The synchronization of pancreatic β-cells by gap junction coupling

Transcription

1 The synchronization of pancreatic β-cells by gap junction coupling G14MBD Mathematics MSc Final Project Report Summer 2009 School of Mathematical Sciences University of Nottingham Kyle Wedgwood Supervisor: Professor S. Coombes Division: Applied August 2009 I have read and understood the School and University guidelines on plagiarism. I confirm that this work is my own, apart from the acknowledged references. 1

2 We formulate a simple planar nonlinear integrate and fire model to describe the bursting behaviour of pancreatic β-cells. We demonstrate that, although simple, the model can support a variety of bifurcations, and that it may recapture the dynamics of more complicated and higher dimensional models. Upon building networks of these IF cells with gap junction coupling, we find that we can simulate experimental data, with the network showing degrees of synchronicity even where single cells are heterogeneous and exhibit irregularity. 2

3 Contents 1 Introduction 5 2 Existing Models Chay-Keizer Chay-Keizer with Endoplasmic Reticulum A Glycolytic Mechanism Phantom Burster Networks Models with Stochasticity Introducing Heterogeneity Absolute Integrate-and-Fire Phase Plane Analysis Phase Response Curve Fast-Slow analysis Firing Map Gap Junction Coupling Numerical Simulations Coupling Strength Coupling Architecture Noise Inhomogeneities Experimental Data Discussion 80 6 Acknowledgments 85 3

4 A Proof that period is independent of drive 86 B XPP Code for Single Cell 86 C Functional relation between V and A for section D MATLAB Code for Network 87 E W 1 from section

5 1 Introduction Pancreatic β-cells, comprising 65-80% of the islets of Langerhans, are responsible for the production and secretion of the hormone insulin into the bloodstream. [1] The presence of insulin encourages cells in the liver, muscle and fat tissue to take up glucose in the bloodstream and store it as glycogen. Conversely, the absence of insulin causes cells to use fat as an energy source. Through this system, the body can regulate blood glucose levels. Dysfunction or destruction of β-cells can lead to both Type I and Type II diabetes. [2] β-cells, like many other secretory cells, display oscillations in membrane potential. In the regulatory system, high levels of glucose lead to repetitive oscillations in transmembrane potential, similar to action potentials in neuronal cells. The change in membrane potential leads to a reduction in intracellular calcium levels, which in turn leads onto the release of insulin. [3] Isolated cells are observed to fire spikes irregularly, but clusters of cells which are coupled together are seen to synchronise behaviour in the form of bursting. We define bursting as a dynamic state in which a cell fires two or more spikes, termed the active phase, followed by a period of quiescence, termed the silent phase [4]. Experimental data suggest that clusters of size 20 or greater can exhibit this synchronisation between cells. The coupling between cells is thought to be due to gap junction coupling. [5][6] Mathematical models currently used to describe the bursting behaviour of β-cells have a high dimensionality, making them difficult to analyse with the standard techniques. Many of the more recent models are based on a standard template of 5

6 the Chay-Keizer model [3], which we shall discuss more thoroughly in the following section. The Chay-Keizer model is preferred as it can capture the dynamics of the cells, but is a relatively simple model. Based upon work by John Rinzel [7], progress may be made in these higher dimensional systems by taking advantage of the inherent differences in timescales between different variables in the model to reduce the dimensionality of the problem. Developed by Lapicque in 1907, the integrate-and-fire (IF) model is a computationally cheap 1-dimensional model to describe the behaviour of neuronal cells [8]. The IF model may however be generalised to fit the behaviour of any cell exhibiting spike like oscillations in membrane potential. The basic IF model is a linear model, in which voltage grows subject to some external drive until it reaches some threshold, at which point it is reset and the process repeats. As such, the model supports periodic orbits with a single discontinuity. Adaptive IF models introduce a second, coupled variable to the model which may give rise to more interesting behaviour [9]. Upon reset, the the adaptive variable is changed by some amount and integration proceeds again. The adaptive models can exhibit more interesting types of solution, bursting being the one we are most interested in. The Izhikevich model [10], formulated as a model to describe the behaviour of cortical neurons, is a simple IF model in which the linear voltage variable is replaced with a quadratic one. This model can replicate the behaviour of the cortical neurons, but we may change parameters in this model to obtain fits to the β-cells. Although the model itself is simple, due to the quadratic term in the Izhikevich model, we cannot find analytical solutions in the model. Furthermore, such mod- 6

7 els have been shown to support chaotic behaviour for certain parameter sets [11]. Based on work by Stephen Coombes and Margarita Zacchariou [12], we introduce a different kind of non-linearity into the basic adaptive IF model. In our model we will approximate the quadratic term of the Izhikevich model using a continuous, piecewise linear term. Using this approximation, we can now find analytical solutions to the model. Dr. Paul Smith has provided me with some experimental data on a cluster of five cells. In his experiment Dr. Smith monitored the levels of calcium levels in each of these cells over a sampling period in which the cells were exposed to glucose. Dr. Smith went on to infer coupling architecture between cells using statistical techniques. The time course of the average Ca 2+ across the cells is shown in figure 1. Figure 1: Time averaged calcium levels across 5 pancreatic β-cells exposed to glucose in Dr. Smith s cultured cells In this project we first aim to develop a nonlinear integrate and fire model which 7

8 captures the bursting behaviour of pancreatic β-cells. We then look to find synchronicity in networks of cells built through gap junction coupling. Our main aim here is to replicate the experimentally observed behaviour whereby individual cells are heterogeneous and may oscillate irregularly, but networks of cells act in a more regular manner. Finally, we look to reproduce, using our model, the qualitative behaviour of the data sets provided by Dr. Smith. Upon variation of parameters we find that the model supports a number of different solution types, including single spike oscillations, subthreshold oscillations and periodic bursts. We proceed to use phase plane techniques to analyse the possible bifurcations of such orbits, and find that the model, although simple, displays a variety of different smooth and non-smooth bifurcations. We use Malkin s theorem to find the response of the phase of a singe spike, and subthreshold orbits to external perturbations. We briefly perform analysis in the fast-slow limit based on [7] in order to write an explicit expression for the period of single spike and bursting orbits, whilst demonstrating we do lose some of the overall bifurcation structure of the model in this limit. We construct a firing map for the adaptive variable in the model and find the same qualitative structure as for the map for the Chay-Keizer model [13], and consider the bifurcations of this map. At the network level we perform numerical simulations looking for phase locked states. We introduce heterogeneity and stochasticity into the network and demonstrate that under these conditions we can find solutions for which individual cells oscillate irregularly but that the network displays more regularity, and that we may even reproduce more complex bursting behaviour in the network. 8

9 In section 2, we will review some of the existing models used to describe the behaviour of islets, and look at some of the more recent work in this field. In section 3 we put forward the model, and carry out analysis at the single cell level, including studies of the bifurcation structure, phase response curves and the firing map. In section 4 we build gap junction coupled networks of such cells and perform numerically analysis to look for synchronicity in the network with possible irregularity at the single cell level. 2 Existing Models 2.1 Chay-Keizer In 1983 Teresa Chay and Joel Keizer developed a model [3] to describe the bursting behaviour of clusters of pancreatic β-cells. Their model was based around a Hodgkin-Huxley mold, including a current balance equation incorporating two K + channels, a Ca 2+ and a leakage channel and typical Hodgkin-Huxley gating dynamics. The Chay-Keizer model also describes the evolution of the cytosolic Ca 2+ concentration. In the first model the dynamics of the Ca 2+ variable are defined by the flux of calcium across the membrane. The Chay-Keizer model describes the dynamics of 5 variables, and as such it is difficult to make any analytical progress. Progress may be made by first reducing the system to a 3 dimensional one and then by noting that the timescale of the voltage and recovery variable dynamics is much shorter than that of the Ca 2+ one. In the fast-slow limit [7], the full system may be studied by dividing it into a fast subsystem and a slow one. If we then consider the Ca 2+ to be quasi-steady, it may be treated as a parameter in the fast subsystem, which is now a planar 9

10 one. The fast subsystem may then be analysed by phase plane techniques, using c as a bifurcation parameter. A similar approach is to base the model on a Moris-Lecar framework, and then add an extra variable to describe the calcium dynamics as in [14]. The Moris- Lecar model is a planar system and so we now have a 3-dimensional system. The model may again be studied using the fast-slow analysis, as the voltage and gating dynamics are much faster than those of the calcium dynamics. Since the model is now a planar one, we may use phase plane techniques to analyse its behaviour. The voltage, v-nullcline has a characteristic Z-shape, whilst nullcline of the gating variable n is a monotonically increasing function of n. Variation of the bifurcation parameter c sweeps the v-nullcline to the left (increasing c) and right (decreasing c). Where c is low and negative, then exists only one stable fixed point on the upper branch of the v-nullcline. Increasing c causes this fixed point to lose stability via a Hopf bifurcation and so we get stable limit cycle solutions. Increasing c further still we have a saddle node bifurcation where the n-nullcline tangentially crosses the lower knee of the v-nullcline. This now gives two further fixed points, with the one on the lower branch of the v-nullcline being stable, and the one on the middle branch being a saddle point. The system now exhibits bistability. The limit cycle is still a stable solution and trajectories starting above the saddle point are attracted towards it. Trajectories starting below the fixed point are attracted towards the stable fixed point. As c is increased further still, there is a homoclinic bifurcation as the limit cycle touches the saddle point and is destroyed. The only stable solution is the stable fixed point and so now all trajectories tend towards this point, however large enough 10

11 Figure 2: Bifurcation diagram for the simplified Chay-Keizer model based on a Moris-Lecar framework. As c is increased, the system moves through a Hopf bifurcation (HB), followed by a saddle-node bifurcation. As c increases further, the limit cycle is destroyed by a homoclinic bifurcation(hc) perturbations cause the system to undergo one oscillation and so the system is excitable. Since the qualitative behaviour of the system changes from periodic orbits to quiescence, we expect that sufficiently large oscillations in c will cause the system to exhibit bursting. This behaviour is summarised in figure 2. We note that that the Hopf bifurcation occurs where c is negative. Biophysically c represents the intracellular level of calcium and so cannot take negative values. It suffices to say that oscillations are present until the homoclinic bifurcation. We also note that the amplitude of the oscillations seems relatively unchanged by c, but the period of oscillations increases dramatically as we approach the homoclinic bifurcation. 11

12 2.2 Chay-Keizer with Endoplasmic Reticulum One of the criticisms of the Chay-Keizer model is that in its original form, it is unable to account for some of the variation in calcium dynamics; namely that experimental evidence has suggested that calcium levels plateau quickly, rather than in the stepped fashion seen in the model. Further to this, it appears that calcium levels may actually start to decrease before the end of the active; a phenomenon not seen in the model, suggesting that cytosolic calcium may not in fact be the driver of bursting in β-cells. In 1996, Teresa Chay added an endoplasmic reticulum calcium component to the cytosolic component [15]. Here, through SERCA pumps, the ER acts as a calcium sink during the active phase of a burst, and as a source during the silent phase. The actions of the ER now add a slow component to the cytosolic calcium and make for better fit to experimental data. The model now predicts that cytosolic calcium plateaus quickly and then oscillates during a burst; a better mechanism for driving bursting. The addition of the ER adds more flexibility to the dynamics of bursting as the period may now range from a few seconds to minutes based on the conductance of the calcium activated potassium channel. 2.3 A Glycolytic Mechanism The bursts produced in the Chay-Keizer model are periodic, but bursts, at least in mice islets have been observed to be modulated by some underlying rhythm. In 2004, Keola Wierschem and Richard Bertram considered a glycolytic mechanism for this underlying rhythm whereby oscillations in glycolysis, the first stage of aerobic respiration, give rise to the complex bursting patterns through the mediation of intracellular ATP levels [16]. In this glycolytic mechanism, the conductance of 12

