Pioneering models for intracellular Ca²⁺ signaling include the Goldbeter et al. model (Santello and Volterra, 2009), which predicts the occurrence of periodic spikes of the ion in the absence of IP₃ oscillations, indicating that repetitive Ca²⁺ spikes do not necessarily require a concomitant periodic change in IP₃ and can be induced by external stimulation. The model assumes the existence of two distinct internal stores, one sensitive to IP₃ and the other sensitive to Ca²⁺. The IP₃ produced by agonist stimulation leads to a release of Ca²⁺ from the IP₃-sensitive store via the IP₃Rs. The released Ca²⁺ will stimulate a further release from the Ca²⁺ sensitive store (see Figure 1), which self-amplifies above a threshold value for cytosolic Ca²⁺ concentration (C), representing a model for Induced Calcium Release (CIRC). Depletion of the Ca²⁺-sensitive pool (C_ER) limits the release. This model makes the critical assumption that the Ca²⁺ in the IP₃-sensitive store remains constant as the extracellular medium rapidly replenishes it. The model lacks a mechanism for IP₃-dependent Ca²⁺ inhibition. The two variables in the model are the concentration of free Ca²⁺ in the cytosol and in the IP3-insensitive repository (e.g., the endoplasmic reticulum or sarcoplasmic reticulum); these variables are denoted Z and Y, respectively. Assuming that buffering is linear with respect to Ca²⁺ concentration, the time evolution of the systems is governed by the two kinetic equations:

In Equation 1, the ν₀ parameter, which is assumed take constant, relates to the Ca²⁺ input from the extracellular medium into the cell; k_Z, which is assumed to be linear, pertains the outflow of Ca²⁺ into outflow from the cell, which occurs even in the absence of external stimulation. ν₁β denotes the InsPs-modulated release of Ca²⁺; ν₂ indicates the rate of ATP-driven pumping of Ca²⁺ from the cytosol into the InsP₃-insensitive store, while ν₃ represents the rate of transport from this pool into the cytosol; finally, the term k_fY refers to a nonactivated transport of C into C_ER.

When the cell receives an external signal, this triggers an increase in InsP₃, which leads to a rise in the saturation function β and, subsequently, to an increase in cytosolic Ca²⁺.

Were V_M2 and V_M3 denote, respectively, the maximum rates of Ca²⁺ pumping into and release from the intracellular store; these processes are described by Hill functions whose cooperativity coefficients are taken as n and m; p denotes the degree of cooperativity of the activation process; K₂, K_R, and K_A are threshold constants for pumping, release, and activation.

The Goldbeter model assumes that two different types of pools are required for Ca²⁺ oscillations, some of which are sensitive to InsP₃ and others with RyR and thus sensitive to Ca²⁺. Due to the InsP₃R’s inherent sensitivity to both Ca²⁺ and InsP₃, this proved unneeded. Subsequently, Dupont and Goldbeter formulated a version of the model that assumes the existence of a single pool in which Ca²⁺ and IP₃ are co-agonists for the induction of Ca²⁺ release (Dupont and Goldbeter, 1993; Table 1).

In 1992, the De Young-Keizer model (Young et al., 1992) studied the properties of the IP₃ receptor/ Ca²⁺ channel; in particular, it examined the biphasic response of the IP₃ receptor/channel to cytosolic Ca²⁺ and how this could be sufficient to induce Ca²⁺ oscillations. The rate constants in the equations were fitted to the kinetic and equilibrium data and the model successfully reproduced a series of in vivo and in vitro experiments (Berridge and Irvine, 1989; Mouillac et al., 1990; Smrcka et al., 1991; Taylor and Exton, 1987). The model incorporates a positive Ca²⁺ feedback mechanism on IP₃ production by phospholipase-C (PLC). It was noted that this enriches the properties of oscillations and leads to Ca²⁺ oscillations accompanied by IP₃ oscillations (see Figure 2). They created a simplified model of the IP₃ receptor/channel by assuming that Ca²⁺ conduction is mediated by three equivalent, independent subunits, all of which must be in a conducting state before the receptor allows Ca²⁺ to flow. There are three binding sites on each subunit, one for IP₃, one for Ca²⁺ activation and one for Ca²⁺ inactivation. Consequently, each subunit can exist in eight states, with transitions controlled by first and second order rate constants for association and dissociation, respectively. Each state is labeled with S_ijk the first index refers to the IP₃ binding site, the second to the Ca²⁺ activation site and the third to the Ca²⁺ inactivation site; i,j,k take the value 0 or 1 depending on whether the binding site is unoccupied or occluded (see Figures 3, 4).

