsations [20]. 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 [3].

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 [21]. For instance, the amplitude of Ca^2 + oscillations in response to external stimuli can be highly variable, depending on the intensity of stimulation [2, 11, 22]. 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 [23]. 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 [1, 3].

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 [24] 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 Ca^2 + oscillations, FM of Ca^2 + pulsations, or combined AM and FM (AFM) Ca^2 + 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 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 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 Ca^2 +, via IP₃ regulation and IP₃-dependent Ca^2 +-induced Ca^2 + 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 Ca^2 + 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 [31] 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 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 intercellular Ca^2 + signaling, such as heart cells, pancreas cells, and liver cells.

2 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 IP₃ regulation. Given the relative intricacy of this signaling pathway (see Fig. 1), each basic building block of the model is described separately in the next sections.

Fig. 1

Block diagrams of a production and b degradation of inositol 1,4,5-trisphosphate (IP₃), summarize the complexity of the signaling network underlying glutamate-induced intracellular dynamics of this second messenger. A peculiar feature of IP₃ metabolism is its coupling with intracellular calcium (Ca^2 +) dynamics which, in astrocytes, primarily occurs through c Ca^2 +-induced Ca^2 + release from intracellular stores. Production of IP₃ is brought forth by hydrolysis of the highly phosphorylated membrane lipid phosphatidylinositol 4,5-bisphosphate (PIP₂) by PLCβ and PLCδ, two isoenzymes of the family of phosphoinositide-specific phospholipase C. d PLCδ signaling is agonist independent and modulated by Ca^2 +. e The contribution of PLCβ to IP₃ production instead depends on agonist binding to G-protein coupled metabotropic receptors (mGluRs) found on the surface of the cell. Degradation of IP₃ mainly occurs through phosphorylation into inositol 1,3,4,5-tetrakisphosphate (IP₄), catalyzed by IP₃ 3-kinase (3K), and dephosphorylation by inositol polyphosphate 5-phosphatase (5P). The activity of IP₃-3K is regulated by Ca^2 + in a complex fashion which may be approximated as depicted in f. For simplicity, inhibition of IP-5P by Ca^2 +/CaMKII-dependent phosphorylation [32] and competitive binding of IP₄ to IP-5P are not considered in this study. Legend of the different arrows is below f

2.1 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 [33]. In astrocytes, though, IP₃-dependent CICR from the ER is considered the primary mechanism responsible of intracellular Ca^2 + dynamics [34].

Calcium-induced Ca^2 + release (see Fig. 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 IP₃ receptor (IP₃R) 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 [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) [4]. Binding of glutamate to mGluRs increases cytosolic IP₃ concentration and promotes the opening of a few IP₃R channels [37]. As a consequence, intracellular Ca^2 + slightly increases. Since the opening probability of IP₃R channels nonlinearly increases with Ca^2 + concentration [38], such an 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 IP₃R channels takes place, leading to CICR termination [39]. In parallel, SERCA pumps, whose activity increases with cytoplasmic Ca^2 + [40], quickly sequester excess cytoplasmic Ca^2 + by pumping it back into the ER lumen. The intracellular Ca^2 + 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 [41].

The SERCA pump rate can be taken as an instantaneous function of cytoplasmic [Ca^2 +] (denoted hereafter by C) by assuming a Hill rate expression with exponent 2 (see Appendix 1):

$$ \label{eq1} J_{{\rm pump}} \left( C \right)=v_{{\rm ER}} \cdot {\rm Hill}\left( {C^2,K_{{\rm ER}} } \right) $$

(1)

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 [42].

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:

$$ \label{eq2} J_{{\rm leak}} \left( C \right)=-r_{\rm L} \left( {C_{{\rm ER}} -C} \right) $$

(2)

where C _ER is the Ca^2 + concentration inside the ER stores [36].

IP₃R channels can be thought of as ensembles of four independent subunits with three binding sites each: one for IP₃ and two for Ca^2 +. The latter sites include an activation site and a separate site for inactivation [36]. IP₃-binding sensitizes the receptor toward activation by Ca^2 + but only if both IP₃ 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 IP₃-binding > > Ca^2 +-activation > > Ca^2 +-inactivation, Li and Rinzel proposed the following equation for the Ca^2 + current through the IP₃R channels [24]:

$$ \label{eq3} J_{\rm {chan}} \left( {C,h,I\,} \right)=r_{\rm C} p^{{\rm open}}\left( {C_{{\rm ER}} -C\,} \right) $$

(3)

with the channel open probability given by $p^{{\rm open}} = m_\infty ^3 n_\infty ^3 h^3$, where m _∞ = Hill( I,d ₁ ), $n_\infty ={\rm Hill}\left( {C,d_{5} } \right)$, and h account for the three gating reactions, respectively, IP₃-binding, activating Ca^2 +-binding, and Ca^2 +-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 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 [35, 43, 44], 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 as C _ER = (C ₀ − C)/c ₁ 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:

$$ \begin{array}{rcl} \label{eq4} J_{{\rm chan}} &=&r_{\rm C} m_\infty ^3 n_\infty ^3 h^3\left( {C_0 -\left( {1+c_1 } \right)C\,} \right) \\[6pt] J_{{\rm leak}} &=&r_{\rm L} \left( {C_0 -\left( {1+c_1 } \right)C\,} \right). \end{array} $$

(4)

