Anti-phase, asymmetric and aperiodic oscillations in excitable cells—I. Coupled bursters

Pergamon

Bulletin of Mathematical Biology, Vol. 56, No. 5, pp. 811-835, 1994 Elsevier Science Ltd

�9 1994 Society for Mathematical Biology Printed in Great Britain. All rights reserved

009~8240/94 $7.00 + 0.00

0092-8240(93)E0019-T

A N T I - P H A S E , A S Y M M E T R I C A N D A P E R I O D I C O S C I L L A T I O N S I N E X C I T A B L E C E L L S - - I . C O U P L E D B U R S T E R S

ARTHUR SHERMAN Mathematical Research Branch, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD 20892, U.S.A.

(E.mail: [email protected]. )

I seek to explain phenomena observed in simulations of populations of gap junction-coupled bursting cells by studying the dynamics of identical pairs. I use a simplified model for pancreatic //-cells and decompose the system into fast (spike-generating) and slow subsystems to show how bifurcations of the fast subsystem affect bursting behavior. When coupling is weak, the spikes are not in phase but rather are anti-phase, asymmetric or quasi-periodic. These solutions all support bursting with smaller amplitude spikes than the in-phase case, leading to increased burst period. A key geometrical feature underlying this is that the in-phase periodic solution branch terminates in a homoclinic orbit. The same mechanism also provides a model for bursting as an emergent property of populations; cells which are not intrinsic bursters can burst when coupled. This phenomenon is enhanced when symmetry is broken by making the cells differ in a parameter.

1. Introduction. This paper will present the mathematical basis of some previously reported and some new phenomena of coupled burst oscillators. Bursting is a slow alternation between a silent phase and an active phase of fast oscillations. Important interactions occur between the fast and slow processes.

Our motivation is to understand the genesis and synchronization of bursting electrical activity in the fi-cells of the islets of Langerhans, which secrete insulin in response to blood glucose. There are 10 6 islets in the human pancreas, each an ellipsoidal mass of small, round (hence isopotential) cells coupled to their nearest neighbors by low resistance electrical pathways (gap junctions). In simulations this is modeled as discrete diffusion of voltage on a lattice.

The islet is similar to cardiac muscle and other excitable media in that it depends on gap junctions to produce a coordinated response via wave propagation. Its temporal dynamics are more complex, however: bursting vs simple oscillation. A key question, still unresolved experimentally, is whether individual fi-cells can burst when isolated from the islet. Thus, coupling may be necessary for the existence of bursting as well as synchrony.

811

812 A. SHERMAN

Bursting controls the influx of C a 2+ needed for secretion, although metabolic factors may play a Ca2§ role (Gembal et al., 1992). Experimentally, plateau fraction (proportion of time spent spiking vs silent) correlates well with secretion (Meissner and Schmelz, 1974), and other experiments (Santos et al., 1991) imply that mean Ca 2§ levels correlate with plateau fraction. Coupling appears to be necessary for a normal secretory response to glucose (Halban et al., 1982), but it is not known whether this property is mediated by bursting.

Because the ionic currents in/~-cell models are similar to those in neural models, the examples here may also be viewed as the simplest form of neural network: a two-cell, resistively coupled network. Although vastly oversimpli- fied, they illustrate the increased complexity of neural behavior caused by coupling even two cells.

We have previously reported simulation studies of synchronization and bursting in large cell clusters which revealed that moderate coupling (in the physiological range) increases burst period 50-100% over that of isolated cells (Sherman and Rinzel, 1991). This prolongs the high Ca 2§ plateaus, possibly significant for secretion. Complex spike patterns, somewhat like experimental records, have also been noted (Smolen et al., 1993).

In order to understand these phenomena, we considered an idealized case, pairs of identical cells, and showed (Sherman and Rinzel, 1992) that weakly coupled cells could burst synchronously, but with the spikes in the two cells 180 ~ out of phase (anti-phase or AP) and of smaller amplitude than those of uncoupled cells. Strongly coupled cells reverted to spikes that were in-phase (IP) and equivalent to those of isolated cells. Here I will show, in addition, bursting with spikes that are phase-locked with unequal amplitude (asymmetric or AS) or non-phase-locked and non-periodic (likely quasi-periodic or QP).

Such out-of-phase solutions are typical of weakly coupled oscillators. Kawato et al. (1979) studied AP oscillations in a van der Pol system. Ashkenazi and Othmer (1978) studied pairs of chemical oscillators using perturbation methods. Taylor and Kevrikidis (1991) studied another chemical oscillator and drew analogies to forced oscillator systems. Morita (1986, 1987), Kuramoto (1984) and Ermentrout (1985) have derived general conditions for instability of the IP solution for rings of coupled oscillators (including our degenerate two-cell rings) near a Hopf bifurcation. Lewis (1968) found AP spiking in an electronic (analog) motorneuron model. Hansel et al. (1993) have analysed AP solutions in the Hodgkin-Huxley equations with weak synaptic coupling.

Our focus is on the implications when the steady states of the system exhibit hysteresis with respect to a parameter that gives rise to bursting when the parameter is allowed to vary slowly. I give a geometrical, numerical analysis of

COMPLEX OSCILLATIONS, COUPLING AND BURSTING 813

the AP, AS, QP and IP behaviors and their relation to bursting by extending Rinzel's (1985) decomposition (reviewed in Section 2) of the bursting system into fast and slow subsystems to pairs of identical cells. I numerically compute bifurcation diagrams of the fast subsystem with the slow variable(s) viewed as parameters and use these to predict the trajectory of the full system when the slow variable(s) are allowed to vary. As we switch back and forth between the two points of view, bear in mind that all the bifurcation diagrams are computed with the slow variable(s) frozen and all trajectories are solutions of the full system.

