## Critical Size of a Pacemaker

The SA node is a small clump of self-oscillatory cells in a sea of excitable (but nonoscilla-tory) cells whose function is to initiate the cardiac action potential. SA nodal cells have no contractile function and therefore no contractile machinery. Thus, when viewed in terms of contractile efficiency, SA nodal cells are a detriment to contraction and a waste of important cardiac wall space. On the other hand, the SA node cannot be made too small because presumably, it would not be able to generate the current necessary to entrain the rest of the heart successfully. Thus it is important to have some measure of the critical size of the SA node.

An ectopic focus is a collection of cells other than the SA node or AV node that are normally not oscillatory but that for some reason (for example, increased extracellular potassium) become self-oscillatory and manage to entrain the surrounding tissue into a rapid beat. In some situations, particularly in people with scar tissue resulting from a previous heart attack, the appearance of an ectopic focus may be life-threatening.

We model the behavior of a clump of oscillatory cells in an otherwise nonoscillatory medium in a simple way, using FitzHugh-Nagumo dynamics:

— = e(v - yw - a(rla)), dt where v represents the membrane potential and w the recovery variable for our excitable medium. The function f(v) is of typical "cubic" shape (cf. Chapter 4). The function a(r) is chosen to specify the intrinsic cell behavior as a function of the radial variable r. The number a is a scale factor that measures the size of the oscillatory region. We take e to be a small positive number and require y > 0,f'(v)y < 1 for all v. This requirement on y guarantees that the steady-state solution of (14.19)-(14.20) is unique. If the domain is bounded, typical boundary conditions are Neumann (no-flux) conditions. Notice that space has been scaled to have unit space constant. We assume radial symmetry for the SA node as well as for the entire spatial domain.

When there is no spatial coupling, there are two possible types of behavior, exemplified by the phase portraits in Figs. 4.17 and 4.19. In these examples, the system has a unique steady-state solution that is globally stable (Fig. 4.17) or has an unstable steady-state solution surrounded by a stable periodic orbit (Fig. 4.19), depending on the location of the intercept of the two nullclines.

The transition from a stable to an unstable steady state is a subcritical Hopf bifurcation. The Hopf bifurcation is readily found from standard linear analysis. Suppose v* is the equilibrium value for v (v* is a function of a). Then the characteristic equation for (14.19)-(14.20) (with no diffusion) is where X is an eigenvalue of the linearized system. There is a Hopf bifurcation (i.e., X is purely imaginary) when provided that cy2 < 1. If f'(v*) > cy, the steady-state solution is an unstable spiral point, whereas if f'(v*) < cy, the steady-state solution is linearly stable. If c is small, most of the intermediate (increasing) branch of the curve f (v) is unstable, with the Hopf bifurcation occurring close to the minimal and maximal points. Thus, there is a range of values of a, which we denote by a* <a <a*, for which the steady solution is unstable.

We wish to model the physical situation in which a small collection of cells (like the SA node or an ectopic focus) is intrinsically oscillatory, while all other surrounding cells are excitable, but not oscillatory. To model this, we assume that a(r) is such that the steady solution is unstable for small r, but stable and excitable for large r, so that limr^œ a(r) = a < a* and limr^œ f'(v*(r)) < cy. As an example, we might have the bell-shaped curve

with a < a* < b < a*. The scale factor R was chosen such that a(1) = a*, so that cells with r < 1 are self-oscillatory and the cells outside unit radius are nonoscillatory.

Another way to specify a(r) is simply as the piecewise constant function

a, for r > 1, with a < a* < b < a*. The specification (14.25) is particularly useful when used in combination with the piecewise linear function

1 — v, for v > 3, since then all the calculations that follow can be done explicitly (see Exercises 11 and 12).

There are two parameters whose influence we wish to understand and that we expect to be most significant, namely, a, the asymptotic value of a(r) as r ^ro, and a, which determines the size of the oscillatory region. Note that as a decreases, the cells become less excitable and the wavespeed of fronts decreases. We expect the behavior to be insensitive to variations in b, although this should be verified as well.