Adding together the above terms (1) and (4), the cytoplasmic Ca^2 + balance is given by:

$$ \label{eq5} \dot{C}=\left( {r_{\rm C} m_\infty ^3 n_\infty ^3 h^3+r_{\rm L} } \right)\left( {C_0 -\left( {1+c_1 } \right)C\,} \right)-v_{{\rm ER}} \frac{C^2}{C^2+K_{{\rm ER}}^{2} }. $$

(5)

This equation is coupled with an equation for h that accounts for the kinetics of IP₃Rs [24]:

$$ \label{eq6} \dot{h} =\frac{h_\infty -h}{\tau _h }, $$

(6)

where:

$$ h_\infty =\frac{Q_2 }{Q_2 +C},\quad \tau _h =\frac{1}{a_2 \left( {Q_2 +C\,} \right)},\quad {\rm and}\quad Q_{2} =d_2 \frac{I+d_1 }{I+d_3 }. $$

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.

2.2 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 [33, 45, 46]. In the case of Ca^2 + signaling in astrocytes, however, the information usually arrives as a nonoscillatory stimulus at the plasma membrane and is translated into intracellular Ca^2 + 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 [47]. 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 Ca^2 +. In physiological conditions glutamate-induced astrocyte Ca^2 + signaling is synaptically evoked [2–4]. 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 AM, 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 [18, 19]. 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 one. When this happens, however the effects of CICR are large and the increase of intracellular Ca^2 + is fast and remarkable. Accordingly, inactivation of IP₃R channels is also faster and basal Ca^2 + 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 Ca^2 + uptake 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 IP₃, whereas their frequency does not, as it is essentially fixed by IP₃R channel recovery from Ca^2 +-dependent inactivation [19].

From a dynamical systems perspective, AM and FM encoding are associated with well-distinct bifurcation diagrams. Amplitude modulations of Ca^2 + 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. 2a–c). Alternatively, if the Hopf bifurcation is subcritical, AFM might be found [18]. On the contrary, in FM (Fig. 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 [19].

Fig. 2

Both AM-encoding or FM-encoding Ca^2 + oscillations can be generated by the Li–Rinzel model for CICR, depending on the value of K _ER, the Ca^2 + affinity of sarco-endoplasmic reticulum Ca^2 +-ATPase pumps. For example, AM encoding can be found at a–c K _ER = 0.1 μM whereas FM encoding exists for smaller K _ER, such as d–f K _ER = 0.05 μM. a In the phase plane, AM encoding is associated with a single intersection between the C-nullcline (orange) and the h-nullcline (green). These are the curves for which $\dot{C} =0$ and $\dot{h} =0$, respectively. Accordingly, the only possible bifurcations that can be found are connected with loss/gain of stability, i.e., they are Hopf bifurcations. b The associated bifurcation diagram indeed shows that oscillations arise via supercritical Hopf bifurcation (H₁) at [Ca^2 +] ≈ 0.15 μM and [IP₃] ≈ 0.36 μM, whereas they die at [Ca^2 +] ≈ 0.32 μM and [IP₃] ≈ 0.64 μM via a subcritical Hopf bifurcation (H₂). The fact that H₁ is supercritical accounts for the occurrence of oscillations of arbitrarily small amplitude that increases with IP₃ yet with almost constant period (c). d In FM-encoding conditions, the C-nullcline is sharply N-shaped and there exists a small range of IP₃ values where it can intersect the h-nullcline at three points. e This region is delimited by two “knees” shown by the fixed-point continuation curve, which correspond to a saddle-node bifurcation at [Ca^2 +] ≈ 0.13 μM and [IP₃] ≈ 0.48 μM and a saddle-node homoclinic bifurcation at [Ca^2 +] ≈ 0.07 μM and [IP₃] ≈ 0.53 μM. Pulsatile oscillations arise and die via subcritical Hopf bifurcations respectively at H₁ ([Ca^2 +] ≈ 0.05 μM, [IP₃] ≈ 0.51 μM) and H₂ ([Ca^2 +] ≈ 0.39 μM, [IP₃] ≈ 0.86 μM). While their amplitude is essentially constant, their period can be arbitrarily long (f) due to the saddle-node homoclinic bifurcation. b, e Conventions: stable equilibria are shown as solid lines, respectively for low (black) and high (blue) IP₃ concentrations. Unstable equilibria are displayed as red dashed lines. Oscillations are located in the diagram as min (green)–max (black) envelopes, with stable oscillations as filled circles and unstable ones as empty circles. Parameter values for the L–R model as in Table 1

Table 1 Parameter values for the ChI and G-ChI models

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. 1) immediately hints that IP₃ concentration indeed depends on both intracellular Ca^2 + 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.

2.3 IP₃ regulation: the ChI model

2.3.1 PLCδ production

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 [48]. 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 Ca^2 + levels (Fig. 1d) [49]. Structural and mutational studies of complexes of PLCδ with Ca^2 + and IP₃ revealed complex interactions of Ca^2 + with several negatively charged residues within its catalytic domain [50–52], a hint of cooperative binding of Ca^2 + to this enzyme. In agreement with these experimental findings, the PLCδ activation rate can be written as [27, 53]:

$$ \label{eq7} v_{\updelta } \left( {C,I} \,\right)=v_{\updelta }^{\prime } \left( I \,\right)\cdot {\rm Hill}\left( {C^2,K_{{\rm PLC\updelta }} } \right) $$

(7)

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 [54]. Accordingly, assuming competitive binding [55], the maximal PLCδ-dependent IP₃ production rate can be modeled as follows:

$$ \label{eq8} v_{\updelta }^{\prime } \left( I \,\right)=\frac{\bar {v}_{\updelta } }{1+\frac{I}{\kappa _{\updelta } }} $$

(8)

where $\kappa _{\updelta }$ is the inhibition constant of PLCδ activity.

