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S. Schuster, Max Delbrück Centre for Molecular Medicine, Department of Bioinformatics, Robert-Rössle-Str. 10, D-13092 Berlin-Buch, Germany. Fax: + 49 30 94062834, Tel.: + 49 30 94063125, E-mail: firstname.lastname@example.org
This review provides a comparative overview of recent developments in the modelling of cellular calcium oscillations. A large variety of mathematical models have been developed for this wide-spread phenomenon in intra- and intercellular signalling. From these, a general model is extracted that involves six types of concentration variables: inositol 1,4,5-trisphosphate (IP3), cytoplasmic, endoplasmic reticulum and mitochondrial calcium, the occupied binding sites of calcium buffers, and the fraction of active IP3 receptor calcium release channels. Using this framework, the models of calcium oscillations can be classified into ‘minimal’ models containing two variables and ‘extended’ models of three and more variables. Three types of minimal models are identified that are all based on calcium-induced calcium release (CICR), but differ with respect to the mechanisms limiting CICR. Extended models include IP3–calcium cross-coupling, calcium sequestration by mitochondria, the detailed gating kinetics of the IP3 receptor, and the dynamics of G-protein activation. In addition to generating regular oscillations, such models can describe bursting and chaotic calcium dynamics. The earlier hypothesis that information in calcium oscillations is encoded mainly by their frequency is nowadays modified in that some effect is attributed to amplitude encoding or temporal encoding. This point is discussed with reference to the analysis of the local and global bifurcations by which calcium oscillations can arise. Moreover, the question of how calcium binding proteins can sense and transform oscillatory signals is addressed. Recently, potential mechanisms leading to the coordination of oscillations in coupled cells have been investigated by mathematical modelling. For this, the general modelling framework is extended to include cytoplasmic and gap-junctional diffusion of IP3 and calcium, and specific models are compared. Various suggestions concerning the physiological significance of oscillatory behaviour in intra- and intercellular signalling are discussed. The article is concluded with a discussion of obstacles and prospects.
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Many processes in living organisms are oscillatory. Besides quite obvious examples such as the beating of the heart, lung respiration, the sleep-wake rhythm, and the movement of fish tails and bird wings, there are many instances of biological oscillators on a microscopic scale, such as biochemical oscillations, in which glycolytic intermediates, the activities of cell-cycle related enzymes, cAMP or the intracellular concentration of calcium ions exhibit a periodic time behaviour. Calcium oscillations had been known for a long time in periodically contracting muscle cells (e.g. heart cells) and neurons , before they were discovered in the mid-1980s in nonexcitable cells, notably in oocytes upon fertilization  and in hepatocytes subject to hormone stimulation [3,4]. Later, they have also been found in many other animal cells (cf. [5–10]) as well as in plant cells , with many of these cells not having an obvious oscillatory biological function. The oscillation frequency ranges from ≈ 10−3 to ≈1 Hz.
A striking feature of the investigation of calcium oscillations is that almost from its beginning, experiments have been accompanied by mathematical modelling [12–18]. In recent years, much insight has been gained into the processes involved in calcium dynamics at the subcellular, cellular and intercellular levels and, accordingly, the models have become more elaborate and diversified. In particular, bursting oscillations and chaotic behaviour, various types of bifurcations, and the coupling between oscillating cells have been analysed. Moreover, the role of mitochondria as organelles, which are, besides the endoplasmic reticulum (ER), capable of sequestering and releasing calcium, has been studied. These developments are here put into the context of the various simpler models developed previously. Although focussing on the modelling aspect, we will always aim at relating the model assumptions and theoretical conclusions to experimental results.
A scientific model is a simplified representation of an experimental system. It should meet two criteria often contradicting each other: First, it should describe the features of interest as adequately as possible. Second, it should be simple enough to be tractable and interpretable. We believe that, in model construction, guidance should be sought primarily from the experimental data. For example, the occurrence of self-sustained calcium oscillations can be described by relatively simple, ‘minimalist’ models (e.g. the two-variable model by Somogyi & Stucki , and see Cacyt/Caer models, below). However, if, for example, the detailed gating characteristics of the calcium release channel is also to be described, more comprehensive models are needed (e.g. the eight-variable model by De Young & Keizer , and see Detailed kinetics of the Ca2+ release channels section). Of course, the models should be in accord with physico-chemical laws such as the principle of detailed balance.
This review on calcium dynamics is focussed primarily on deterministic models of the temporal behaviour. Spatio-temporal aspects such as calcium waves (cf. ) will be treated in relation to coupled cells (see Coupling of oscillating cells). In the deterministic approach, the mathematical variables are the concentrations of relevant substances and possibly the transmembrane potential; the fluctuations of these variables are neglected. In comparison to stochastic modelling, this approach has the advantage that the mathematical description is simpler. The results derived from deterministic models of calcium oscillations are already in good, and sometimes excellent, agreement with experiment. However, in small volumes, fluctuations may not be negligible. For example, in a cell organelle with a volume of 1 µm3, a free Ca2+ concentration of 200 nm implies the presence of only 120 unbound ions. On the other hand, the binding of Ca2+ ions to proteins brings about that a much larger number of ions are present in total. Thus, it is worth investigating whether fluctuations can be assumed to be buffered under these conditions. Stochastic models have been developed for single Ca2+ channels , intracellular wave propagation [21–25] and intracellular oscillations [26,27].
The deterministic modelling of biological oscillations and rhythms is based on a well-established apparatus to describe self-sustained oscillations in chemistry and physics by nonlinear differential equation systems [28–32]. The same apparatus has been used for the modelling of cell cycle dynamics [33,34], heart contraction and fibrillation , glycolytic oscillations [36,37] and cAMP oscillations .
The models of calcium oscillations are based on a description of the essential fluxes (Fig. 1). The cytoplasmic compartment is linked with the extracellular medium and several intracellular compartments, most notably the ER and mitochondria, through exchange fluxes. In microorganisms, special compartments may exist, such as the acidosomal store in Dictyostelium discoideum. The cascade of events underlying calcium oscillations has often been described (e.g [5,39]). A central process is the release of Ca2+ ions from the ER via channels sensitive to inositol 1,4,5-trisphosphate (IP3), termed IP3 receptors (IP3R) (compare [40–42]). IP3 and diacylglycerol (DAG) are formed from phosphatidyl inositol 4,5-bisphosphate (PIP2) by phosphoinositide-specific phospholipase C (1-phosphatidylinositol-4,5-bisphosphate phosphodiesterase, PLC, EC 184.108.40.206). Different isoforms of (phosphoinositide-specific) PLC are activated by hormone-receptor coupled G-proteins (PLCβ), protein kinases (PLCγ) and calcium (PLCδ) . Another ER calcium release channel, particularly prominent in muscle cells, is the ryanodine receptor (RyR), whose physiological activator appears to be cyclic ADP ribose . Opening of the IP3R, in the presence of IP3, and of the RyR is also stimulated by calcium binding (calcium-induced calcium release, CICR) [39,41,45,46]. Several isoforms of both receptors have also been shown to be inhibited by high calcium concentrations . (As for oocytes, the signalling pathway via IP3 is subject to debate [8,47].) Additionally, many other processes may play a role in the signalling cascade in various cell processes, such as activation of protein kinase C (PKC) by DAG and calcium (cf. [41,48]), phosphorylation of the IP3R by PKC (cf. ), ‘cross-talk’ of the G-protein with this kinase [49,50] and the contribution of the RyR activated by cyclic ADP ribose [44,51].
The steep calcium gradient across the ER membrane is sustained by active pumping through the sarcoplasmic reticulum/ER calcium ATPase (SERCA, EC 220.127.116.11). In hepatocytes, for example, the baseline concentration in the cytosol is about 0.2 µm and rises to about 0.5–1 µm during spikes, while the level in the ER is about 0.5 mm. A similarly high gradient exists across the cell membrane. Various entrance pathways, chiefly calcium store-operated [42,52] and receptor-operated , have been described. Ca2+ ions are also bound to many substances such as proteins, phospholipids and other phosphate compounds.
For these various reactions and transport processes, flux balance equations can be formulated. Throughout the paper, italic symbols of substances will be used for concentrations while Roman symbols stand for the substances themselves. The general balance equations for the variables of Fig. 1, the concentrations of IP3 (IP3), cytoplasmic calcium (Cacyt), ER calcium (Caer), mitochondrial calcium (Cam), and occupied calcium binding sites of the buffer species j in the cytosol (Bj) are:
where ρer and ρmit are the cytosol/ER and cytosol/mitochondria volume ratios and the rate expressions have the same meaning as in the legend of Fig. 1. Equations similar to Eqn (5) can also be written for the buffers in the ER and mitochondria. Furthermore, the transitions between different states of the IP3R can play a role in IP3-evoked calcium oscillations [18,54–57]. Of particular relevance is the desensitization of the IP3R induced by calcium binding, which can be expressed by the following balance equation
Ra denotes the fraction of receptors in the sensitized state; vdes and vrec stand for the rates of receptor desensitization and recovery, respectively.
Moreover, several models include, as a variable, the cell membrane potential [58–60]. This may be of importance when calcium oscillations and action potential oscillations interact. However, we restrict this review to the core mechanisms of cytoplasmic calcium oscillations that apply both to electrically nonexcitable and excitable cells.
Most models of calcium oscillations fit into the general system of balance equations (Eqns 1–6). To our knowledge, no model that includes all of the six equations has so far been published, although various combinations of processes have been used. In the Minimal models section, we discuss all classes of minimalist models involving two out of the six variables entering Eqns (1–6) suggested up to now. The section Higher-dimensional models is devoted to more complex models involving three or four out of the six variables mentioned above or additional variables such as the various states of the IP3R or the concentration of active subunits of the G-protein. The overview of models given in Minimal models and Higher-dimensional models updates and corrects the classification given previously .