In Section 3, I show that the increase in burst period results from an increase in the range of the slow variable. Further, as coupling strength increases, cell pairs go from AP to IP oscillations through a regime that includes QP oscillations and AS periodic oscillations. The AP branch is seen to arise as a secondary Hopf bifurcation from the steady state, AS as a pitchfork bifurcation from IP, and QP as a torus bifurcation from AP. Bursting can occur with QP spikes even when there are no stable periodic solutions of the fast subsystem for the range of parameter values traversed during a burst (Fig. 5). QP and AS spiking are important because they appear to be the dominant modes in multi- cellular (especially heterogeneous) simulations. All of these out-of-phase behaviors exist only for weak coupling. I show that the Hopf bifurcation points that give rise to AP are annihilated as coupling becomes infinite.

In addition to increased burst period, the range of parameters of the slow subsystem that permits bursting is extended. This is treated in Section 4 from the perspective of bifurcations of the full system as well as of the fast subsystem. In an extreme case, one can couple two tonically oscillating cells and obtain bursting. I analyse a previously reported example (Sherman and Rinzel, 1992) and show a new phenomenon: aperiodic alternation between tonic oscillations and bursts (Fig. 11).

I conclude with two examples that lead beyond the limits of this study, which assumes that the oscillators' slow variables have the same time course. Both non-identical cells (Fig. 12) and identical cells (Fig. 13) can burst with slow variables that differ. Bursts can be obtained more robustly from non- intrinsically bursting cells when the cells are not identical.

A companion paper (Sherman and Zhu, 1994) will apply singular perturbation methods to further explore properties of the underlying fast dynamics and begin to address non-identical cells. Certain features of our model are shown in the more general setting there not to be obligatory, but to depend on coefficients of the normal form of a single-oscillator and the form of the coupling. For example, AS may bifurcate from AP rather than IP, and there may not exist a range of coupling values for which AP is stable while IP is unstable. The numerical results obtained here, however, stand independent of that work.

814 A. SHERMAN

2. Single Cell Dynamics . Mot iva t ed bo th by a par t icu la r interest in burs t ing in pancreat ic /?-cel ls and by the desire to derive general results appl icable to neurons and o the r osci l la tory systems, I employ the fol lowing mode l (Sherman and Rinzel, 1992) ob ta ined by simplifying a biophysical /?-cel l mode l (Sherman et al., 1988):

d V dt - I c a ( V ) - I K ( V ' n ) - g s S ( V - - VK) (1)

d n r ~ t = 2 ( n ~ ( V ) - n ) (2)

d S = s (v)-s (3)

Ica(V) = gcamm(V) ( V - Vca)

1K(V) = gKn(V - VK)

xoo(v) = 1

v ) / o x = m , n, S. 1 + e x p ( ( V x -

Pa rame te r s are chosen to expose key features of the system, no t to achieve biophysical fidelity, bu t are plausible. Mos t of the p h e n o m e n a descr ibed here were first seen in realistic examples. P a r a m e t e r values are listed in Table 1 and figure legends.

Table 1. Parameter values

gca 3.6 gK 10 gs 4

20 msec z s 35 sec Vc, 25 mV V K - 75 mV 0 m 12 mV V m - 20 mV 0, 5.6 mV V, - 16 mV 0 s 10 mV V s -45 to - 2 5 mV* 2 0.8-1.1" gc 0.0~.5"

* Values for specific cases are listed in figure legends.


The model includes the minimal features needed for a biophysical model of square-wave bursting: a fast inward current (/Ca), a slower outward current (IK) and a very slow variable (S) that switches the active, spiking phase on and off. The slow variable has been postulated to be Ca 2§ (Chay and Keizer, 1983; Sherman et al., 1988) or ADP (Keizer and Magnus, 1989; Smolen and Keizer, 1992), activating an inhibitory K § current; S is an abstraction of these mechanisms. S could also be replaced by a variable representing the inactivation of an excitatory Ca 2 + current (Chay and Cook, 1988; Keizer and Smolen, 1991). More idealized models have also been proposed in which the total ionic current is represented by a polynomial (Hindmarsh and Rose, 1984).

We study the system by direct numerical integration using the Gear (1967) algorithm. Bursts were calculated with the FORTRAN subroutine LSODE (Hindmarsh, 1974), and analysis of spikes and steady states in the V-n phase plane was carried out using the program PhasePlane (Ermentrout, 1990). Unstable solutions and bifurcations were numerically calculated with the program AUTO (Doedel, 1981); a version with enhanced Floquet multiplier routines was used (Fairgreave and Jepson, 1991).

Capacitance does not appear in equations (1)-(3) because they have been scaled by a typical conductance to remove dependence on cell size. The values of coupling strength, 9c, below are then relative to a typical ionic conductance. This scaling also highlights the separation of time scale. Because z ~ "c s, we can decompose the system into a fast subsystem, equations (1) and (2), and a slow subsystem, equation (3) (Rinzel, 1985).