Figure 3 shows the behavior of this term when Ca^2 + and corresponding IP₃ levels obtained from the bifurcation diagrams of the L–R model in Fig. 2 are substituted into (7). We have set $K_{{\rm PLC\updelta }}$ 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 $v_{\updelta }$ that could preserve the main AM/FM properties.

Fig. 3

Bifurcation diagrams for PLCδ-dependent IP₃ production are drawn by substituting into (7), [Ca^2 +] and [IP₃] values obtained from the bifurcation diagrams in Fig. 2b (panel a above) and in 2e (panel b above), respectively. Colors as in Fig. 2b, e

2.3.2 IP₃ degradation

Two major IP₃ degradation pathways have been described so far (Fig. 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 Ca^2 + dependent [63].

The rate of both IP-5P dephosphorylation (v _5P) and IP₃-3K phosphorylation (v _3K) of IP₃ can be considered as of Michaelis–Menten type [56, 64, 65]. Therefore:

$$ \label{eq9} \begin{array}{c} {v_{5{\rm P}} \left( I \right)}={\bar {v}_{{\rm 5P}} \cdot {\rm Hill}\left( {I,K_5 } \right)} \\[6pt] {\nu _{3{\rm K}} \left( {C,I} \right)}={\nu _{3{\rm K}}^\ast \left( C \right) \cdot {\rm Hill}\left( {I,K_3 } \right)} . \end{array} $$

(9)

Since $K_{5}> 10~\upmu$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:

$$ \label{eq10} \nu _{{\rm 5P}} \left( I \right)\approx \bar {r}_{{\rm 5P}} \cdot I $$

(10)

where $\bar {r}_{{\rm 5P}}$ is the linear rate of IP₃ degradation by IP-5P and can be defined by parameters in (9) as $\bar {r}_{5{\rm P}} ={\bar {v}_{{\rm 5P}} } \mathord{/{\kern.9pt} {\vphantom {{\bar {v}_{{\rm 5P}} } {K_5 }}} \kern-\nulldelimiterspace} {K_5 }$.

In basal conditions, phosphorylation of IP₃ by IP₃-3K is very slow. The activity of IP₃-3K is substantially stimulated by Ca^2 +/calmodulin (CaM) via CaMKII-catalyzed phosphorylation (Fig. 1b) [67]. However, other experimental reports have suggested that Ca^2 +-dependent PKC phosphorylation of IP₃-3K could have inhibitory effects [68]. Notwithstanding, evidences for this latter possibility are contradictory [69]. 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. 1f).

Phosphorylation of IP₃-3K by active CaMKII (CaMKII*) only occurs at a single threonine residue [67, 70], therefore we can assume that $v_{3{\rm K}}^\ast \left( C \right)$ ∝ [CaMKII*]. 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 [71]. 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 [72]. 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 [58]. Hence, if we surmise the following kinetic reaction scheme for CaMKII phosphorylation:

$$ \begin{array}{rcl} \label{eq11} 4{\rm Ca}^{2+}+{\rm CaM}&\mathop\rightleftarrows\limits^{k_b}_{k_u}& {\rm CaM}^+ \\[10pt] {\rm KII}+{\rm CaM}^+&\mathop\rightleftarrows\limits^{k_1}_{k_{-1}}&{\rm CaMKII}\mathop\rightleftarrows\limits^{k_2}_{k_{-2}}{\rm CaMKII}\mbox{*} , \end{array} $$

(11)

it can be demonstrated that [CaMKII*] ∝ Hill(C ⁴, K _D) (see Appendix 2).

Accordingly, $v_{3{\rm K}}^\ast \left( C \right)\propto {\rm Hill}\left( {C^4,\;K_{\rm D} } \right)$ and the equation for IP₃-3K-dependent IP₃ degradation reads:

$$ \label{eq12} v_{{\rm 3K}} \left( {C,I}\, \right)=\bar {v}_{3{\rm K}} \cdot {\rm Hill}\left( {C^4,K_{\rm D} } \right) \cdot {\rm Hill}\left( {I,K_3 } \right). $$

(12)

Experimental observations show the existence of three regimes of IP₃ metabolism [57]. At low [Ca^2 +] and [IP₃] (<400 nM and <1 μM, respectively), IP-5P and IP₃-3K degrade roughly the same amounts of IP₃. Then, at high [Ca^2 +] (≥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 [Ca^2 +].

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 Ca^2 + 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 Ca^2 + dependence of IP₃-3K. In particular, by opportune choice of parameters such as $\bar {v}_{3{\rm K}} >K_3 \bar {r}_{5{\rm P}}$, theoretical investigations showed that these two regimes are essentially generated by the Ca^2 +-dependent Hill term in the expression of v _3K irrespectively of the assumption of Michaelis–Menten kinetics for IP₃ dependence of IP₃-3K (Fig. 4). Accordingly, a linear approximation for v _3K such as:

$$ \label{eq13} v_{3{\rm K}} \left( {C,I} \,\right)=\bar {r}_{3{\rm K}} \cdot {\rm Hill}\left( {C^4,K_{\rm D} } \right) \cdot I $$

(13)

where $\bar {r}_{{\rm 3K}} ={\bar {v}_{3{\rm K}} } \mathord{/{\kern.9pt} {\vphantom {{\bar {v}_{3{\rm K}} } {K_3 }}} \kern-\nulldelimiterspace} {K_3 }$ could also be considered instead of (12), in agreement with previous investigations found in the literature [26, 57].