Different experimental results were obtained concerning the question whether Ca2+ outside the cells is necessary for the maintenance of oscillations. Removal of external Ca2+ leads to a cessation of oscillations in most cases in endodermal cells  and HeLa cells . In other cell types, such as salivary gland cells, external Ca2+ is not required . For hepatocytes, Woods et al.  found that external Ca2+ was necessary for oscillations while others found that it was not [66,67] or that inhibition of the plasma membrane Ca2+ pump does not prevent oscillations . If oscillations occur in the absence of external Ca2+, they are usually slower and eventually fade away (cf. ).
It has often been argued that in calcium oscillations, information is encoded mainly by their frequency [5,12,70–72]. However, a possible role of amplitudes in signal transduction by calcium oscillations has also been discussed [73–75]. Frequency and amplitude encoding will be reviewed in Frequency encoding, based on an analysis of the local and global bifurcations by which calcium oscillations arise (subsections Hopf bifurcations and Global bifurcations). The models addressing the questions of how the oscillatory calcium signal is transformed into a nearly stationary output signal and how the target proteins sense the varying frequency are reviewed in the subsection entitled Modelling of protein phosphorylation driven by calcium oscillations. In the subsection Chaos and bursting, complex temporal phenomena will be discussed. Coupling of oscillating cells allows intercellular communication based on calcium signals, as described in the relevant section below. In the Conclusion, we will review the suggestions concerning the possible physiological significance of oscillatory calcium dynamics in comparison with adjustable stationary levels. Moreover, we will discuss some obstacles and give an outlook on the further development of the field. In particular, we will suggest a possible ‘networking’ of different modelling approaches in biochemistry. Mathematical fundamentals necessary for the review are outlined in the Appendix.
To simulate self-sustained oscillations by a system of kinetic equations, at least two variables are needed (see Appendix). The free cytosolic calcium concentration should be taken as a dynamic variable, because this is the quantity most frequently measured. The only model not including Cacyt as a dynamic variable published so far is a simplified, two-variable version of a model involving the G-protein . Cacyt can then be calculated by an algebraic equation (based on quasi-steady-state arguments) from IP3. In our opinion, this model is not sufficiently supported by experimental data. Experiments show that changes in the activity of the SERCA [77,78] and in receptor-activated calcium influx  affect the frequency and spike width of Ca2+ oscillations, thus arguing for a participation of Ca2+ in the mechanism of oscillations.
Five minimal, two-variable systems including Cacyt can be conceived from the basic equations (Eqns 1–6), three of which have indeed been studied in the literature (Table 1). Models that include the remaining combinations exist, but are not minimal because they involve also additional variables (see subsections Consideration of the IP3 dynamics and Inclusion of mitochondria). The following three subsections discuss each class of two-variable models in turn, referred to by the names of the variables involved: Cacyt/Caer, Cacyt/IP3R, and Cacyt/protein.
Table 1. Rate laws for three types of minimal models of Ca2+ oscillations. In each case, the positive feedback is provided by CICR.
Cytoplasmic and ER Ca2+ (Cacyt, Caer)
Cytoplasmic Ca2+, active IP3R (Cacyt, Ra)
Cytoplasmic Ca2+ and Ca2+ buffer (Cacyt, B)
a In the original model, an effect of the ER membrane potential was included in vrel; however, oscillations are also obtained with the simpler expression given (cf. [115,162]).
Dupont & Goldbeter 
Li & Rinzel 
Marhl et al.  a
Ca2+ exchange with extracellular medium
Ca2+ binding to proteins
Total cellular Ca2+
v0 + v1β
k3 (1 − Ra)
k+(B0 − B) Cacyt − k_B
Related 2D models
To construct a kinetic model, in the balance equations the dependencies of the flux rates on the model variables must be specified (rate laws). For one representative of each model class, rate laws are given in Table 1, together with references to related models. Although all of these models are minimal in the sense of containing two dynamic variables, there are considerable differences with respect to the complexity of the rate laws. This will be explicitly discussed for the Cacyt/Caer models below.
The analysis of two-dimensional models shows that self-sustained oscillations can only occur if one of the model variables exerts an activatory effect on itself (autocatalysis, feedback activation; see Appendix). A prominent feedback loop is CICR exhibited both by RyR and IP3R Ca2+ release channels. Indeed, all three types of minimal models involve CICR. By contrast, a putative activation of Ca2+ release by Caer would not suffice to generate oscillations.
A model for self-sustained Ca2+ oscillations that is not only minimal with respect to the number of variables but also very simple with respect to the rate laws is the `one-pool model' proposed by Somogyi and Stucki . As shown by Dupont & Goldbeter , it can be derived by simplifying a ‘two-pool model’, in which IP3-sensitive and IP3-insensitive stores were considered [14,15,81]. Interestingly, recent findings show that in Dictyostelium discoideum, indeed both IP3-sensitive and IP3-insensitive stores exist .
The following processes are included in the one-pool model (Fig. 1): vin, vout, vrel, and vserca. IP3 plays the role of a parameter entering the rate expression of vrel and can be set to different values, according to the level of agonist stimulation. We shall discuss the Somogyi–Stucki model here in some detail by way of example, because several interesting features can be seen relatively easily from it. The influx into the cell is assumed to be constant. The transport of Ca2+ both out of the cell and into the store is modelled by functions linear in the cytosolic Ca2+ concentration, kiCacyt. The only nonlinear function is that for the channel flux of Ca2+ from the intracellular store. Together with a leak through the ER membrane (or a background conductance of the channel), this reads:
The rate function in Eqn (7) is a simple description of the cooperative behaviour found in CICR (and represents a higher nonlinearity than simple mass action kinetics, kCaerCacyt; see Appendix). In principle, however, one could simplify the model by using a function quadratic in Cacyt, in which case the model would coincide with the Brusselator . The system even oscillates if the kinetics of vrel is a product of two Michaelis–Menten terms for Cacyt and Caer, and also vserca obeys a Michaelis–Menten kinetics for Cacyt. In many models [80,83], Caer enters the rate laws for vrel and vserca through a Hill equation with Hill coefficient two (see Table 1). Friel  proposed a model for neurons that is similar to the Somogyi–Stucki model , yet with a somewhat more realistic rare law for vrel in that Caer in Eqn (7) was replaced by (Caer–Cacyt) because the release flux is driven by the Ca2+ gradient. Moreover, smaller values for the Hill coefficient were used.
For a mathematical analysis of the one-pool model [17,80], it is convenient to sum up the two differential equations, giving
Thus, in any steady state of the system, we have the unique solution:
The stationary value of Caer in turn is a unique function of Cacyt. Therefore this model allows exactly one stationary state.
Roughly speaking, the cause for the oscillation is an overshoot phenomenon due to the nonlinearity of CICR. Upon opening of the IP3R, Caer is released. However, Cacyt cannot remain permanently elevated by this flux, cf. Eqn (9). During release, Caer and therefore also the driving force for the release flux decrease. At some instant, Ca2+ extrusion from the cell and Ca2+ pumping into the ER overtake release and thus Cacyt declines. Upon continued stimulation, the process could repeat, giving rise to oscillations. It is an important feature of this model that the total free Ca2+ concentration in the cell, Cacyt + Caer/ρer, oscillates in the course of Cacyt oscillations. From this, one can conclude that the essential mechanism counteracting the autocatalytic release is the subsequent depletion of the total Ca2+ in the cell. Note that complete depletion of the calcium stores is not required for this mechanism to work (cf. ).
To determine the exact requirements for oscillations, intuition is, however, insufficient and we do need modelling. To establish these requirements, a stability analysis is instrumental (see Appendix). A major advantage of the simplicity of the model equations is that the stability calculations can be performed analytically [17,86]. The parameter range in which the steady state is an unstable focus can be determined. In this parameter range, the oscillations can easily be found by numerical integration of the differential equations. The dynamics of Cacyt exhibits the repetitive spikes found in experiment.
A biologically relevant bifurcation parameter is the rate constant of the channel, kch, because it increases upon hormone stimulation of the cell mediated by IP3. For low values of kch, the steady state is stable. As it increases, a point is reached where stable limit cycles occur. When kch is increased even further, the oscillations eventually vanish and the steady state becomes stable again. (For a discussion of the bifurcations in this model, see Hopf bifurcations.) From Eqn (9), it can be seen that the steady-state concentration Cacyt does not depend on the rate constant of the channel. This appears to be in disagreement with experimental observations showing that at very high hormone stimulation, elevated stationary Cacyt levels occur [17,66,87]. It has been reported for some cell types that hormone stimulation, besides causing IP3 synthesis, also leads to activation of Ca2+ entry into the cell. This can be mediated by store-operated [42,52] and receptor-operated  calcium entry. Dupont & Goldbeter  modelled the latter effect by including, in the influx rate, a function expressing the occupancy of the cell membrane receptor with hormone, so that the steady-state concentration Cacyt is indeed increased. This has recently been followed up . The other possible mechanism involves Ca2+ entry from the external medium into the cytosol stimulated by emptying of the Ca2+ stores [52,88]. However, the mechanism for this phenomenon, called ‘capacitative Ca2+ entry’, via a Ca2+ release-activated current (CRAC) is not yet clear .
