THE EFFECT OF COUPLING, NOISE, AND HETEROGENEITY ON BURSTING OSCILLATIONS

Transcription

1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 11, Number 1, Spring 2003 THE EFFECT OF COUPLING, NOISE, AND HETEROGENEITY ON BURSTING OSCILLATIONS GERDA DE VRIES ABSTRACT. Bursting refers to a complex oscillation characterized by a slow alternation between spiking behaviour and quiescence. In this paper, we give an overview of the bifurcation analysis of coupled models of square-wave bursting oscillations. The bifurcation analysis is facilitated by the decomposition of the system of equations into a fast and a slow subsystem. An investigation into the bifurcation structure of the fast subsystem reveals the different mechanisms underlying the modification of bursting characteristics by coupling itself and heterogeneity in the model parameters. Both weak and strong coupling regimes are discussed. Emergent bursting, that is, the genesis of bursting oscillations within a population of individuals that are incapable of bursting themselves, is reviewed as well. The bifurcation analysis reveals that emergent bursting is aided by heterogeneity or noise, depending on the strength of the coupling. Implications of the results obtained are discussed in the context of bursting oscillations observed in the membrane potential of pancreatic β-cells. 1 Introduction Bursting oscillations are complex oscillations characterized by two timescales. On the slow time scale, silent and active phases alternate. The silent phase is relatively quiescent, while the active phase exhibits oscillatory behaviour on a much faster time scale (rapid spiking). Bursting oscillations have received a lot of attention in recent years, in particular in the context of physiology. For example, bursting is observed in the potential difference across the membrane of many cells, from different types of nerve cells to endocrine cells, in a large variety of animal species, including humans. Bursting oscillations have provided fertile ground for mathematical investigations on several levels, including the development of detailed biophysical models with the aim of making predictions about the biological system, the classification of bursting models, and the analysis of Copyright c Applied Mathematics Institute, University of Alberta. 29

2 30 GERDA DE VRIES minimal models with the aim of gaining a deeper understanding of the mathematical structure underlying the oscillations. In this paper, the focus will be on providing an overview of the efforts made to understand the role of coupling on bursting oscillations, in the context of pancreatic beta cells. The pancreas is a secretory organ in mammals. Its function is to synthesize and release a number of substances, including hormones such as insulin and glucagon. These hormones are secreted into the bloodstream and regulate the blood glucose (sugar) level. The hormones are released from micro-organs called Islets of Langerhans, scattered throughout the pancreas. Each islet contains several thousand cells, the majority of which are β-cells. The function of the β-cells is to secrete insulin in response to elevated levels of glucose in the blood. Insulin serves to signal target tissues (liver, muscle, and fat cells) to utilize or store the glucose, causing glucose levels to drop. The membrane potential of pancreatic β-cells undergoes bursting oscillations. During the active phases, calcium enters the cells, which in turn allows the cells to release insulin. The stronger the glucose signal, the longer (shorter) the active (silent) phases, and the higher the release of insulin. The significance of bursting lies in the fact that higher peak and mean levels of intracellular calcium levels can be achieved than with a continuous spiking solution (see de Vries and Sherman (2000)). Sato et al. (1999) recently have shown that secretion of insulin from pancreatic β-cells is proportional to the average intracellular calcium concentration. Thus, bursting may serve to optimize insulin secretion. In Section 2, we will concentrate on the analysis of a simple but representative model of the electrical activity of a single pancreatic β- cell. We will review the decomposition of a bursting model into a fast and a slow subsystem. The bifurcation diagram for the fast subsystem, using the slow variable as the bifurcation parameter, will reveal bistability between a stable steady state and a stable limit cycle, corresponding to the silent and active phases of the bursting oscillation, respectively. The slow subsystem is driven by the fast variables, resulting in a hysteresis loop that causes the alternation between the two stable attractors. A good understanding of the fast-slow decomposition method will provide the basis for the analysis of coupled systems in the remainder of the paper. Beta cells do not act in isolation. Instead, they are in close electrical contact with their nearest neighbours. A single-cell model can be viewed as representing the behaviour of a single cell situated within a wellcoupled cluster of cells, such as the Islet of Langerhans. In recent years,

3 BURSTING OSCILLATIONS 31 a series of papers (Sherman and Rinzel, 1991, 1992; Sherman, 1994; de Vries et al., 1998; de Vries and Sherman, 2000, 2001) has appeared describing the role of electrical coupling in synchronizing the electrical activity of such a population. The remainder of the paper will be devoted to reviewing the results of this series of investigations, highlighting the role of bifurcation analysis of the coupled fast subsystem in elucidating the mechanisms underlying the modulation of bursting properties via coupling, noise, and heterogeneity in the model parameters. Attention will be focussed on two-cell systems, for which a bifurcation analysis is tractable. Numerical simulations have demonstrated that the analytical results for two-cell systems carry over to many-cell systems. In Section 3, we review results for coupled bursting models. We begin with a system of two identical cells, and show the influence of coupling strength on the modification of bursting properties, especially under weak to moderate coupling conditions. When coupling strength is strong, the two-cell system is shown to behave exactly as the individual cells. Next, we investigate the effect of introducing heterogeneity in the model parameters in the case of strong coupling, and show that heterogeneity serves to increase the robustness of the bursting solution. The results from Section 3 form the basis for Section 4, in which bursting is discussed as an emergent phenomenon, that is, bursting in a population of cells each of which is incapable of bursting individually. Emergent bursting can occur spontaneously in systems of weakly coupled cells, but can be facilitated by the addition of noise or heterogeneity, depending on the strength of the coupling. Finally, in Section 5, we compare results from these coupling studies with some earlier studies. We close with a discussion of the significance of the results in the context of the biological problem. 2 Single-cell model of a burster Models of the electrical activity observed in β-cells are based on the Hodgkin-Huxley equations for neuronal electrical activity (Hodgkin and Huxley, 1952). A good introduction to these types of equations can be found in the text on mathematical physiology by Keener and Sneyd (1998). Essentially, it is assumed that the cell membrane can be modelled as a capacitor in parallel with an ionic current, and that the capacitive current is in balance with the ionic current. The ionic current is the result of ions flowing through protein channels (pores) in the membrane. The channels control the flow of ions by opening and closing in response to certain stimuli. Thus, to complete a model for electrical activity, the current balance