Fig. 4

a Experimental observations suggest the existence of three regimes of IP₃ metabolism: one for low [Ca^2 +] and [IP₃] in which IP₃-3K (Ca^2 +-dependent color curves) and IP-5P (black curve) activities are similar; an intermediate one for higher [Ca^2 +] in which IP₃ degradation by IP₃-3K is predominant; and a third one for [IP₃] > 8 μM in which IP₃ is degraded mainly by IP-5P in a Ca^2 +-independent fashion. Both enzymes can be assumed Michaelis–Menten. b, c Physiological IP₃ concentrations suggest only the first two regimes. Notably, these latter regimes can be mimicked either b by keeping the hypothesis of Michaelis–Menten kinetics for IP₃-3K (9) or c by a linear approximation of this dependence (13)

Indeed, the behaviors of v _3K in (12) and (13) for IP₃ and Ca^2 + concentrations obtained from the corresponding Li–Rinzel bifurcation diagrams are qualitatively similar (Fig. 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 [Ca^2 +] is reached, such as in FM conditions (Fig. 5c–d). Notwithstanding, the Michaelis–Menten constant of IP₃-3K for its substrate is experimentally reported to be $K_{3} \approx 1~\upmu$M [56, 59] and it is likely that intracellular IP₃ levels can reach such micromolar concentrations in vivo [73]. Therefore, in the following, we will keep the Michaelis–Menten formulation for v _3K (12).

Fig. 5

Bifurcation behaviors of IP₃-3K-dependent IP₃ degradation in a, b AM and c, d FM conditions are compared for a, c Michaelis–Menten (12) or b, d linear approximations (13) of the IP₃ dependence of IP₃-3K rate. Despite qualitatively similar behaviors, the linear approximation is not further taken into account in the present study, because IP₃-3K activity may saturate in physiological conditions, thus invalidating the linear approximation

Finally, experimental measurements show that for [Ca^2 +] > 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 < < K ₃ (i.e., $v_{3{\rm K}} \left( {C,I}\, \right)\approx \bar {r}_{3{\rm K}} \;\times \;I\,)$, then v _3K ≈ 20v _5P. Accordingly, we set the maximal degradation rates in the following such that $\bar {v}_{3{\rm K}} \approx 20K_3 \bar {r}_{{\rm 5P}}$.

2.3.3 Model analysis

In summary, our model of Ca^2 + 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)):

$$ \label{eq14} {\text{\it\.{I}}}\mathop =\frac{\bar {v}_{\updelta } }{1+\frac{I}{\kappa _{\updelta } }}{\rm Hill}\left( {C^2,K_{{\rm PLC\updelta }} } \right)-v_{{\rm 3K}}{\rm Hill}\left( {C^4,K_{\rm D} } \right){\rm Hill}\left( {I,K_3 } \right)-r_{{\rm 5P}} I\,. $$

(14)

Equation 14 together with (5) and (6) define our three-variable “ChI” model, whose name is composed of the letters denoting its state variables.

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 $\dot{I} =0$ and C → 0 in (14) and solved for I as a function of C in the resulting equation. In parallel, we set $\dot{C} =0$ 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 $K_{{\rm PLC\updelta}} \le {\rm H}_1$, K _D ≈ H₂, K ₃ > H₂, where H₁ and H₂ denote Ca^2 + and IP₃ concentrations at the two Hopf bifurcations in the L–R bifurcation diagrams (Fig. 2). Such choice of parameters together with the others given in Table 1 ensures the existence of Ca^2 + and IP₃ oscillations with amplitudes that are in agreement with those reported in the literature ([73]; see Fig. 3 of Online Supplementary Material).

Fig. 6

Consistency of the equation for the endogenous IP₃ metabolism with respect to the L–R core model can be tested as follows: at resting physiological conditions: $\dot {C}=\dot {I}=0$, $h=h_\infty \left( C \right)$ and C →0 so that $v_{{\rm 3K}} \left( {C,I_{\rm s} } \right) \approx$. Hence, for steady IP₃ values (I _s) such as $I_{\rm s} <\!< \kappa _{\updelta}$, one gets $v_{\updelta } \left( {C,I_{\rm s} } \right)\approx \bar {v}_{\updelta} \cdot {\rm Hill}\!\left( {C^2,K_{{\rm PLC\updelta}}} \right)$. Accordingly, (14) can be solved for I _s, yielding $I_{\rm s} \left( C \right)\approx r_{{\rm 5P}}^{-1} \cdot \bar {v}_{\updelta } \cdot {\rm Hill}\!\left( {C^2,K_{{\rm PLC\updelta}}} \right)$ (magenta curve). The latter curve must be compared with the corresponding I(C) curve (black) obtained by solving for I the equation $\left. {\dot {C}} \right|_{h=h_\infty \left( C \right)} =0$ in the original L–R model (5). By changing $\bar {v}_{\updelta}$, $\bar {r}_{{\rm 5P}}$, $K_{{\rm PLC\updelta }}$, and $\kappa _{\updelta}$ according to their experimental values, we seek consistency when I _s(C) ≈ I(C). In these conditions, in fact, our mathematical description of IP₃ metabolism and the L–R model predict equivalent steady intracellular IP₃ levels

An important feature of our model is that despite the coupling between Ca^2 + and IP₃, the equation for Ca^2 + 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. 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 $\dot{C} =0$, $\dot{h} =0$, and $\dot{I} =0$ (Fig. 7) onto the C–I plane for different values of h and C (Fig. 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 Ca^2 + 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 Ca^2 + 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.