In Fig. 1, we dissect the behavior of equations (1) and (2) using S as a bifurcation parameter. For each value of S, the fast subsystem goes to a steady- state, oscillates (spikes), or both. The Z-curve, labeled SS, is a projection into the V-S plane of the curve I~= 0, ri = 0. The lower branch consists of stable node points. The middle branch consists of unstable saddles delimited by saddle- node bifurcations at the knees. The upper steady-states lose stability through a supercritical Hopf bifurcation at S ~ 0.04, giving rise to a cylinder of limit cycle oscillations. The projection of the cylinder is bounded by the curve LC, indicating the maxima and minima of the oscillations. The periodic branch terminates at a homoclinic orbit, HC.

For SFtB<S<S v (Fig. 1), LC is globally attracting. For S>Suc , the hyperpolarized steady-state is globally attracting. For Sv < S < SHC, the fast subsystem is bistable.

We now view S as a slow variable. SS is the projection of the null curve for V and n, and the curve Z (o 0 = 0) is the projection of the null surface for S. If Z crosses the middle branch of SS, S increases when Vis high and decreases when Vis low, and the full system, equations (1)-(3), bursts (Fig. 2). If 57 is shifted downward to intersect the lower branch of SS, the resulting steady-state of the full system is stable, and the solution is trapped on the alower branch. This can

816 A. SHERMAN

be done by decreasing V s, which we call a "burst bifurcation parameter" to distinguish it from S, which controls bifurcations of the fast subsystem. If V s is increased sufficiently, the system is trapped in the upper oscillatory state, and spikes continuously. Terman (1992) has proved a conjecture of Rinzel (1985) that in the limit r/Zs--.O, this occurs when Z intersects HC. Thus, S~ and Snc determine the extent of the bursting regime. As the full system steady-state moves from S~ to Snc, the duration of the active phase relative to the burst period (plateau fraction) increases. Pernarowski et al. (1992) have used singular perturbation methods to calculate the location of Suc and thereby the dependence of plateau fraction on a parameter.

In Fig. 2, the system bursts even though E intersects LC. S decreases when the spikes dip below 2, but has a net increase during each spike, allowing the burst to terminate normally at HC. If I2 is raised further by increasing V s to ~ - 3 5 , bursting goes to continuous spiking.

Since changes in the S equation do not affect the fast dynamics, spike amplitude is unchanged. This mimics the effect of glucose to increase plateau fraction in/~-cells without changing spike amplitude (Rinzel, 1985; Himmel and Chay, 1987).

Another parameter whose effect can be interpreted by the phase-plane representation is 2. The fast oscillations depend on the slow response of n and, hence, I K to changes in V, compared to the fast response of/ca (here assumed instantaneous). Speeding up n by increasing 2 tends to reduce spike amplitude, raising the minimum Vand lowering the maximum. This shifts Snc to the right

-10

LC

.-. -30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...,

o SN

-50 LC ~ ) n / H C

SS

-70 i 0.0 SHB S v SHC 0.3

S

Figure 1. Single cell bifurcation diagram. 2= 0.9, V s = - 38. Here and hereafter, dashed = unstable, solid = stable. SS = steady state; LC = limit cycle; HB = Hopf

bifurcation; SN = saddle-node bifurcation; HC = homoclinic orbit.


and tends to increase burst period. By Terman's result, this also enlarges the burst regime with respect to V s .

As 2 is increased, a second HB emerges out of the right knee [via a Takens-Bogadanov bifurcation (Wiggins, 1990)] and moves leftward along the upper steady-state branch. This branch does not play a role in bursting but completes the dynamic picture. It ends in a homoclinic orbit on the middle branch, which merges with the homoclinic orbit of the main periodic branch to form a single branch that begins and ends on HBs. At large enough 2, the two HBs also merge and the upper branch becomes completely stable (Rinzel, 1985). Then bursting is no longer seen, only slow waves without spikes.

3. Pairs of Identical Cells. I now extend the fast-slow decomposit ion to the case of identical pairs of cells coupled by a resistance. The variables V, n and S are indexed by cell number and a coupling term is added to the V equation:

E :>

r~

-20

-45

-70 0.20

0.17 0 25

-20

T (see)

g -45

S=0

SS

-70 0.165 0.185 0.205

S

Figure 2. (Top panels) Time courses for V and S corresponding to Fig. 1. Bottom) Projection of trajectory into V - S plane for single cell superimposed on expanded bifurcation diagram from Fig. 1. Heavy, long-dashed curve is the S nuU-cline

(S = 0). Directed curve is trajectory. SS is V null-cline (l?= h = 0).

818 A. SHERMAN

dye_ dt

Ic,(V~)-/K(V~, n,)-gsS~(V ~- VK)--g~(V ~- Vj), i , j= 1, 2. (4)

This is conceptually simple because it is a discrete form of diffusion and also biologically relevant, since fl-cells and many neurons are connected by gap junctions. There is no coupling on the auxiliary variable n or on S. For simplicity, we assume that the junct ional conductance is constant, independent of V.

Adding two more degrees of freedom substantially increases the complexity of the system. In considering the fast subsystem, we simplify by assuming that both cells have the same value of S, so that we can cont inue to employ one- parameter bifurcation diagrams (Fig. 3). In the calculations of trajectories, where the Sz are free to differ, in fact S~ ~ S 2 in many cases. In Figs 12 and 13, however, this condit ion is violated and the method is inadequate.

Because the equations are symmetric with respect to interchange of indices, we can use one of the V~ to record the state. For every solution with V~ > V 2 there is a corresponding solution with V 2 > 171, so plott ing only Va effectively displays all solutions.