The 24 not-all-independent speed constants of the model were reduced to 10 constants, k ± 1 , … , k ± 5 by introducing the following two assumptions:

Since experimental data indicate that the receptor subunits act cooperatively, for the channel to be open and in conduction, all three subunits must be in the S₁₁₀ state (one bound to IP₃ and one to activating Ca²⁺). The gives rise to seven differential equations for the receptor states. Although there are eight states, only seven are independent. As far as mass-action kinetics are concerned, the Ordinary Differential Equations (ODEs) for the receptor states have the present form:

where p denotes [IP₃] and c denotes [Ca²⁺].

The DeYoung and Keizer model consists of seven ODEs for receptor states with the following Equations 5–7 that describing the [Ca²⁺] handling of the IP₃-sensitive Ca²⁺ pool and the IP₃ production:

where c is the cytosolic free Ca²⁺ concentration, J₁ is the outward flux of Ca²⁺ and J₂ is the inward flux (see Figure 2).

J₁ has two components, the Ca²⁺ flux through the IP₃ receptor/channel and a constant leak flux. c₁ is the ratio between the volume of the ER and the volume of the cytosol. c_ER and c are the Ca²⁺ in the ER and cytosolic Calcium, respectively; ν ₁ is the max Ca²⁺ channel flux, ν ₂ is the Ca²⁺ leak flux constant; ν ₃ is the Max Ca²⁺ uptake and K₃ is the Activation constant for ATP-Ca²⁺ pump (Table 2).

In 1993, Atri et al. constructed a minimalist model for Ca²⁺ wave oscillations (Smrcka et al., 1991). The model, which served as the basis for a number of other models, proved simple enough to allow an understanding of the oscillatory phenomena underlying the spatio-temporal properties of Ca²⁺. A single intracellular Ca²⁺ pool that releases Ca²⁺ through the IP₃R is included in the model. It is believed that Ca²⁺ modulates the IP₃R in a biphasic manner, with intermediate Ca²⁺ acting to increase Ca²⁺ release while low and high Ca²⁺ act to block it (see Figure 5). The model takes its cue from Finch et al. (1991), and distinguishes between the time scales of channel activation and inactivation, where inactivation proceeds at a slower rate than activation. This temporal separation is critical for the spatial propagation of the Ca²⁺ signal, as inactivation must occur more gradually than activation to ensure the effective transmission of waves.

The model equation is:

According to Atri et al., there are three binding domains on the IP₃ receptor, the first of which binds IP₃ and the other two bind Ca²⁺; when IP₃ is linked to domain 1 Ca²⁺ is attached to domain 2, but Ca²⁺ is not bound to domain 3, the receptor merely passes the Ca²⁺ current. Consequently, Ca²⁺ binds to domain 2 of the receptor to activate it and to domain 3 to deactivate it. Based on functionality, each binding domain consists of a certain number of binding sites. Assuming domain independence, the steady-state Ca²⁺ flux through the IP₃ receptor, J₁, is given by:

Where in Equation 9 p 1 is the probability that IP₃ is bound to domain 1, p 2 is the probability that Ca²⁺ is bound to domain 2 and 1; p 3 is the probability that Ca²⁺ is bound to domain 3; k f is a constant and represents the maximum total Ca²⁺ influx through the IP₃ receptors.

Thus, if we let c denote [Ca²⁺] can P denote [IP₃] then the following Equations 9–19 result:

Note that the expression of p 3 assumes that Ca²⁺ binds to the inactivating domain in a cooperative manner and while p 1 and p 2 are instantaneous functions of [Ca²⁺] and [IP₃], p 3 acts on a slower time scale, therefore:

The dimensionless variable n represents the proportion of IP₃ that have not been closed by Ca²⁺ and it is described by:

n ∞ c is the steady-state value of n as a function of the intracellular Calcium concentration c,

τ n is the time constant for the dynamics of n (Table 3).