In the light of the reasoning about minimal models given in the Introduction, it is of interest to investigate whether the one-pool model may be simplified further. Neglecting particular fluxes would perturb the Ca2+ balance. In particular, neglecting the influx into the cell is interesting in view of experiments where external Ca2+ was removed (see Introduction). If both influx and efflux were completely disregarded in the model, the total amount of calcium in the cell would be conserved: Caer/ρer + Cacyt = constant. Thus, the equation system would effectively be one-dimensional, unless additional dynamic variables are included, such as the open probability of the channel  or the Ca2+ level in an intermediate domain near the mouth of the channel .
The flux through the ER membrane channel is pivotal due to its autocatalytic nature. Interestingly, although the leak seems to be negligible in comparison to the CICR flux, it is not. A bifurcation analysis (cf. Frequency and amplitude behaviour) shows that if the leak rate is set equal to zero, the model can indeed give rise to oscillations. However, there is no parameter range with small values of the rate constant of the channel for which a steady state is obtained . This is in disagreement with experiment, because for very low agonist stimulation, no oscillations were found [3,4,17,66]. In conclusion, the one-pool model cannot be simplified any further.
For subtypes I and II of the IP3R, the dependence of vrel on Cacyt is more complex than is expressed by Eqn (7) in that at higher values of Cacyt, this rate decreases . This does not principally alter the behaviour of Cacyt/Caer models [83,92].
Cacyt/IP3 receptor models
Experimental studies on the IP3R indicate that the inhibition of this receptor by Cacyt can play a role in the generation of oscillations if it occurs on a time-scale of seconds compatible with the time-scale of the oscillations while the activation is much faster [55,93,94]. In the Cacyt/IP3 receptor models, spikes terminate because the IP3R is inhibited at high Cacyt and remains inhibited for some time so that the released Ca2+ can be transported back into the ER. Thus, the mechanisms causing the oscillatory behaviour are localized in or near the ER membrane. In contrast to the Cacyt/Caer models, the Cacyt/IP3R models work without (as well as with) Ca2+ exchange across the plasma membrane. Two hypotheses have been put forward (see Detailed kinetics of the Ca2+ release channels): (a) transition of the receptor into an inactive conformation upon Ca2+ binding [56,93,95,96]; (b) inactivation of the receptor by phosphorylation .
The first of these possibilities was studied in two-dimensional models [97–99] with Cacyt, Eqn (2), and Ra, Eqn (6), being the model variables. As in several other Cacyt/IP3R models Eqn (6) was specified to have the form:
motivated by analogy to the Hodgkin-Huxley model of nerve excitation [100,101]. Eqn (10) can be interpreted as a relaxation to the steady state with time constant 1/k. , the steady-state fraction of receptors in the sensitized states, is a decreasing function of Cacyt. In the models of Poledna [97,98] and Atri et al., this function was chosen to be K/(Cacyt + K), and , respectively, where K denotes the equilibrium constant of Ca2+ binding. Note that Ra is not the fraction of open receptor subunits per se but of the subunit form that can be in the open state if Ca2+ is bound at an activating binding site. The essential positive feedback is again provided by CICR modelled by a Hill equation in the kinetics of Cacyt.
A more mechanistic, eight-dimensional model was developed by De Young & Keizer  (see also Detailed kinetics of the Ca2+ release channels). This model was simplified, by using time scale arguments, to two-dimensional models [89,102,103]. For the model by Li & Rinzel , the specific form of the rate law entering Eqn (10) as well as the other rate laws are given in Table 1. Also the Cacyt/IP3R models obtained by simplification of larger models have a structure reminiscent of the Hodgkin–Huxley models. Accordingly, the Ca2+ dynamics can be interpreted as an ER membrane-associated excitability [89,104], so that the term nonexcitable cells often used for hepatocytes, oocytes and other cells exhibiting Ca2+ oscillations appears no longer to be appropriate. Moreover, Li & Rinzel  also considered a three-dimensional system, in which the Ca2+ exchange across the plasma membrane is taken into account.
Cacyt /protein models
In addition to the sensing of the calcium signal (see Modelling of protein phosphorylation driven by calcium oscillations), Ca2+-binding proteins can exert a feedback on the process of Ca2+ oscillations itself. Provided that (a) Ca2+ binding to proteins is very fast, and (b) the dissociation constant is well above the prevailing (free) Cacyt, the overall effect of such buffers is an increase in the effective compartmental volume. In several models, a rapid-equilibrium approximation for Ca2+ binding to proteins is used [105–108], which only requires condition (a) to be fulfilled. For example, Wagner & Keizer  modified the Cacyt/IP3R model of Li & Rinzel . However, the rapid-equilibrium approximation is not always justified [109,110]. Accordingly, several mathematical models [71,106,107,111–115] include the dynamics of Ca2+ binding to proteins, showing that the cytosolic proteins can be essential components of the oscillatory mechanism and can play an important role in frequency and amplitude regulation. We have shown earlier by mathematical modelling that, in the presence of Ca2+-binding proteins, Ca2+ oscillations can arise even in the absence of an exchange across the plasma membrane and of an intrinsic dynamics of the IP3R . In Cacyt/protein models, the role of alternating supply and withdrawal of Ca2+ is played by the fluxes of the dissociation and binding of Ca2+ to and from binding sites.
Ca2+-binding proteins (as well as Ca2+-binding phospholipids) show a wide range of values of the binding and dissociation rate constants [109,110,116]. Roughly, two types of proteins can be distinguished [116–119]. The first class represents the so-called buffering proteins (also known as ‘storage’ proteins) such as parvalbumin, calbindin, and also C-terminal domains of calmodulin or troponin C, which bind calcium relatively slowly but with a high affinity [109,116]. The second class, which is referred to as the signalling proteins (also known as ‘regulatory’ proteins) comprises binding sites that have very high rate constants of binding and dissociation with respect to calcium, but low affinity. Examples are provided by the N-terminal domains of calmodulin or troponin C. Some of these signalling proteins interact with proteins (e.g. CaM kinase II) that transfer the calcium signal by phosphorylating other proteins (see Modelling of protein phosphorylation driven by calcium oscillations). The interplay between buffering and signalling proteins has been examined by modelling studies, using the rapid-equilibrium approximation only for the signalling proteins [71,114,120]. A transfer of Ca2+ from the rapid, low affinity, to the slow, high affinity, binding sites, has been mimicked. This is in agreement with observations both in Ca2+ oscillations and Ca2+ transients, even within one protein molecule as in the case of calmodulin. In skeletal muscle, for example, the Ca2+ released into the cytosol first binds to troponin C and, after a brief lag phase, the bound Ca2+ population shifts to parvalbumin [116,121]. There, the buffering proteins have the function of terminating the Ca2+ transients evoking muscle contraction. Likewise, this mechanism may play a role in the termination of spikes in oscillations.
In the Cacyt/protein models, the positive feedback necessary for two-dimensional models to generate limit cycles is provided again by CICR. Additional nonlinearities enter the model by the consideration of the transmembrane potential across the ER membrane. While in the model of Jafri et al., the transmembrane potential is considered as a dynamic variable, so that the model is three-dimensional (an extended model  including the cytosolic counterion concentration is even four-dimensional), the quasi-electroneutrality condition has been used in [71,113,114] to express this variable into the others. The models (directly or indirectly) including the ER transmembrane potential give slightly asymmetric spikes where the upstroke is somewhat faster than the decrease. During the upstroke, the potential is depolarized, which implies that the driving force of the Ca2+ efflux from the store is diminished both by the decreasing Ca2+ gradient and the decreasing electric gradient.
It should be noted that the magnitude of the ER transmembrane potential is not well known. Because of the high permeability of the ER membrane for monovalent ions it has often been argued that the potential gradient due to Ca2+ transport is rapidly dissipated by passive ion fluxes [104,121–123]. An opposing view is that the highly permeant ions directly follow the potential without depleting it, as described by the Nernst equation. An interesting model prediction is that the value of the potential depends on the effective volume of the ER accessible to Ca2+.
Consideration of the IP3 dynamics
In the Cacyt/Caer models, the IP3 concentration is considered as a parameter which can be set equal to different, fixed values. This approach is supported by findings showing that IP3 oscillations are not required for Ca2+ oscillations . However, a coupling between oscillations in IP3 and oscillations in Cacyt seem to be of importance in some cell types [16,72,76,125–127]. Mechanisms for this coupling are the activating effect of Cacyt on the δ isoform of PLC [43,63] and on the IP3 3-kinase (EC 18.104.22.168) , and Cacyt feedback on the agonist receptor .
This inspired the idea of the IP3–Ca2+ cross-coupling (ICC) models, in which a stimulatory effect of Cacyt on the activity of PLC [12,13,18] or on the consumption of IP3[130,131] are taken into account, in addition to IP3 induced Ca2+ release. IP3 is a system variable in these models and oscillates with the same frequency as Cacyt. Meyer & Stryer  first studied a model in which, in addition to IP3, only two Ca2+ pools are considered: Cacyt and Caer. As these are then linked by a conservation relation (Cacyt + Caer = constant), the model is two-dimensional. It gives rise to bistability rather than oscillations, which is understandable because the cross-coupling between IP3 and Cacyt does not fulfil the condition that the trace of the Jacobian be positive (see Appendix). Next, Meyer & Stryer  included a Ca2+ exchange between cytosol and mitochondria. As the conservation relation now includes Cam, the system is three-dimensional, even though Cam does not occur explicitly as a variable because the efflux out of the mitochondria is assumed to be constant. In three-dimensional systems, the trace of the Jacobian need not be positive in order to obtain oscillations (in fact, at the Hopf bifurcation, it must be negative, cf. ). Thus, violation of the conservation relation Cacyt + Caer = constant is not an error, as assumed previously , but a prerequisite for the ICC models to generate oscillations. In a later version of the model, Meyer & Stryer  proposed to consider, as a third independent variable, a parameter describing the inhibition of the IP3R by Cacyt and did not include mitochondria.