3.1. Anti-phase oscillations. With weak coupling (e.g. gr two periodic solutions arise in the fast subsystem via Hopf bifurcation on the upper

-10

- 3 0 ZIIII AS2 ""i" ........... 52:;~::~'::~;;5"::2"~: ....

\ A S i ...... X ' p \ . . . " - . "<XN Ap / / ,

> . ........... . > / / 7

IP -50 - - - _ /SO>'

cO

-70 , , ----_____~s 0.00 0.10 0.20 ).30

S

Figure 3. Two-cell bifurcation diagram (anti-phase spiking). Same parameters as Fig. 1, except g~=0.06. Solution branches: SSS = symmetric steady state; ASS = asymmetric steady state; IP=in-phase periodic; AP=ant i -phase periodic; AS1, 2 = asymmetric periodic. Minima of periodics are labeled; corresponding maxima are located roughly symmetrically across SSS. AS2 cylinder is contained within AS 1. Bifurcation points: open squares, pitchfork of steady-states or periodics; filled

squares, Hopf; diamonds, torus.


steady-state branch (Fig. 3). One is an in-phase solution (IP), for which the cells behave identically. Since the coupling current is then 0, the behavior is also identical to that of an uncoupled cell. The IP branch is mathematical ly distinct from the single-cell limit cycle, LC in Fig. 1, but its projections in the V~-S i planes are equivalent to LC. A new (supercritical) Hopfbifurcat ion gives rise to an anti-phase periodic branch (AP).

Following Ashkenazi and Othmer (1978) and Kawato et al. (1979), we can establish some relationships between the IP and AP branches by linearizing the fast subsystem equations. We express equations (1) and (2) formally as:

X= F(X, S), (5)

where X = (V, n) r. The coupled fast subsystem is then:

J(1 = F(X1, S)+D(Xz-XI) (6)

f(z= F(Xz, S) + D(X1- Xz), (7)

where the coupling matrix D is:

01 D = . (8)

0

If J = dF i s the Jacobian of a single oscillator, then the Jacobian of the coupled system is:

o

This can be block diagonalized by the t ransformation Yl=(Xl+X2)/2, Y2 = ( x l - x 2 ) / 2 to:

[: J-2D

The IP HB occurs when trace (J) = 0, i.e. at the same value of S as the HB of the single-cell system. The AP HB occurs when trace ( J - 2 D ) = 0, or gc = ~/2 trace ( J ) . This relation is double-valued because trace ( J ) increases from 0 to a max imum and then decreases back towards 0 as S exceeds SH~. Therefore, there are two AP HBs that merge and disappear as gc increases, and AP only exists for relatively weak coupling. If a single cell has two IP HBs (for large enough 2; see Section 2), then the two AP HBs of the coupled cells coincide with the IP HBs when g~ = 0.

For the case of Fig. 3, there is only one IP HB, but there is a second AP HB

820 A. S H E R M A N

near the right knee of the Z-curve. It is not shown here because it dies nearby via a homoclinic orbit and plays no role in bursting. As gc-~0 it collides with the right knee of the Z-curve and disappears in a Takens-Bogdanov bifurcation.

There are four other bifurcations off the steady-state Z-curve. Two occur when det ( J ) = 0 and reproduce the left and right knee saddle-nodes of the single cell. The other two occur when det ( J - 2 D ) vanishes and are pitchfork bifurcations (open squares in Fig. 3) on the middle branch. For our case, det ( J - 2 D ) = 0 implies gc = - r 2 / 2 2 det (J). Analogously to the trace, det ( J ) = 0 at the knees and is negative between the knees with a single minimum. Therefore, the pitchforks must be on the middle branch; they coincide with the knees when gc = 0 and move towards each other and coalesce as gc increases.

From the two BPs, two new Z-curves bifurcate representing asymmetric steady-state (ASS) solutions of the four-variable fast subsystem. There are two HBs (not shown) that are related by interchange of cell index on ASS. Here they are uninteresting, but in other cases they give rise to stable periodic solutions with unequal amplitudes.

Determining the stability of IP and AP and the existence of secondary bifurcations analytically requires knowledge of higher order terms and will be taken up in Sherman and Zhu (1994). Here, we numerically compute Floquet multipliers and continue solutions with AUTO.

The IP multipliers include a pair inherited from the uncoupled system and a new pair. The inherited pair makes IP stable to symmetric perturbations (perturbations that affect the two cells identically); the new pair makes it unstable to non-symmetric perturbations.

The IP branch is initially stable but loses stability through a secondary pitchfork bifurcation (a Floquet multiplier goes through t). This gives rise to two new periodic solution branches in which the cells have slightly unequal amplitude and are slightly out of phase. These are labeled AS 1 (cell 1 has large amplitude, cell 2 has small amplitude) and AS2 (vice versa). Alternatively, one can view these curves as the amplitudes of the two cells in either of the periodic solutions. These solutions are initially stable, but when coupling is weak they quickly become unstable via a torus bifurcation (a pair of multipliers enters the unit circle) and play no role in bursting. As we will see, however, with stronger coupling they can be important.

The AP branch is initially unstable [proved by Kawato et al. (1977)] but becomes stable through a torus bifurcation. It has smaller amplitude than IP. Geometrically, the AP cylinder lies inside the IP cylinder. Biophysically, a cell at a spike maximum tends to depolarize its neighbor, which is at a minimum, and vice versa. As shown in Aronson et al. (1990) and Kawato et al. (1979), the AP amplitude is smaller locally near a Hopf bifurcation. Here we see numerically that this property holds globally, and that, as a consequence, the termination of the AP branch lies to the right of that of the IP branch.


