Conditions for synchronization and chaos in networks of β-cells

Conditions for synchronization and chaos in networks of β-cells J.M.W. van de Weem ∗ J.G. Barajas Ram´ırez ∗∗ R. Femat ∗∗ H. Nijmeijer ∗ ∗

Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. ∗∗ Instituto Potosino de Investigaci´ on Cient´ıfica y Tecnol´ ogica, IPICyT, Camino a la Presa San Jos´e 2055, Col. Lomas 4a. secci´ on, San Luis Potos´ı, San Luis Potos´ı, 78216, M´exico Abstract: The proper function of β cells is characterized by a spiking-bursting activity in their membrane potential. These cells are electrically coupled within the Langerhans islets of the pancreas, the topology of their interactions induce synchronization and the activation of switchoff cells. This fact is relevant to the processes of insulin production and release, as such these phenomena are relevant in the mechanisms of glucose homeostasis in the body. However, not much is known about the topology of β-cell network in the real-world, nor about conditions of their synchronization. In particular, about the conditions for the emergence of chaotic bursting activation in networks of inhibited β-cells not much is known. We investigate these phenomena in terms of complex network theory, and numerically show under what conditions synchronization and chaotic bursting emerges for β-cell network models. Keywords: Coupled systems; Pancreatic β-cells; Complex networks, Chaos, Synchronization. 1. INTRODUCTION The electrically coupled networks of β cells are the insulin producing part of the Langerhans islets in the pancreas; as such, they play a fundamental part in the homeostasis of glucose in the body. The insulin liberation process in β cells is directly related to the electrical spiking-busting activity of their membrane. A particularly significant aspect of their operation is that in isolation, a single β cells can be on an activated state, which is characterized by a periodic spiking-busting behavior, or alternatively, the isolated β cells can be inhibited, that is, with its insulin liberation mechanism switch-off and the electrical activity of its membrane on a stable equilibrium point. However, when both activated and inhibited β-cells are naturally electrically-coupled via gap junction channels within the pancreatic islets of Langerhans, experimental studies show that complex, possibly chaotic, busting activity is generated, as the β-cells of the entire network become activated due to their interactions; even if in isolation most of the β cells are inhibited [Smolen et al. (1993)]. In other studies, the synchronization of pancreatic cells was associated with their electrical connectivity as a force that coordinates their electrical bursting and metabolic oscillations [Pedersen et al. (2005)]. However, there is not much known about the topology and coupling strengths between βcells in real-world conditions inside the Langerhans islets, nor about conditions for the chaotic activation of β-cell networks. So we ask: Can the coupling between β-cells 1 2 3 4

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induce chaos?, and when is such network in synchrony? Using knowledge of complex networks might help to get a deeper comprehension on these phenomena. In this paper we are interested in conditions under which β-cells synchronize or produce chaotic behavior. Our particular interests lie in numerical and experimental modeling of network of these cells. 2. A SINGLE β-CELL The model that is used in this research to simulate the electrical activity of β-cells is given by the following ordinary differential equations [Pernarowski (1998)]: du = f (u) − w − k(c) dt dw 1 (1) = (w∞ (u) − w) dt τ¯ dc = ε(h(u) − c). dt where u refers to the membrane potential, w refers to an ionic channel activation, and c refers to the glucose concentration of the cell. The terms f(u), k(c), h(u), and w∞ (u) that are used in (1) are defined as: f (u) = f3 u3 + f2 u2 + f1 u, f3 = − a3 , f2 = aˆ u, f1 = τ1¯ − a(ˆ u2 − η 2 ), 3 h(u) = β(u − uβ ), k(c) = τ¯c, w∞ (u) = w3 u + w2 u2 + w1 u + w0, w3 = τ¯ − a3 , w2 = aˆ u, w1 = τ1¯ − a(ˆ u2 − η 2 ) − 3¯ τ, w0 = −3¯ τ . with parameters for an activated β cell were adjusted from experimental data to be: a = 41 , η = 34 , u ˆ = 23 , β = 4, ε = 0.0025, τ¯ = 1, and uβ = −0.954. The trajectories of single β-cell model are shown in Fig. 1.