4 32 GERDA DE VRIES equation must be complemented by a set of equations modelling the dynamics of the ionic currents. There are many different models of the electrical activity in pancreatic β-cells (see Bertram and Sherman (2000), Sherman (1995), and Sherman (1996) for reviews). Although the physiological details included in these models differ, the dynamics underlying the bursting oscillations follow the same mathematical principles. For the purposes of this review, we will examine the following representative model (Sherman and Rinzel, 1992), consisting of three ordinary differential equations (already scaled for ease of presentation): (1) (2) (3) τ dv dt = I ion(v, n, s) = [I Ca (v) + I K (v, n) + I s (v, s)], τ dn dt = λ [n (v) n], τ s ds dt = s (v) s. Equation (1) is the current balance equation for the membrane potential, v, with the right-hand side representing the potassium and calcium currents thought to play an important role in the bursting oscillations. Here, I Ca (v) represents the current flowing through fast voltage-gated calcium channels, and I K (v, n) represents the current flowing through slower voltage-gated potassium channels. Finally, I s (v, s) represents a slowly modulated current. Equations (2) and (3) govern the dynamics of the ionic currents, with n and s representing open probabilities, with n, s [0, 1], and λ, τ, and τ s are parameters governing the time scales of the dynamics. The physiological identity of the slow current remains controversial (Sherman, 1996). In this particular model, it is assumed that this current is a potassium current that slowly activates with increasing s, but it is straightforward to modify the model to incorporate a different slow current. Each of the ionic currents is written in the form I X = g X (v v X ) (cf. Ohm s Law), where g X represents the conductance of the channel (the reciprocal of the resistance, that is, the ease with which an ion of species X is able to pass through the channel), and v X is the Nernst potential for ion species X (the membrane potential at which the electrical and concentration gradient across the cell membrane are in balance, that is, at which there is no net flow of the ion species in question). The currents

5 BURSTING OSCILLATIONS 33 used in are (4) (5) (6) I Ca (v) = g Ca (v v Ca ) = ḡ Ca m (v)(v v Ca ), I K (v, n) = g K (v v K ) = ḡ K n(v v K ), I s (v, s) = g s (v v K ) = ḡ s s(v v K ), where ḡ X represents the maximal conductance of the channel, and the factors m (v), n, and s represent the fraction of channels open for the three currents, respectively. The model is completed by specifying the functional form of the steady state functions, 1 (7) x (v) = ( ), 1 + exp vx v θ x for x = m, n, s, and the values of all model parameters, listed in Table 1. The solution to the model is shown in Figure 1. ḡ Ca = 3.6 v Ca = 25 mv v m = 20 mv θ m = 12 mv τ = 20 msec ḡ K = 10 v K = 75 mv v n = 17 mv θ n = 5.6 mv λ = 0.9 ḡ s = 4 v s = 38 mv θ s = 10 mv τ s = 35, 000 msec TABLE 1: Parameter values for the β-cell model, (1) (7), unless specified otherwise in the figure captions. The bursting oscillation shown can be understood with a geometrical perturbation analysis, following the pioneering work of Rinzel (1985). We review the idea here, since we will use it extensively in the analysis of coupled systems. We begin by noting that τ s >> τ. That is, on a short time scale, s can be considered to be constant. This motivates an analysis of the fast subsystem, consisting of the equations for v and n, treating s as a bifurcation parameter: (8) (9) τ dv dt = I ion(v, n; s), τ dn dt = λ [n (v) n].

6 34 GERDA DE VRIES v (mv) s t (sec) FIGURE 1: Behaviour of the β-cell model, (1) (7). The solid curve represents the membrane potential v, and the dotted curve represents the slow activation variable s. The bifurcation diagram for the fast subsystem is shown in Figure 2(a). The curve of steady states, as a function of the bifurcation parameter s, traces out a smooth Z-shaped curve. The knees of the curve represent saddle-node bifurcations (SN1 and SN2), at s = s SN1 and s = s SN2. Also indicated is the stability of the steady states (solid = stable; dashed = unstable). The stability of the steady states on the upper branch changes at a supercritical Hopf bifurcation (HB) at s = s HB. The C- shaped curve represents the branch of stable limit cycles emanating from the Hopf point (the top half of the curve indicates the maximum of v along the limit cycle; the bottom half indicates the minimum of v). At s = s HC, the limit cycle contacts the saddle point on the middle branch of the Z-shaped curve, creating a homoclinic orbit with infinite period, and the system undergoes a homoclinic bifurcation (HC) (also known as a homoclinic connection or a saddle-loop bifurcation). For values of s [s SN1, s HC ], the fast subsystem exhibits bistability between a low-v steady state on the bottom branch of the Z-curve, and a stable limit cycle, corresponding to the silent and active phases, respectively. This bistability, together with the slow dynamics of s, is key in explaining the bursting behaviour. Figure 2(b) shows the same bifurcation diagram, with the nullcline of the slow subsystem, ds/dt = 0, superimposed. During the silent phase then, ds/dt < 0, and so the solution of the full system of equations closely follows the lower branch of the Z-shaped curve leftward. A switch to the active phase is made when SN1 is reached. At that time, the solution of the full system of equations begins to follow the branch of limit cycles. Since ds/dt > 0