Fig. 7

Surfaces for $\dot{C}=0$ (orange), $\dot{h} =0$ (green), and $\dot{I} =0$ (red) for the ChI model described by (5), (6), and (14)

Fig. 8

Projections of the surfaces for $\dot{C} =0$ (orange), $\dot{h} =0$ (green), and $\dot{I} =0$ (red) onto the I–C plane for different values of a, c h or b, d C, both a, b in AM and c, d FM conditions, and allow one to appreciate the nature of coupling between IP₃ metabolism and Ca^2 + dynamics in the ChI model. In particular, since none of the parameters of the equation for IP₃ metabolism (14) are found in the equations for $\dot{C} =0$ and $\dot{h} =0$, the latter surfaces are not affected by inclusion of IP₃ dynamics into the L–R core model. It follows that IP₃ dynamics may influence Ca^2 + dynamics only through modulations of the dynamics of h, i.e., Ca^2 +-mediated deactivation of CICR IP₃R/channels

3 Modeling glutamate regulation of IP₃ production: the G-ChI model

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 Ca^2 + signals in astrocytes are mediated by group I and II mGluRs [74]. Metabotropic GluRs are G-protein coupled receptors associated with the phosphotidylinositol signaling-cascade pathway [75]. Although it is likely that the type of mGluRs expressed by astrocytes depends on the brain region and the stage of development [76], 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. 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:

$$ \label{eq15} v_{{\rm glu}} \left( {\gamma ,C} \right)=\bar {v}_{\upbeta} \cdot R\left( {\gamma ,C} \right) $$

(15)

where $\bar {v}_{\upbeta }$ 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., $\bar {v}_{\upbeta }$ in (15)) is also dependent on intracellular Ca^2 + [49]. Notwithstanding, such dependence seems to occur for [Ca^2 +] > 10 μM, hence out of our physiological range [54]. Therefore, Ca^2 + 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 [60]. In the current study, we choose 0.7, yielding:

$$ \label{eq16} R\left( {\gamma ,C}\, \right)={\rm Hill}\left( {\gamma ^{0.7},K_\gamma \left( {\gamma ,C} \,\right)} \right). $$

(16)

In (16), R(γ,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 PLCβ signaling that occurs essentially through two reaction pathways [48]: (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, K _γ (γ,C ), such that the effective Hill midpoint of R(γ,C ) increases as PLCβ termination takes over, namely:

$$ \label{eq17} K_\gamma \left( {\gamma,\!C} \,\right)=K_{\rm R} \left( {1+\frac{K_{\rm p} }{K_{\rm R} }{\rm Hill}\left( {\gamma ^{0.7},K_{\rm R} } \right){\rm Hill}\left( {C,K_\pi } \right)} \right). $$

(17)

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 PLCβ terminating signals.

Hill(γ ^0.7, K _R) 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$\left( {C,{\rm }K_\pi } \right)$ 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 [79]. 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 [80]. All these effects indeed are lumped into (17), as explained below.

PKC is activated in a complex fashion (Fig. 1e). Indeed, its activation by mere intracellular Ca^2 + is minimal [81], 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 [79] so that $\left[ {\rm PKC\mbox{*}} \right]\propto {\rm Hill}\left( {\left[ {{\rm DAG}} \right],K_{{\rm DAG}}^{\prime } } \right) \cdot {\rm Hill}\left( {C,K_\pi } \right)$. Remarkably, [DAG] can itself be related to intracellular Ca^2 + concentration [82] so that [PKC*] can be rewritten as $\left[ {\rm PKC\mbox{*} } \right]\propto {\rm Hill}\left( {C,K_{{\rm DAG}} } \right) \cdot {\rm Hill}\left( {C,K_\pi } \right)$. Finally, K _DAG < < K _π [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: $\left[ {\rm PKC\mbox{*}} \right]\propto {\rm Hill}\left( {C,K_\pi } \right)$, which accounts for the second Hill function in (17).