Another combination of variables was chosen by De Young & Keizer . The PLC is again assumed to be activated by Cacyt. A model for Ca2+ waves with the same set of variables but a simpler IP3 dynamics was presented in . The model of Swillens & Mercan  involves, as a variable, the level of IP4 (which is formed from IP3 by phosphorylation) (see Table 2). In order that this model generates oscillations, these authors included, in addition to the effects mentioned above, an inhibition of vrel by Caer, an assumption which has not been followed up in later models. In the model of Dupont & Erneux , the desensitized receptor is included as a fourth variable. As it involves CICR and receptor desensitization, the IP3–Ca2+ cross-coupling is here not necessary for the generation of Ca2+ oscillations.
Table 2. Overview of some three-dimensional models of Ca2+ oscillations.
a Using the conservation relation Cacyt + Caer/ρer + Cam/ρmit = const. bUsing the conservation relation Cacyt + Caer/ρer = const. cUsing the conservation relations Cacyt + Caer/ρer + Cam/ρmit + B = const. and
In a three-dimensional model , the G-protein is explicitly considered as an important part in the signalling pathway from the agonist to IP3 formation via PLCβ. The conversion of G-proteins to their active form is described by a separate differential equation, with DAG (which is set equal to IP3) and Cacyt being the other variables. (In a follow-up model , which was also studied in , active PLC was included as a fourth variable.) A direct effect of Cacyt on PLC is not considered. Rather, the model includes an inactivation of G-protein via PKC, activation of PKC by Cacyt and a putative positive effect of IP3 (or DAG) on PLC. In principle, the latter feedback can be used for constructing a two-dimensional model without CICR . However, so far there is no experimental evidence for this mechanism.
Detailed kinetics of the Ca2+ release channels
As introduced above, one class of models centre on the dynamics of the IP3R. Different states of this receptor (e.g. two states , five states , eight states  or 125 states ) are distinguished according to the binding of Ca2+ and/or IP3, and the occupancies of the various states are taken as dynamic variables. The transitions between the states are modelled by mass-action kinetics. In most of these models, Ca2+ exchange across the plasma membrane is not considered. The models lead to Ca2+ oscillations at fixed IP3 concentration. As a comprehensive overview of these models has been given , we will review them here only briefly.
The functional IP3R consists of four identical subunits [41,133]. Each subunit appears to be endowed with at least one IP3 binding site and at least one Ca2+ binding site. To explain the biphasic effect of Cacyt, various hypotheses have been put forward. The most commonly shared view is that two Ca2+ binding sites exist, with one of these being activating and the other being inhibitory [18,54,99,134]. In the case of independent subunits, this gives rise to seven (23−1 = 7) independent differential equations for the fractions of the receptor subunit states. The eighth variable is Cacyt. In the kinetic model of the IP3R proposed by De Young and Keizer , it is assumed that the ligands can bind to any unoccupied site on the receptor irrespective of the binding status of other sites. In the model of Othmer and Tang , a sequential binding scheme is proposed: IP3 has to bind at the IP3 site before Ca2+ can bind to the channel, and Ca2+ has to bind to the positive regulatory site before it can bind to the inhibitory site. All of these models reproduce the result that the steady-state fraction of open channels vs. log(Cacyt) is a bell-shaped curve.
A difficulty in the detailed models of the IP3R is the uncertainty about the values of the rate constants for the transitions between receptor states. The more different receptor states are considered, the more redundant is of course the parameter identification problem. This is a further motivation, besides the reduction of model dimension, for simplifying the models by the rapid-equilibrium approximation, leading to the models discussed above (cf. ). This simplification is feasible if Ca2+ binding to the positive regulatory site is a fast process compared with that of binding to the inhibitory site.
The dual effect of Cacyt and IP3 on the IP3R can be considered as an allosteric effect. Along these lines, an alternative approach to describing the kinetics of the IP3R, based on the Monod model of cooperative, allosteric enzymes was presented . This model is again able to mimic the bell-shaped curve of the dependence of Ca2+ release from the vesicular compartments on Cacyt, whereas the IP3 binding process itself is not cooperative. The model is less complicated than the De Young–Keizer model  (in which a sort of Hill equation is derived because it is assumed that three subunits have to be in the activated state in order that the channel opens) in that it involves a smaller number of variables (Table 2), but more sophisticated in that a conformational change in the IP3R is assumed. Further models describing the kinetics of IP3-sensitive Ca2+ channels include those presented in [56,90,135].
The IP3R can be phosphorylated (with one phosphate per receptor subunit) by protein kinases A and C and Ca2+/calmodulin-dependent protein kinase II (CaM kinase II) . Sneyd and coworkers [94,136] presented models including phosphorylation of subtype III of the IP3R. The model proposed for pancreatic acinar cells  includes four different states of the receptor with one of these being phosphorylated. Moreover, the model includes Cacyt as a variable. The open probability curve of the IP3R is calculated to be an increasing function of Cacyt, as found for type-III IP3R . The model can explain long-period baseline spiking typical for cholecystokinin stimulation, which is accompanied with receptor phosphorylation, as well as short-period, raised baseline oscillations. It is worth taking into account the existence of three different subtypes of the IP3R in modelling studies in more detail because experimental work points to a physiological significance of the differential expression of IP3R subtypes [56,137–139].
Inclusion of mitochondria
It has been known for several decades that mitochondria contribute significantly to Ca2+ sequestration [140–143]. Besides the Ca2+ uniporter there are several other Ca2+ transport processes across the mitochondrial inner membrane, most notably the permeability transition pore (PTP) [144,145] and the Na+/Ca2+ and H+/Ca2+ exchangers [146,147] which appear to function primarily as export pathways. Over a long time, the accumulation of Ca2+ was believed to start at Ca2+ concentrations of about 5–10 μM (cf. ), which is much higher than physiological Cacyt. Accordingly, except for the model of Meyer & Stryer , mitochondria had first been neglected in studying Ca2+-mediated intracellular signalling. Later experiments re-evaluated the role of mitochondria in this context, showing that mitochondria start to take up Ca2+ via the Ca2+ uniporter at cytosolic concentrations between 0.5 and 1 μm[145,147,148]. This apparent contradiction with the earlier experiments can be resolved by the fact that, in a number of cells, mitochondria are located near the mouths of channels across the ER membrane [149,150]. In these small regions (the so-called microdomains) between the ER and mitochondria the Ca2+ concentrations could be 100- to 1000-fold larger than the average concentration in the cytosol [144,151]. It was found that mitochondria indeed sequester Ca2+ released from the ER [146,147,152–155]. For example, in chromaffin cells, around 80% of the Ca2+ released from the ER is cleared first into mitochondria . In the light of these findings, the role of mitochondria in Ca2+ oscillations was studied [148,157–159]. In particular, it was shown that a change in the energy state of mitochondria can lead to modulation of the shape of Ca2+ oscillations and waves, which are generated by autocatalytic release of Ca2+ from the ER.
These results have stimulated the inclusion of mitochondria in the modelling of Ca2+ oscillations [12,71,115,160–162] and Ca2+ homoeostasis [163–165]. In the early model of Meyer & Stryer , mitochondria are essential for the occurrence of oscillations (see above). The mitochondrial Ca2+ efflux is modelled to be constant. However, this assumption is questionable because the efflux must tend to zero as Cam tends to zero.
Selivanov et al. modelled the so-called mitochondrial CICR (m-CICR) through the PTPs in the inner membrane as observed experimentally [157,158]. They showed that Ca2+ oscillations could arise even in the absence of Ca2+ stores other than mitochondria. It remains to be seen whether this is physiologically relevant. While PTPs clearly play a role in the Ca2+ dynamics in gel suspensions of mitochondria  and in apoptosis in intact cells , this is less clear for cells under normal physiological conditions [166,167].
In the model presented previously , two basic Ca2+ fluxes across the inner mitochondrial membrane are taken into account. The Ca2+ uptake by mitochondria is, in agreement with experimental data (see above), modelled by Hill kinetics with a large Hill coefficient to describe a step-like threshold function. For the Ca2+ release back to the cytosol, the Na+/Ca2+ and H+/Ca2+ exchangers [146,147] but not PTPs are taken into account and described by a linear rate law. The model shows that mitochondria play an important role in modulating the Ca2+ signals and, in particular, could regulate the amplitude of Ca2+ oscillations . Ca2+ sequestration by mitochondria leads to highly constant amplitudes over wide ranges of oscillation frequency, due to clipping the peaks at about the threshold of fast Ca2+ uptake (see also ). This is in agreement with the idea of frequency-encoded Ca2+ signals (see Frequency encoding). Moreover, keeping the global rise of Cacyt below 1 μm may be of special importance in preventing the cell from apoptosis. Inclusion of mitochondria can also give rise to a dynamics more complex than simple oscillations (see Chaos and bursting).
Frequency and amplitude behaviour
For a better understanding of biological oscillations, it is of interest to analyse the dependence of frequency and amplitude on certain parameters (e.g. hormone concentration). In particular, this can help elucidate the role of oscillatory dynamics in information transfer. A straightforward method is by numerically integrating the differential equation system for different parameter values [18,80,113]. However, if several parameters are of interest, this method is very time-consuming. A more systematic way, which is, however, restricted to certain parameter ranges, is the analysis of the neighbourhood of the bifurcations from stable steady states leading to oscillations. The behaviour of oscillations near a bifurcation can often be established analytically. For example, so-called scaling laws exist, which give relevant quantities such as frequency and amplitude as functions of a bifurcation parameter.