13 the ATP mediated K + channel varies through oscillations in glycolysis. As this process is only, at most, weakly dependent on cytosolic calcium levels, trajectories may be dragged outside of the region of bistability, which may mean that oscillation may no longer be resettable under variations in membrane potential [14] and so we may no longer see bursts. Another ATP related idea which has been put forward is that rises in cytosolic calcium inhibit mitochondrial ATP production and this gives rise to the same change in ATP mediated potassium channel conductance. The major drawback in the hypotheses of the SERCA pumps and glycolytic mechanisms, is that, although the results match nicely to the data, it is difficult to test them experimentally. For example, compartmentalising the intracellular calcium experimentally is difficult [14]. However, these models do present experimentalists with more ideas to test in the long run. 2.4 Phantom Burster Another model worth mentioning is the so-called Phantom Burster model presented in 2000 by Bertram et al. [17]. The authors considered a mechanism which would allows single cells to fire on both slow and fast timescales, whilst allowing networks of cells to burst on a medium timescale, which the authors term medium bursting. Here the authors split the slow variable into two variables so that the system now comprises both a planar fast subsystem and a planar slow subsystem. They postulate that the medium bursting behaviour is due the interaction between these two variables which have time constants of 1-5s and 1-2min respectively. These timescales are still large in comparison to the fast voltage dynamics and so the authors argue that the similar fast-slow analyses may be used here. 13

14 The authors find that they can reproduce some of the medium burst behaviour, and indeed turn fast bursters into medium bursters, by changing the conductance of the faster of the two slow variable s 1. The authors do however point out that they cannot postulate to what this channel may represent biophysically, due to the lack of availability of pharmological blockers for possible channels. Instead, they model the dynamic voltage clamp they used in their biological experiments to represent it. Whilst this may represent conditions for bursting in a laboratory setting, it does not represent an in vivo situation, and so more consideration is needed to identify this channel. 2.5 Networks Individual β-cells have been seen to oscillate irregularly, with some cells oscillating, some bursting and some staying silent. Even those cells bursting may do so on different timescales to one another. Furthermore, these timescales are reported to be faster than those of the bursts seen at the network level [17]. Due to this irregularity at the single cell level, modelling these individual β-cells is difficult, and the models presented thus far are a much better fit for a synchronised population of cells. Arthur Sherman [18] considered 2 Chay-Keizer like cells coupled together. He demonstrated that out of phase solutions can only exist where the coupling strength is low. As this increases, the synchronised state stabilises, and that the period of bursts increases and then decreases as this strength increases. Burst period increases are associated with spike amplitude decreases. The mechanism for this is dependent on the existence of a homoclinic bifurcation in the model. 14

15 This is something studied in more detail by Postnov et al.[19] who considered synchronisation of modified Van der Pol oscillators. They considered how different types of phase locked states between oscillators exchanged stability as parameters changed. They also found a period doubling route to chaos under certain parameter sets. However, these sets required some parameters to be negative, which would not be appropriate in a biophysical model of β-cells. Sherman [18] also highlighted the importance of gap junction coupling for synchronisation as synaptic coupling may not necessarily be zero even if identical cells are at the same phase. Importantly, Sherman demonstrated that coupling a silent cell with a cell undergoing periodic oscillations may result in bursting at the network level as the network passes between both solution types. Whilst studies of 2 coupled cells allow us to better understand network behaviour, real islets contain many more cells, with different intrinsic dynamics, and so the results may not be applicable once we scale up the network. In the remainder of this section, we will consider larger networks with more complicated structure. 2.6 Models with Stochasticity Although we may not understand entirely the processes involved in the oscillations at the single cell level, it may be useful to introduce some stochasticity into the model. We note here that the stochastic process refers to the stochastic nature of one or more of the individual processes in the model rather than simple probabilistic noise which would also cause the model to behave more irregularly. In a series of papers addressing this by Sherman et al. [20][21] the authors consider the process of the opening and closing of the Ca 2+ activated K + channel to be a stochastic one, by assigning opening and closing probabilities to the chan- 15

16 nels. As such the the fraction of open channels in a network is then a Markov chain process. Due to the stochastic nature of the model, it can reproduce some of the irregular behaviour seen in small clusters of cells, however the authors admit that the dynamics in isolated cells cannot be described by their model. The authors then use gap junction coupling to build networks of cells. Upon simulation they find that they can produce more regular behaviour at the network level, with regular bursting behaviour with fluctuations in the burst and silent phase duration. Further regularity can be achieved with clusters of 167 cells. Whilst there are approximately 2000 β-cells per islet, and so these results are favourable compared to the size of islets, bursting behaviour has been experimentally observed with clusters of size 20 and above. The authors here also point out that they have not considered the nature of ATP mediated K + channels and that the model ignores the gap junctional resistance between cells which may result in delays in the influence of one cell on another. In a later paper by Arthur Sherman and John Rinzel [21], they apply these techniques to what they term a multicell model, now taking into account finite gap junctional resistances. Modelling a multicell arranged as a cube with 125 cells, they show that these cells are able to synchronise their bursting behaviour, though they make the distinction between burst synchronisation and spike synchronisation, which is not observed in their model. They also find that in their stochastic model for multiple cells, that there is an optimal coupling strength, for which the cells exhibit the highest degree of synchronisation, in both spike and burst behaviour. They also find that their model supports traveling waves of electrical 16

17 activity, and that this behaviour does not require any pacemaker cells. To fully replicate the behaviour of inherently stochastic processes in β-cells, adopting a stochastic approach to modelling the intracellular behaviour seems a good strategy to adopt. The authors of the paper above have already indicated that they may seek to adopt a stochastic approach to modelling the ATP mediated K + channels. It is very encouraging that these stochastic models are able to display the high levels of burst synchronicity observed in pancreatic islets, as well as irregularity at the small network level. However, as we start to introduce more stochasticity into the model, it quickly becomes harder to perform analysis on using the techniques that are currently available. In addition to this, owing to the small time steps needed to evolve the model, the models become computationally expensive and hence much slower to run. When taking into account more than one type of channel, the relative speed on the ion channels and their response to membrane potential become even more important, and may further reduce the time step needed for accurate simulation. 2.7 Introducing Heterogeneity Another way of introducing more irregular behaviour at the single cell level is to introduce heterogeneity into the cells. In their 2008 paper Benninger et al. [22] studied numerically, coupled cells based on a phantom burster model. They introduced heterogeneity into their network by randomly distributing the electrical coupling strength between adjacent cells. They find that the model predicts traveling waves of calcium levels amongst the network. In this paper, the authors refer to the cells as being synchronised in this case, but we are careful with this notion as we expect zero time lag between oscillations in cells in the network. 17

18 The authors also seem to suggest that it is the heterogeneity of coupling between cells which drives the calcium waves, although we would expect to see travelling waves for homogeneous adjacent coupling of homogeneous cells. Upon decreasing the mean coupling strength, they observe that a number of cells desynchronise with the rest of the population and that the propagation speed of the travelling wave is significantly affected. The authors compared their simulated results with those they obtained experimentally, and found a decent fit for the most part, with some discrepancies. The authors hypothesise, based on these results that the loss in insulin pulsatility in type II diabetes may be caused in a downregulation in gap junction forming connexin proteins. They support this claim by showing experimentally that knocking out such a connexin disrupts wave propagation. We may also consider a more basic form of heterogeneity, by simply adding a heterogeneous leakage current to each cell. For two cells, such heterogeneity may cause a shift away from a synchronous solution to one presenting phase lags. In [6] the authors consider 2 coupled cells with heterogeneous drive in which they find that raising the coupling parameter reduces both the phase lag and magnitude of the bursts produced. Perhaps the most important result comes from work by Smolen et al. [5]. They show that the gap junction coupling profoundly shapes and modifies electrical activity. Bursting therefore results from the cooperation of cells with different properties, and that this acts to make the behaviour in islets more robust and perhaps more suited to their biological role. We have presented some of many models and techniques built to describe the bursting mechanics of β-cells. As with most mathematical modelling, there is a 18

19 payoff between how accurately the model fits the data and how much analysis can be done on the model. The Chay-Keizer model is the preferred template for β-cell models due to its relative cheapness compared to other models. However, it too is a high dimensional model and so we cannot find any closed form solutions for it. The fast-slow analysis pioneered by John Rinzel allows for significant progress in analysing the bifurcation structure of higher dimensional models. However, such analysis requires us to be in the singular limit, where one set of timescales is infinitely fast compared to another. Since we are primarily dealing with electrical signals and intracellular dynamics, this is not necessarily the case. Additionally, when dealing with bifurcation structures of the reduced model, we may miss more interesting bifurcations of the full model; bifurcations of the reduced model may not correspond to the bifurcation structure of the full model [6]. Work looking at stochasticity and heterogeneity at the network allows us to capture some of the more interesting behaviour at the single cell level whilst demonstrating synchronicity at the network level. However this adds even more complexity to the model and analysis becomes incredibly hard to make, and so we must resort to numerical techniques to make any progress. 3 Absolute Integrate-and-Fire We now introduce a different kind of model to capture the dynamics of pancreatic β-cells. Following studies done by Stephen Coombes and Margarita Zacchariou [12] we will use a non-linear integrate-and-fire model with constant drive: C dv dt τ da dt = f(v)+i γa (3.1) = α(v v s ) δa (3.2) 19

20 α, γ, C, τ, δ > 0 I R With reset conditions v v reset,a a + g τ when v = v thresh (3.3) This model exhibits spike adaptation; when the voltage reaches threshold, the calcium variable is increased by a constant amount. Upon variation of parameters, the model can exhibit tonic firing, burst firing, subthreshold oscillations or may be quiescent. For the function f(v), we chose the piece-wise linear function: s 1 (v v s ) v>v s f(v) =, s 1,s 2,v s > 0 (3.4) s 2 (v s v) v<v s Here, v s represents a switch in the dynamics, and the line v = v s may be thought of as a switching manifold [24]. For both v>v s and v<v s, the dynamics are governed by a linear system. We can solve the equations in each region and then ensure continuity of the solution across v = v s to find a solution to the full system. It should be noted that not all orbits cross v s and so the evolution of these orbits will simply be defined by a system of two linear equations. We begin by non-dimensionalising the system. We first note that the derivative dv/dt is invariant under the transformation ˆv = v v s. Substituting this into our system yields: C dˆv dˆt τ da dˆt = ˆf(ˆv)+I γa (3.5) = αˆv δa (3.6) 20

21 Figure 3: Top: Bursting orbits produced by the full Chay-Keizer model, Bottom: Bursting orbits produced by the absolute integrate-and-fire model (AIF) s 1ˆv, ˆv >0 ˆf(ˆv) = s 2ˆv, ˆv <0, s 1,s 2 > 0 (3.7) Setting V = ˆv δs 1, A = a s 1 and ˆt = s 1 C t, we obtain the following system: dv dˆt 1 da ω dˆt = F (V )+µ βa (3.8) = αv A (3.9) Where ω = C τ δ s 1, β = γ δs 1, µ = I and: δs 2 1 V, V > 0 F (V )= SV, V < 0, S > 0 (3.10) Where S = s 2 s 1. The reset conditions are now: V V r, A A + g τ when V = V th (3.11) With V r =(v r v s )/s 1 and V th =(v th v s )/s 1. After dropping the hats, we may now recast the equations in each of the two regions (now V > 0 and V < 0) in 21