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We are interested in showing how the connections of the network can force an inhibited β cell to become chaotically activated. To this end, we set the parameter uβ of the model (1) to uβ = −1.300, leaving the other parameters as above. For these new parameter values the electrical dynamics of the β cell move towards an stable equilibrium point at ([¯ u, w, ¯ c¯] = [−1.0472, −1.1043, 1.0071]). Then, we characterized the dynamical behavior of the inhibited β cell calculating its Lyapunov exponents over 10,000 seconds. As shown in Figure 2, the exponents along the directions of u, w and c, are h1 = −0.0277, h2 = −0.3621 and h3 = −1.1233. All exponents are negative, which is consistent with a stable equilibrium point behavior. To corroborate our results experimentally, the system of equations (1) was implemented on an electronic circuitry, in this more realistic realization, all cells are different due to uncertainties in the electronics.

Fig. 1. States u,w and c of an activated β-cell in isolation. 3. COMPLEX NETWORKS: SYNCHRONIZATION AND EMERGENCE

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The N node networks that will be discussed are linearly and diffusively coupled only in their first state variable (u), i.e. the coupling between a pair of cells is dependent on the difference in output voltage of these cells. Then, the entire network can be written as: N X x˙ i = f (xi ) + c aij Γxj , i = 1, 2, . . . , N (4) j=1

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where xi = (ui , wi , ci ) are the state variables of ith node, the constant c ∈ R represents the coupling strength, and the internal coupling matrix equals Γ = diag(1, 0, 0) in our situation. If there is a connection between node i and node j (i 6= j), then aij = aji = 1; otherwise, aij = aji = 0. aii can be defined using the degree ki of node i, which is the total number of connections of node i: N X aii = −ki = − aij , i = 1, 2, . . . , N. (5)

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Fig. 2. Lyapunov exponents for an inhibited β-cell A measure for chaotic behavior of a cell in isolation is the Lyapunov exponent. It describes the long-term development of a direction in the phase space of a system, where a positive exponent represents expansion and a negative exponent represents contraction. When there exists at least one positive Lyapunov exponent, the system can be considered as chaotic, with the magnitude of the exponent reflecting the time scale on which the system dynamics become unpredictable [Wolf et al. (1985)]. The Lyapunov exponents can be calculated using numerical methods form time series or from the system description by: 1 hi = lim |J(t, x0 )ui |, i = 1, 2, 3 (2) t→∞ t where hi is the ith Lyapunov exponent of the isolated cell along the direction ui ; J(t, x0 ) is the Jacobian matrix of equations (1) evaluated at a randomly selected initial condition x0 , with {u1 , u2 , u3 } is a set of orthonormal vectors in the tangent space of the system. Then, the Lyapunov exponents can be ordered as: h1 ≥ h2 ≥ h3 .

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A connectivity matrix A = (aij ) ∈ RN ×N can be created, which represents the coupling configuration of the network. There are no isolated clusters considered inside the network, meaning that A is a symmetric, irreducible matrix having a zero eigenvalue with multiplicity 1 and all other eigenvalues strictly negative [Wang and Chen (2002)]: 0 = λ1 > λ2 ≥ λ3 ≥ . . . ≥ λN . (6) 3.1 Synchronization According to [Wu and Chua (1995)], a network will synchronize if the largest non-zero eigenvalue λ2 of A is negative enough, i.e. T λ2 ≤ − . (7) c where c > 0 is the coupling strength of the network and T > 0 is a positive constant feedback gain such that zero is an exponentially stable point of a node in isolation.

3.2 Emergence of chaotic behavior The dynamical behavior of the entire network can be characterized in terms of its transverse Lyapunov exponents (tLes), which describe the deviation of the network trajectories from its synchronized solution. As presented in [Barajas-Ram´ırez and Femat (2008)], the tLes of a linearly and diffusively coupled network of identical systems are determine from the Lyapunov exponents of a node in isolation (hi ), and the eigenvalues of the connectivity matrix (λk ) as: (a)