While extensive bifurcation analysis has been carried out for models of nerve excitation [168–170], this is not the case for models of Ca2+ oscillations. (One paper pursuing this aim is ). Nevertheless, several papers deal with special aspects of bifurcations in Ca2+ oscillations. These will be reviewed below.
The most frequent transition leading to self-sustained oscillations in the models developed so far is the Hopf bifurcation (see Appendix). Let ε denote some dimensionless parameter measuring the distance from the bifurcation. For Eqn (7), a convenient parameter is with being the rate constant of the channel flux at the bifurcation. It can be shown analytically that near a supercritical Hopf bifurcation, the frequency remains nearly constant while the amplitude grows proportionally to the square root of ε, (Hopf Theorem, cf. ). However, it should be acknowledged that Ca2+ oscillations often represent so-called relaxation oscillations, which is due to the presence of both slow and fast processes. If the Ca2+ channel is open, Ca2+ release is much faster than the pump rate or the leak. Intuitively speaking, in relaxation oscillations, the concentration gradient across the ER membrane accumulated during a slow buildup is dissipated during a sudden discharge. The slow build-up is performed during the intermediate phases between spikes, while the discharge occurs during the first part of the spike (upstroke). The second part of the spike is, depending on the system, fast as well or somewhat slower. Changes in oscillation period are mainly due to variation in the duration of the interspike phase.
In relaxation oscillations, the supercritical Hopf bifurcations (as well the subcritical counterparts) have the striking feature that the growth of the oscillation amplitude near the bifurcation occurs in an extremely small parameter range. Numerical calculations for the subcritical Hopf bifurcation in the Somogyi–Stucki model  show that this change is confined to less than 10−5% of the value of kch. As the trajectories occurring in this range have, in the phase plane, the shape of a duck (canard in French), they are called canardtrajectories [31,169]. In fact, for various models, in diagrams depicting the amplitude vs. a bifurcation parameter [80,89,92,107,171], the emergence of periodic orbits is seen as a virtually vertical line (Fig. 2A), irrespective of whether the Hopf bifurcation is subcritical or supercritical. This implies that, practically, Ca2+ oscillations often appear to arise with a finite amplitude even at supercritical Hopf bifurcations.
Upon further increase of the bifurcation parameter, in many models, the oscillations eventually disappear at another Hopf bifurcation with a gradually decreasing amplitude (Fig. 2A). This is because the increase in the parameter reduces time hierarchy. While the bifurcation with a steep increase in amplitude was found more often in experiment [3,4,66] and is certainly physiologically more important because the signal can then be better distinguished from a noisy steady state, also smooth transitions have been observed [17,63]. Some authors have studied situations with parameter values for which time hierarchy is less pronounced at both Hopf bifurcations, so that they both are smoother [18,94,98,125,126].
Hopf bifurcations are not the only type of transition by which Ca2+ oscillations can arise. For example, in a model including the electric potential difference across the ER membrane and the binding of Ca2+ to proteins  (see Cacyt/protein models), a so-called homoclinic bifurcation (see Appendix) was found . For a model of the IP3R, a homoclinic bifurcation has been discussed briefly in Chapter 5, Exercise 12 in the monograph . A characteristic of the homoclinic bifurcation is that the oscillation period tends to infinity as the bifurcation is approached (see Appendix). In the case of Ca2+ oscillations, this is related to a very long duration of the ‘resting’ phase between spikes, while the shape of spikes remains almost unaltered. It is indeed often found in experiment that spike form is practically independent of frequency. Interestingly, homoclinic bifurcations have also been found for the Hodgkin–Huxley models of nerve excitation, and are important for the generation of low-frequency oscillations .
In a model including the binding of Ca2+ to proteins, the ER transmembrane potential and the sequestration of Ca2+ by mitochondria  (see Inclusion of mitochondria), an infinite-period bifurcation (see Appendix) was found . This bifurcation is also called saddle-node on invariant circle (SNIC) bifurcation . An example is shown in Fig. 2B. As the two newly emerging steady states require an infinite time to be approached or left, the period again diverges to infinity at the bifurcation, while the amplitude remains fairly constant.
As mentioned in the Introduction, a widely held hypothesis is that in Ca2+ oscillations, information is encoded mainly by their frequency [5,12,70–72,173]. This view is substantiated by the experimental finding that, upon varying hormone stimulation, frequency usually changes more significantly than amplitude. Moreover, Ca2+ oscillations usually display a typical spike-like shape with intermediate phases where Cacyt remains nearly constant. Li et al.  found in experiments with caged IP3 that artificially elicited Ca2+ oscillations induced gene expression at maximum intensity when oscillation frequency was in the physiological range. On the other hand, the level of activated target protein (see below) is likely to depend also on oscillation amplitude. Accordingly, a possible role of amplitudes in signal transduction by Ca2+ oscillations has also been discussed [73–75]. It was shown experimentally that upon pulsatile stimulation of hepatocytes by phenylephrine, not only the frequency but also the amplitude of Ca2+ spikes depends on the frequency of stimulation . It was argued that amplitude modulation and frequency modulation regulate distinct targets differentially .
For the phenomenon of frequency encoding, it is obviously advantageous if the oscillation frequency can vary over a wide range, while the amplitude remains nearly constant. This is particularly well realized in situations where the period diverges as a bifurcation is approached, while the amplitude remains finite, as it occurs in homoclinic and infinite-period bifurcations. It can be shown that near a homoclinic bifurcation, the period increases proportionally to the negative logarithm of ε, where ε is again some dimensionless distance from the bifurcation, (cf. ). In an infinite-period bifurcation, the scaling law reads . However, it should be checked whether the parameter range in which a significant change in frequency occurs is wide enough to be biologically relevant.
The subcritical Hopf bifurcations in various models do not lead to a diverging period. Nevertheless, time-scale separation in the system and, hence, the relaxation character of the oscillations often become more pronounced near the bifurcation, so that the frequency is indeed lowered drastically (cf. ). For the model developed by Somogyi & Stucki , for example, an approximation formula for the period, T, as a function of the parameters in the form was derived . In general, it may be argued that time hierarchy facilitates frequency encoding. This may be another physiological advantage of such a hierarchy besides the improvement in stability of steady states and the reduction of transition times .
It should be acknowledged that in the one-pool models, not only frequency but also amplitude changes significantly depending on agonist stimulation (Fig. 2A). This effect is less pronounced in the two-pool models . As pointed out in Inclusion of mitochondria, the constancy of amplitude is granted particularly well if the height of spikes is limited by sequestration of Ca2+ by mitochondria [12,71]. Another mechanism restricting oscillation amplitude is the biphasic dependence of the IP3R on Cacyt. Indeed, models including this exhibit fairly constant amplitudes [83,92].
Hopf bifurcations with an extremely steep increase in amplitude share with global bifurcations the abrupt emergence of the limit cycle and the absence of hysteresis. It may be argued that this behaviour is of physiological advantage. A small change in a parameter (e.g. a hormone concentration) can give rise to a distinct oscillation with a sufficiently large amplitude. Thus, misinterpretation of the signal is avoided because, in the presence of fluctuations, a limit cycle with a small amplitude could hardly be distinguished from a steady state. So far, there is no evidence that hysteresis, which would imply that the signal depends on the direction in which the bifurcation is crossed, would be physiologically relevant. Hysteresis occurs, for example, in a subcritical Hopf bifurcation without time-scale separation (Fig. 2B).
Sometimes, it has been argued that the information transmitted by Ca2+ oscillations is encoded in the precise pattern of spikes (temporal encoding) rather than in the overall frequency . It is an interesting question whether temporal encoding can be understood as a sequence of frequency changes or whether new concepts are necessary to understand it. In this context, it would be helpful to adopt methods for analysing information in neuronal spike trains (e.g ).
Modelling of protein phosphorylation driven by calcium oscillations
Interestingly, the effect caused by the oscillatory Ca2+ signal is usually a stationary output, for example, upon fertilizing oocytes, generating a stationary endocrine signal or enhancing the transcription of a gene. In some instances, however, the final cellular output is oscillatory as well, as in the case of secretion in single pituitary cells . The models discussed above provide a sound explanation for the fact that a change in a stationary signal (agonist) can elicit the onset of oscillations. What has been studied much less extensively is how these oscillations can produce an approximately stationary output.
De Koninck & Schulman  performed experiments showing that CaM kinase II can indeed decode an oscillatory signal. As this enzyme can phosphorylate a variety of enzymes, the Ca2+ signal can be transmitted to different targets. Of particular importance is the autophosphorylation activity of CaM kinase II, because in the phosphorylated form, the enzyme traps calmodulin and keeps being active even after the Ca2+ level has decreased. This amounts to a ‘molecular memory’, by which the oscillatory input is transformed into a nearly stationary output.
It was shown that CaM kinase II activity increased with increasing frequency of Ca2+/calmodulin pulses in a range of high frequencies (1–4 Hz) . However, in electrically nonexcitable cells, the frequency of Ca2+ oscillations is usually below this range. To model the decoding of low-frequency signals, Dupont & Goldbeter [70,180] proposed a model based on an enzyme cycle involving a fast kinase, which is activated by Cacyt, and a slow phosphatase, which is Cacyt-independent. Intuitively, it is clear that an integration effect can be achieved in such a system, because the phosphorylation following a Ca2+ spike will persist for a while (cf. ). The model of Dupont & Goldbeter  indeed predicts, with appropriately chosen parameter values, that the mean fraction of phosphorylated protein is an increasing function of frequency. The dependence on frequency is more pronounced if zero-order kinetics for phosphatase and kinase are chosen (cf. the phenomenon of zero-order ultrasensitivity in enzyme cascades [181,182]).