To complete the model, it can be shown by numerical analysis of $K_\gamma \left( {\gamma ,C} \right)$ (17) that the term related to the GTPase-dependent PLCβ termination pathway, i.e., Hill(γ ^0.7, K _R), can be neglected to a first approximation (Fig. 9). Hence $K_\gamma \left( {\gamma ,C\,} \right)$ can be simplified as $K_\gamma \left( C\, \right)$:

$$ \label{eq18} K_\gamma \left( C\, \right)\approx K_{\rm R} \left( {1+\frac{K_{\rm p} }{K_{\rm R} }{\rm Hill}\!\left( {C,K_\pi } \right)} \right). $$

(18)

Using (15), (16), and (18), our final expression for the glutamate-dependent IP₃ production reads:

$$ \label{eq19} v_{{\rm glu}} \left( {\gamma ,C\,} \right)=\bar {v}_{\upbeta } \cdot {\rm Hill}\!\left( {\gamma ^{0.7},K_{\rm R} \left( {1+\frac{K_{\rm p} }{K_{\rm R} }{\rm Hill}\!\left( {C,K_\pi } \right)} \right)} \right). $$

(19)

Substituting (19) into (14), we obtain

$$ \label{eq20} {\begin{array}{rcl} {\dot{I}}&=&{\bar {v}_{\upbeta } \cdot {\rm Hill}\!\left( {\gamma ^{0.7},K_{\rm R} \left( {1+\frac{K_{\rm p} }{K_{\rm R} }{\rm Hill}\!\left( {C,K_\pi } \right)} \right)} \right)+\dfrac{\bar {v}_{\updelta } }{1+\frac{I}{\kappa _{\updelta } }}{\rm Hill}\!\left( {C^2,K_{{\rm PLC\updelta }} } \right)+} \\[6pt] &&{-v_{3{\rm K}} {\rm Hill}\!\left( {C^4,K_{\rm D} } \right){\rm Hill}\!\left( {I,K_3 } \right)-r_{5{\rm P}} I}\,. \\ \end{array} } $$

(20)

This equation, combined with (5) and (6), defines our G-ChI model of glutamate-dependent intracellular Ca^2 + dynamics in astrocytes.

Fig. 9

a, b Numerical investigation shows that the term related to the GTPase-dependent PLCβ termination pathway in the expression of the agonist-dependent IP₃ production (15 and 17) can be neglected so that $K_\gamma \left( {\gamma ,C}\, \right)\approx K_\gamma \left( C\, \right)$. c, d Bifurcation behaviors of v_glu(γ,C ) (19), obtained by substituting γ and C with their values derived from bifurcation diagrams of the agonist-dependent model (see also Figs. 10a and 11a)

Fig. 10

Bifurcations diagrams for AM-derived parameter sets of the G-ChI model (5, 6, 17), show c, f that the inclusion of IP₃ dynamics remarkably affects the frequency of oscillations. a, d In particular, Ca^2 + oscillations are essentially AFM encoding rather than merely AM encoding. d–f Low values of the glutamate-dependent maximal rate of IP₃ production, $\bar {v}_{\upbeta }$, extend the range of oscillations to arbitrarily high glutamate concentrations. In these conditions, phase-locked Ca^2 +/IP₃ oscillations and pulsations can be observed. Namely, there is a threshold glutamate concentration (which can equivalently be described by a threshold frequency of a pulsed stimulation), for which the frequency of oscillations (pulsations) locks to a particular value and does not change for further elevations of glutamate concentration. Parameters as in Table 1 except for d–f where $\bar {v}_{\upbeta } =0.05\;{\rm \upmu M}$ s^− 1

Fig. 11

Bifurcation diagrams of the G-ChI model for FM-encoding sets of parameters. d–i In analogy with Fig. 10, reduced values of $\bar {v}_{\upbeta }$, the maximal rate of PLCβ-dependent IP₃ production, extend to infinity the range of oscillations, leading to phase-locking of Ca^2 +/IP₃ pulsating oscillations. d–f There is also an intermediate range of $\bar {v}_{\upbeta }$ values for which oscillations and fixed concentrations of [Ca^2 +] and [IP₃] can coexist. b, c, e, f, h, i Unlike Ca^2 + oscillations, IP₃ oscillations are always AFM encoding with respect to the concentration of agonist (see also Fig. 10b, c, e, f). Parameters as in Table 1 except for d–f where $\bar {v}_{\rm \beta } =0.2\;{\rm \upmu M s}^{-1}$ and g–i where $\bar {v}_{\rm \beta } =0.05{\rm \upmu M s}^{-1}$

4 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 the bifurcation diagrams in Figs. 10 and 11. We note that the choice of $\bar {v}_{\upbeta }$, 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 $\bar {v}_{\upbeta }$ 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. 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. 11d–f).

As $\bar {v}_{\upbeta }$ 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 Ca^2 + is smaller. Then Ca^2 +-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 Ca^2 + dynamics.

The value of $\bar {v}_{\upbeta }$ at which intracellular Ca^2 + dynamics locks onto stable oscillations also depends on $\bar {v}_{\updelta }$, the strength of the endogen PLCδ-mediated IP₃ production. To some extent, increasing $\bar {v}_{\updelta }$ decreases the minimal $\bar {v}_{\upbeta }$ 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 Ca^2 + dynamics in the G-ChI model might have important implications for the encoding of the stimulus. Bifurcation diagrams in Figs. 10 and 11 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 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 [19]. Here, if we consider the AM-derived bifurcation diagrams for Ca^2 + and IP₃ dynamics (Fig. 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. 10a, b, d, e). But the period of oscillations (Fig. 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, Ca^2 + oscillations become AFM encoding. Notably, IP₃ dynamics appears to be always AFM encoding both in the AM (Fig. 10b, e) and in the FM-derived bifurcation diagrams (Fig. 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 Fig. 11d with Fig. 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, 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 Ca^2 + 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 Ca^2 + oscillations, as a consequence of the numerous Ca^2 +-dependent feedbacks on IP₃ metabolism. Smooth Ca^2 + oscillations such as those obtained in AM-like conditions (Fig. 12a, AM) are coupled with small zigzag IP₃ oscillations (Fig. 12b, AM). Under FM conditions instead, pulsating large-amplitude Ca^2 + variations (Fig. 12a, FM) can be lagged by analogous IP₃ oscillations (Fig. 12b, FM), with the difference that whereas Ca^2 + pulsations are almost fixed in their amplitude, IP₃ ones can substantially vary.