In 1994, Yue-Xian Li and John Rinzel deduced a model that reduces the De Young-Keizer model to a two-variable system to describe Calcium dynamics. This was mainly done by identifying the binding rates involving IP₃ and activating Ca²⁺ molecules as faster rates than the binding rate of deactivating Ca^2+. This made it possible to essentially split the model into two halves, with and without deactivating Ca²⁺ binding. The two dynamic variables of the LR model are the concentration of free cytosolic Ca²⁺ (C) and the fraction of open subunits of the inositol triphosphate receptor (h) (see Figure 6; Li and Rinzel, 1994); this result was obtained by using the method of multiple scales to solve the equations of the De Young-Keizer model on a succession of faster time scales to reduce it to a 2D system:

and

with J1, J2, and J3 given by the equations:

Were J₁ is a release of Ca²⁺, mutually controlled by Ca²⁺ and by IP₃ concentration; J₂ is a passive loss of Ca²⁺ from the endoplasmic reticulum (ER) to the cytosol; and J₃ an active absorption of Ca²⁺ in ER due to the action of the pumps. Again, in Equation 22, h = x 000 + x 100 + x 010 + x 110 is the fraction of channel not yet inactivated by Ca²⁺.

Along with the gating variables:

Therefore, the level of IP₃ is directly controlled by the signals affecting the cell from its external environment. In turn, the level of IP₃ determines the dynamic behavior of the LR model. The Calcium signal can therefore be considered as coded information relating to the level of IP₃ (Table 4).

Most models for Ca²⁺ dynamics are derived from the two-variable models mentioned so far. Since the realization of the pioneering models mentioned above, the intracellular dynamics of Ca²⁺ and IP₃ have been characterized much more comprehensively, and above all, specific and more sophisticated models for intracellular and extracellular Ca²⁺ dynamics in astrocytes have been realized. When astrocytes respond to stimulation, they register a variety of spatiotemporal dynamics of Ca²⁺ elevation, each of which may have its own coding. Understanding the biophysical mechanisms underlying the rich Ca²⁺ dynamics in astrocytes is important because distinct coding patterns may correspond to different downstream signaling, including gliotransmission and consequently control of synaptic function.

More recently, models have also been created for subcellular Ca²⁺ increases linked to metabotropic glutamate receptors (mluRs). Here, the models offer the possibility of establishing a link between the properties of mGluRs and their implication in intracellular Ca²⁺ dynamics.

Glutamate is the most abundant excitatory neurotransmitter in the brain and plays a crucial role in various physiological processes, including learning, memory, and synaptic plasticity.

As demonstrated by electron microscopy the outer surface of the plasma membrane of astrocytes has specific receptors for glutamate. Smith et al. (2014) showed that cultured astrocytes responded to extracellular glutamate with rapid and oscillatory elevations of intracellular free Ca²⁺ concentration (Innocenti et al., 2000; Dani et al., 1992; Charles et al., 1991). In 1994, Mennerick and Zorumski experimentally demonstrated that astrocytes are able to uptake and transport 90% of glutamate from the extracellular space (Mennerick and Zorumski, 1994); Parpura and Haydon subsequently demonstrated that astrocytes modulate neuronal excitability through the release of glutamate linked to physiologically relevant increases in Ca²⁺ (Shao and Mccarthy, 1994; Parpura and Haydon, 2000).

Metabotropic glutamate receptors (mGluRs) are membrane proteins capable of responding to glutamate, the central nervous system’s main excitatory neurotransmitter as a result, they are crucial in the transmission of signals between cells in the nervous system. Research employing in situ hybridization and immunocytochemistry reveals that mGluR3 is the most often expressed mGluR subtype in glia (Smith, 1992; Petralia et al., 1996; Wroblewska et al., 1998). Astrocytes express Group I mGluR subtypes, which includes mGluR1 and 5, reviews can be found in Barres, 1991 (Barres, 1991; Steinhäuser and Gallo, 1996; Parpura et al., 1994; Testa et al., 1995; Miller et al., 1995; Winder and Conn, 1996; Hermans and Challiss, 2001; Verkhratsky et al., 1998; Biber et al., 1999; Cai et al., 2000; Aronica et al., 2003; Perea and Araque, 2007; Araque and Navarrete, 2010; Sun et al., 2013).

It is interesting to note that cell lines that express the mGluR5 receptor are the primary source of concurrent InsP3 and Ca²⁺ oscillations. These glutamate-induced Ca²⁺ oscillations have unusual characteristics, so it is plausible that different oscillatory mechanisms prevail depending on the receptor type (Kummer et al., 2000; Lemon et al., 2003; De Pittà et al., 2009).