A more detailed model was presented for the liver glycogen phosphorylase . This enzyme includes calmodulin as a subunit. For the Michaelis-type rate law of the phosphorylase kinase, it was assumed that both the maximal activity and Michaelis constant are highly nonlinear functions of Cacyt. The model shows, both for a sinusoidal input and for oscillations generated by the two-pool model , that a given level of active glycogen phosphorylase can be elicited by a lower average Cacyt level when Ca2+ oscillates than when it is stationary.
A mechanism for decoding Cacyt signals by PKC involving also DAG was proposed by Oancea & Meyer  but has not yet been formulated as a mathematical model. A model describing the phosphorylation of CaM kinase and a target protein after cooperative binding of Ca2+ to calmodulin as well as the autophosphorylation of CaM kinase was developed by Prank et al.. It predicts an increase in activation of target proteins with increasing frequency of the Ca2+ signal.
Chaos and bursting
Experimental results very often show more complex forms of Ca2+ dynamics than simple, regular oscillations [67,72,185] (for review, see ). The most common pattern of such complex oscillations is a periodic succession of quiescent and active phases, known as bursting (Fig. 3). Bursting can be periodic or chaotic. It has been studied intensely in the case of transmembrane potential oscillations in electrically excitable cells [5,60,101,160,172,187]. However, an important difference is worth noting. While often in electric bursting, each active phase comprises several consecutive, large spikes with nearly the same amplitude, in Ca2+ bursting, single large spikes are followed by smaller, ‘secondary’ oscillations.
Complex Ca2+ oscillations may arise by the interplay between two oscillatory mechanisms; this is not, however, the only possibility . The underlying molecular mechanisms as well as the biological significance for intracellular signalling are not yet understood in detail (cf. Conclusions). Different agonists may induce different types of dynamics in the same cell type. For example, while hepatocytes exhibit regular Ca2+ oscillations when stimulated with phenylephrine, stimulation of the same cells with ATP or UTP elicits regular or bursting oscillations depending on agonist concentration [67,72,185].
Several combinations of three equations out of the system (Eqns 1–6) have been suggested to explain bursting in Ca2+ oscillations. Shen & Larter  demonstrated regular bursting and transition to chaos in a model involving Cacyt, Caer and IP3. Both the activatory and inhibitory effects of Cacyt on vrel are included. Moreover, Cacyt is assumed to activate IP3 production. Three combinations of variables giving rise to bursting have been studied by Borghans et al.. The first model extends the one-pool model based on Eqn (2) and Eqn (3) by considering the fraction of sensitized IP3R as a third variable and, accordingly, including Eqn (6). The second model extends an ICC model  by including the CICR mechanism. This model can generate not only bursting but also chaotic behaviour. It was further analysed mathematically  and shown to admit birhythmicity (i.e. the coexistence of two stable limit cycles, cf. ). The third model is based on the two-pool model [14,15] with the Ca2+ level in the IP3-insensitive pool being the third variable. For the first two proposed models, the cause for the transitions between active and quiescent phases can be studied by considering the difference in time scales between the fast, spike-generating subsystem made up of Cacyt and Ra, or Cacyt and IP3, and the slow dynamics of Caer.
Another explanation of complex intracellular Ca2+ oscillations has been proposed recently [115,162]. In addition to the ER, also Ca2+ sequestration by mitochondria and the Ca2+ binding to cytosolic proteins is taken into account. These studies extend earlier work  on modelling the possible mitochondrial modulation of Ca2+ signals. As the Ca2+ exchange across the plasma membrane is neglected, there is a conservation relation involving Cacyt, Caer, Cam, and B, so that the model is three-dimensional. Simple Ca2+ oscillations, periodic and aperiodic bursting and chaos can be obtained with appropriate parameter values (Fig. 3). In all of these regimes, single large-amplitude spikes are followed by small oscillations of nearly constant amplitude. Such small-amplitude oscillations during the quiescent phase are indeed found in experiment, although it is difficult to distinguish them from noise. The transition from a limit cycle to chaos via a folded limit cycle (Fig. 3D) and repeatedly folded limit cycles is known as the period-doubling route to chaos (and was also found in ). Interestingly, in other parameter ranges, the succession of behaviours follows the intermittency route to chaos . Besides complex dynamics also birhythmicity and even trirhythmicity can be found . The model predicts that spike amplitudes in the active phases of bursting are remarkably insensitive to changes in the level of agonist. This is due to the fact that mitochondria clip the peaks in Cacyt, as observed already in the earlier models generating simple oscillations [12,71].
A model proposed by Kummer et al. involves the variables Cacyt, Caer, and the concentrations of active Gα subunits of the G-protein and active PLC. IP3 is assumed to be proportional to the latter variable due to quasi-steady-state arguments. The model assumes the presence of two different receptors, for phenylephrine and ATP, both of which activate PLC through the Gα subunit. The rate of Gα activation is modelled as k1 + k2*Gα, with k1 and k2 being proportional to the concentrations of phenylephrine and ATP, respectively. (The term k2*Gα describing an autocatalytic activation can be regarded as a linear approximation of k′2*G*(GTPtotal − G), with the latter complying with the conservation relation for total G-protein.) The model is in particularly good agreement with experimental observations in two respects . First, each oscillation period starts with a large, steep spike followed by a number of pulses of decreasing amplitude around an elevated mean value. Second, varying the parameters k1 and k2 independently, one finds that stimulation by ATP can induce (periodic or aperiodic) bursting, while stimulation by phenylephrine can only elicit regular oscillations. From a more theoretical point of view, it is interesting that Kummer et al. were able to reduce this model to three dimensions by just excluding Caer and the fluxes vserca and vrel. The reduced model can still generate chaotic behaviour although the nonlinearities involved are simple Michaelis–Menten rate laws, so that it represents one of the simplest models generating chaos.
In the three-dimensional model of Chay [60,160] (the variables are Cacyt, Caer and the cell membrane potential), the essential nonlinearities reside in the ion fluxes across the cell membrane. The model establishes a link between the electrical bursting and calcium bursting in excitable cells. However, experiments indicate that Cacyt is not likely to be the slow variable underlying electrical bursting in pancreatic β-cells .
Coupling of oscillating cells
The models discussed so far focus on the temporal evolution of the Ca2+ concentration. However, cellular Ca2+ transients also have a spatial dimension. In the cytoplasm of single cells, Ca2+ gradients can be observed when Ca2+ release from the ER is excited at particular subcellular locations [6,42,130]. The local excitation can spread through the cell as a concentration wave, which appears to be propagated by Ca2+ diffusion and CICR. In hepatocytes, periodic Ca2+ waves are seen that originate from a particular region within a cell . Moreover, in the intact liver and in hepatocyte multiplets, Ca2+ waves can spread from cell to cell [191–194]. In contrast to isolated hepatocytes, which exhibit substantial variations of Ca2+ oscillation periods between cells when stimulated by hormone, coupled hepatocytes oscillate with the same period , or nearly the same period . There are fixed phase relations in that the cells oscillating faster in isolation peak before the slower cells. Thus the intercellular coupling leads to a (near) 1 : 1 entrainment, or synchrony, of the oscillations in adjacent cells. On the larger scale of the liver, periodic Ca2+ waves propagate from the periportal to the pericentral region of each liver lobulus independent of the direction of perfusion . The direction of wave propagation may correlate with a gradient in hormone receptor density . Intercellular entrainment of Ca2+ oscillations has also been observed in other cell types, such as pancreatic acinar cells , articular chondrocytes [199,200], kidney cells , and in the blowfly salivary gland . This phenomenon can be viewed a particular instance of the intercellular propagation of Ca2+ waves observed in many systems [202–204].
Two pathways have been implicated so far in intercellular Ca2+ signalling: (a) the diffusion of cytoplasmic messenger molecules through gap junctions [205–208] and (b) the release of paracrine messengers into the extracellular space and their diffusion to neighbouring cells [209,210]. In the systems in which intercellular entrainment has been observed so far, cells have also been shown to be coupled by gap junctions. In hepatocytes, entrainment is disrupted by gap-junctional uncouplers but not by exclusion of paracrine signalling [195,211].
To capture the spatial propagation of Ca2+ signals, diffusion fluxes of Ca2+ and IP3 must be included in the general balance equations (Eqns 1–6). For Cacyt, the balance equation then reads
where the vi denote the Ca2+ exchange fluxes with the various compartments, cf. Eqn (2), and Dc is the cytoplasmic Ca2+ diffusion coefficient. The Ca2+ concentration is now a function of time and spatial position, . Likewise the kinetic terms vi depend on spatial location as functions of . The spatial dependence of the vi can also explicitly reflect the subcellular organization of the Ca2+ transport processes. A similar balance equation holds for the IP3 concentration in place of Eqn (11):
where Dp denotes the diffusion coefficient of IP3. In Eqn (11) and Eqn (12), ∂2/∂x2 is the Laplace or diffusion operator. For simplicity, we have given a spatially one-dimensional formulation of the diffusion terms (that can be generalized to two and three dimensions). The x-axis is considered to lie along the direction of Ca2+ wave propagation.
As in the case of Eqns (2–4), Eqn (11) can be understood to implicitly contain the effect of fast Ca2+ buffering. In addition to the definition of effective rate constants, one can now also define, under certain conditions, an effective diffusion coefficient for Ca2+ that includes the effect of Ca2+ buffering . In Eqn (11), Dc is understood as such an effective diffusion coefficient. It is generally about an order of magnitude lower than the Ca2+ diffusivity in water and also the cytoplasmic IP3 diffusivity . Moreover, the value of Dc is influenced by the diffusivities and concentrations of Ca2+ buffers, which can thus have a decisive impact on the spatial propagation of Ca2+ signals [105,108,110].