22 matrix form as follows: For V>0 For V<0 d V = 1 β V + µ (3.12) dt A αω ω A 0 d V dt A = S αω β V + µ (3.13) ω A 0 The numerical simulations shown in figures in this report are solutions to the full, dimensional equations (3.1, 3.2). Unless otherwise stated, the parameters used in simulations are shown in the following table: Parameter Value Parameter Value s α 0.11 s γ 4 v th -20 δ 1 v r -40 I 1 v s -50 g 5 C 1 τ 30 To solve equations (3.12) and (3.13), we first look for solutions for the complementary function, that is the solution to the homogeneous system of equations, in which we do not consider the constant vector on the righthand side. Defining X =(V,A) T the solution to a general linear equation X = MX where M is a 2x2 matrix is: X = c 1 ν 1 e λ 1t + c 2 ν 2 e λ 2t (3.14) Where λ 1,2,ν 1,2 are the eigenvalues and eigenvectors of the matrix M. For complex eigenvalues the general solution can be simplified into a solution involving trigonometric functions. 22

23 Figure 4: Blue: 1-spike oscillation with γ =0.11, Red: 1-spike burst with γ =0.05. For both orbits C =1 and g =4. Alsopicturedontheleftistheswitchingmanifold. Ontherightweobservethedifferenceinperiod for the two kinds of orbit We differentiate between two kinds of 1-spike periodic orbit. Orbits which remain solely in the V positive region will be referred to as 1-spike oscillations, Orbits which cross over into the region where V is negative shall be referred to as 1-spike bursts. Although the traditional definition of a burst is an orbit containing more than than spike, we see that there is a greater time difference between spikes for our 1-spike bursts than the 1-spike orbits, analogous to the silent phase in traditional bursts. Shown in figure 4 are two examples of such orbits. In order to solve for the 1 spike burst orbit in 4 we must solve the dynamics of each regions and then ensure continuity across V = 0. We take initial data to be V (0) = V r,a(0) = Ā = A( ) + g/τ, where is the period of the burst. Before solving the system, we shall first define t 1 and t 2 as the times which trajectories 23

24 cross between regions, namely: t 1 = inf{t : t>0,v(t) 0} t 2 = inf{t : t>t 1,V(t) 0} We also denote the matrices in 3.12 and 3.13 by M 1 and M 2 respectively. The eigenvalues of M are given by 2λ 1,2 = Tr(M) ± Tr(M) 2 4det(M) and thus for V>0: λ 1,2 = (1 ω) ± (1 ω) 2 4ω(αβ 1) 2 (3.15) We can verify easily that for V>0, these eigenvalues are real. The eigenvectors scale as ν 1,2 (β,1 λ 1,2 ) T. Note that that the form given for ν 1,2 here may not be normalised, but this simply provides an expression of the proportionality. The constants c 1 and c 2 will pin down the exact solution to the initial conditions. We now find the particular integral of the solution. The non-homogeneous part of the system is constant, so we look for a constant PI, X PI = d =(d 1,d 2 ) T. The derivative of this is zero, so substituting this into the equations, we obtain: 0 = 1 β d 1 + µ (3.16) 0 αω ω 0 Thus X PI = A 1 ( µ, 0) T. Finally, to pin down the constants c 1 and c 2, we use d 2 the initial conditions detailed earlier: V r d 1 Ā d 2 = β β 1 λ 1 1 λ 2 c 1 c 2 (3.17) And we take the matrix product of the inverse of the above matrix with the vector on the left hand side to determine c 1,c 2. 24

25 We find that the eigenvalues of M 2 are complex, and so we expect oscillatory dynamics. Writing λ = x + iy, we find the solution for t t 1 to be: X(t) = c 3 e x(t t 1) (ω + x)cos(y(t t 1)) y sin(y(t t 1 )) αω cos(y(t t 1 )) + c 4 e x(t t 1) y cos(y(t t 1)) + (ω + x) sin(y(t t 1 )) αω sin(y(t t 1 )) + d 3 d 4 Where c 3,c 4,d 3,d 4 are found in a similar fashion to 3.16 and 3.17 using [0,A(t 1 )] T as initial data at t = t 1 After the orbit crosses over in V>0 region, its dynamics are again described by 3.12 and so the solution will be: X(t) =c 5 ν 1 e λ 1(t t 2 ) + c 6 ν 2 e λ 2(t t 2 ) + d (3.18) Where we now use initial data at t = t 2, namely [0,A(t 2 )] T, to compute c 5 and c 6. We have now computed the solution in each portion of the orbit and thus solved for the one spike orbit in 4. To close the system, we have to find initial data such that (V ( ),A( )) T = V th, Ā g T τ. Construction of multispike bursting orbits can be performed by piecing together solutions to equation (3.12), taking initial data to be [V r,a(t i )] T where the t i are the times of the successive spikes during a burst and by then repeating the above process as V crosses 0. As we will see, it is useful to write the solutions to our system in matrix exponential form. The general solution to a system of linear differential equations X = M X + b is given by [23]: X(t) =G(t) X(0) + K(t) b where G(t) =e Mt, K(t) = t 0 G(s)ds (3.19) For full details of the form of G and K, we refer to reader to Appendix A of [23]. 25

26 To write our solution in this form, we define the following matrices: G 1 (t) =e M1t, K 1 (t) = G 2 (t) =e M2t, K 2 (t) = t 0 t 0 G 1 (s)ds (3.20) G 2 (s)ds (3.21) Our solution then becomes: G 1 (t) X(0) + K 1 (t) b, 0 t<t 1 X(t) = G 2 (t) X(T 1 )+K 2 (t) b, t 1 t<t 2 G 1 (t) X(T 2 )+K 1 (t) b, t 2 t< (3.22) The T i now refer to the time spent each region, as opposed to the t i, which refer to the times from the start of the orbit, t = 0. More rigorously T i = t i t i 1, i =1, 2, 3 where t 3 = and t 0 = 0. The period of the burst is now given by = 3 i=1 T i. We may write the second and third components of 3.22 in terms of the initial data, by noting that: X(T 1 ) = G 1 (T 1 ) X(0) + K 1 (T 1 ) b (3.23) X(T 2 ) = G 2 (T 2 ) X(T 1 )+K 2 (T 2 ) b = G 2 (T 2 ){G 1 (T 1 ) X(0) + K 1 (T 1 ) b} + K 2 (T 2 ) b = G 2 (T 2 )G 1 (T 1 ) X(0) + {G 2 (T 2 )K 1 (T 1 )+K 2 (T 2 )} b (3.24) and then by substituting these identities back into our solution These identities ensure that the solution is matched across the different regions. To close the system, we must find the value of Ā. Writing Γ = G 1(T 3 )G 2 (T 2 )G 1 (T 1 )andb = G 1 (T 3 ){G 2 (T 2 )K 1 (T 1 )+K 2 (T 2 )} b, we have that Ā =(1 Γ 12) 1 (Γ 21 V r + B 2 +g/τ). As can be seen in figure 5, the model also exhibits subthreshold period oscillations, in which the model now never reaches threshold, and thus these orbits are continuous. We note here that the period of these oscillations is invariant under 26

27 Figure 5: Multiple subthreshold oscillations. Here τ =3,C =1,β =0.2. As I is increased in integer values from 1 (black) to 4 (blue) the amplitude of oscillation increases, but the period is unchanged. Also shown is the switching manifold v = v s changes in the drive. Details of this can be seen in Appendix A. For initial data in this situation, we choose X(0) = [0,A 0 ] for some, as yet undetermined A 0, and evolve first in the region V > 0. We may then write down the equation of these orbits in matrix exponential form as: G 1 (t) X(0) X(t) + K 1 (t) b 0 t t 1 = G 2 (t) X(T 1 )+K 2 (t) (3.25) b t 1 t t 2 Where we now enforce the condition X(T 2 )= X(0) G 2 (T 2 )G 1 (T 1 ) X(0) + {G 2 (T 2 )K 1 (T 1 )+K 2 (T 2 )} b = X(0). We may find the value of A 0 in a similar fashion as for the bursting orbit, and hence close the system. 3.1 Phase Plane Analysis In order to better understand the behaviour that we see in the model, it is useful to look at the (V,A) phase plane. The nullclines of the system are found by setting dv/dt, da/dt = 0. Thus the V -nullcline has V shape and the A-nullcline 27

28 Figure 6: Left: Voltage trajectory corresponding to the red orbit in figure 4. Right: Corresponding orbit in the phase plane, along with nullclines (light blue). Also pictured are the threshold and reset values is simply a straight line. The equations of the nullclines are given by: 1 (V + µ) V>0 β A = and A = αv (3.26) 1 ( sv + µ) V<0 β Figure 6 shows the phase plane corresponding to the 1-spike burst in figure 4. We note that in the the current situation, the two nullclines do not intercept anywhere, and thus there are no fixed points in the system. We know that the A- nullcline must pass through the origin. V will always increase and reach threshold in this situation. The reset conditions ensure that we have a stable limit cycle. The type of limit cycle will be determined by the parameter values; we may have either a bursting or tonic solution. Upon variation of parameters the model may have one, two or no fixed points. In this section we hope to classify these fixed points and establish the bifurcations of the model. We shall denote the fixed point, where it exists, on the right hand branch of the V -nullcline by ς +, and the fixed point on the left branch by ς. We may find the 28

29 values of these fixed points as ς + = µ, αµ αβ 1 αβ 1 and ς = µ, αµ αβ+s αβ+s. Due to the discontinuous nature of our system, ς + only exists as a fixed point provided that its V -value does not exceed V th. We must therefore have µ V th (αβ 1) for the existence of ς + as a fixed point. At this point we shall also define two important points, firstly, where the line V = V r intercepts the V -nullcline: (V r,a crit )= V r, 1 (V β r + µ) and secondly, where the two branches of the V - nullcline meet: (0,A knee )= 0, µ. β We shall first consider a single fixed point on the right branch of the V -nullcline. Stability can be inferred by considering the eigenvalues of the Jacobian, M 1 at this point, which which have already and is given in equation (3.15). At a Hopf bifurcation we require that Tr(M 1 )=0 and that det(m 1 ) > 0. Setting ω = 1 gives us Tr(M 1 )=0 and det(m 1 )=4(αβ 1). For the Hopf bifurcation we thus require that αβ > 1, but we have this automatically by the condition for the nullclines to intercept. Thus we have a Hopf bifurcation at ω = 1, which may be seen in figure 7. This gives us a condition for stability on the timescales in the model, namely: C τ < s 1 δ (3.27) We may create a fixed point of the left branch of the V -nullcline by allowing µ to be negative. In this situation, there may or not be another fixed point on the right hand branch. If the gradient of the A-nullcline is too steep, namely if αβ > 1 µ V th the Jacobian M 2 at ς. then there is no fixed point. We calculate the eigenvalues λ +, of 2λ +, = ω S ± (1 + S) 2 4ω(S + αβ) (3.28) We know that S, β, α, ω > 0, and so ω s<0 and 4ω(s + αβ) > 0 and so this fixed point has (λ +, ) < 0 and is stable. Since this can be used to de- 29