When glutamate binds to its membrane receptor, a sequence of events is set off: the ethorotrimeric G-protein, which is named for its three distinct polypeptide subunits, α, β, and γ, interacts with the receptor to create a receptor-G-protein complex on the inner membrane surface. When the α subunit interacts with the receptor, it undergoes a conformational shift that releases the GDP attached to it and replaces it with GTP. This, in turn, activates the phospholipase C-β (PI-PLCβ) that is specific to phosphatidyl-inositol. PI-PLCβ is located on the inner surface of the membrane, linked to the interaction between its PH domain and a PIP₂ molecule immersed in the bilayer. The PI-PLCβ enzyme catalyzes a reaction that cleaves PIP₂ into two molecules, inositol 1,4,5-triphosphate (IP₃) and diglycerol (DAG). The resultant IP₃ molecules diffuse into the cytoplasm and attach to a particular IP₃ receptor found on the smooth endoplasmic reticulum surface (Rosa et al., 2022). DAG stimulates PKC activity, which in turn phosphorylates the mGlu5 receptor at Ser-839. This phosphorylation leads to the uncoupling of the receptor from the G protein signaling cascade.

Modeling studies have not always been conducted in tandem with experimental research on mGlur receptor-mediated Ca²⁺ signaling; although mGlur receptors are highly expressed in the central nervous system (CNS) and have been linked to several pathophysiological processes as well as neuro-psychiatric disorders (Nicoletti et al., 2011; Spooren et al., 2001).

Young et al. (1992), Li and Rinzel (1994), and Höfer et al. (2002) models as a starting point, in De Pittà et al. (2009) constructed a generic model for glutamate-induced Ca²⁺ (Glu) dynamics in astrocytes, including additional biochemical processes relevant for a more realistic description of astrocyte activity. Such extensions include the production and degradation of IP₃ within the astrocyte cell, mediated by two membrane-associated enzymes, PLCβ and PLCδ (see Figure 7). Later, De Pittà and Berry (2019) further refined their model by focusing on the rate of IP₃ production following activation of glutamate receptors mGluRs, building a new model. The De Pittà model for IP₃/Ca²⁺ signaling is constituted by three ODES, respectively, for intracellular Ca²⁺ (C), the IP₃R gating (h), and the mass balance equation for intracellular IP₃ lumping terms. Regarding the differential equations for the variables C and h above, the De Pittà model considers the original Li-Rinzel model formulation described for the CICR and provides a more detailed description of IP₃ production and degradation, proposing a three-variable model for glutamate-induced intracellular Calcium dynamics caused by synaptic activity in astrocytes.

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 (Rebecchi and Pentyala, 2000). 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. PLCδ is the enzyme responsible of endogenous IP₃ production in astrocytes, it is essentially activated by increased intracellular Ca²⁺ levels (Rhee and Bae, 1997). The model proposed for PLCδ -mediated IP₃ production (J_δ) (Equation 29) derived from structural and mutational studies (Höfer et al., 2002; Pawelczyk and Matecki, 1998).

where O_δ is the maximal rate of IP₃ production by PLCδ and K_δ is the inhibition constant of PLCδ activity. According to experiments, PLCδ activity is inhibited by high IP₃ concentrations (> 1 μM) because they compete with PIP₂ for the enzyme’s binding (Allen and Barres, 2009).

In astrocytes there are two several pathways for IP₃ degradation: the dephosphorylation of IP₃ by inositol polyphosphate 5-phosphatase (IP-5P), and the phosphorylation of phosphorylation of IP₃ by the IP₃3-kinase (IP₃-3 K). For the description of the two IP₃ degradation dynamics we use the relations given by Equations 30, 31:

where O_5p is the maximal rate of IP-5P mediated IP₃ degradation in the linear approximation.

For IP₃-3 K degradation we can write:

where O_3k is the maximal rate of IP₃ degradation by IP₃-3 K.

In summary, the De Pittà model of Ca²⁺ dynamics with endogenous IP₃ metabolism (Equation 32) is based on the two LR equations but the IP₃ concentration (I) is now provided by a third coupled differential Equations 20a, 20, 21, 21b, 33.

The model highlights the complex biochemical reactions coupled with Ca²⁺ dynamics via the different second messengers (Table 5).