µi (λk ) =hi + cλk , (8) for i = 1, 2, 3; and k = 2, . . . , N ; with µi (λk ) the tLe associated to the ith local direction of a single cell and the k th node along the corresponding eigenvector of the coupling matrix A. The synchronized solution of the network is stable if all the transverse direction are contracting, that is, if all the tLes of are negative. From (6) and (3), for positive coupling strength c < 0, the synchronized solution becomes more stable as the connectivity increases (|c| → ∞). However, since we are interested in the emergence of chaos, the coupling strength is chosen to be negative (c < 0), in this way, as the coupling strength increases, the synchronized solution becomes locally unstable. The smallest value of c such that this occurs, is when the largest tLe becomes positive, i. e., µ1 (λN ) > 0 or equivalently: h1 > |c|. λN

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The emergence of chaotic solutions require an extra condition, to ensure that all the trajectories of that move away from the unstable synchronized solution remain bounded. In [Barajas-Ram´ırez and Femat (2008)] this is expected when the overall sum of tLes associated with each node are negative: m X

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Nevertheless, these limits on (9) and (10) are not concrete boundaries for the emergence of chaos, since bounded chaotic behavior may not occur when only the largest tLes is positive. The distribution of positive tLes due to the network topology model plays a decisive roll on the emergence characteristics of the network. Furthermore, for chaotic behavior to emerge the dynamics of most of the nodes in the network must combine: positive tLes; and its compensation, in the form of a negative overall sum of Lyapunov exponents. In the following section, these results are use to determine when a network of inhibited β-cells produces chaotic bursting, by virtue of their interactions. 4. NETWORK TOPOLOGY MODELS To answer the questions stated in the introduction, we consider some benchmark network topologies. Including regular and complex models:

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Fig. 3. Regular networks: (a) Globally coupled topology. (b) Nearest neighbor topology. [Wang and Chen (2002)]

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Fig. 4. Complex networks: (a) Small-world topology. (b) Scale free topology. [Wang and Chen (2003)] 4.1 Globally coupled In the globally coupled model, all cells are linearly and diffusively connected to each other. The eigenvalue spectrum of the connectivity matrix for this network model has all its non-zero eigenvalues equal to -N. Accordingly, the condition for synchronization (7) becomes: T (11) c so for a given positive coupling strength, network synchronizes is much easier for networks with a large number of nodes. λ2gc = −N ≤ −

4.2 Nearest-neighbor coupled In the nearest-neighbor network model (Fig. 3b), the N cells are arranged in a ring, where each cell is connected only to its 2l nearest cells, with l > 0 being an even integer. The largest non-zero eigenvalue of this model is λ2nn =
Pl/2 −4 j=1 sin2 jπ [Wang and Chen (2002)], which means N that λ2nn goes to zero for large networks. As such, the nearest-neighbor network will be very difficult to synchronize when the number of nodes is large. 4.3 Small-world The small-world model proposed by Newman and Watts is a transition between above regular and totally random networks (Fig. 4a). The starting point for this model is

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The above topologies can also be experimentally realized on electronic circuitry using multiple cells on a circuit board and a computer to realize the appropriated connection scheme among the cells. 5. RESULTS 5.1 Synchronization The threshold value of the system is numerically determined as T = 1.6. Because there are no analytical formulas present to calculate λ2 for some network topology models, they are determined numerically, with the minimum coupling strength (c > 0) for synchronization calculated from (7). This is done 20 times for network sizes of 4 to 100 cells of each topology to get a good average. As expected, the minimum coupling strength to synchronize a nearest-neighbor network increases rapidly when the network grows (Fig. 5a). For l = 2, this results in a coupling strength c = 405.4 for a network of 100 cells. The needed coupling strength is reduced significantly to c = 29 when each cells is connected to its six nearest neighbors, but is still very large. From equation (11), it would be expected that the coupling strength of a globally coupled network is inversely proportional to the number of cells. This is verified by Fig. 5b, where a network of 4 cells needs c = 0.4 to synchronize and c = 0.016 is enough for a network of 100 cells. In general, the small-world topology is showing a decreasing line with respect to the number of cells (Fig. 5c). Reducing the number of nearest-neighbors in this model will give a different result mainly in small networks: a stronger coupling is needed. But when the network is sufficiently large, it can be noticed that the network with l = 2 gets close to the network with l = 6. On the other side,

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Another complex network can be found in the scale-free model of Barab´asi and Albert (Fig. 4b). The construction algorithm of this models starts with a small number of cells m0 . Then, at every time step, a new node introduced and connected to m ≤ m0 of cells already-existing in the network. The connections are chosen using the preferential attachment mechanism: the probability Πi→j that a new ith cell be connected to already-existing j cell depends on the degree ki of ith cell related to the total number of connections in the network: Πi = ki /Σj kj [Wang and Chen (2003)]. This algorithm results on networks with a few cells having many connections (“hubs”) and many cells with only a few connections. There is no analytical formula present for calculating the eigenvalues, the only thing that can be said for this network is that the largest non-zero eigenvalue tends to −1 for large N and dependent on m0 and m.