30 Figure 7: Bifurcation diagram with C =1,α =0.2 andincreasingτ. We see a Hopf bifurcation (HB) as well as a grazing bifurcation to 1-spike oscillations. Also shown is a plot of the period of the oscillations as τ is increased. scribe the stability of any fixed point on the left branch, we infer that the entire branch is stable. In the regime where this is the only fixed point, all trajectories will tend towards this point. The system in this case is excitable: perturbations which results in trajectories starting to the right of the right hand branch of the V -nullcline will cause the system to undergo one oscillation before returning to ς. If we now allow αβ < 1 µ V th we create a second fixed point on the right hand branch of the V -nullcline. The Jacobian M 1 is unchanged, but we now have that det(m 1 ) < 0, and so ς + is now a saddle node. In this regime, the model now exhibits bistability. Trajectories on the right of ς + will be pushed rightward up to threshold and, provided parameters are set up correctly, will repeat this after reset. We thus have a limit cycle in the right region of the phase plane. Clearly 30

31 the solution to which trajectories are attracted to depends on the initial conditions of the trajectory. Given that we have a saddle point separating a stable fixed point and a stable limit cycle, we expect a homoclinic bifurcation to occur at some point in parameter space, and we shall consider this in due course. As we increase the value of µ through 0, the two fixed points come together and annihilate one another in a saddle node bifurcation, whereafter there are no fixed points. As discussed, now there exists only a stable limit cycle, the dynamics of which are decided by other parameters. If αβ > 1 we longer have this saddle node bifurcation as there is only one fixed point for any value of µ. As µ passes through 0 the fixed point moves from the left to the right hand branch of the V -nullcline. Where ω>1, there is no change in stability as the fixed point changes branch. When ω<1, the fixed point on the right hand branch is now an unstable focus. The real part of the eigenvalues of ς is given ω S<0, whereas the real part of the eigenvalues for ς + are given by 1 ω>0, thus as µ passes through 0, there is a discontinuous change in the sign of this real part and so we have a discontinuous Hopf bifurcation where ω<1, αβ > 1atµ =0. Work carried out by D.J. Simpson and J.D. Meiss [24] allows us to make more statements about the bifurcation described above. The main findings of their paper are that the Hopf cycle consists of two spiral segments, one either side of the switching manifold. We can infer the criticality of the Hopf bifurcation using by calculating the difference between the ratios of the real and imaginary parts of the Jacobians either side of the switching manifold. Finally, they show that the amplitude of the oscillation grows linearly with the bifurcation parameter µ, rather than with µ as in traditional Hopf bifurcations. This can be seen more clearly in figure 8, where we vary I in the full system. 31

32 Figure 8: Bifurcation diagram with C =1,τ =3,α =0.2 andincreasingi. We see a Hopf bifurcation at I =0. TheoscillationsproducedgrowlinearlywithI and their period is invariant under changes in I until we see a grazing bifurcation to 1-spike oscillations. Also shown is a plot of the period of oscillations as I is increased. 32

33 Figure 9: Bifurcation diagram showing the homoclinic (HC) and saddle node (SN) bifurcations under variation of I with C =1 As mentioned, in the case where αβ < 1 µ V th and µ<0 we expect a homoclinic bifurcation as the limit cycle comes into contact with the saddle node, destroying the limit cycle (see figure 9). After the homoclinic bifurcation the only stable solution which exists is ς and all trajectories will tend towards this point. Variation in a number of parameters may result in this homoclinic bifurcation. Increases in the magnitudes of µ, β or α will shift ς + up towards the limit cycle. In addition to this, increases in g and decreases in τ may result in a homoclinic bifurcation as, after the reset conditions are enforced, the limit cycle is pushed towards to saddle node. As so many parameters may result in this bifurcation, it is difficult to form explicit expressions which tell us the point in parameter space where this occurs. We may form an implicit relationship between the parameters by finding a relationship 33

34 between V and A on the limit cycle. We do this by solving: dv da = V + µ βa αωv ωa (3.29) Subject to the initial condition (V,A) =(V r, Ā). To find out which parameter sets result in the homoclinic bifurcation, we then simply substitute the coordinates of ς + into the orbit equation. This gives us a relationship of the form H = H(V r, Ā) where details of the form of H may be found in ap- µ, αµ αβ 1 αβ 1 pendix C. We note that the parameters g and τ are tied up in the definition of Ā = A( ) + g/τ. We have seen in section 3 that we may have two qualitatively different 1-spike orbits. The change between these orbits occurs where αβ > 1 as the 1-spike oscillation grazes the line V = 0, that is, where the governing equations of the system change. Since s 2 <s 1, trajectories crossing into the left hand region will be forced leftwards and track along the left hand branch of the V -nullcline until A<A knee and the voltage may reach threshold once more. This will cause the period of the oscillations to increase, and thus we have a grazing bifurcation resulting in a period increase at this point. The bifurcation diagram, figure 10 suggests that the crossing of the orbits themselves at this point is discontinuous. We find that in a narrow window as we vary parameters, the 1-spike oscillation becomes a doublet, before becoming a 1-spike burst. We may find the parameter sets for which this bifurcation occurs implicitly using the relation found above. The orbit grazes V = 0 at the point (0,A knee ), and so we simply substitute this point into the relationship we found previously to obtain these parameter sets. As the amplitude of the subthreshold oscillations increasing, V becomes closer to reaching threshold. Where the the orbit grazes the line V = V th we have a bifurcation from smooth subthreshold oscillations to 1-spike bursting orbits. At 34

35 Figure 10: Grazing bifurcation resulting in period addition (PA) as orbits cross the switching manifold under variation of α, withc =1,g =3. Weseethatthe1-spikeoscillationbecomesadoubletbeforebecoming a1-spikeburst.thismanifestsitselfasadiscontinuityinthebifurcationdiagram. this point the invariance of the period on the drive is destroyed. The period is now much shorter, as the discontinuity in the orbit means trajectories are covering shorter distance. As we increase the drive further we see that the 1-spike orbit then becomes a multispike-orbit, the number of spikes increasing as we increase the drive. In order to find the parameter sets for which this occurs we must find a relation analogous to the one we found previous, but taking into account that the trajectory now enters the left region. To this we must solve dv/da =( SV + µ βa)/(αωv ωa) and then match solutions where V =0. Due to the complicated nature of this matching, we shall not do this here, but this may be seen in figure 8. Where αβ < µ V th +1, ς + exists on the right of the line V = V th. As such, trajectories do not feel this fixed point. As we increase αβ beyond µ V th +1 we may therefore expect a discontinuous bifurcation to occur. If ω>1, ς + will be 35

36 Figure 11: Non-smooth bifurcation as ς + crosses v = v th where ς + is stable under variation unstable, and there will be no qualitative change in behaviour at αβ = µ V th +1. If ω<1, ς + is now stable and trajectories will be attracted to this point. We therefore have a bifurcation to a stable solution, analogous to a discontinuous tangential bifurcation. Where µ is negative, ς + is always unstable, and thus we require µ>0 for this type of bifurcation. The bifurcation diagram in this case is shown in figure 11. We now move to consider multispike bursting orbits. These orbits necessarily require a large difference in timescales. Due to the fast voltage dynamics, trajectories are projected quickly rightwards until they reach threshold. The reset conditions are applied and depending on the value after reset, trajectories will either be projected to threshold once more, or be projected leftwards, towards the left branch of V -nullcline. In this case, they track along this nullcline until A<A knee and trajectories are once more projected rightwards. The portion of an orbit spent tracking along the V -nullcline corresponds to the silent phase of bursting, where calcium levels in the cell are falling, and the oscillations between 36

37 Figure 12: Top: Voltage evolution during a burst. Parameters are as in the table in section 3. Bottom: Corresponding calcium evolution reset and threshold correspond to the active phase of bursting, in which cytosolic calcium builds up in the cell. We note that the qualitative form of the calcium build up is similar to the stepped buildup of calcium in the original Chay-Keizer model. A plot of the calcium evolution during a burst may be seen in figure 12. Changes in the number of spikes in a bursting orbit can occur through 2 different non-smooth bifurcations. We highlighted at the beginning of this section the important of the point (V r,a crit ). After reset, trajectories will restart at a point on the line V = V r.ifa r <A crit, where A r is the value of A at reset, the system is free to fire again and we obtain another spike. If A r >A crit + then trajectories will be projected leftwards and the system will not fire another spike. We are careful to include a small O( C ) as trajectories may still fire another spike τ if A crit <A r <A crit +, under certain conditions. This is because close to the V -nullcline, A varies faster than V. For values of A to the left of the A-nullcline, da/dt < 0. This means that if A r is within a distance of to A crit, trajectories 37

38 will move downwards across the V -nullcline. The trajectory will now be on the right of the V -nullcline, where dv/dt > 0 and so will be able to spike again. Based on the above, we expect to be able to add a spike n +1 to an n-spike orbit upon variations in parameter space so that the the reset value of spike n now has A r <A crit +. Variation of any of the parameters in our model and so we shall not find sets in which these occur here, though we do note that C and g are particularly important in this type of spike adding. Close to this spike-adding bifurcation point, we find more exotic orbits, which present themselves as being chaotic. Figure 13 shows an example of such an orbit. Where parameters are varied so much that, upon resetting, the value of A is never less than A crit, the model can no longer support bursting and instead we only see tonic bursting. The easiest way to do this is by increasing µ, shifting the V -nullcline upwards, or by reducing β, increasing the gradient of the nullcline. For use later, we make note of the minimum value of drive which precludes bursting (in dimensional form), I c = γa r s 1 v r. If we reduce µ to a value which allows bursting, we see the spontaneous emergence of bursting orbits from tonic ones. The other mechanism for adding an n +1 th spike is by another grazing bifurcation, this time, where dv/dt =0atV = V th. In this case, we may have n spikes, followed by a subthreshold peak which does not reach threshold. As we vary parameters we may see this peak grow until it reaches threshold, at which point another spike is generated. To find an expression relating the parameters, we must construct a relation between V and A as before, but now also take into consideration the reset conditions for all n spikes. Once we have found this, we may substitute the point at which we know the grazing bifurcation will occur, namely (V,A) = V th, 1 (V β th + µ) to give us our relation. A perhaps more interesting 38

39 Figure 13: Chaotic orbit close to a spike adding bifurcation point. Left: Voltage trajectory, Right: Corresponding orbit in the phase plane orbit is one in which we have n spikes where each interspike interval contains a subthreshold peak. In this situation, it appears that the orbit is attempting to reach some stable limit cycle, but one that is beyond the line V = V th. Each time the system resets, the trajectory is projected towards the limit cycle, but is interrupted as it is reset at threshold. 3.2 Phase Response Curve It is often useful to consider the phase response of an oscillator in response to perturbations. For periodic solutions, we may consider the flow of the variables as dynamics on a circle, such that the phase of the oscillator is given by a single variable θ [0, 1), such that θ = 1 [12]. For weak external perturbations, such that ( V, A) ( V, A)+ε(δ 1 (t),δ 2 (t)) and ε small, we have, from Malkin s theorem [23], that: dθ dt = 1 + ε Q T (δ 1 (t),δ 2 (t)) (3.30) 39