7 BURSTING OSCILLATIONS 35 here, the solution moved rightward. A switch back to the silent phase is made when HC is reached. The hysteresis loop is self-sustaining, and results in the bursting oscillation shown in Figure (a) v (mv) 40 HB SN2 60 SN1 HC (b) v (mv) ds/dt = 0 HC SN s FIGURE 2: Bifurcation analysis of the fast subsystem, (8) (9). (a) Bifurcation diagram. Solid curves indicate stable branches, dotted curves indicate unstable branches. Open circles represent saddle-node bifurcations (SN1 and SN2), the filled square represents the Hopf bifurcation (HB), and the filled circle represents the homoclinic bifurcation (HC). (b) As (a), focusing on the region of bistability, with the projection of the slow nullcline, ds/dt = 0, superimposed (thick dashed curve). Also superimposed is the projection of the numerical solution from Figure 1. The bursting solution shown here belongs to the class of square-wave (Bertram et al., 1995; Rinzel, 1987) or fold-homoclinic bursters (Izhikevich, 2000). For a introduction to the literature on classifying bursting oscillations, see Izhikevich (2000). Other types of bursting oscillations can be obtained in different parameter regimes, or with slight modifications to the model equations. The discussion below is applicable to

8 36 GERDA DE VRIES bursting oscillations of the square-wave type. Another widely studied model of this type is the Hindmarsh-Rose model (Hindmarsh and Rose, 1984), developed in the context of bursting oscillations observed in thalamic neurons. 3 Two-cell systems of coupled bursters Pancreatic β-cells are in close electrical contact with their nearest neighbours through gap junctions. Gap junctions are protein channels connecting one cell to another, so that ions can flow from one cell to the other without passing through extracellular space. A single-cell model can be viewed as representing the behaviour of a single cells situated within a well-coupled cluster of cells, such as the Islet of Langerhans. In this section, we will review efforts to understand the effect of gap-junctional coupling in synchronizing the electrical activity of a population of cells. In a coupled-cell system, the variables v, n, and s are indexed by cell number, and a coupling term is added to the equation for v, (1). The resulting system of equations is as follows: τ dv i dt = I (10) ion(v i, n i, s i ) g c (v i v j ), j Ω i (11) τ dn i dt = λ [n (v i ) n i ], ds i (12) τ s dt = s (v i ) s i, where g c represents a constant junctional conductance, i = 1... N, N is the number of cells in the system, and Ω i is the set of indices of the cells that are coupled directly to cell i. The focus will be on extending the bifurcation analysis for single-cell models to two-cell models, that is, i, j = 1, 2. Simulations of larger clusters of cells have shown that the results obtained for two-cell models can be carried over to large populations. In Section 3.1, results are reviewed for two-cell systems of identical bursting cells (nonbursting cells will be coupled in Section 4). It will become apparent that coupling strength (the size of g c ) plays a crucial role in modifying bursting properties. Under strong coupling conditions, the two-cell system behaves exactly as the single cells. In this case, heterogeneity in the model parameters can increase the robustness of the bursting solution, and results for this situation are reviewed in Section 3.2.

9 BURSTING OSCILLATIONS Identical cells Sherman and Rinzel (1991) examined the role of gap-junctional coupling in synchronizing electrical activity within small clusters (5 5 5 cells) of identical cells through numerical simulations. They found that moderate coupling conductances are sufficient for synchrony of the bursts. Moreover, the burst period in the coupled-cell model can be % longer than that of the single-cell model if the coupling is not too strong, showing that diffusive coupling can modify the dynamics of oscillation in addition to promoting synchronization. Motivated by these results, Sherman and Rinzel (1992) studied an idealized case, namely a coupled system consisting of two identical bursting cells. It was shown that synchrony on the level of bursts occurred over a wide range of coupling conductances, but that synchrony on the level of the spikes within the active phase depends on coupling strength. In particular, with weak coupling, numerical simulations indicate that the spikes have smaller amplitude and are phase-locked 180 out-of-phase, and that the burst period is longer. With strong coupling, the system behaves as a single cell, and the spikes are in-phase. With intermediate coupling strengths, asymmetrically phase-locked and quasi-periodic oscillations can be observed. To discover the reasons underlying these observations, Sherman (1994) extended the fast-slow decomposition method to the two-cell case. In this case, the fast subsystem is four-dimensional, and there are two candidates for the bifurcation parameter, corresponding to the presence of two slow variables, namely s 1 and s 2. The observation that s 1 s 2 in numerical simulations justifies the study of the following coupled fast subsystem, with a single bifurcation parameter, s: (13) τ dv i dt = I ion(v i, n i ; s) g c (v i v j ), (14) τ dn i dt = λ [n (v i ) n i ], for i, j = 1, 2, i j. Figure 3(a) shows the pertinent branches of the bifurcation diagram for this system for weak coupling. Instead of one Hopf bifurcation on the upper branch of the Z-shaped curve of steady states, there now are two Hopf bifurcation. The left-most Hopf bifurcation (HB1) gives rise to a branch of in-phase oscillations, while the other Hopf bifurcation (HB2) gives rise to a branch of anti-phase oscillations. In-phase oscillations refer to those oscillations in which both cells spike at precisely the same time, they are perfectly synchronized. Anti-phase oscillations refer to those oscillations in which the two cells spike precisely