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Fig. 5. Minimum coupling strength needed for synchronization of β cells on different network topologies: (a) Nearest-neighbor, (b) Globally coupled, (c) Smallworld, and (d) Scale-free

reducing the probability of a new connection significantly increases the coupling strength needed for synchronization. In Fig. 5c we can see the differences between p = 0.20 and p = 0.05, where c = 0.12 and c = 0.55 respectively are needed for synchronization of a network of 100 cells. The scale-free network shows an increasing coupling strength which remains constant for sufficiently large networks. This is expected because, when the network is sufficiently large, the addition of a new cell doesn’t change significantly the topology of the network. Increasing the number of connections per new cell from m = 3 to m = 5 decreases the minimum coupling strength for synchronization from 1.05 to 0.5 (Fig. 5d). It can be concluded that all topologies behave differently. While the minimum coupling strength for the nearestneighbor and the scale-free topologies increase with the number of nodes in the network, the opposite happens for the globally coupled and small-world topologies. Fig. 6 is a numerical simulation of where nine β cell models are linearly and diffusively interconnected into a globally coupled network. Initially, the cells are uncoupled and bursting is asynchronous. The cells are coupled after 1000 seconds with c = 0.18, which is stronger than the minimum coupling strength for synchronization satisfying the synchronization criterion of c ≥ − 1.6 −9 = 0.1778. Then, after a small transitory phase the entire network synchronizes to the bursting behavior of a single isolated cell (fig. 1). An interesting point is how fast the synchronization is achieved, it happens within two bursting periods, even though the coupling strength is only slightly stronger than necessary for synchronization.

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Fig. 7. 1st-, 2nd- and 3rd-largest tLes and the line when the largest sum of tLes is zero for different topologies of networks ranging from N = 4 to N = 100. a) Nearest neighbor (l = 6), b) Globally coupled, c) Small-world (p = 0.05, l = 6) and d) Scale-free (m0 = m = 3). 5.2 Chaotic behavior The conditions on (9) and (10) give guidelines for the emergence of chaos in networks of inhibited β-cells: the values of the coupling strength when the largest tLe becomes positive. We identify the transition value, when this tLe is zero, and calculated it for networks of different sizes coupled in different topologies. The transition values become a constant for the nearestneighbor network in an early stage of the network growing process (Fig. 7a). According to the results on subsection 3.2, synchronization to the equilibrium point would be expected only with a very small negative coupling strength of c = −.003 for the 6-nearest-neighbor network. Bounded chaotic behavior can be expected between c = −0.003 and c = −0.042, whereupon for more negative values

Fig. 8. Trajectories of a globally coupled network of nine cells for different coupling strengths. a) all cells go to the equilibrium point for c = −0.03, b) asynchronous bursting at c = −0.08 (two cells shown) and c) continuous bursting at c = −0.50 (one cell shown). unbounded behavior may exist. These values get closer to zero as the number of nearest-neighbors is increased. In the globally coupled model, the tLes becomes positive more easily for large networks (Fig. 7b). The coupling strength needed to get the first tLe, the second tLe and the largest sum of tLes positive, converges to c = −0.004 for a network of 100 cells. The third tLe gets positive for c = −0.011 in the same network. The results for the small-world model (Fig. 7c) look similar to the previous model. The coupling values when the tLes become positive are more separated and need a more negative coupling strength to get the cells activated in this example. But a higher probability for new connections or more nearest-neighbors in the initial network will result in less negative tLes, thus its behavior is more or less similar to that of the globally coupled network. Also the scale-free model behaves in the same manner as the previous two models: when the network is larger, it is easier to activate the cells (Fig. 7d). When the number of connections for a newly introduced cell is increased, the

the synchronized bursting behavior is not perfect followed by all the nodes, these effects are due to the differences and uncertainties associated with the electronic realization of the cells and their interconnections, which are not considered in the theoretical analysis presented. 6. DISCUSSION