40 The phase response curve (PRC) is then given by the so-called, adjoint equation: d Q dt = DF T (t) Q (3.31) Where DF(t) is the Jacobian of the dynamical systems evaluated along an orbit. To find the phase response curve for a periodic orbit crossing over into the V<0 region, we must again consider the form of the Jacobian in each region and enforce continuity across the change in dynamics. We must also enforce conditions on the initial conditions Q(0) such that Q T ( V, A) = 1/. In our model, we have a discontinuity at reset, and thus Q will not be continuous. We therefore need to ensure that Q( + )= Q(0). As in [12], we shall introduce components of Q(t) =(q 1 (t),q 2 (t)) and ensure that dq 1 /dq 2 is continuous across reset. Using the G i (T i ), i =1, 2, 3 from section 3, where G 3 (t) =G 1 (t), we may write the solution of the adjoint equation as [23]: Q i (t) =G T i (T i t) Q i (T i ) (3.32) To impose the continuity of Q across the separate domains we enforce that Q 2 (0) = Q 1 (T 1 ), Q 3 (0) = Q 2 (T 2 ). The normalisation condition on Q gives: q 1 (T 3 )(V th + µ βa )+q 2 (T 3 )(αωv th ωa )= 1 (3.33) Where A = A(T 3 ). To close the system we shall impose that the derivative dq 1 /dq 2 is continuous across reset. Here, we shall define two ratio quantities: q = q 1 (T 3 )/q 2 (T 3 ) and q = q 1 (0)/q 2 (0). Using the Γ from section 3, we may then express q in terms of q as follows: q = Γ 11q +Γ 12 Γ 21 q +Γ 22 (3.34) 40

41 We also have that: dq 1 dq 2 = and so our continuity condition is: dq 1 dt dq 2 dt = q αω βq + ω (3.35) q αω βq + ω q αω = β q + ω (3.36) Numerical analysis shows us that this has two solutions for q, both of which are negative. We choose the larger of the two of these solutions, q = Υ q 1 (T 3 )= Υq 2 (T 3 ). We now substitute this into our normalising condition 3.33 which gives us our initial condition: Q(T 3 )= q 1(T 3 ) = 1 Υ((Υ αω)v th +(ω βυ)a + µυ) 1 q 2 (T 3 ) (αω Υ)V th +(βυ ω)a µυ) 1 The full solution for the adjoint equation can now be written as: G 1 (T 1 t)g 2 (T 2 )G 1 (T 3 ) Q(T 3 ) 0 t t 1 Q(t) = G 2 (T 2 t)g 1 (T 3 ) Q(T 3 ) t 1 t t 2 G 1 (T 3 t) Q(T 3 ) t 2 t (3.37) (3.38) We turn our attention briefly back to the sub-threshold oscillations discussed in the previous section. Proceeding as in [22] we can again construct PRC s in a similar fashion as for 1-spike bursts, the only difference from the previous construction are the initial conditions. Subthreshold oscillations are continuous and so we need not consider discontinuities in the PRC. Instead, we enforce periodicity, setting Q(T 2 )= Q(0). Upon introducing Σ = G T 2 (T 2 )G T 1 (T 1 ), the periodicity, and the normalisation condition give us a linear pair of equations for Q(T 2 ): Π q 1(T 2 ) q 2 (T 2 ) = 1 0 Π= µ βa 0 ωa 0 Σ 11 1 Σ 12 (3.39) 41

42 Figure 14: Left: Phase response curve for the 1-spike bursting orbit in figure 4. Right: PRC for the black orbit shown in figure 5. Also pictured (dashed) are the voltage trajectories corresponding to the PRC Using the A 0 we found in section 3. As in [22], we may solve this using Cramer s rule, giving q i (T 2 ) = det(π i )/det(π), where: Π 1 = 1 ωa 0 0 Σ 12 1 Π 2 = µ βa 0 Σ 11 0 (3.40) We may compute the solutions to the adjoint equation using the adjoint routine in XPP at low computational cost. For the subthreshold orbit, we integrate over one period but for the 1-spike bursting orbit we must ensure however that our integration interval does not contain the discontinuity. We integrate from initial conditions on typical orbit, starting with V (0) = V r, up to time T such that V (T )=V th. Shown in figure 14 are examples of phase response curves for both types of orbit In principal, the PRC for a bursting orbit may be found by repeatedly enforcing the continuity of dq 1 /dq 2 each time a spike is fired. The algebra required to do this quickly becomes complicated, although still tractable as the equation of the curve is fixed by the normalisation and continuity conditions. 42

43 3.3 Fast-Slow analysis One of the strengths of the model, other than its computational cheapness, is that we need not rely on fast-slow analysis in the singular limit to make analytical progress. With our model in this limit we can find an explicit formula for the period of an oscillatory solution. We shall now carry out this analysis, for completeness, on our original model (3.5, 3.6). We are thus either in the limit as τ, or in the limit as C 0. In the limit as τ we have da/dt = 0 and so a is simply equal to the initial condition on a for all t. As such we may now use it as a parameter in the v equation and so have collapsed the system into just one ODE. This ODE has two fixed γa I points, given by (v +,v )= s 1, I γa s 2. Shown in figure 15 is a plot of dv/dt against v. We see that v is stable and v + is unstable. Since da/dt = 0, we have a family of solutions which are, up to a point, neutrally stable to perturbations in a. Positive perturbations in a (or negative shifts in I) will cause the graph to move downwards. In figure 15 we show the effect of reducing the value of a (solid line) from 2 to 1 (dashed line). We have also pictured the line v = v r. For the solid blue line, there is only one stable solution, namely the negative fixed point, but as we reduce a to 1 (s γ 1v r + I) =a crit, so that v + = v r, we see bistability emerge through a homoclinic bifurcation. Since dv/dt is positive for all v > v +, trajectories starting with v>v + will simply increase until v reaches threshold. Since we are in the limit as τ, g/τ 0 and so the system will be reset to (v r,a) and the process will repeat and thus we have a stable limit cycle between the points (v r,a) and (v th,a). Where v + >v r, after being reset the voltage is now forced down to v, where it remains. Perturbations in v larger than v + v will allow 43

44 Figure 15: dv dt against v in the limit as τ. We see there are two fixed points, one stable and one unstable. For the solid line, a =2andthereisonlyonestablesolution,forthedashedlinea =1,therearenow two stable solutions, the fixed point to the left and a limit cycle to the right. Also pictured is the value of the reset the voltage to reach threshold once more and so the system is excitable for large enough perturbations. Upon reducing a to a value of I/γ, v + = v and we have a saddle node bifurcation as the two solutions annihilate one another. Now, for all v, dv/dt > 0 and so the system will reach threshold and enter the stable limit cycle described above. We see then that some of the bifurcation structure of the full system is preserved under the assumption τ, but that we may miss some bifurcations under this approximation: that the reduced 1D model cannot burst gives tells us that bifurcations of bursting orbits will be missed. In this limit we may now find an explicit formula for the period of an oscillation in this limit cycle by using separation of variables and then integrating between 44

45 v r and v th. The period turns out be: vth dv = I γa + s 1 v = 1 I γa + s1 v th ln s 1 I γa + s 1 v r v r (3.41) We know that this cycle is neutrally stable to perturbations in a, provided that a>a crit. The period of oscillation will change, but the amplitude will remain fixed. By recasting the dynamics of our equation in terms of a phase variable θ: θ mod 1 = 1 v(t) 0 dv I γa + s 1 v (3.42) We may may find a PRC, R(θ) for for some non-zero external input I s (t) tobe: R(θ) = e s 1, R(θ + k) =R(θ) k Z (3.43) (I γa) Where is as in We now turn our attention to the limit C 0. Here we may or may not have a fixed point in the plane, but due to the large difference in C and τ it is necessarily unstable. In this situation, a still varies, but for the majority of the orbit, it is infinitesimally slower than v. We can use this fact to explicitly construct a solution for the period of a bursting solution. For v>0, trajectories get pushed infinitely quickly to threshold, during which time a varies infinitesimally, v is reset and a is increased as per the reset conditions. This process is repeated until a is beyond a crit, after which trajectories are projected towards the left hand branch of the v-nullcline. The number of spikes, n per burst in this situation can be found as: acrit a knee n = = g τ s1 τv r γg (3.44) Where [ ] denotes the integer part and a knee = I/γ. When v enters a distance of O( C τ ) of this nullcline, the timescales of variables are comparable and trajectories track along the v nullcline, for non-zero time, until a reaches a knee are trajectories are projected rightwards and bursting resumes. We thus have that the orbit spends infinitesimal time off the left v-nullcline and so we may approximate the 45

46 period of the burst by the time spent on this nullcline. We know that v varies as 1 s 2 (I γa) on this nullcline and we substitute this into (3.6), the governing equation for a. We then solve using separation of variables, integrating between a crit and a knee to give us the period. = = aknee τda α a crit s 2 I ( αγ s 2 δ)a s 2 τ αγs1 v r + δs 1 s 2 v r + δs 2 I ln αγ + δs 2 δs 2 I (3.45) 3.4 Firing Map Due to the nature of the non-smooth dynamics of the system at reset, it is useful to consider a map of the calcium variable at successive firing times. This will collapse the dynamics of the full system to a one-dimensional difference equation. We have, from (3.22), the closed form solution for the subthreshold evolution of a 1-spike bursting orbit. To find the firing map, we assume that the voltage reaches threshold at time T n, and we wish to find the value A n+1 at time T n+1 : the time at which the voltage next reaches threshold. We evolve the solution from initial data (V (T n ),A(T n )) T =(V r,a n ) T. We also have that the system will reach threshold when V = V th. For simplicity, we shall only consider orbits staying in the region V > 0 in this section, for which the dynamics are governed solely by (3.12). In what follows in this section we let G(t) =G 1 (t) and K(t) =K 1 (t). Thus, we must solve the equation: V th = G(T n+1 T n ) V r + K(T n+1 T n ) µ 0 (3.46) A n+1 A n Without loss of generality, we will take T n to be 0 in what follows. Expanding the bottom row to solve for A n+1 we obtain the equation: A n+1 = G 21 ( T )V r + G 22 ( T )A n + µk 21 ( T ) (3.47) 46

47 Where T is the time of the first return (to threshold) of the system. Defining C( T )=G 21 ( T )V r + µk 21 ( T ) we then have A n+1 = G 22 ( T )A n + C( T ). In order to close this equation, we must now find an expression for T. We define T as inf{t : t>0,v(t) V th }. Again, taking T n = 0 and expanding the top row of 3.46 we now have the equality: V th = G 11 ( T )V r + G 12 ( T )A n + µk 11 ( T ) (3.48) In our simulations, we keep V th and µ fixed, and thus we have that T is uniquely determined by the value of A n. We now divide through by G 12 ( T ) to obtain: V th G 12 ( T ) G 11( T ) G 12 ( T ) V r K 11( T ) G 12 ( T ) µ = A n (3.49) Defining W ( T )=(V th G 11 ( T )V r µk 11 ( T ))/G 12 ( T ), we have that W ( T )= A n = T = W 1 (A n ) where W 1 exists. For the function W to be invertible we require that it is a one to one mapping on our domain. We find that W is a monotonically increasing function and is thus invertible. A typical plot of W 1 may be found in Appendix E. We now have that A n+1 = G 22 (W 1 (A n ))A n + C(W 1 (A)) which is now a function of only A n. We may then define composite functions F(A) =G 22 (W 1 (A)), G(A) =C(W 1 (A)) and finally P (A) =F(A)A + G(A) to obtain our firing map for V>0, A n+1 = P (A n ). For orbits crossing over into V<0the firing map becomes more difficult to find analytically due to the switch in dynamics. We may proceed as before to solve equations in both regions, ensuring continuity at the switching manifold, however the algebra required to find explicit forms for the firing times becomes complicated. 47