10 38 GERDA DE VRIES 180 out-of-phase. Both branches of in-phase and anti-phase oscillations terminate with a homoclinic bifurcation (HC1 and HC2). The other bifurcations labelled do not play a direct role in the bursting oscillations, but are discussed in detail in the original paper by Sherman (1994). For parameter values near s = s SN1, the in-phase oscillations are unstable, while the anti-phase oscillations are stable. Thus, there again is bistability in the fast subsystem, in this case between a stable steady state and a stable anti-phase oscillation. Superimposing the dynamics of the slow subsystem then gives rise to bursting as before, but now the active phase is carried by the branch of anti-phase oscillations, resulting in spikes with a smaller amplitude, and a longer burst period (due to a lengthening of both the active and silent phases), thus explaining the observations noted by Sherman and Rinzel (1992). As coupling strength increases, the branch of anti-phase oscillations destabilizes and completely disappears, while the branch of in-phase oscillations completely stabilizes. That is, for strong coupling, the behaviour of the two-cell system is indistinguishable from that of the singlecell (cf. Fig. 3(d)). For intermediate coupling strengths, asymmetrically phase-locked oscillations can be observed (cf. Fig. 3(c)). These oscillations arise from a pitchfork of periodics bifurcation (PP) on the branch of in-phase oscillations. Asymmetrically phase-locked oscillations are characterized by a fixed phase difference between spikes of the two cells, with the spikes of one cells having larger amplitude than those of the other cell. When none of the phase-locked oscillations are stable (cf. Fig. 3(b)), quasi-periodic oscillations can be observed. These are characterized by rhythmic variations in spike amplitude and phase difference, and are thought to arise from torus bifurcations (TR). For the purposes of this paper, the role of electrical coupling in systems of identical cells can be summarized succinctly as follows. Weak and moderate coupling strengths modulate the bursting properties of the cells. In particular, burst period is maximized with an intermediate coupling strength. With strong coupling, the two-cell system behaves as a single cell. 3.2 Effect of heterogeneity under moderate to strong coupling conditions Two-cell systems of non-identical cells were considered by de Vries et al. (1998), motivated in part by numerical simulations of clusters of heterogeneous cells by Smolen et al. (1993). In the latter study, it was shown that physiologically plausible coupling conductances can synchronize the bursting pattern of the cells even in the presence of substantial heterogeneity of the cell parameters. In the two-cell study by

11 BURSTING OSCILLATIONS 39 de Vries et al. (1998), heterogeneity was introduced ideally, namely by the addition of the parameter β i to the equations governing the dynamics of s i, (12): (15) τ s ds i dt = s (v i ) s i + β i (a) (b) TR v 1,2 (mv) HB1 HB2 SN1 SN2 HC1, TR (c) PP (d) v 1,2 (mv) PP s s FIGURE 3: Bifurcation analysis of the coupled fast subsystem for identical cells, (13) (14). (a) g c = 0.03; (b) g c = 0.1; (c) g c = 0.15 and λ = 1; (d) g c = 1. Saddle-nodes (SN) are represented by open circles, Hopf bifurcations (HB) by filled squares, homoclinic bifurcations (HC) by filled circles, pitchfork of periodics bifurcations (PP) by open squares, and torus bifurcations (TR) by open diamonds. The branch of limit cycles emanating from the left-most Hopf bifurcation (HB1) carries in-phase oscillations, while the one emanating from the right-most Hopf bifurcation (HB2) carries anti-phase oscillations. Branches of limit cycles emanating from the pitchfork of periodics bifurcation (PP) carry asymmetrically phase-locked oscillations. Effectively, the parameter β i shifts the relative position of the slow nullcline for s i with respect to the Z-shaped curve of steady states in the

12 40 GERDA DE VRIES bifurcation diagram for the fast subsystem. Note that the linear transformation s i = s i β i transfers the heterogeneity to the fast subsystem, namely to the v i equation, where it appears as an additional linear potassium conductance. Thus, the heterogeneity is more general than it appears at first glance. We focus on the moderate to strong coupling regime, for which a twocell system with identical cells behaves as the single cell (cf. Fig. 3(d)), and ask what is the effect of heterogeneity on the bursting solution. The numerical solution for a specific case of β 1 β 2 is shown in Figures 4(a) and (b). As before, bursts are synchronized, but now the spiking during the active phase appears to be asymmetric (the spikes of the two cells no longer are in-phase, nor of equal amplitude). In addition, the burst period is lengthened relative to the single-cell burster (not shown). These observations can be explained with a bifurcation study of the coupled fast subsystem. Note that the approximation s 1 s 2 no longer holds (cf. Fig. 4(b)). Nominally, a two-parameter bifurcation diagram would be required to explain the observations. However, s 2 s 1 constant. Thus, if we let (16) (17) δ = (s 2 s 1 )/2, s = (s 2 + s 1 )/2, then we can write s 1 = s δ and s 2 = s + δ, and this motivates the computation of the bifurcation diagram for the coupled fast subsystem with respect to s, holding δ fixed. In particular, we study the following coupled fast subsystem: (18) τ dv 1 dt = I ion(v 1, n 1 ; s δ) g c (v 1 v 2 ), (19) τ dn 1 dt = λ [n (v 1 ) n 1 ], (20) τ dv 2 dt = I ion(v 2, n 2 ; s + δ) g c (v 2 v 1 ), (21) τ dn 2 dt = λ [n (v 2 ) n 2 ], where now δ is taken to be the average value of (s 2 s 1 )/2, based on many iterations of the full two-cell system. The primary bifurcation parameter is s, and the secondary one is δ. From the bifurcation diagrams for v 1 and