Fig. 9. Experimental realization of the synchronization of a globally coupled network of seven cells with coupling strength c = 0.6. For viewing purposes only four cells are shown (the number of inputs in the oscilloscope). tLes will be less negative. In overall, it can be concluded that it is easier to attain chaotic bursting behavior for a large network with many connections than for a small network with few connections. The results of all topologies are very close to each other, especially for large networks. The only exception is the nearest-neighbor model. Numerical simulations of a globally coupled network of nine inhibited cells is used to illustrate the different kinds of chaotic bursting induced by the network interconnections. The largest-, second largest- and third largest tLe become positive at c = −0.003, c = −0.04 and c = −0.125 respectively. The largest sum of tLes is positive for values below c = −0.058. It would be expected that all cells go to the equilibrium point until the largest tLe becomes positive. However, this behavior is still encountered until c = −0.034 (Fig. 8a). The oscillatory responses during the first seconds are due to the choice of the initial values. Asynchronous bursting is found for c = −0.035 and below (Fig. 8b). The bursting period and amplitude increase for more negative values of c till eventually all cells continuously burst (fig. 8c), but the network trajectories remain always bounded. This example illustrates that the tLe method can create certain regions to predict bounded chaos busting, although it doesn’t give concrete boundaries. 5.3 Experimental results An experimental realization of the β cell model in (1) in electronic circuitry was use to validate the theoretical results of presented above. The results shown in Fig. 9 correspond to a computer-based interconnection scheme where seven electronically realized activated β cells were globally coupled, for a coupling strength of c = 0.6 the entire network synchronizes to the same busting behavior. Is worth noting that in this experimental realization, the coupling strength needed to achieve synchronization is three times as high as theory says it should be, also

In this report we used four different network topologies for ensembles of β-cell models and investigated conditions for synchronization and emergence of chaotic bursting on activated and inhibited cells, respectively. An electronic setup to experimentally validated the theoretical results was successfully realized. The results presented have potential applications on explaining some aspects of biological processes. In particular, the electronic realization may help biologists with experimental setups of β-cells, due to its ease of interpretation and manipulation. In this contribution only deals with uniform coupling, i. e. the same coupling strength for all connections, with changes in the connectivity caused only by the network topology. For future research, it maybe possible to have different coupling strengths for different connections. Although this might give a more accurate representation of the real coupling between β-cells in the Langerhans islet, it will also be more difficult to analyze. Another remark is that partial synchronization, that is, where only a part of the network synchronizes, was observed numerically and experimentally for coupling strengths smaller than the coupling strength needed for complete synchronization. Further knowledge about this phenomenon maybe of significance for some specific biological purpose, however, this will be subject for future research. REFERENCES Barajas-Ram´ırez, J. and Femat, R. (2008). Transition to complex behavior in networks of coupled dynamical systems. 17th IFAC World Congress, Seoul Korea. Pedersen, M., Bertram, R., and Sherman, A. (2005). Intraand inter-islet synchroization of metabolically driven insulin secretion. Biophysical journal, 89, 107–119. Pernarowski, M. (1998). Fast and slow subsystems for a continuum model of bursting activity in the pancreatic islet. SIAM J. Appl. Math., 58, 1667–1687. Smolen, P., Rinzel, J., and Sherman, A. (1993). Why pancreatic islets burst but single β-cells do not: The heterogeneity hypothesis. Biophys. J., 54, 1668–1680. Wang, X. and Chen, G. (2002). Synchronization in scalefree dynamical networks: robustness and fragility. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49, 54–62. Wang, X. and Chen, G. (2003). Complex networks: small-world, scale-free and beyond. IEEE Circuits and Systems Magazine, 3(1), 6–20. Wolf, A., Swift, J., and Wolf, H. (1985). Determining lyapunov exponents from a time series. SIAM J. Appl. Math., 16, 285–317. Wu, C. and Chua, L. (1995). Synchronization in an array of linearly coupled dynamical systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42(8), 430–447.