48 Figure 16: Left: Firing map for a multispike bursting trajectory where τ = 5,C = 0.2,g = 0.2,α = Right: Flight times corresponding to the firing map Instead, we use MATLAB to find the firing map. To do this we seed the integrateand-fire model with initial data corresponding to different values of A n, with the orbit always starting from V = V r. We can then track the value of A at the next point the system reaches threshold as well as the flight time of the trajectory between these points. Shown in figure 16 is a plot of the firing map together with a plot of the return times. It is reassuring to note that our firing map has very similar qualitative features as the map for the full model Chay-Keizer model produced by Georgi Medvedev [13] as it indicates that our simplified model may capture the dynamics of the more complicated model. The MATLAB simulations show us that the firing map, as in [13] can be separated into three distinct regions, one in which the map is approximately linear with slope less than one, one in which the the value rapidly decreases followed by a third region in which all values of A n map onto approximately the same 48

49 value. We may understand this portion of the map by returning to the phase plane. The model is unable to fire whilst trajectories are tracking the left branch of the slower nullcline. Once the trajectory reaches the knee of this nullclines, trajectories move quickly to the right due to the difference in timescales between the voltage and calcium variable. As such, we have that the value of A n+1 to be approximately that of the value of A knee and further, that A n+1 A knee as ω 0. The first part of the map corresponds to orbits which stay in V > 0 and may correspond to repetitive spiking during a burst. The inner region is of more interest. We see that there is a marked change in the dynamics of the firing map in this narrow region. We know that trajectories starting close to (V r,a crit ) evolve more slowly as dv/dt starts close to zero. We find that the return time increases monotonically as trajectories start closer to this point. Since the return time is greater, there is more variation in A and this accounts for the curvature in the firing map. The marked change in the return times occurs when the value of A n >A crit + and trajectories are now projected leftwards, and so we are in the rightmost region of the map. We then see qualitatively different behaviour dependent on the difference between the two timescales. For high differences, we see that the first return time increases with A n, but that it decreases for low differences. As the difference in timescales becomes larger, trajectories are pushed further left quickly as V varies much faster than A whereupon trajectories are forced to track the left V -nullcline. For smaller differences in timescales, trajectories may spend less time tracking this nullcline and so return times will be smaller. As A n increases, trajectories have travel further before they reach this nullcline and this explains 49

50 the increase in return time for high timescale differences. Now that we have characterised the first return map, it is useful to consider fix points of it, namely where A = P (A ). Fixed points of this map correspond to periodic solutions. Graphically we can that with the parameters used here that there exists just one such fixed point of this map. In the present situation this is located in the inner region of the map. We now consider perturbations to this fixed point of the form A n = A + δ n, with δ n 1. Substituting this into our equation yields: A + δ n+1 = P (A + δ n ) (3.50) We may then use Taylor s Theorem to expand the function on the right hand side as follows: A + δ n+1 = P (A )+P (A )δ n + h.o.t δ n+1 = P (A )δ n (3.51) Thus we have that δ n = P (A ) n δ 0. Thus we have that the fixed point is stable if P (A ) < 1. We now need to find the value of P (A ). Using results from before, we may work out that for orbits only in the region V > 0, letting J(A) =H 1, we have that: dp da = d(af(a)) + dg(a) da da = G 22 (J(A)) + d(g 22(J(A))) A + dc(j(a)) da da = G 22 (J(A)) + dg 22 dj dj da A + dg 21 dj dj da V r + dk 21 dj dj da µ = G 22 (J(A)) + dg 22 da dj dw A + dg 21 da dj dw V r + dk 21 da dj dw (3.52) Where da dw =(dw da ) 1. As can be imagined, finding analytical values for P (A) will be difficult for orbits in the full system. Here we can once again call on MATLAB 50

51 to find numerically estimate the value of P (A ). After MATLAB constructs the full firing map, we then ask it to find the value of A. After this, we compute the derivative at this point using the estimate P (A )= P (A +ε) P (A ε). We note 2ε here the need for ε to be very small to obtain a more accurate estimate for P (A ). We observe that the firing map has a large negative gradient > 1 on the middle part of the curve and so periodic solutions here are unstable. These fixed points are known as snap-back repellers [11]. Solutions are repelled away from the fixed point, but the rightmost portion of the map causes trajectories to snap-back into the left portion where they then move upwards, towards the fixed point once more. Thus, snap-back repellers correspond to bursting solutions. Now that we have found the firing map, its fixed point and can infer the stability of the fixed point we can examine how the map changes under the variation of the parameters. We note that nowhere is the gradient of the map greater than 1, so we look for bifurcations where P (A )= 1. We look first at the response of the map to variation in the external drive µ. For large values of µ the fixed point is in the left portion of the curve, where, since the gradient of the curve is less than 1, it is necessarily stable. Looking at our phase plane, we have seen that increases in µ shift the V -nullcline upwards, and so the value of A at reset never reaches A crit and so this corresponds to tonic firing. Medvedev found a period doubling bifurcation in the Chay-Keizer model where variation of his parameter g KC shifted the fixed point to the middle region where the slope of the map is greater than 1. Reducing the value of µ stretches the map leftwards, and thus shifts the fixed point into the middle region, as in Medvedev s map, however we find different qualitative behaviour. Variation of ω here only affects the slope of the firing map, but not the qualitative features, or the posi- 51

52 tions of the left, middle and right regions. As we decrease µ beyond the point where P (A )= 1, we may have a bifurcation to many different types of solution, dependent on the values of C, τ and g in the original system. Returning the phase plane we see that decreasing µ shifts the nullclines downwards so that A at reset may exceed A crit and thus we may find tonic, doublet spiking or full bursting solutions at this bifurcation. Looking at the system in reverse, sufficient increases in µ may cause bursting solutions to die into tonic solutions. We move next to the parameter β. We know that increases in β cause A crit to decrease. This results in the systematic annihilation of spikes. Thus as β increases we have successive non-smooth bifurcations such that a burst with n spikes will becomes a burst with n 1 spikes. In our firing map this corresponds to a shift of the fixed point down the middle portion of the map, where the gradient of the map is large and negative. If we continue to increase β further, we see that the fix point moves to the right portion of the map, where the gradient is now small and negative. This portion of map corresponds to a 1-spike burst, as all of the other spikes have been annihilated in the cascade of bifurcations. Moving in the other direction, sufficient decrease in β will have the same effect as in increase in µ and will shift the V -nullcline upwards so that A does not reach A crit and hereafter the model supports only tonic firing with no bursts. We also note that the although sufficient decreases in β will add spikes to a burst, the maximum number of spikes per burst is controlled by the difference in the timescales of the two variables. We also note that increases in β results in qualitative change in the middle portion of the map. For low values (figure 16), the map is concave, but as β increases (figure 17) this portion becomes convex. We have established that the curvature 52

53 Figure 17: Increasing β to 8 we now see that the middle portion of the map is convex instead of concave of this portion of map is due to trajectories starting near (V r,a crit ) T, allowing A to vary more as V reaches threshold. Initially A falls in this region, but as the return time increases, A begins to rise. As we increase β, V grows more slowly and so the return time, and the value of A n+1 increase. We find very similar qualitative behaviour for the parameter α, including the qualitative effect of the shape of the map. Decreases in α will cause the system to cease any bursting in favour of tonic solutions. Increases in α shift the fixed point of the map downwards along the middle branch of the map, where it is inherently unstable, and then finally onto the right branch, corresponding to a burst of 1 spike, where the fixed point is stable again. The cascade of spike removing bifurcations of α and β highlight the dependence of the voltage dynamics on the calcium variable. We have already seen that when C and τ in the full model are comparable, and 53

54 Figure 18: Firing map for the black orbit in figure 5 for sufficiently high value of α or β that the model will only undergo subthreshold oscillations. For these solutions, we use the return map described earlier. Now we just have a standard oscillatory orbit and we see that the slope of the branch of the firing map corresponding to the fixed point is less than 1, that the fixed point of this map is stable. As we increase τ, the firing map shifts up, and the slope of the righthand branch increases. When the slope of the map is equal to 1, we lose the fixed point in a tangential bifurcation. Now the map has no fixed points and we see the emergence of a stable bursting orbit. As we reduce τ the opposite process occurs and we see a Hopf bifurcation at the value of τ when the fixed point is at the origin A = 0. Similar results are found on decreasing either α or β. A more rigorous method of obtaining the stability of the subthreshold orbits is to find the Floquet exponents of the orbit. We have found already the system of equations governing perturbations to periodic orbits ref. For simplicity we now take our start and end-points to be such that V (0) = 0. From [23], since the 54

55 Jacobians of our system are piecewise constant, we may write the fundamental matrix solution of these equations as Σ = G 2 (T 2 )G 1 (T 1 ), where T = T 1 + T 2. Letting γ 1,2 be the eigenvalues of the matrix Σ we denote the Floquet exponents by σ 1,2 = ln(γ 1,2 )/T mod(2πi). We note that one of these exponents will always be equal to zero as this corresponds to perturbations along the orbit, and thus to γ k = 1. We have the the orbit will be stable if the real part of the remaining exponent is negative. t Following the work in [23], we use the fact that γ 1 γ 2 =exp Tr(DF(s))ds and 0 that one of the multipliers is equal to one. We thus have that (σ 1,σ 2 )=(0,σ) where σ is given by: σ = 1 T T1 0 TrM 1 ds + T2 = 1 T (T 1 TrM 1 + T 2 TrM 2 ) = T 1(1 ω)+t 2 ( S ω) Ť = T 1 ST 2 T ω = 1 (1 + S)T 2 T 0 TrM 2 ds ω (3.53) Remembering that T 1 + T 2 = T. Requiring that σ<0 results in the following condition on T 2 : T 2 > 1 ω 1+s T (3.54) Which gives a lower bound for the time that the orbit must spend in V < 0 in order for the cycle to be stable. We have that T 1, andt 2 are invariant under changes in µ so it is sufficient to find the value for any value of I. This turns out to be negative. Plots of the multipliers against β and τ are shown below. We note here that σ is a decreasing function of β and so increases here make the limit cycle more strongly attracting. We see that the cycle becomes unstable 55

56 Figure 19: Left: Floquet multipliers under variation of τ. Right: Floquet multipliers under variation of γ. For both plots C =1,α =0.5 as τ increases. This corresponds to the grazing bifurcation discussed in section 3.1 as the limit cycle touches the threshold line. When calculating the Floquet multipliers, we did not consider the reset condition of the full model. It is therefore interesting to note the the loss of stability in the limit cycle coincides with this bifurcation, although, since V th does not appear in the equations governing subthreshold evolution, we see that this is not the case for general V th. Plots of the Floquet multipliers against τ and γ are shown respectively in figure Gap Junction Coupling We now wish to consider networks of coupled β-cells. We consider gap junction coupling, namely of the form: K ab (V b V a ) (4.1) where a and b are respectively the indices of two different cells, and K ab is the associated strength of coupling between the two cells. In section 3.2 we computed 56

57 phase response curves for both subthreshold orbits and 1-spike bursts. However, we wish to study multispike bursting at the network level and so our previous analyses do not apply here. Much of the work in this section is based on simulations run in MATLAB where we make observations about the types of solutions we observe. We outlined how to construct PRCs for multispike bursting orbits at the end of section 3.2, and in principal, we may use this to analyse solutions at the network level. We will briefly outline the way in which we may achieve this, but will not progress any further with this. We consider a network comprising N cells. Indexing the cells over i we now have that the system is governed by the equations: C dv i dt = f(v i )+I βa i + 1 N N K ij (v j v i ) (4.2) j=1 τ da i dt = γv i δa i (4.3) for i =1...N, where we have now scaled the the coupling strength with a factor of 1 N. K ij = K ji. We also demand symmetry of the gap junction coupling; namely that We may rewrite this system of equations as a system of phase equations: dθ i dt = 1 T + 1 N N K ij H(θ j θ i ) (4.4) j=1 for i =1...N. H(θ) is the phase interaction function, which is defined for gap junction coupling as [23]: H(θ) = 1 T T 0 Q T (t)(v(t + θ T ) v(t), 0)dt (4.5) Where v(t) is a periodic solution of our system and Q, the associated adjoint. For identical oscillators with frequency Ω = 1/ T we may move to a rotating by introducing a new variable θ i =Ωt + ψ i, whereupon 4.4 becomes: 57