13 BURSTING OSCILLATIONS 41 v 2 with respect to s, for a fixed value of δ computed from the numerical solution shown in Figure 4(a) and (b), the diagrams for v 1 with respect to s 1 and v 2 with respect to s 2 can easily be obtained. If the trajectories from Figure 4(a) and (b) then are overlayed, the bifurcation diagram shown in Figure 4(c) results. v 1,2 (mv) (a) v 1,2 (mv) t (sec) (c) s 1, t (sec) s 1, (b) FIGURE 4: (a) and (b) Behaviour of the 2-cell model, (10) (11) and (15), with g c = 1, β 1 = 0, and β 2 = 0.1. The solid curves are the traces for cell 1; the dotted curves the traces for cell 2. The average value of (s 2 s 1)/2 calculated over many bursts is approximately (c) Bifurcation diagrams of the coupled fast subsystem, (18) (21), computed with δ = On the left (solid curves) is the diagram for cell 1; on the right (dotted curves) the diagram for cell 2. Superimposed are the projections of the numerical solutions from (a) and (b). The agreement between the numerical solution of the full system and the bifurcation diagram for the coupled fast subsystem is quite good, a posteriori justifying the symmetric treatment of s 1 and s 2. Moreover, and perhaps more importantly, it indicates that a study of the effects of varying δ in (18) (21) is appropriate. Since δ increases as β 2 β 1 increases, δ can be interpreted as a measure of the heterogeneity

14 42 GERDA DE VRIES present in the system. Using δ as a secondary bifurcation parameter, we can address how varying δ affects the relative position of the pertinent bifurcations in the one-parameter bifurcation diagram. Geometrically, for a fixed value of δ, we collapse the bifurcation diagram of v 1 or v 2 with respect to s onto a line, recording only the locations of the bifurcations (from left to right, those are HB, SN1, HC, and SN2). The two-parameter bifurcation diagram then shows how these positions change as δ is varied. The resulting bifurcation diagram is shown in Figure δ HB SN1 SN FIGURE 5: Two-parameter bifurcation diagram of the coupled fast subsystem, (18) (21), with g c = 1. SN1 and SN2 indicate the curves of saddle-nodes, HB indicated the curve of Hopf bifurcations, and HC indicates the curve of homoclinic bifurcations. The bullet indicates the location of a codimension-2 Takens-Bogdanov bifurcation. s HC For a given horizontal cut through this diagram, the distance between the intersections with the SN1 and HC curves indicates the size of the region of bistability in the fast subsystem. Increasing δ, that is, increasing the heterogeneity in the system, enlarges the region of bistability (by lifting the homoclinic bifurcation (HC) away from the left saddlenode bifurcation (SN1)). Thus, the active phase and, subsequently, the silent phase are lengthened, increasing the burst period, as observed in Figure 4(a). Increasing δ further leads to a loss of the homoclinic and Hopf bifurcations (at the codimension-2 Takens-Bogdanov bifurcation labelled with a bullet), and a concurrent loss of spikes during the active phase (the resulting oscillations resemble relaxation oscillators). This suggests that while a moderate amount of heterogeneity appears beneficial in that it generates more robust bursting with longer burst period,

15 BURSTING OSCILLATIONS 43 a large amount of heterogeneity is harmful in that it eliminates spiking activity (in the context of pancreatic β-cells, this would translate into a loss of calcium entry, needed for the release of insulin). 4 Emergent bursting in two-cell systems of coupled nonbursters The experimental finding by Kinard et al. (1999) that approximately 1/3 of isolated pancreatic β-cells exhibit large-amplitude spiking instead of bursting oscillations motivated the following questions. Can spiking cells contribute to the bursting phenomenon or, pushing this idea further, can bursting be an emergent phenomenon, obtained in a population of cells, none of which is capable of bursting individually? de Vries and Sherman (2000) and de Vries and Sherman (2001) addressed the above questions in detail by studying populations of nonbursting cells, motivated in part by earlier reports of emergent bursting published in Sherman and Rinzel (1992) and Sherman (1994). In the discussion so far, the models for bursting have depended fundamentally on two elements, namely (1) bistability in the fast subsystem between a stable steady state and a stable limit cycle, and (2) a slow process with a time constant comparable to the period of bursting. Inherent in the assumption of both elements for each cell within a population is that each cell is capable of bursting provided that its slow subsystem is tuned appropriately. Bistability can be eliminated from the fast subsystem of each cell by moving the homoclinic bifurcation (HC) so that it coexists with the left saddle-node bifurcation (SN1) in a codimension-1 saddle-node-on-an-invariant-circle bifurcation (SNIC), as shown in Figure 6. Thus, individual cells exhibit either a spiking solution or a steady-state solution, depending on the location of the nullcline of the slow subsystem relative to the bifurcation diagram for the fast subsystem. Emergent bursting then can be obtained under different coupling conditions for different reasons. The mechanisms underlying the phenomenon can be understood in terms of the results obtained in the previous section. Emergent bursting under weak coupling conditions is reviewed in Section 4.1, while emergent bursting under moderate to strong coupling conditions is reviewed in Section Emergent bursting with weak coupling: help from noise Emergent bursting in a two-cell system of identical cells under weak coupling conditions first was reported by Sherman and Rinzel (1992). In Figure 4 of that paper, it was shown that weak coupling can convert spikers to bursters, with the bursts of the two cells being in-phase, and