Although it is generally more difficult to obtain and analyse solutions for the reaction-diffusion Eqns (11,12) than for systems of ordinary differential equations, such as Eqns (1–6), a number of numerical and analytical tools exist [213,214]. In particular, models based on equations of this type can describe the propagation of intracellular Ca2+ waves [101,215,216].
If cells are coupled by gap-junctions, in addition to Eqns (11,12) the junctional fluxes must be included in a model. In the absence of membrane potential differences between the cells, these can be assumed proportional to the concentration differences across the junctions for each substance. For example, for a pair of coupled cells the junctional fluxes from cell 1 to cell 2 of Ca2+, jc, and of IP3, jp, can be written as:
Pc and Pp are the gap-junctional permeabilities for Ca2+ and IP3, respectively. Both Ca2+ and IP3 have been shown to permeate gap junctions in various cells [217,218]. The effect of fast Ca2+ buffering on the gap-junctional fluxes can be accounted for in a similar fashion as for the kinetic terms, and Pc can accordingly be defined as an effective gap-junctional permeability for Ca2+. Ca2+ buffering reduces the effective intercellular Ca2+ permeability. Direct measurements of Ca2+ and IP3 permeabilities are not available in the literature; however, permeability coefficients for various other molecules have been determined in some systems [219,220]. Gap-junctional conductivities (determining the electrical current through the junctions) are also available for many cell types, yet their relation to permeabilities for particular ionic species is not straightforward [83,221].
Comparison of models and experiments
Synchronization and, more generally, entrainment are common phenomena in systems of coupled oscillators. In the case of the intercellular entrainment of Ca2+ oscillations, the participating mechanisms and, specifically, the messenger molecules exchanged between cells have been a focus of the experimental work [217,218,222]. Recently, two models relating to experiments in hepatocytes were proposed. They study two specific entrainment mechanisms: intercellular coupling mediated by diffusion of IP3 and Ca2+, cf. Fig. 4.
Two extreme possibilities for intercellular coordination can be envisaged. (a) Transient, agonist-induced coordination: intercellular coordination of Ca2+ oscillations is a transient phenomenon that is caused by the initial application of hormone and afterwards slowly decays until cells become uncoordinated again. Such a mechanism can in principle work also without any intercellular coupling, though coordination is enhanced by coupling. (b) Active entrainment through coupling: the coordination is inherently caused by the intercellular coupling. As a consequence, cells being uncoordinated in the absence of gap-junctional coupling may become coordinated when coupling is restored, under otherwise constant conditions (e.g. the hormonal stimulus is not changed). In this case, the putative coupling messenger must clearly be sensitive to phase differences of the oscillations in adjacent cells. In the model proposed by Dupont and coworkers , Ca2+ activation of one of the IP3-degrading enzymes, IP3 3-kinase, causes IP3 oscillations to occur sumperimposed on the Ca2+ oscillations (cf. subsection Consideration of the IP3 dynamics). IP3 diffusion across gap junctions coordinates the Ca2+ oscillations in adjacent cells, but does not lead to stable 1 : 1 entrainment. Immediately after agonist application, there is a transient 1 : 1 coordination which subsequently disappears. The model predictions compare well with a number of experimental results. If there is no Ca2+ feedback on IP3 dynamics, entrainment cannot be brought about by IP3 diffusion. As shown previously , gap-junctional Ca2+ fluxes can lead to active 1 : 1 entrainment. Such an autonomous entrainment has been found in experiments with application and subsequent washing out of gap-junctional uncouplers, or transient block of ER Ca2+ release .
Whether active 1 : 1 entrainment is obtained depends crucially on the gap-junctional permeability. For the hypothesized Ca2+ coupling, it was shown that the permeability must lie within certain bounds to obtain correspondence of model simulations and experimental results  (a related study was made on intercellular Ca2+ waves ). If the permeability falls below the critical value for 1 : 1 entrainment (synchronization), entrainment of heterogeneous cells will still occur, but with ratios of the oscillation periods different from 1 : 1. If the frequency ratio of coupled oscillators equals a rational number, this phenomenon is also called phase locking.
Recently, the influence of a number of other processes on intercellular Ca2+ wave propagation has been studied for a simple, nonoscillatory model system, including cytoplasmic Ca2+ buffering and level of agonist stimulation .
What is the point in oscillations?
A question immediately arising in the context of calcium signalling is why the signal is transmitted by oscillations rather than by adjustable stationary calcium concentrations. This question has often been discussed [12,69,70,86, 113,179,183,224] but surprisingly little work on modelling has been presented so far [70,183,184]. First, it is worth mentioning that not every biological phenomenon necessarily needs to have a reason in terms of evolutionary advantage. It may well be that oscillations just arise because it is hard to avoid them under certain circumstances due to the nonlinearities involved. Moreover, it has even been argued that in certain cells, e.g. cochlear Hensen cells, oscillations are closely related to pathophysiological conditions such as noise-induced hearing loss . The nonlinearities, in turn, are likely to be necessary for a high amplification of signals. The phenomenon of frequency encoding could then be explained by the fact that the mean Cacyt level increases with increasing frequency due to the special form of the oscillations characterized by spikes and interspike phases of varying length.
Nevertheless, it is of course interesting to speculate about the physiological advantages of oscillatory behaviour. Already the switch between stationary and pulsatile regimes may serve as a (digital) signal, while changes in oscillation frequency may serve as analogue signals. The latter, in turn, may be manifold: they may be encoded by frequency, amplitude, or spike form. It has also been argued that frequency encoded signals could prevent long-lasting receptor desensitization  and are more robust to noise [183,226]. Discrete events (spikes) can be recognized as signals better than potentially spurious wanderings of the steady-state concentration [88,207]. Moreover, oscillations are a suitable means for switching on different processes with one and the same second messenger. For example, Dolmetsch et al. were able to show that the expression of three different transcription factors in T-lymphocytes was specifically triggered depending on the frequency of Ca2+ oscillations.
A special property of Ca2+ ion is that concentrations elevated over a longer period are lethal to the cell due to formation of unsoluble Ca2+ salts. This harmful effect can be avoided by an oscillatory behaviour. As corroborated by a recent model , an oscillatory regime can increase the sensitivity of the Ca2+ sensing enzymes to this second messenger because Ca2+ can periodically exceed the threshold for enzyme activation even if the average Ca2+ level remains below the threshold. Moreover, a very wide range of signal strengths (notably several orders of magnitude) may be achieved. A comparable variation in steady-state levels would imply severe problems with respect to osmotic balance and solvent capacity.
Another advantage arises from the spatial aspect: Coupled oscillators are able to exhibit a wide range of possible behaviours such as synchronization with or without phase shift, phase locking, quasiperiodicity and chaotic regimes. Thus, many more types of different signals could be transmitted from cell to cell than by stationary states.
A further point is the binding to proteins. If the Ca2+ level were constant (at different adjustable values), this binding would be in equilibrium, so that the fraction of bound Ca2+ were only be determined by the equilibrium constants. In an oscillatory regime, however, also the on and off rate constants are relevant so that the system has more degrees of freedom for fine-tuning regulation.
In view of the models describing bursting and chaos (see earlier), it is interesting to speculate about the physiological role of these phenomena. Again, they might be hard to avoid due to the underlying nonlinearities, as soon as more than two variables are involved. The three-dimensional models showing bursting with small secondary oscillations [72,115,186] show that the effect of one variable can approximately be neglected. Its effect is just a small fluctuation around a regular oscillation. On the other hand, dynamics with two superimposed oscillatory patterns (Fig. 3) clearly provides more possibilities to encode information. It is interesting to investigate whether this has a physiological significance. For some other biological systems it has been proposed that a possible role of complex (chaotic) oscillations could be the detection of weak signals within cells because of the extreme sensitivity of a chaotic state to periodic forcing . The physiological relevance of chaotic behaviour has been intensely discussed in the case of cardiac chaos [228–230].
Obstacles and prospects
The modelling of Ca2+ oscillations is complicated by the wide diversity of the nature of this phenomenon. Their generation in different cell types may not be due to one and the same mechanism (cf. ). However, the extensive experimental and theoretical studies on this subject point to a central role of the CICR. Other mechanisms such as the effect of Ca2+ on IP3 turnover or the sequestration of Ca2+ by mitochondria play a modulatory role and may be cell-type specific. Accordingly, if not only the occurrence of spike-shaped oscillations in general but more specific phenomena are to be described, specific models must be developed for different cell types, as exemplified by the work on hepatocytes , pancreatic acinar cells  or pituitary gonadotropes . Relatively little work has been carried out so far on discriminating different models on the basis of experimental data, for example with respect to the mechanisms of spike termination (see Minimal models). Interestingly, even in a given cell, the form and width of spikes may vary depending on the type of agonist used [72,185]. Moreover, the spike form and frequency may vary between different single hepatocytes although being reproducible on the same cell , indicating heterogeneity of cellular parameters.
Several problems that arose in the beginning of the work in this field are still unsolved. For example, it is still not clear under what circumstances IP3 follows a significantly oscillatory regime and whether this is important for modelling Ca2+ oscillations. This might depend on oscillation frequency because it was found that PLC is activated by Cacyt with a saturation at frequencies below the maximum . Moreover, it is still a matter of debate under which conditions frequency encoding or amplitude encoding play the most important role, or whether a more complex mechanism (temporal encoding) is relevant that may have developed during biological evolution.