58 N dψ i dt = K ij H(ψ j ψ i ) (4.6) j=1 Asynchronous and synchronous solutions to 4.6 may be defined where in the former, the ψ i are constant, but may be different and in the latter where ψ i = ψ for some constant phase ψ. We may check the stability of these solutions by linearising about the steady state Ψ = (φ 1,...,φ N ), substituting perturbations of the form ψ i = ψ i + ψ i into 4.6. This results is a system of equations defined by: d ψ i dt = 1 N N H if (Ψ) ψ N j, Hij (Ψ) = K ij H (ψ j ψ i ) δ i,j K ik H (ψ k ψ i ) j=1 k=1 (4.7) Where H (ψ) =dh(ψ)/dψ. One of the eigenvalues of the the Jacobian H is zero. The phase locked solution will be stable provided that all remaining eigenvalues are negative. For the synchronous state, H (ψ j ψ i )=H (ψ k ψ i )=H (0). We have thus outlined how to determine the stability of phase locked solutions, but we shall not progress any further with this due to the complexity of finding H for multispike bursting orbits. Where N = 2 by symmetry, both a synchronous φ = θ 2 θ 1 = 0 and an antisynchronous solution φ = 1/2 are guaranteed to exist, and we expect one such solution to be stable, and the other unstable. The stability of the two solutions is given by the odd part G(φ) = H(φ) H( φ) 2 [23] of the phase interaction function H. The sign of the derivative G where G = 0 will give the stability of these solutions. Where G > 0 we have an unstable solution, and where G < 0 we have a stable solution. In figure 20, we plot G(φ) for the periodic and 1-spike burst. Here we observe that for both orbits, the synchronous solution is stable and the 58

59 Figure 20: G(θ) for: Left: Periodicsolutionwhereτ =3,C =1andβ =0.25. Right: 1-spike burst where τ =5,C =1,β =0.11, g =4. Inbothcasesthesynchronousstateisunstable anti-synchronous solution is unstable. 4.1 Numerical Simulations There are four main network associated areas in which we focus the remaining work in this section. These are: the coupling strength between the cells the architecture of the coupling noisy drive inhomogeneities in the gap junctions and the cells themselves. In order to solve 4.3 in MATLAB, we first define a vector z such that: 59

60 z =(V 1,...,V N,A 1,...,A N ) T (4.8) which contains all of the output data we are interested in. We note that the voltage and calcium variable of cell i are N positions apart. We now define a 2Nx2N matrix R which will take into account the coupling between cells and the interaction between the voltage and calcium variables in each cell. R is made up of 4 NxN matrices which we shall denote X, Y 1,Y 2 and Y 3 so that: R = X Y 1 (4.9) Y 2 Y 3 The matrices Y 1, Y 2, Y 3 are given by β I C N, γ I τ N, δ I τ N respectively, where I N is the NxN identity matrix. The matrix X is the connection matrix which encompasses the coupling between the cells. The exact form of X will vary depending on which coupling architecture we choose to impose, however X has two properties that are true for all such architectures. Firstly, due to the symmetry of the coupling, X will be a real symmetric matrix, and due to the form of the coupling, the diagonal entries of X will be equal in magnitude but opposite in sign to the sum of the remaining row elements: N X(i, i) = X(i, j) for i =1,... N (4.10) j=i We note here that setting X be a matrix of zeros recover a system of N AIF uncoupled cells. We shall make the definition of X more precise in due course. Next, we define a 2N vector w : w =(f(v 1 )+µ),...,f(v N )+µ, 0,...,0) (4.11) and the system becomes: z = Rz + w (4.12) 60

61 This system is analogous to the system for a single cell, and we may solve this system in MATLAB using one of the built in ODE solvers. Details of the code used may be found in Appendix D. In all simulations, we distribute initial data randomly upon the Cartesian space [ 80 v i 20] [0 a i 2]. Figures shown are for N = 30, for ease of viewing, but results have been collected for N = 50 and N = Coupling Strength We first look to gain insight into the network by considering a homogeneous coupling strength, setting K ij = K i, j so that we may rewrite the coupling term in equation (4.3) as K N N j V j V i. For high values of K we see that the cells no longer burst, and we have that the cells oscillate, but only reach threshold once per oscillation. This is similar effect to raising I beyond I c from section 3.1. In this regime the cells do not synchronise. We note that the relative strength of coupling will depend on both the values of K, N and the coupling architecture we impose. For global coupling, we refer to high coupling as K>1, weak coupling K<0.001and intermediate coupling for values in between. For high values of K which do not result in burst death we observe that the cells like to synchronise, albeit over a long timescale. Upon closer inspection of the burst patterns of the cells, we find that that although the bursting envelopes are in synchrony, we do not observe this same synchrony at the spike level. When K is much smaller we are in the regime of weak coupling. Here we observe that the cells tend to cluster into subpopulations. Within each of the subpopulations, the bursting envelopes are synchronised but the spikes are not, and the 61

62 Figure 21: Voltage evolution with: Left: Strong coupling K = 1. Right: Intermediate coupling K = 0.01 bursting of the clusters is in anti-phase. When one cluster is in the active phase, the other is in the silent phase and vice versa. This is analogous to the two cell system for weak coupling. The sizes of the clusters formed are not necessarily equal, and it seems that the the distribution of clusters is dependent on the initial conditions of the system. We may see this by prescribing gradated rather than random initial conditions. We observe that cells with an initially high value of a tend to cluster together as do cells with low initial a values. From this we may ascertain that these initial values are important in determining the number of clusters and which cluster cells belong to. Shown in figures 21 and 22 are large time solutions where K high, intermediate and low. Where K is low, we set up initial conditions such that the first half of cells have low initial values of a and the second half have high initial values. This clearly shows two distinct clusters, although many more clustered solutions are possible. 62

63 Figure 22: Calcium levels for network with weak coupling, K = Coupling Architecture We now consider the different ways in which the cells may be coupled. Since gap junction coupling requires cells to be adjacent, we would not expect global coupling in an islet. The extreme opposite to global coupling is to consider that cells are only coupled to their nearest neighbours. We consider here a ring of cells, so that that cells 1 and N are coupled to each other, so that each cell is coupled to exactly two others. To do this in MATLAB, X takes a form similar to a tridiagonal matrix: X = K N (4.13) 63

64 Where all other entries to the matrix are zeros. For strong coupling we again find that the cells are unable to burst and we simply have that the cells undergo 1-spike oscillations in phase with one another. For weaker coupling however we now see the emergence of waves of bursting. The cells which are in the active phase pull those around them up to the bursting point, but with some time lag. We also observe that there may be more than one of the cells and so we then have waves starting in several positions which propagate in both directions. This wave-like behaviour exemplifies the diffusive nature of gap junction coupling. We could, but have not done so here, measure that speed of these waves for comparison with the wavespeeds found in [22]. To do this we would need all cells to start at a steady state (I <0) and then perturb one of the cells with a sufficiently strong Dirac delta function to initiate the wave, and then measure the phase lag over the network. It is most likely that the populations will not be in either of these two cases, rather somewhere in between. In an islet, we may expect that the cells on the boundary are coupled to the fewest number of cells, whereas the cells in the centre of the islet are very densely coupled to the cells around them. β-cells comprise between 65-80% of the cells in an islet [1], meaning that the remaining proportion is comprised of other types of cell, namely α- and γ-cells. We do not wish to consider coupling between these cells. As a result of this, the density of coupling that is the number of cells and individual cells is coupled to divided by the total number of cells may range between zero and one. We may investigate the effect of low and high densities in our network by gradating the connection density of the cells. To do this, we now rewrite X in the 64

65 Figure 23: Waves of bursting for a network of 100 cells. Left: Voltage, Right: Calcium following way: X = K N N (N 1) (4.14) The first cell is now be coupled to all other cells, the second cell is coupled to all but the last cell and so on until the final cell is only coupled to the first cell. We note that two of the cells, namely those cells in the middle will have the same number of connections. For high K values, densely connected cells now no longer reach threshold and undergo sub-threshold oscillations. Less densely connected cells may still undergo 1-spike oscillations. Clearly the notion of a synchronous solution is not applicable here as the cells are following qualitatively different orbits. For intermediate 65

66 Figure 24: Wave tail like behaviour for cells gradated by coupling density. Cells to the left are more densely connected values of K, densely connected cells now synchronise, but even after long times cells with the lowest connection density are not synchronised to the more densely connected cells. Instead the cells exhibit a phase lag and trail behind the large cluster of synchronised cells, like the tail of wave. As the connection density is reduced in these cells, the coupling they experience is more and more like that of purely diffusive coupling. However, the cells are not coupled to each other, so it is interesting to note that these cells lag behind each other in a seemingly ordered fashion. For weak coupling, we again see the cells clustering, although the pattern in which they do so is not immediately obvious upon considering the connection densities. Here we observe that cells with similar connection densities may cluster together, but equally that they may be in a different and possibly anti-synchrony to the 66

67 Figure 25: Unpredictable clustering of cells gradated by coupling. Left: Voltage, Right: Calcium cell next to them. We may also consider a more random connection density for the network. Returning to the global connection matrix, we now allow a proportion of K ij to be zero. We define the network connection density as the number of connections between cells divided by the number of possible connections between cells, namely ρ = K ij=0. We may now prescribe the connection density, but allow the position N(N 1) of the connections to be random. We do however, ensure that there are no cells which are uncoupled, as we have already considered the behaviour of a single cell. Here we may see the appearance of clusters either due to weak coupling between sets of cells, or due to the random nature in which the coupling between cell was established, clusters may not in fact be coupled together at all. In this instance we may observe clustering at levels are K where we expect synchronicity to be the stable solution. 67

68 4.4 Noise As discussed in section 2, individual cells beat irregularly; it is only when coupled to other cells in a population then we observe regular bursting behaviour. In [20] the authors considered making the opening and closing of one of the ion channels and stochastic process, so that their model now had noise in the system, derived from the inherently randomness in the cells. Whilst the AIF model is not based on any biophysical realisations, it would be nice to see if it can produce the same kind of behaviour. We adopt a similar, albeit more crude methodology of introducing noise into our system. In our model, the drive I will be our stochastic variable. We allow I to make a random walk on the interval [ 1, 4]. Here we allow I to take both negative values and values >I c which would prevent cells from bursting. For simplicity, we set the transition rates to be symmetric. We shall first investigate the effect of noisy drive in a single cell. Every time the cell reaches threshold, or after every second the cell spends sub-threshold, MAT- LAB ceases the integration and generates a random number p. If the transition rate q is larger than p, I will increase to I +. If the transition rate is smaller than 1 p, I will decrease to I, otherwise I will remain unchanged. The details of the random walk are summarised in the transition table below. State I I + I 1 1 q q State I + I I 1 I 4 q 1 2q q State I I I 4 q 1 q As expected, the trajectories follow a more unpredictable path, with both varied 68