If we now let S i be slow variables, the full system bursts with reduced spike amplitude and increased amplitude of S (Fig. 4). Increased S amplitude will tend to lengthen the bursts, provided the rate of change of S does not increase too much. In fact, S does increase faster along AP than IP during the active phase because V, and hence So~(V), are larger on average, but the net active phase duration increases. For the silent phase trajectories only the increased length of the path matters; the speed is the same because they follow the same manifold in both AP and IP cases. Consequently, the silent phase duration increases proportionally more than the active phase duration, and plateau fraction decreases.

Figure 4 (bottom) shows the superimposed projections of the cell trajectories for one burst in the V1-S 1 and V2-S 2 planes. The anti-phase spikes follow the AP branch rather than the unstable IP branch, but shadow the unstable IP branch transiently at the beginning of the active phase.

The S values of the two cells are close, but not identical. Indeed, with each

Figure 4. (Top panels) Time courses for V and S corresponding to Fig. 3. (Bottom) Projection of trajectory into V~-S i planes superimposed on bifurcation diagram from Fig. 3. Note anti-phase spikes. There are two bursts per period; cells alternate

in initiating the active and silent phases. Directed curve is trajectory.

822 A. SHERMAN

burst they alternate in being larger, so that there are two bursts in every period of the full-system oscillation. The V-S projections show that the cell which has smaller S in the current burst phase (solid curve) will have larger S in the next burst. The symmetry of the full system is reflected in the symmetry of the trajectory around the 1-1 diagonal in the V1-V 2 and S~-S z planes (not shown).

3.2. Quasi-periodic spiking. We have seen that the AP branch can only exist for weak coupling strength. In the limit of very strong coupling the spikes must be in-phase. We now look at the effects of varying gc between those extremes. Another motivation is that the AP and IP Hopf bifurcations occur at the same value of S when gc = 0: two pairs of conjugate eigenvalues cross the imaginary axis simultaneously, gc is a natural second parameter to unfold this co-dimension two bifurcation.

As g~ increases, the AP HB and HC of the fast subsystem move to the right and the AP amplitude decreases. The burst period of the full system is more than twice that of an uncoupled cell at g~ = 0.08.

With further increase in gc, AP merges with the second AP branch near the right knee to produce a single AP branch that begins and ends at HBs (Fig. 5). By g~ = 0.1, the torus points from the two original AP branches have collided and disappeared, and the AP solution is unstable everywhere. Meanwhile, the IP branch gains stability as the secondary pitchfork bifurcation moves to the right.

For the range of S traversed by the bursting trajectory, however, none of the fast-system periodic branches is stable. The full system bursts (Figs 5, bottom, and 6) with spikes that vary rhythmically, but not periodically, in amplitude (Fig. 6, bottom). First one cell has larger spike amplitude then the other, but the time-averaged amplitudes are equal. The phase differences are not locked but oscillate around n. We conjecture that the spikes are carried by quasi- periodic solutions that bifurcate from the AP branch, or possibly by tertiary branches.

It is difficult to completely characterize this case because AUTO cannot follow the quasi-periodic solution branches. Fourier spectra (not shown) of the spike solutions for some fixed values of S show multiple, distinct peaks. The Poincare sections (V~, V 2 and n~ when n 2 crosses 0.04 from below; not shown) are invariant circles, sometimes with cusps that lead to harmonic resonances (multi-peaked periodic solutions).

Note also that the bursts are not periodic in this case. The peak values of S vary irregularly (Fig. 6, middle), and the cells alternate irregularly in initiating active and silent phases.

3.3. Asymmetric spiking. If gc is increased further, the AP HBs merge and the AP branch disappears, while IP and AS1, 2 gain stability. For the


parameters of Figs 5 and 6, the BP on the IP branch moves right and a second BP emerges at the HC point and moves left. With respect to bursting, it is the latter bifurcation that matters. By 9c=0.18, the IP branch is stable in the hysteresis region and the full system bursts with in-phase spikes. Eventually, the two bifurcation points meet and the entire IP branch becomes stable.

By changing one parameter slightly (increasing 2 to 1.0) we can arrange for the AS branches to be stable in the hysteresis region, while the IP branch remains unstable there. Such a case is shown in Fig. 7 with 9c = 0.15. The full system can burst in this configuration too (Figs 7, bottom, and 8). At first glance, the spikes appear to be quasi-periodic as in Fig. 5, but this is a transient effect resulting from the weak attraction of AS 1, 2 (a pair of conjugate Floquet multipliers has magnitude near 1). In Fig. 7 (bottom), the spikes approach AS2 towards the end of the burst. The other cell (not shown) approaches the curve labeled AS1. In the next burst they switch. If S is frozen, the spikes asymptotically approach one of the AS periodic solutions, but only after a time

-10

g ;>

-30

-50

-70 0.00

I P ~/,/

_ _ , i , i S ~

0.05 0.10 0.15 0.20 0.25

]~ I

-30

-50

-70 0.16 0.20 0.24

S

Figure 5. Quasi-periodic spiking. When coupling is increased to gc = 0.10, AP loses stability. Bursting now occurs with quasi-periodic spikes. (Top) Bifurcation diagram. (Bottom) Projection of one burst (first burst in Fig. 6) into V~S i plane. Asymmetric steady-state branch still exists, but is omitted for clarity. Symbols as in

Fig. 3.