16 44 GERDA DE VRIES 0 20 HB v (mv) 40 SN2 60 SNIC FIGURE 6: Bifurcation diagram of the single-cell fast subsystem, (8) (9), with λ = 0.8. Projections of two nullclines of the slow subsystem, (3), are superimposed for v s = 22 (upper heavy dashed curve) and v s = 35 (lower heavy dashed curve). s the spikes within the active phase being anti-phase. In his extension of the fast-slow decomposition method to a system of two coupled identical cells, Sherman (1994) showed that the emergent bursting solution is carried by a branch of anti-phase oscillations, present only under weak coupling conditions (cf. Fig. 3(a); in this case, the in-phase branch of oscillations does not terminate at HC1, but at a SNIC). However, the bursting solution exists only for a small range of coupling strengths, and is lost well before the branch of anti-phase oscillations becomes irrelevant due to destabilization, before it disappears altogether. Careful computation of bifurcation diagrams for the coupled fast subsystem revealed that the early loss of emergent bursting is due to the stabilization of the branch of in-phase oscillations near the SNIC as coupling strength increases. Trajectories of the full system that leave the bottom branch of the curve of steady states are captured by the stable portion of the in-phase branch of oscillations, and trapped there. de Vries and Sherman (2000) demonstrated that the addition of small amounts of noise enhances this form of emergent bursting by significantly increasing the range of coupling strengths over which bursting is seen.

17 BURSTING OSCILLATIONS 45 Noise was included by considering the stochastic opening and closing of one of the ion channels represented in I ion (cf. equation (1)). The resulting model was contracted to a Langevin description with the method of Fox and Lu (1994), and solutions were computed numerically. It was shown that the addition of noise aids emergent bursting by perturbing the basin of attraction for the in-phase oscillations so that trajectories of the full system land in the basin of attraction for the anti-phase oscillations, a prerequisite for the bursting solution. Further, a type of stochastic resonance can be demonstrated for this system. Sufficient noise is needed to draw the trajectories of the full system away from the in-phase spiking solution and towards a bursting solution, but too much noise obliterates the deterministic burst dynamics. Using mean intracellular calcium levels as a measure of the efficacy of the electrical activity (motivated by the correlation of intracellular calcium levels with insulin secretion), de Vries and Sherman (2000) showed that the system indeed is optimized with an intermediate amount of noise. Under moderate to strong coupling conditions, the anti-phase branch of oscillations no longer plays a role (it either is unstable, or absent), and this form of emergent bursting no longer is possible. 4.2 Emergent bursting with moderate to strong coupling: help from heterogeneity In this case, coupling is sufficiently strong such that a two-cell system of identical nonbursting cells behaves as the individual cells do. That is, a two-cell system of identical nonbursters is incapable of bursting, no matter how the slow subsystems of the individual cells are tuned. A brief mention of emergent bursting under these coupling conditions, together with heterogeneity in the model parameters, appeared in Sherman (1994). In particular, in Figure 12 of that paper, a bursting solution was shown for a two-cell system of nonbursters. Individually, one of the cells would be a spiker, while the other would be silent. Robust bursting was obtained with a relatively strong coupling strength. As demonstrated in the previous section, under these coupling conditions, heterogeneity in the model parameters affects the relative position of the fast subsystem bifurcations. In particular, heterogeneity lifts the homoclinic bifurcation (HC) away from the left saddle-node bifurcation (SN1), thereby enlarging the region of bistability. In the case of nonbursters, the introduction of heterogeneity first creates a region of bistability by destroying the SNIC bifurcation through a codimension-2 saddle-node loop bifurcation (SNL), essentially splitting the SNIC bifurcation into its two constituent parts, namely a saddle-node bifurcation