69 Figure 26: Voltage, Calcium and I evolution for a single cell where I is a stochastic variable active and silent phase durations. Figure 26 shows the time evolution of the voltage, calcium and stochastic variable. Longer silent periods correspond to points where I is negative, whereas longer duration active phases correspond to high values of I. We now follow the same steps for our network. Here the stochastic drive is updated each time any cell reaches threshold or after every second where no cell reaches threshold. Where I remains positive for the entire integration, we have similar to results as for the case where I is constant.if the coupling is too strong, we see no bursting behaviour, or synchronicity. For intermediate values of K the system will tend to synchronise the bursting behaviour, though the stochastic nature of I may cause the timescale over which synchronisation takes place to increase. For low 69

70 values of K, where we expect to see clustering, this phenomenon may cause the behaviour to never reach a steady state and instead the phase differences will drift. In the case where I now is allowed to become negative we now sometimes observe perfect synchrony, at both the bursting and the spiking level. As the cells are homogeneous, each has the same stable fixed point for a given value of I<0. When I becomes negative all cells will tend towards this point. Provided I remains negative for long enough, all cells will reach this point and so have the same value. When I becomes positive they will effectively all have the same equations of motion, with the same initial conditions and so will be perfect synchrony. We thus have a sufficient but not necessary condition which will ensure perfect synchrony. We require the supremum of the times taken for each cell to reach the fixed point when I<0tobeless than the duration for which I is negative then the cells will become perfectly synchronised. This condition is not necessary for synchronisation because the cells may synchronise over several periods of negative I. Each time I becomes negative the orbits will move closer to one another, and perfect synchrony may then be achieved through the regular coupling K with I>0. For weak coupling we have seen that cells tend to cluster, and this infers that the globally synchronised state is unstable. For the situation described above, even in the weak coupling regime, if the cells become perfectly synchronised, they will remain that way. If there is no difference between the voltages in the cells then the coupling has no effect and the cells simply follow their orbits at the same rate. Here, the synchronous state is metastable. The cells remain synchronised 70

71 Figure 27: Perfect synchrony after a long period with I<0. This behaviour is independent of the value of K despite the fact that a clustered state is more stable. 4.5 Inhomogeneities Our assumption that the cells in the islets are homogeneous is a particularly strong one. We now relax that assumption by allowing the cells to have different drives, as in [6], but impose a globally coupled structure. We consider three different types of heterogeneous drive to the cells: constant, periodic, as in [25], and stochastic. For constant drive, each cell is prescribed a value of I on the interval [ 1, 4]. The only change we have to make to our system is to change the vector w as follows: w =(f(v 1 )+I 1,...,f(V N )+I N, 0,...,0) T (4.15) Where I i = I j i, j. 71

72 We find, unsurprisingly that for weak coupling strength, those cells with negative drive do not oscillate, whilst cells with a very high drive cannot burst and undergo 1-spike oscillations. However, if the coupling strength is strong enough, here about K =0.05, we find that the cells with negative drive are pulled up sufficiently, and that cells with high drive are pulled sufficiently down, so that they may now burst. Qualitatively, the bursts of the cells are different. Where K = 0.05 the cells with negative drive display only 1-spike bursts, whereas other cells have mutlispike bursts. However, the termination of bursting is the same for all cells. For periodic drive, we consider a functional form of I as: I = ξ sin(σt)+η (4.16) It is useful to observe the behaviour of a single cell to this type of drive. We find that the pattern of bursting in this instance looks more irregular than for constant drive. The cell now displays longer duration bursts and silent phases. In fact, the orbit is periodic, but, as the period of the drive is not the same as the period of an orbit, the periodicity is now the lowest common multiple of the period of the drive and the period of a bursting orbit. If we now apply the same periodic drive to a network of cells, we find similar solutions. For 2 cells, we have anti-synchronous bursting for weak coupling and bursting synchronicity for stronger coupling. We may observe the same kinds of irregular bursting at the network level. If we allow ξ>η, that is to allow the drive to becomes negative, we may recapture more regular bursts at the network. Again, these bursting patterns are synchronised at the burst level, but not at the 72

73 Figure 28: Irregular bursting for a cell with periodic drive I =3sin(t)+4 spike level. If we now allow, σ to take on smaller values, in effect increasing the wavelength of the drive, we find waves of synchronisation. Where the drive is negative for long periods of time, the cells synchronise almost perfectly. They stay this way until I reaches its maximum value. As I decreases the cells start to desynchronise, and the bursting patterns become more irregular until the drive becomes negative once more. It appears as if in the decreasing portion that an asynchronous state is more stable than the synchronous one, and so we observe this unusual pattern. This behaviour is similar to the complex bursting behaviour observed experimentally [16]. This suggests that the our now periodic drive could be used to represent intrinsic ATP oscillations due to oscillations in intracellular glycolysis. 73

74 Figure 29: Waves of synchronisation. Here K = 0.01,ξ = 2,η = 1 For γ,σ,η 0. We consider two types of heterogeneity with this form for µ, we first is phase difference, and the second is period difference. For phase difference we change the vector w to: w =(f(v 1 )+ξ sin(σ(t u 1 )+η,..., f(v N )+ξ sin(σ(t u N )+η, 0,...,0) T (4.17) Where the u i are distinct and take values on the interval [0, 2π]. Burst envelopes seem to drift around one another, never becoming settled at constant phase lag, but rather moving around one another. This suggests that we may not have a steady state solution for constant phase differences in this situation, but instead that we have a steady periodic solution for phase differences. These results hold true whether we allow the drive to enter negative or high regions. If we allow I to take on negative values, we observe the same kind 74

75 of synchronisation waves as described above. This is more marked for smaller σ as the drives spend more time in the negative regions. If we increase the coupling strength, all cells will burst for long periods of time, then all enter a silent phase together. The termination of bursting here is caused by a sufficient number of drives being low enough. We observe periodic bursting behaviour, where the length of the bursts are different, but are periodic. The resumption of bursting is due to a sufficient number of drives exiting the low region, and as the drives are at a constant phase lag, the silent phases are all of the same length. The periodicity of the burst lengths is determined primarily by the value of σ and appears to be related in a linear fashion, but also on the range over which we allow I to vary. Increasing K still further, up to O(10) we see the emergence of more regular, and synchronised bursting behaviour as coupling now dominates the drive. For period difference, w becomes: w =(f(v 1 )+ξ sin(σ 1 t)+δ,..., f(v N )+γξ(σ N t)+δ, 0,...,0) T (4.18) Where the σ i are distinct and take values on the interval [0.001,1]. In this case we find that for weak and intermediate coupling K 0.1 the cells do not synchronise. In fact the drive here dominates the coupling and for the most part the cells simply travel along their bursting orbit as described by the drive to each of them, independent of the coupling strength. However, when the coupling is strong K 1/N, and the cells are precluded from bursting, we find that although the cells do not synchronise, they all cease bursting at the same point 75

76 Figure 30: Bursting solutions where K =1,I i =0.8sin(t u i )+η. Thelengthsoftheburstsarenotequal but exhibit periodicity, in this case with period 2000s and resume again at the same point. Once the drive with the longest wavelength becomes negative, all the drives will, at some point, simultaneously be negative and so all trajectories will now be attracted towards the stable fixed point. As the shortest wavelength drive becomes positive once again, that cell can now oscillate and pulls all other cells around it up to threshold quickly and now the cells undergo 1-spike oscillations until all drives are simultaneously negative again. As we increase the coupling strength further, K O(10) coupling dominates the drive and we begin to see the emergence of more regular bursting, with matching bursting envelopes. Finally we consider stochastic drive to each variable. As in section 4.4 I undergoes a random walk on the interval [ 1, 4] but, now the I i are distinct. We note that this does not truly represent the stochastic nature of indiviudal cells as the transition rates for any particular cell depend on the phase of the all of the other 76

77 Figure 31: Left: Voltage, Right: Calcium across all cells. Cells exhibit no form of synchronisation. Here K =0.01 Figure 32: All cells terminate a long period period at the same time, where all drives are simultaneously negative. The cells resume bursting when the drive with shortest wavelength becomes positive 77

78 Figure 33: Cells with separate stochastic drives adopt an almost synchronised, regular bursting pattern. Here q =0.1, =0.2, K =0.3 cells in the network. Since we allow the stochastic variables to change on each successive spike, we expect more fluctuation in the I i where one or more cells are in the active phase. Now for intermediate coupling, we see no synchronous behaviour. In effect, by introducing inhomogeneous stochastic drives, we have destroyed a steady solution as the governing equations are changing very frequently. This can be overcome by increasing the coupling strength however. For stronger coupling, the cells do now display more regular bursting. The bursting envelopes are not quite synchronous, but the duration of active and silent phases are quite regular. As we increase the coupling strength further, the cells can no longer burst. This is due to the high drives in some of the cells coupled with high values of K. As a result, we no longer see bursting for lower values of K than we might expect. 78

79 Figure 34: Calcium levels across all cells with gradated coupling strengths. Eventually all cells become synchronised at a rate dependent on their collective coupling strength to the network As well as the cells themselves being inhomogeneous, we may expect the coupling between them to be inhomogeneous as well. In order to see the effect of homogeneity here, we assign, remembering that all of the coupling is symmetric, a coupling strength to every possible connection in the network. We gradate these strengths so that the strongest coupling is between the first and second cells and the weakest between cell N and N 1. We find that this network can synchronise. The rate at which each cell becomes synchronised with the rest of the network is dependent on the collective strength it has with the rest of the network. 79

80 4.6 Experimental Data The data provided to me by Dr. Smith were based on experiments with a cluster of five cultured, interacting cells. Statistical analysis on the raw data performed by Dr. Halliday suggests significant coupling between cells based upon a frequency analysis of the calcium fluorescence. From this, Dr. Smith has hypothesised that the cells are coupled as if on a string. These hypotheses were based on phase difference betweens between cells, but as we have seen we may observe clustering within the network. We shall briefly attempt to emulate these results by coupling cell with inhomogeneous drives as described above with inhomogeneous, possibly zero, coupling strengths. We find that the best qualitative fit to the data is unsurprisingly for stochastic drives, with intermediate levels of coupling. Here the cells will burst irregularly, but the coupling will allow for more regular bursting in networks. We find that for I [0, 3] that although the cells are now oscillating regularly, the averaged calcium levels across all 5 cells is more irregular. This is due to the fact that the bursts are not synchronous, as the bursts of the cells with higher drive contain more spikes. In both the stochastic and constant case, the bursting envelopes are almost, but not quite in synchrony. However, we do not see this as a problem as we would not expect such a small cluster to synchronise. We do not have locally available experimental data to suggest whether or not the evolution of the voltage is synchronised to verify this. 5 Discussion In this project we have formulated a non-linear integrate-and-fire model to describe voltage and calcium oscillations in pancreatic β-cells, and in larger networks 80

81 Figure 35: Average Calcium levels across 5 cells where each cell has a distinct drive Figure 36: Average Calcium levels across 5 cells where each cell has a distinct stochastic drive 81

82 Figure 37: Calcium evolution of one cell in: Left: Dr. Smith s cultured cells, Right: Our network with heterogeneous stochastic drive of cells coupling by gap junctions. We observe that under variation of parameters the model can exhibit burst firing, tonic firing or may simply be silent and remain at a steady state. The biggest advantage models of this kind have over the models discussed in section 2 is this we may find closed form solutions for the trajectories of our variables. In addition to this, we have not had to make an assumption that the ratio of timescales is in a singular limit to make analytical progress. Upon enforcing this assumption, we may find closed form solutions for the period of oscillations. As well as forming closed form solutions, we have constructed the phase response curves for 1-spike and subthreshold periodic orbits, and indicated how to construct PRCs for bursting orbits, and then use these to find phase interaction functions at the network level. Although we have not explicitly calculated PRCS 82