824 A. SHERMAN

equal to several burst periods. If the coupling is weakened, AS1, 2 becomes unstable by torus bifurcation. Thus, the time course in Fig. 8 is analogous to the damped, oscillatory approach to a steady-state near a Hopf bifurcation.

Our identification of the spikes as AS rather than QP is also supported by the observations that one cell's amplitude is always larger than the other's (Fig. 8, bottom), and that the phase difference of the spikes in a burst is near 0, rather than oscillating around zc as in Fig. 6. Finally, the full-system bursts are periodic, as in the anti-phase case, with the S amplitudes alternating and repeating every two bursts and with symmetrical trajectories in the S1-S z and V1-V 2 planes (not shown). (If 9c is increased to 0.2, however, this symmetry is broken.)

4. Full System Bifurcations. The bifurcation diagrams of the previous section (Figs 3, 5 and 7) show that the fast subsystem can oscillate at values of S for which the only attractor is a resting steady-state when uncoupled. This leads to

Figure 6. (Top panels) Time courses of V (one cell) and S (one cell solid, one dashed) corresponding to Fig. 5. Bursts are now aperiodic, with irregular alternation pattern, unlike Figs 3 and 4. (Bottom) Expanded view of spikes from both cells in second burst. Spike amplitude varies slowly and phase difference oscillates around

~. Larger vs smaller amplitude alternates between the cells.


an increase in burst period when the S i are unfrozen. Now I show that the extension of the oscillatory range of the fast regime also allows an extension of the bursting range of the full system.

As described for single cells (Section 2), increasing V s causes the full system to bifurcate from bursting to continuous spiking at some point after the S null- surface, Z, first intersects LC (Rinzel, 1985; Terman, 1992; Wang, 1993). Consequently, the reduction in spike amplitude in the fast subsystem induced by weak coupling delays the onset of continuous spiking in the full system, enlarging the range of V s for which the system bursts.

This quantitative effect of coupling suggests a qualitative effect: if 2 is reduced sufficiently the spike amplitude is increased so that Z cannot intersect the middle branch of SSS without also intersecting LC. Then, the fast subsystem is mono-stable and it may hold that the full system can spike or be silent, but not burst. With coupling, however, the anti-phase periodic cylinder

> E ;>

-20

-45

-70

IP / / /

0.05

-20

0.15 0.25

-45

-70 0.17 0.22

S

Figure 7. Asymmetric spiking. Coupling increased to gc = 0.15 and 2 increased to 1.0. Bifurcation diagram and V - S projection. AP branch has disappeared. AS branches are only stable periodic solutions in hysteresis region. This is not an obligatory pattern; with 2=0.9, IP restablizes from right to left before AS can appear. Symbols as in Fig. 3. Asymmetric steady-state branches omitted for clarity.

826 A. SHERMAN

can still lie above Z. Thus, bursting may exist for a range of values of V s for the coupled system, but fail to exist for any value of Vs for the isolated system.

An example [-first reported in Sherman and Rinzel (1992)] is shown in Fig. 9. Uncoupled, the system spikes continuously or beats, but coupled, it bursts. The bifurcation diagram (Fig. 10) shows that AP is stable and IP is unstable, except for a tiny interval near the knee where the two solutions are bistable. The latter feature can lead to complex interactions. If V s is reduced, 2; intersects the 1P cylinder near the bistable region. Then, all trajectories of the full system that leave the lower branch of the Z-curve are captured by an in-phase beating solution, which is the same as that of an uncoupled cell. Similarly, if 9c is increased to 0.08, the IP stability interval grows to the left from Sv and again the full system beats in-phase.

At the border between in-phase beating and bursting with anti-phase spikes (e.g. 9c=0.08, but V s = - 3 4 ) there can be persistent, irregular alternation

Figure 8. Time courses of V (one cell; top) and S (both cells; middle) corresponding to Fig. 7. Bursting is periodic, with two bursts per period as in Fig. 4. Spikes (both cells; bottom) are asymmetric and slightly out-of-phase rather than anti-phase. For the duration of each burst one cell has larger spike amplitude than the other, but they alternate from burst to burst. Spikes appear quasi-periodic as in Figs 5 and 6, but are actually damped, oscillatory transients to asymmetric periodic solutions.


between the two (Fig. 11). After the first burst in Fig. 11 (top), there is an episode of beating that spontaneously ends in a second burst. Then there are two more bursts with anti-phase spikes followed by more beating. Figure 11 (bottom) shows that, depending on how the trajectory emerges from the silent phase, it may either go directly to the AP branch and initiate a new burst or follow the IP branch leftward until it loses stability. Note that, even in Fig. 9, the bursts are not periodic, and some bursts (e.g. the second one) have additional large spikes as the trajectory transiently samples the unstable IP branch. This may be analogous to the transition from bursting to continuous spiking in single cells studied by Terman (1992).

The example of Figs 9 and 10 is delicate; 2 and V s must be carefully chosen, and 9c cannot be too strong. The spikes are large, so the period and S amplitude cannot be large. More robust bursting can be obtained from cells with different values of V s . If V s = - 3 8 and - 4 5 , the uncoupled cells beat and remain silent,

g~=0

;~ -45 >

-70

r.~

>

>

0.20

0.17

g~ = 0.04

-20

-45

-70

0.20

0.17 0 5 10 15 20 25

T (sec)

Figure 9. Bursting from beating. With 2 = 0.8, bistability is lost and an isolated cell can only be silent or beat continuously, depending on V s . With V s = - 38, the cell beats (upper panels). If the cells are weakly coupled (9c=0.04), they burst with

anti-phase spikes (lower panels). Only one cell of the pair is shown.