18 46 GERDA DE VRIES (SN1) and a homoclinic bifurcation (HC). Increasing heterogeneity further then enlarges the region of bistability as before. The two-parameter bifurcation diagram of the heterogeneity parameter δ versus s for this situation is almost identical to the one shown in Figure 5 (the only difference is that the SN1 and HC curves intersect tangentially at δ = 0), and is not shown here (see Fig. 5(a) in de Vries and Sherman (2001)). The larger the coupling strength, the more heterogeneous the two cells need to be in order to obtain the bistability required for bursting solutions. However, there is a tradeoff, since as the heterogeneity increases, so does the incongruity between the bursting solutions of the two cells (for example, the burst amplitudes of the cells may be very different when one of the cells lies deep in the parameter regime where the cells are silent; in this case, the coupling strength is just enough to cause the cell to burst, but not enough to do so in a significant manner). That is, heterogeneity is beneficial in generating a bursting solution, but only up to a certain point. 5 Discussion and perspectives In this paper, we have brought together related results from a series of papers investigating the role of coupling in synchronizing, modifying, and even generating bursting oscillations in a population of cells. The focus has been on reviewing bifurcation studies of two-cell systems. These bifurcation studies rely heavily on extensions of the fast-slow decomposition method for single-cell models (Rinzel, 1985) to two-cell systems of both identical and nonidentical cells. The extension of the method to a two-cell system of identical cells was made possible by the observation from numerical solutions of the coupled two-cell systems that the slow variables, s 1 and s 2, are approximately the same over the entire period of a bursting oscillation. This suggests the use of s = s 1 = s 2 as the primary bifurcation parameter in the coupled fast subsystem. Similarly, the subsequent extension of the method to a two-cell system of nonidentical cells was made possible by the observation that the difference between the two slow variables is almost constant. This suggests a symmetric treatment of the system, with s 1 = s δ and s 2 = s + δ, where s is the long-time average of s 1 and s 2 and the primary bifurcation parameter, and δ is the long-time average of half the difference between s 1 and s 2. That is, δ can be viewed as a measure of the heterogeneity in the system. It is used as a secondary bifurcation parameter, and allowed a systematic investigation of the effect of heterogeneity on bursting oscillations. Under weak to moderate coupling conditions, coupling serves to in-

19 BURSTING OSCILLATIONS 47 troduce additional branches of limit cycles in the bifurcation diagram for the coupled fast subsystem, in particular branches of anti-phase and asymmetrically phase-locked limit cycles. When stable, these additional branches can carry bursting solutions with active phase oscillations that have smaller amplitudes than those seen in single-cell systems. Also, these additional branches terminate (through homoclinic bifurcations) later than the original branch of limit cycles. That is, the corresponding region of bistability is enlarged. Thus, when an active phase is carried by one of these additional branches, the active phase and, subsequently, the silent phase are lengthened, resulting in an extended burst period. Therefore, coupling can be viewed as not only enriching the dynamics of bursting, but also increasing its robustness. Under strong coupling conditions, the additional branches of limit cycles no longer are present, and the structure of the bifurcation diagram for the coupled fast subsystem is precisely the same as that for the single cell. However, numerical simulations show that the characteristics of the bursting solution for the full system are modified when heterogeneity is included in the slow subsystem. In particular, the burst period is lengthened, and the active phase oscillations have smaller amplitude. These modifications can be understood through a bifurcation analysis of a related fast subsystem, namely one to which the heterogeneity is transferred as described above, with a symmetric treatment of the slow variables s 1 and s 2, and the inclusion of the heterogeneity parameter, δ. Heterogeneity also enlarges the region of bistability. Thus, heterogeneity also can be viewed as increasing the robustness of bursting. Finally, the genesis of bursting from nonbursting cells by coupling was reviewed. The phenomenon of emergent bursting can be understood in terms of the previous results. The key idea here was to first eliminate the region of bistability so that none of the cells is capable of bursting individually. Under weak coupling conditions, the additional branches of limit cycles create regions of bistability that can carry a bursting solution. Similarly, under moderate to strong coupling conditions, heterogeneity first creates and then enlarges a region of bistability necessary for bursting. In the case of weak coupling conditions, emergent bursting is found over a range of coupling strengths smaller than expected on first glance because of the appearance of a second type of bistability when coupling strength is increased: bistability between two limit cycles, namely a small-amplitude anti-phase limit cycle and a large-amplitude in-phase limit cycle. For such coupling strengths, the solution of the full system is trapped on the branch of in-phase limit cycles, even though the branch of anti-phase limit cycles is still present and stable. The branch

20 48 GERDA DE VRIES of in-phase limit cycles is not bistable with the lower branch of the curve of steady states, and hence there is no bursting oscillation anymore, but a large-amplitude spiking oscillation instead. By including a moderate amount of noise into the system, the solution can escape from the branch of in-phase limit cycles to the branch of anti-phase limit cycles, upon which bursting can ensue. Thus, noise can enhance this form of emergent bursting by significantly increasing the range of coupling strengths over which bursting is seen. The investigation into the role of heterogeneity in the model parameters in the genesis of bursting from nonbursting cells reviewed here is similar to a previous study by Smolen et al. (1993), but there is a very important difference. In the latter study, a population of heterogeneous cells was considered as well. There, the heterogeneity was present in parameters of both the fast and slow subsystems. Most of those cells were incapable of bursting individually, either being silent of spiking continuously. Provided the coupling between the cells is sufficiently strong, the population behaved roughly like a cell with average properties. That is, as long as the fast subsystem of the average cell of a population exhibits bistability, and the corresponding slow subsystem is tuned appropriately, then the whole population will burst. In the emergent bursting studies discussed here, an additional condition was imposed, namely that the average cell is a nonburster, by removing the bistability from the fast subsystem. Here, bursting can occur provided the coupling is sufficiently strong, but not too strong. The inclusion of noise in channel dynamics has been investigated previously as well. Since recording from individual cells isolated from islet revealed noisy electrical activity, with many cells showing noisy spiking activity rather than bursting, Atwater et al. (1983) hypothesized that the intrinsic burst dynamics of individual cells are disrupted by stochastic channel fluctuations. In intact islets, the effect of these fluctuations would be shared, and bursting recovered. The simulations by Chay and Kang (1988) and Sherman et al. (1988) supported the channel-sharing hypothesis. They used stochastic ion channels, and demonstrated that noise could cause premature transitions between the active and silent phases in individual cells, shortening the bursting period. With sufficient noise, bursting was obliterated. Bursting was recovered when the cells were members of a population, provided the population was sufficiently large, and coupled sufficiently strongly. In that study, noise in individual cells can be viewed as an undesirable intruder that can be overcome by coupling. In the study of emergent bursting with noise reviewed here, coupling serves to introduce desirable branches of limit