With a nonuniform a(r), the uncoupled medium has a region of cells with unstable steady states and a region with stable steady states. With diffusive coupling, the steady state is smoothed and satisfies the elliptic equation

For each r, the function F(v, r) is a monotone decreasing function of v having a unique zero, say v = v*(r),F(v*(r),r) = 0. It follows that there is a unique, stable solution of (14.29), denoted by v0(r), w0(r). In fact, this unique solution is readily found numerically as the unique steady solution of the nonlinear parabolic equation dy = V2y + F(y,r). (14.30)

This steady-state solution is shown in Fig. 14.12. Here are shown three different steady-state solutions of (14.19)-(14.20); the uncoupled solution (the steady states for the uncoupled medium, i.e., with no diffusion), the solution for a symmetric one-dimensional medium, and the solution for a spherically symmetric three-dimensional medium. The three-dimensional solution with spherical symmetry is not much harder to find than the one-dimensional solution, because the change of variables y = Y/r transforms (14.30) in three spatial dimensions into dY d2Y / Y . ,

Figure 14.12 Three steady-state solutions with f(v) = 10.0v(v - 1)(0.5 - v), y = 0.1, and a(r) given by (14.23), with a = 0.104, b = 0.5, a = 2.25. The short dashed curve shows the uncoupled solution, the long dashed curve shows the solution for a symmetric one-dimensional medium, and the solid curve shows the solution for a spherically symmetric three-dimensional medium.

Figure 14.12 Three steady-state solutions with f(v) = 10.0v(v - 1)(0.5 - v), y = 0.1, and a(r) given by (14.23), with a = 0.104, b = 0.5, a = 2.25. The short dashed curve shows the uncoupled solution, the long dashed curve shows the solution for a symmetric one-dimensional medium, and the solid curve shows the solution for a spherically symmetric three-dimensional medium.

Solutions of this partial differential equation are regular at the origin if we require Y = 0 at r = 0. Diffusion obviously smooths the steady-state solution in the oscillatory region.

The issue of collective oscillation is determined by the stability of the diffusively smoothed steady state as a solution of the partial differential equation system (14.19)-(14.20). To study the stability of the steady state, we look for a solution of (14.19)-(14.20) of the form v(r) = v0(r) + V(r)eXt, w = w0(r) + W(r)elt and linearize. We obtain the linear system

Because of the special form of this linear system, it can be simplified to a single equation, namely,

where f = X + . Equation (14.34) has a particularly nice form, being a Schrodinger equation. In quantum physics, the function -f '(v0(r)) is the potential-energy function, and the eigenvalues f are the energy levels of bound states. In the present context, we are interested in determining the sign of the real part of X through f = X + . Notice that the relationship between f and X here is of exactly the same form as the characteristic equation for individual cells (14.21). This leads to a nice interpretation for the Schrodinger equation (14.34). Because it is a self-adjoint equation, the eigenvalues f of (14.34) are real. Therefore, there is a Hopf bifurcation for the medium whenever f = ey. The entire collection of coupled cells is stable when the largest eigenvalue

satisfies ¡x < ay and unstable if the largest eigenvalue has ¡x > ay.

In Fig. 14.13 is shown the potential function f (v(r)) for the three steady profiles of Fig. 14.12. The largest eigenvalue of (14.34) represents an average over space of the influence of f '(v0(r)) on the stability of the steady state. When this value is larger than ay (the critical slope of f (v) at which Hopf bifurcations of the uncoupled system occur), then the entire medium loses stability to a Hopf bifurcation and gives rise to an oscillatory solution. The condition x > ay is therefore the condition that determines whether a region of oscillatory cells is a source of oscillation. If x < ay, the oscillatory cells are masked by the rest of the medium.

Some observations about the size of the eigenvalues x are immediate. Because limr^TO v0(r) = limr^TO v*(r), it follows that f'(v0(r)) < ay for large r. For there to be a bounded solution of (14.34) that is exponentially decaying at there must be a region of sinusoidal behavior in which x < f '(v0(r)). Thus, the largest eigenvalue of (14.34) is guaranteed to be smaller than the maximum of f (v0(r)). Therefore, if v0(r) < a* (so that f (v0(r)) < ay for all r), there are no oscillatory cells, and the steady solution is stable. Furthermore, since the largest eigenvalue is strictly smaller than the maximum of f (v0(r)) and it varies continuously with changes in v0(r), there are profiles a(r) having a nontrivial collection of oscillatory cells that is too small to render the medium unstable. That is, there is a critical mass of oscillatory cells necessary to cause the medium to oscillate. Below this critical mass, the steady state is stable, and the oscillation of the oscillatory cells is quenched.