828 A. SHERMAN

respectively. When coupled they can burst for 9c = 0.5 (Fig. 12), 10-fold higher than identical cells. For very strong coupling, the cells beat in-phase.

When cells are not identical, it is no longer true in general that their S time courses will be the same (Fig. 12). Even identical cells can burst with different values of S; Fig. 13 shows the cells of Figs 3 and 4 (9c=0.06, 2 = 0.9), but with V s increased to -32 . The spikes are slightly out of phase and differ in amplitude, as in Fig. 12. This asymmetric bursting solution is only attained for some initial conditions; it is bistable with the symmetric AP solution and may bifurcate from it. The intermittent solution of Fig. 11 is also bistable with such an asymmetric bursting solution.

5. Discussion. We have extended the theory of bursting for square wave bursters (Rinzel, 1987), such as pancreatic/~-cells, from single cells to coupled pairs of cells. The model used is a simplification of a biophysical/?-cell model (Sherman et al., 1988). We have found similar results with a variety of/%cell models, including one with eight variables (Smolen and Keizer, 1992), a thalamic neuron model (Hindmarsh and Rose, 1984) and in an adaptation (Rinzel and Ermentrout, 1989) of a model for barnacle muscle (Morris and Lecar, 1981).

-20

> -45

>

s-o . . . . . . ~ . . . . . . . _----2~ /

AP

-70 L J 0.165 0.185

S

Figure 10. Projection of second burst from Fig. 9 into V-S plane, superimposed on partial bifurcation diagram. There are also unstable ASS, AS1 and AS2 branches

which are not shown. Open squares mark pitchfork bifurcation of IP.


With the slow variables, Si, frozen, the IP fast subsystem solution is unstable for weak coupling strength, go. New solution branches (AP and AS periodic solutions and Q P solutions) arise. With S i unfrozen, coupled cells can burst with spikes generated by each of the new solutions (Figs 4, 6 and 8). The spikes have smaller amplitude than the in-phase solution, increasing burst period by 50-100%.

As 9c increases, the out-of-phase solutions lose stability and disappear, while the IP solution stabilizes. Thus, strongly coupled (identical) cells synchronize and behave like isolated cells. Burst period increases and then decreases as 9c increases from 0.

The increase in burst period from reduced spike amplitude depends critically on the fact that the periodic spiking branch of the uncoupled fast subsystem

Figure 11. Intermittent beating and bursting with ,~ = 0.8, gc = 0.08 and V s = - 34. Time courses of V 1 (top) and $1, S 2 (middle). Small spikes are AP, large spikes are IP. Projection of last burst and last beating episode (56<T<76) (Bottom). Trajectories coming out of the silent phase either go directly to the AP branch or backtrack along the partially stable IP branch for a while. Open squares mark

pitchfork bifurcation of IP.

830 A. SHERMAN

ends in a homoclinic orbit (Fig. 1), since then the smaller amplitude out-of- phase branches of the coupled system terminate to the right of IP (Figs 3 and 7). This increases the maximum S attained at the end of a burst by the coupled cells, while the minimum S is little affected. Hence, the range of S increases and, provided there is no compensating increase in the rate of change of S, burst period increases.

Burst period can be increased by any parameter change whose sole effect is a reduction in spike amplitude that shifts the homoclinic point of the single cell limit cycle, LC, to the right, such as an increase in 2. Coupling, however, changes burst period without changing the intrinsic properties of the cells.

Recall (Section 2) that for sufficiently large 2, the two LC homoclinic points merge, and a single LC cylinder is formed that connects the two HB points and surrounds the upper steady-state. The same holds for the IP branch of coupled

Figure 12. Non-identical cells. The cells are mono-stable as in Figs 9 and 10 ()~ = 0.8), but have different values of V s ( - 3 8 and -45) and would be beating and silent, respectively, if uncoupled. Robust bursting exists for relatively large coupling (9c = 0.5). Time course of Vfor cell 1 (top) and cell 2 (middle). Spikes are asymmetric and slightly out-of-phase. (Bottom) Time courses of S for both cells. S 1 -S 2 is

approximately constant.


cells. In the absence of a homoclinic orbit, the active phase of an uncoupled cell terminates at or near the right knee. Coupling then cannot increase the range of S further. Indeed, for our model, the rate of traversal of the upper branch is faster when the spikes are out of phase, so the burst period is reduced slightly.

The mechanism of period extension by spike amplitude reduction is of course not available to relaxation oscillators, such as F i tzhugh-Nagumo (1961), that lack spikes. Diffusively coupled, identical relaxation oscillators synchronize at the same solution as an isolated oscillator. Excitatory synaptic coupling, however, can increase the period of relaxation oscillators (Somers and Kopell, 1993). Note that unlike diffusive coupling, which vanishes when the oscillators are in the same state, synaptic coupling may be non-zero even when the oscillators are perfectly synchronized. Thus, with synaptic coupling, the synchronized solution is not, in general, a solution of the isolated units and does not, in general, have the same period.

Figure 13. Symmetry breaking in bursting for identical cells. With 2=0.9 and 9c=0.06 as in Figs 3 and 4, hut V s increased to -32, a new solution emerges with S 1 r S 2 . This solution coexists with a "standard" symmetric bursting solution that has anti-phase spikes. Here, the spikes differ in amplitude but are nearly anti-phase.