21 BURSTING OSCILLATIONS 49 cycles that may be obscured when coupling is too strong, but that are allowed to play a role when noise is included. Simulation studies have demonstrated that the results obtained with bifurcation analyses of two-cell systems carry over to larger systems (see for example de Vries and Sherman (2000) and de Vries and Sherman (2001)). Scaling arguments for clusters of cells that are sufficiently large so that boundary effects can be ignored show that if one increases the number of cells, identical solutions are obtained for larger clusters by increasing the coupling strength appropriately. In particular, increasing the number of cells by a factor N requires an increase in the coupling strength by a factor of N 2/d, where d is the dimension of the cluster (de Vries and Sherman, 2000). In summary, coupling plays an important role in bursting oscillations, not only in synchronizing individual members within a population, but also in determining the character of bursting, and possibly in the genesis of bursting. Thus, the behaviour of a population of cells may be better represented by a two-cell system than a single-cell model. The coupling studied reviewed here have shown that populations of coupled β-cells exhibit more robust bursting solutions than individual cells, and thus may be more reliable in the secretion of insulin than a large number of individual β-cells. Moreover, heterogeneity and noise, which both are inheritantly present, may further increase the reliability of bursting solutions. REFERENCES 1. I. Atwater and L. Rosario and E. Rojas, Properties of calcium-activated potassium channels in the pancreatic β-cell, Cell Calcium 4 (1983), Richard Bertram,Manish J. Butte, Tim Kiemel and Arthur Sherman, Topological and Phenomenological Classification of Bursting Oscillations, Bull. Math. Biol. 57(3) (1995), R. Bertram and A. Sherman, Dynamical complexity and temporal plasticity in pancreatic β-cells, J. Biosci. 25 (2000), T. R. Chay and H. S. Kang, Role of single-channel stochastic noise on bursting clusters of pancreatic β-cells, Biophys. J. 54 (1988), G. de Vries, Multiple bifurcations in a polynomial model of bursting electrical activity, J. Nonlinear Science 8 (1998), G. de Vries and A. Sherman, Channel sharing in pancreatic β-cells revisited: enhancement of emergent bursting by noise, J. Theor. Biol. 207 (2000), G. de Vries and A. Sherman, From spikers to bursters via coupling: Help from heterogeneity, Bull. Math. Biol. 63 (2001),

22 50 GERDA DE VRIES 8. G. de Vries and A. Sherman and H-R. Zhu, Diffusively coupled bursters: effects of cell heterogeneity, Bull. Math. Biol. 60 (1998), R. F. Fox and Y.-N. Lu, Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels, Physical Review E 49 (1994), J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. Lond. B 221 (1984), A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London) 117 (1952), E. M. Izhikevich, Neural excitability, spiking and bursting, Int. J. Bifurcation and Chaos 10 (2000), J. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, 1998, New York. 14. T. A. Kinard, G. de Vries, A. Sherman and L. S. Satin, Modulation of the Bursting Properties of Single Mouse Pancreatic β-cells by Artificial Conductances, Biophys. J. 76 (1999), bibitemri85 J. Rinzel, Bursting oscillations in an excitable membrane model, Ordinary and Partial Differential Equations, Lecture Notes in Mathematics 1151 (B. D. Sleeman and R. J. Jarvis, eds.), Springer, New York, 1985, John Rinzel, A Formal Classification of Bursting Mechanisms in Excitable Systems, Mathematical Topics in Population Biology, Morphogenesis, and Neurosciences, Lecture Notes in Biomathematics 71 (E. Teramoto and M. Yamaguti, eds.), Springer-Verlag, New York, 1987, Y. Sato, M. Anello and J.-C. Henquin, Glucose regulation of insulin secretion independent of the opening or closure of adenosine triphosphate-sensitive K + channels in β cells, Endocrinology 140 (1999), A. Sherman, Anti-phase, asymmetric and aperiodic oscillations in excitable cells - I. Coupled bursters, Bull. Math. Biol. 56 (1994), A. Sherman, Theoretical aspects of synchronized bursting in β-cells, Pacemaker Activity and Intercellular Communication (J. D. Huizinga, ed.), CRC Press, Boca Raton, FL, 1995, A. Sherman, Contributions of modeling to understanding stimulus-secretion coupling in pancreatic β-cells, Am. J. Physiol. 271 (1996), E362 E A. Sherman and J. Rinzel, Model for synchronization of pancreatic β-cells by gap junctions, Biophys. J. 59 (1991), A. Sherman and J. Rinzel, Rhythmogenic effects of weak electrotonic coupling in neuronal models, Proc. Natl. Acad. Sci. USA 89 (1992), A. Sherman and J. Rinzel and J. Keizer, Emergence of organized bursting in clusters of pancreatic β-cells by channel sharing, Biophys. J. 54 (1988), P. Smolen, J. Rinzel and A. Sherman, Why pancreatic islets burst but single β cells do not: the heterogeneity hypothesis, Biophys. J. 64 (1993), Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1 address: devries@math.ualberta.ca