Fig. 12

a, b Examples of forced burst oscillations exhibited by the G-ChI model, under c a square-wave stimulus protocol. This figure illustrates how stationary glutamate stimulations are encoded as oscillations and pulsations of the second messengers Ca^2 + and IP₃. A closer look at oscillatory patterns in a, b reveals that in our model, IP₃ oscillations always lag Ca^2 + oscillations. Indeed, the adoption of the L–R core model for CICR at constant IP₃ concentration implies that IP₃ oscillations are not a prerequisite for Ca^2 + oscillations to occur. Square-wave stimulus: AMγ _min  = 2 nM, γ _max  = 5 μM; FMγ _min  = 1 nM, γ _max  = 6 μM; AM, FM duty cycle, 0.5. Note that in the FM case, the value of γ _max corresponds in the bifurcation diagrams in Fig. 11a, b to a bistable state (a stable fixed point and a stable limit cycle separated by an unstable limit cycle). This explains why pulsations at high stimulations are of limited duration

Simulations of physiologically equivalent glutamate stimulation and associated astrocyte Ca^2 +–IP₃ patterns are shown in Fig. 13. Real multi-array electrode-recording data were considered as inputs of a single glutamatergic synapse (modeled as in Tsodyks and Markram [83]) 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 Ca^2 + 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 Ca^2 + oscillations. Depending on the inherent cellular properties (Fig. 13, for example, considers two cases associated with different SERCA Ca^2 + affinities), the difference of smoothness between IP₃ and Ca^2 + can be dramatic, more likely in the case of FM encoding Ca^2 + pulsations (compare Fig. 13a–b, AM and FM).

Fig. 13

Simulated Ca^2 + and IP₃ patterns obtained when the G-ChI model is fed with physiologically realistic glutamate stimulations, in the AM and FM case. A striking feature is a remarkable increase of the signal smoothness, when one goes from glutamate stimulus to IP₃ traces (b, c) and from the latter to Ca^2 + traces (a, b). This fact suggests different integrative properties for IP₃ and Ca^2 +, which are likely to be cross-coupled (see Section 5), with respect to the stimulus

5 Discussion

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 Ca^2 + dynamics. Here, we have devised and studied a model for agonist-dependent intracellular Ca^2 + dynamics that captures the essential biochemical features of the complex regulatory pathways involved in glutamate-induced IP₃ and Ca^2 + 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 Ca^2 + 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 [48].

We linked the relative expression of these two isoenzymes to the expression of PKC and to the strength of PLCβ regulation by PKC. Indeed, Ca^2 +-dependent PKC activation can phosphorylate the receptor or PLCβ or a combination thereof, leading to termination of IP₃ production [79]. In astrocytes, this mechanism has been suggested to limit the duration of Ca^2 + oscillations, thus defining their frequencies [82]. 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 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 [84–86] and are often associated with pathological conditions [87, 88]. In our model, persistent pulsating Ca^2 + dynamics that are essentially independent of the level of stimulation are observed for weak maximal rates of IP₃ production by PLCβ (Figs. 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 [90].

We note that, although focusing on stimulus-triggered Ca^2 + oscillations, our study also hints, at 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 interpulse interval of the spontaneous Ca^2 + 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 Ca^2 + 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 Ca^2 + 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 Ca^2 + inhibition or the progressive sensitization of IP₃Rs by Ca^2 + [94]. 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 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 [13]. 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 Ca^2 + oscillations/pulsations are inherently FM encoding, that is, they can be either FM or AFM but not AM [18, 19, 98]. In FM, Ca^2 + 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 Ca^2 + 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 [99]. 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 Ca^2 + signals are triggered primarily by sufficiently ample elevations of IP₃ [35], 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 Ca^2 + signaling in several cells [100, 101] including astrocytes [3]. Notwithstanding, the possibility of AFM-encoding Ca^2 + oscillations has recently come up as a reliable alternative mechanism to explain gliotransmitter exocytosis, which is dependent on a specific agonist that triggers astrocyte Ca^2 + dynamics [21, 102].

The above could be relevant to understanding the origin of the integrative properties of Ca^2 + signaling in astrocytes [103]. 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 Ca^2 + 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. 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.

Appendices

Appendix 1

For the sake of simplicity, we have adopted throughout the text the following notation for the generic Hill function:

$$ {\rm Hill}\!\left( {x^n,K} \right)\equiv \frac{x^n}{x^n+K^n} $$

where n is the Hill coefficient and K is the midpoint of the Hill function, namely the value of x at which Hill$\left. {\left( {x^n,K} \right)} \right|_{x=K} =1 \mathord{/\, {\vphantom {1 2}} \kern-\nulldelimiterspace} 2$.