These cells would be spiking continuously if uncoupled.

832 A. SHERMAN

Bursters can be reduced to an equivalent relaxation oscillator by averaging the periodic branch (Rinzel and Lee, 1986). From this point of view, we have shown that a diffusively coupled burster system has a different period than when isolated because it is equivalent to a different relaxation oscillator: the isolated cell behavior is obtained by averaging IP, while the coupled cell behavior is obtained by averaging, say, AP.

Apart from bursting, the reduced system with S frozen at S* also has interesting behavior when there is bistability and a homoclinic point. If S*> S~c, then an isolated cell would be excitable, but not oscillatory, with a low-voltage steady-state as the only asymptotically stable solution. Two such cells coupled, however, could be bistable between a steady-state and AP, AS or QP oscillations. An example with AP oscillations was given in Sherman and Rinzel (1992).

We have studied pairs of cells in order to gain insight into the behavior of large clusters, the natural environment of fl-cells. Our results suggest that the maximum in burst period seen at intermediate coupling strength in simulations of homogeneous (Sherman and Rinzel, 1991) and heterogeneous (Smolen et al., 1993) cell clusters is also caused by reduced spike amplitude. The main differences between large clusters and pairs seem to be that QP or AS, rather than AP, spikes are typical (Rinzel et al., 1992, Fig. 2; Smolen et al., 1993), and that period extension persists for much larger values of coupling strength [gc = 0(1)].

In light of the paucity of experimental reports of bursting in single cells, we have been exploring whether fl-cell bursting is a property of individual cells or populations. In a previous modeling study (Smolen et al., 1993), we proposed the "heterogeneity hypothesis", that most cells cannot burst because the parameter regime for bursting is narrow. It is easy, however, to find model parameters that generate continuous spiking or silent behavior, the responses most commonly observed experimentally in isolated cells. We showed that coupling a roughly equal mixture of such non-bursting cells readily leads to bursting (provided the cells have the basic prerequisites of a Z-curve and a slow variable).

I have addressed this question here with an alternative mechanism that exploits the property that bursting in single cells goes to continuous spiking roughly when the slow null-surface lies above the homoclinic orbit and intersects the periodic cylinder (Rinzel, 1985; Terman, 1992; Wang, 1993). Coupling, by reducing spike amplitude, can allow bursting when the uncoupled system spikes continuously. Thus, the range of the burst bifurcation parameter (here Vs) for which the system bursts is extended at the upper end, and the interval for bursting is lengthened (by ~ 50% with the parameters of Figs 3 and 4). In an extreme case, the uncoupled system can (e.g. by reducing 2) be made mono-stable and unable to burst for any value of V s, while coupling


permits bursting (Figs 9 and 10). (Of course, the uncoupled cell could burst if)~, or possibly some other parameter, were changed.)

This example works only for a limited range Ofgc and with other parameters carefully selected; even moderate 9c synchronizes the cells at an in-phase beating solution. If the cells differ, however, they can burst robustly over a greatly broadened range of 9c (Fig. 12).

In Smolen e t al. (1993), the cells were not mono-stable and could burst, but at different values of the burst bifurcation parameter, there a parameter, R, related to the ATP synthesis rate. Retrospectively, we can call that the "weak heterogeneity hypothesis": coupling allows ensemble bursting at a single value of R, even if no cells can burst uncoupled at that value. Figure 12, then, is an example of the "strong heterogeneity hypothesis": the cells cannot burst for any value of V s . This mechanism can be extended to large clusters (unpublished calculations). As in the two-cell example, as 9c~ ~ the population ceases to burst, but spikes continuously or remains silent, depending on the distribution of parameters.

Our bifurcation diagrams predict the behavior of cell pairs when they have the same values of S. This is a good approximation in many cases. If cells differ, however, the S values typically differ (Fig. 12). Identical cells can also break symmetry and exhibit this behavior (Fig. 13).

Abbott and LeMasson (1993) have given an example of non-symmetric bursting in a Morris-Lecar based model with conductances regulated by in t racel lular Ca 2 +. There, cell parameters become different after coupling as the augmented system consisting of the burst and channel regulation mechanisms undergoes a symmetry-breaking bifurcation. In Fig. 13 the (unregulated) cell parameters remain identical throughout.

These cases cannot, in general, be analysed by the present method. Note that in Figs 12 and 13, the S values differ by nearly a constant value over time. The solutions could be predicted by a one-parameter bifurcation diagram if the constant were known; this requires treating the two slow variables as independent parameters and will be addressed in future investigations.

Joel Keizer, Paul Smolen and, especially, John Rinzel were involved in the modeling work on which this study is based. I thank Maureen Rush and Richard Bertram for careful reading of the text. Many other people have found this subject interesting and have contributed valuable advice and ideas, including Bard Ermentrout, Nancy Kopell, Yoshihisa Morita, Xiao-Jing Wang, Tim Kiemel, Hsiu-Rong Zhu, David Terman and Yue-Xian Li. Thomas Fairgreave provided enhanced Floquet multiplier routines for use with AUTO. Computer time was provided by the National Cancer Institute Biomedical Supercomputing Center and the National Institutes of Health Division of Computer Research and Technology Convex group.

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Rece ived 17 J u n e 1993

Rev i sed 29 N o v e m b e r 1993

Anti-phase, asymmetric and aperiodic oscillations in excitable cells—I. Coupled bursters

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