Suppose f'(v) is a monotone increasing function of v in some range v < v+, and suppose that a(r) is restricted so that v0(r) < v+ for all r. Suppose further that a(r) is a monotone increasing function of its asymptotic value a and a monotone decreasing function of r. Then the steady-state solution v0(r) is an increasing function (for each point r) of both a and a. Therefore, the function f'(v0(r)) is an increasing function of a and a for all values of r, from which it follows—using standard comparison arguments for eigenfunctions (Keener, 1988, or Courant and Hilbert, 1953)—that y(a,a), the largest eigenvalue of (14.34), is an increasing function of both a and a. As a result, if a(r) is restricted so that vo(r) < v+ for all r, there is a monotone decreasing function of a, denoted by a = £(a), along which the largest eigenvalue /x(a, a) of (14.34) is precisely ey.

This summary statement shows that to build the SA node, one must have a sufficiently large region of oscillatory tissue, and that the critical mass requirement increases if the tissue becomes less excitable or if the coupling becomes stronger. Strong coupling inhibits oscillations, because increasing coupling increases the space constant, and a was measured in space constant units. Therefore, an increase of the space constant increases the critical size requirement of the oscillatory region. In Fig. 14.14 is shown the critical Hopf curve a = £(a) for a one-dimensional domain and for a three-dimensional domain (taking e = 0), both found numerically.

Having established that there is a critical size for a self-oscillatory region above which oscillations occur and below which oscillations are prevented, we would like to examine the behavior of the oscillations. Two types of oscillatory behavior are possible. If the far field r ^ < is sufficiently excitable, then the oscillations of the oscillatory region excite periodic waves that propagate throughout the medium, as depicted in Fig. 14.15. On the other hand, it may be that there are oscillations that fail to propagate throughout the entire medium, as depicted in Fig. 14.16. In Fig. 14.15, the oscillatory region successfully drives oscillatory waves that propagate throughout the entire medium. Here, a = 0.2, a = 3.0.InFig. 14.16, the oscillatory regionis incapable of driving periodic waves into the nonoscillatory region. For this figure, a = 0.0, and a = 3.0, so that the medium at infinity does not support front propagation.

The issue of whether or not the entire medium is entrained to the central oscillator is decided by the relationship between the period of the oscillator and the dispersion curve for the far medium. Roughly speaking, if the period of the central oscillator is large enough compared to the absolute refractory period of the far medium (the knee

3.0 — 1-d medium 2.52.01.5 —J3-d medium 1.00.5 - stable 0.0-1

unstable

0.05

0.10

0.15

0.20

0.25

Figure 14.14 The critical curve a = £(a) along which there is a Hopf bifurcation for the system (14.19)-(14.20), shown solid for a one-dimensional and dashed for a three-dimensional medium.

Figure 14.15 Waves generated by an oscillatory corethat propagate intothe nonoscillatory region. The nonlinearity here is the same as in Fig. 14.12 with e = 0.1,y = 0.1,a = 0.2,b = 0.5, and a = 3.0.

Figure 14.16 Waves generated by an oscillatory core that fail to propagate into the nonoscillatory region. The nonlinearity here is the same as in Fig. 14.12 with e = 0.1,y = 0.1,a = 0,b = 0.5, and a = 3.0.

of the dispersion curve), then waves can be expected to propagate into the far field in one-to-one entrainment. On the other hand, if the frequency of the oscillation is below the knee of the dispersion curve, we expect partial or total block of propagation. Block of propagation occurs as the excitability of the far field, parametrized by a, decreases.

We can summarize how the oscillations of the medium depend on coupling strength. For a medium with fixed asymptotic excitability, if the size of the oscillatory region is large enough, there is oscillatory behavior. However, this critical mass is an increasing function of coupling strength. With sufficiently large coupling, the oscillations of any finite clump of oscillatory cells (in an infinite domain of nonoscillatory cells) are quenched. If coupling is decreased, the critical mass for oscillation decreases. Thus, any clump of oscillatory cells oscillates if coupling is weak enough. However, if coupling is too weak, then effects of discrete coupling may become important, and the oscillatory clump of cells may lose its ability to entrain the entire medium. It follows that if the medium is sufficiently excitable, there is a range of coupling strengths, bounded above and below, in which a mass of oscillatory cells entrains the medium. If the coupling is too large, the oscillations are suppressed, while if the coupling is too weak, the oscillations are localized and cannot drive oscillations in the medium far away from the oscillatory source. On the other hand, if the far region is not sufficiently excitable, then one of these two mechanisms suppresses entrainment for all coupling strengths.

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