Future efforts might be spent on a more detailed study of the phosphoinositide pathway, of which the hydrolysis of PIP2 into IP3 and DAG is but a tiny part. A number of phosphoinositides linked by kinases and phosphatases have been found to be second messengers . Moreover, it is promising to analyse Ca2+ sequestration by the nucleus. As only a very limited number of models describing this have been developed so far  (for a review on experimental data, see ), we have not included the nucleus in Fig. 1.
It is interesting to discuss the interrelations between bistability and oscillations. The Somogyi–Stucki model  as well as a simple chemical model [32,234] are examples of oscillating systems that do not exhibit bistability (unless some parameters are set equal to zero). In these models, the oscillations arise via Hopf bifurcations. By contrast, in order that a homoclinic bifurcation or an infinite-period bifurcation can be observed, the model must admit at least two stationary states. This explains why in all models allowing only for one steady state [15,17,80], neither homoclinic nor infinite-period bifurcations can occur. In contrast, more complex models [71,101,113,172] exhibit bistability and global bifurcations.
Moreover, there is another interesting relationship between oscillatory behaviour and bistability. When the Cacyt/Caer models are modified in that the exchange fluxes via the plasma membrane are neglected, they cannot give rise to oscillations anymore because the arising conservation relation causes the systems to be one-dimensional. However, the models reduced in this way exhibit bistability. Analogously, the Meyer–Stryer model  exhibits bistability (but no oscillations) when the exchange with mitochondria is neglected. The complex interplay between bistability and oscillations deserves further general studies.
In the modelling of coupled oscillating cells, the phenomenon of entrainment, that is the phase locking between a fast pacemaker oscillator and slower, entrained, oscillators has been studied (see Coupling of oscillating salts). Cells having different intrinsic oscillation frequencies attain, upon coupling, fixed frequency ratios which are quotients of small integers. Up to now, only 1 : 1 entrainment has been studied in some detail. However, the results in  point to the possible relevance of ratios different from 1 : 1. Similar phenomena can be observed when a cell capable of Ca2+ oscillations is stimulated with an oscillating hormone input . Theoretical studies of this type of entrainment are rare [75,76] and worth being extended. Mathematical modelling should further be exploited in conjunction with experimental work to elucidate the control exerted by the various intracellular mechanisms of Ca2+ signalling on the one hand, and the gap-junctional diffusion of Ca2+ and IP3 on the other, on the intercellular coordination of Ca2+ oscillations.
In biochemistry, the theoretical analyses of stationary states and the modelling of oscillations have surprisingly developed as relatively separate strands over the last decades. A number of well-established theoretical tools such as metabolic control analysis [86,235,236], metabolic flux analysis  and structural analysis of metabolic networks [237,238] have been developed to analyse stationary states. Some of these tools are applicable also to oscillatory systems as long as average fluxes are considered, because for these, the stationary balance equations hold true as well. It is certainly of interest to extend Metabolic Control Analysis to oscillatory processes, to answer questions such as: how are frequency and oscillation controlled by the activity of a given enzyme or the permeability of a channel? Although there are a few attempts (cf. ), this extension is far from being complete . Moreover, it is worthwile extending structural analysis, which does not require the knowledge of kinetic parameters, to signal transduction systems. This could help answer questions such as: What structure (topology) of such a system is favourable for a high amplification of signals [182,240] or a signal transmission that is robust to noise?
Information is always linked with a high amplification of some quantity [181,240]. For example, the replacement of one nucleotide in the DNA can have a large effect, or a few hormone molecules may elicit dramatic changes. In view of the small values of the cytosolic Ca2+ concentration and the large-scale effects that may be induced by Ca2+ oscillations, these oscillations fit into the amplification paradigm. It is worth studying in the future the energetic requirements for amplification in relation to information transfer by Ca2+ oscillations.
A general problem in the analysis of chaotic time-series is the difficulty to distinguish deterministic chaos from oscillations superimposed by stochastic noise . The distinction between regular oscillations and bursting is clearly much simpler. Further work could also concern the question whether stochastic resonance (i.e. the amplification of weak signals by noise, cf. ) plays a role in Ca2+ signalling. First results in this direction have been obtained .
Experimentalists sometimes criticize models by saying that these just reproduce what was found earlier in experiment. However, we believe that the quantitative description constitutes a necessary step in the understanding of a cellular system. Mathematical models in cell biology can be very helpful because they explain why a certain phenomenon occurs and may lead to new or deeper insight (such as by distinguishing molecular mechanisms that can give rise to oscillations from those which can not). The molecular interactions involved in Ca2+ oscillations (e.g. the activation and inhibition of the IP3 receptor by its agonists) are so complex that they cannot be understood intuitively. Thus, Ca2+ dynamics constitutes an excellent example demonstrating the use of mathematical models. Hopefully, the interaction between experiment and theory will lead to further progress so that modelling increasingly gains predictive power.
Appendix: mathematical fundamentals
For dynamical systems described by autonomous ordinary differential equations, a system of at least two equations is required to describe oscillations (see Fig. 5). Moreover, it can be shown that, with autonomous ordinary differential equations, the system should at least be three-dimensional to describe chaos. This follows from the Theorem of Poincaré & Bendixson for two-dimensional systems (cf. ). This theorem says that if, and only if, a trajectory remains for all times, starting with a certain time point, within a finite region of the phase plane without approaching a stationary state, this trajectory is periodic or tends to a periodic trajectory as t. To understand this, it is helpful to realize (although this is not a mathematical proof) that a trajectory cannot cross itself because the differential equations,
determine, for each point x1,x2, the direction of the trajectory uniquely. If, in a two-dimensional system, a chaotic trajectory arose, it would have to avoid to tend to a stationary point and to spiral to a limit cycle (Fig. 6). To avoid the latter, it would have to move in opposite directions in increasingly closer positions. This is impossible because, as long as the functions f1 and f2 are smooth enough (which is, in biochemical kinetics, always the case), this direction cannot change dramatically for points lying close together.
To analyse the oscillation models mathematically, it is helpful to begin with an investigation of the potential steady states in the system. This is because they can be found more easily than limit cycles and because the stability analysis of steady states can be instrumental in the detection of oscillations. From the Theorem of Poincaré & Bendixson, it follows that, in two-dimensional systems, the existence of a finite region as described above and of an unstable focus (i.e. a point from which the trajectory spirals away) lying in this region implies the existence of a stable limit cycle in this region. Moreover, the number of steady states is interrelated with the type of potential transitions to limit cycles (see below).
Positive feedback is a potential mechanism for the generation of self-sustained oscillations (‘back-activation oscillator’) [29,86,244]. Mathematically, this can be shown by analysing the Jacobian matrix,
The trace of matrix J, that is, the expression (∂f1/∂x1) + (∂f2/∂x2) as well as its determinant can be positive in the presence of positive feedback. This leads to an unstable focus. It is worth noting that on the basis of a simple mass-action kinetics, reaction systems with only two variables cannot exhibit limit cycles if only monomolecular and bimolecular reactions are involved , while systems with three variables can . Note that the oscillations arising in the well-known Lotka–Volterra model in population dynamics (cf. ), which is two-dimensional and involves, from a chemical point of view, only monomolecular and bimolecular reactions, are not limit cycles because they do not attract neighbouring trajectories. The (two-dimensional) Brusselator model , which gives rise to limit-cycle oscillations, involves a trimolecular reaction.
When a parameter of the system (e.g. a rate constant) is changed, a point may be reached where the dynamic behaviour changes qualitatively (for example, the number of steady states may change). Such a point is called a bifurcation. An example is provided by a transition from a stable focus (i.e. a steady state for which all trajectories starting in its neighbourhood spiral towards it; the trace of the Jacobian matrix is then negative while the determinant is still positive) to an unstable focus with the emergence of a limit cycle. This bifurcation was called after the mathematician E. Hopf (cf. [5,30,86]. In two-dimensional systems, Hopf bifurcations arise when the trace of the Jacobian matrix equals zero.
Hopf bifurcations are supercritical or subcritical according to whether the limit cycle bifurcating from the steady state is stable (points P and Q in Fig. 2A) or unstable (point S in Fig. 2B), respectively (cf. ). At a supercritical Hopf bifurcation, crossing the bifurcation point leads to a smooth transition from a steady state to a limit cycle, if the growth in amplitude is not too steep. Then it is often called soft excitation. At a subcritical bifurcation, a jump from the steady state to infinity or to a coexisting domain of attraction occurs. Very frequently the attractor is a stable limit cycle. Accordingly, the amplitude jumps from zero to a finite value at the bifurcation (hard excitation).
Hopf bifurcations, as well as the transition from monostability to bistability, are called local bifurcations because qualitative changes occur, in the phase space spanned by the system's variables, only in a neighbourhood of the steady state. At global bifurcations, by contrast, a qualitative change occurs in a larger region in phase space. An example is provided by the homoclinic bifurcation (cf. ), at which a limit cycle coalesces with an unstable steady state (more specifically, a saddle point) to form a homoclinic orbit (see Fig. 6B) and disappears beyond the bifurcation. The velocity of the trajectory tends to zero as it approaches this steady state (exactly at this point, the velocity is zero). Therefore, the period of the limit cycle tends to infinity as the homoclinic bifurcation is approached. When the bifurcation is crossed in the opposite direction, the limit cycle emerges all of a sudden with a finite amplitude. Another global bifurcation leading to a diverging period is the infinite-period bifurcation (cf. ). When a limit cycle disappears in an infinite-period bifurcation, a new steady state appears exactly on the limit cycle and starts dividing into two steady states, with one of them being stable and the other being unstable. The trajectory then runs towards the stable steady state, so that a cyclic orbit can no longer be observed.