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:

$$ {\rm Hill}\left( {x^n,K_1 } \right)\cdot {\rm Hill}\!\left( {x^n,K_2 } \right)\approx {\rm Hill}\!\left( {x^n,K_2 } \right) $$

if and only if K ₁ < < K ₂ (Fig. 1, Online Supplementary Material). Indeed, under such conditions, Hill$\left( {x^n,K_1 } \right) \cdot {\rm Hill}\!\left( {x^n,K_2 } \right)>\!> 0$ only when x > > K ₁, hence

$$\begin{array}{lll} {\rm Hill}\!\left( {x^n,K_1 } \right) \cdot {\rm Hill}\!\left( {x^n,K_2 } \right)&=&\frac{x^{2n}}{x^{2n}+\left( {K_1^n +K_2^n } \right)x^n+K_1^n K_2^n }\\ &\approx& \frac{x^{2n}}{x^{2n}+K_2^n x^n}={\rm Hill}\!\left( {x^n,K_2 } \right). \end{array}$$

This result can be extended to the product of N Hill functions, that is:

$$ \prod\limits_{i=1}^N {{\rm Hill}\!\left( {x^n,K_i } \right)} \approx {\rm Hill}\!\left( {x^n,\max \left( {K_1 ,\ldots ,K_N } \right)} \right) $$

provided that K ₁ < < K ₂ < <...< < K _N .

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 ${\rm Hill}\!\left( {\left( {{\rm Hill}\!\left( {x^n,K_1 } \right)} \right)^m,K_2 } \right)$ or ${\rm Hill}\!\left( {x^m,K_1 \cdot {\rm Hill}\!\left( {x^n,K_2 } \right)} \right)$ (see Fig. 2 in Online Supplementary Material).

Appendix 2

We seek an expression for [CaMKII*] based on the following kinetic reaction scheme:

$$ \label{eq21} 4{\rm Ca}^{2+}+{\rm CaM}\mathop\rightleftarrows\limits^{k_b}_{k_u} {\rm CaM}^+ $$

(21)

$$ \label{eq22} {\rm KII}+{\rm CaM}^+\mathop\rightleftarrows\limits^{k_1}_{k_{-1}}{\rm CaMKII}\mathop\rightleftarrows\limits^{k_2}_{k_{-2}}{\rm CaMKII}\mbox{*} . $$

(22)

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:

$$ \label{eq23} \left[ \mbox{CaMKII*} \right]\approx \frac{k_2 }{k_{-2} }\left[ \mbox{CaMKII} \right]. $$

(23)

Then, under the hypothesis of quasisteady state for CaMKII, we can write:

$$ \label{eq24} \frac{d}{dt}\left[ {{\rm CaMKII}} \right]=k_1 \left[ {{\rm KII}} \right][ {{\rm CaM}^+} ]-\left( {k_{-1} +k_2 } \right)\left[ {{\rm CaMKII}} \right]+k_{-2} \left[ {\mbox{CaMKII*}} \right]\approx 0. $$

(24)

It follows that incorporation of (23) into (24) leads to:

$$ \label{eq25} \left[ \mbox{{CaMKII}* } \right]=K_1 K_2 \left[ {{\rm KII}} \right][ {{\rm CaM}^+} ] $$

(25)

where $K_i ={k_i } \mathord{/{\kern1pt} {\vphantom {{k_i } {k_{-i} }}} \kern-\nulldelimiterspace} {k_{-i} }$. Defining $\left[ {{\rm KII}} \right]_{\rm T} =\left[ {{\rm KII}} \right]+\left[ {{\rm CaMKII}} \right]+\left[ {\mbox{CaMKII*}} \right]$ as the total kinase II concentration and assuming it constant, we can rewrite (25) as follows:

$$ \label{eq26} \left[ \mbox{{CaMKII}* } \right]=\frac{K_2 \left[ {{\rm KII}} \right]_{\rm T} }{1+K_2 }\frac{[ {{\rm CaM}^+} ]}{[ {{\rm CaM}^+} ]+K_m } $$

(26)

with $K_m =\left( {K_1 \left( {K_2 +1} \right)} \right)^{-1}$.

The substrate concentration for the enzymatic reaction (22) is provided by reaction (21) according to which:

$$ \label{eq27} [ {{\rm CaM}^+} ]=\left[ {{\rm CaM}} \right]\frac{[ {{\rm Ca}^{2+}} ]^4}{[ {{\rm Ca}^{2+}} ]^4+K_{\rm d} } $$

(27)

with $K_{\rm d} ={k_u } \mathord{/{\kern1pt} {\vphantom {{k_u } {k_b }}} \kern-\nulldelimiterspace} {k_b }$. Therefore, substituting (27) into (26), we obtain:

$$ \label{eq28} \left[\mbox{{CaMKII}*}\right]=\frac{K_2 \left[ {{\rm KII}} \right]_{\rm T} }{1+K_2 }\left( {1+\frac{K_{\rm m} }{\left[ {{\rm CaM}} \right]}} \right)^{-1}\frac{[ {{\rm Ca}^{2+}} ]^4}{[ {{\rm Ca}^{{\rm 2}+}} ]^4+\frac{K_{\rm m} K_{\rm d} }{K_{\rm m} \,+\, \left[ {{\rm CaM}} \right]}} $$

(28)

so that $\left[\mbox{{CaMKII}*}\right]\propto {\rm Hill}\left( {[ {{\rm Ca}^{2+}} ]^4,K_{\rm D} } \right)$ with $K_{\rm D} =\left( {\frac{K_{\rm m} K_{\rm d} }{K_{\rm m} \,+\,\left[ {\rm CaM} \right]}} \right)^{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}$.