Applications of Control Theory to the Dynamics and Propagation of Cardiac Action Potentials

Abstract

Sudden cardiac arrest is a widespread cause of death in the industrialized world. Most cases of sudden cardiac arrest are due to ventricular fibrillation (VF), a lethal cardiac arrhythmia. Electrophysiological abnormalities such as alternans (a beat-to-beat alternation in action potential duration) and conduction block have been suspected to contribute to the onset of VF. This study focuses on the use of control-systems techniques to analyze and design methods for suppressing these precursor factors. Control-systems tools, specifically controllability analysis and Lyapunov stability methods, were applied to a two-variable Karma model of the action-potential (AP) dynamics of a single cell, to analyze the effectiveness of strategies for suppressing AP abnormalities. State-feedback-integral (SFI) control was then applied to a Purkinje fiber simulated with the Karma model, where only one stimulating electrode was used to affect the system. SFI control converted both discordant alternans and 2:1 conduction block back toward more normal patterns, over a wider range of fiber lengths and pacing intervals compared with a Pyragas-type chaos controller. The advantages conferred by using feedback from multiple locations in the fiber, and using integral (i.e., memory) terms in the controller, are discussed.

Appendix

Lyapunov analysis was used to assess the stability of Eq. (3), which is non-autonomous. A number of restrictions were imposed: θ(·) was replaced by θ_c(V − V _n ) = 0.5(1 + tanh(c _a(V − V _n ))) in Eqs. (1)–(3), and it was assumed that i ^d(t) is a continuous function of time, and that $\tilde{i}(t) = -K\tilde{V}(t).$ The single-cell case of the dynamic equations was used. Under all these assumptions, the right-hand sides of Eqs. (1) and (2) satisfy sufficient conditions for existence and uniqueness of solutions, local to a given initial time and set of initial conditions. However, a stronger assumption was made, that the desired trajectories, V ^d(t), and n ^d(t), are bounded (not necessarily periodic) for all times t ≥ t ₀. A shorthand version of the continuous-time error dynamics (Eq. 3), for the closed-loop single-cell case, was written as $\dot{{{\mathbf{s}}}} = \phi({{\mathbf{s}}},t).$ Here, the system state consists of the errors for the membrane potential and gating variable of the single cell, i.e., ${{\mathbf{s}}} = [\tilde{V}\; \tilde{n}]^{\text{T}}.$ The point $(\tilde{V}=0, \tilde{n}=0)$ is an equilibrium point of the system.

The following theorems are equivalent to statements found in Khalil.15 Specifically, Theorem 1 corresponds to Corollary 3.4 of Theorem 3.8, and Theorem 2 corresponds to Theorem 3.11.

Theorem 1

Lets = 0 be an equilibrium point for$\dot{{{\mathbf{s}}}} = \phi_1({{\mathbf{s}}},t),$where ϕ₁is piecewise continuous in t and locally Lipschitz in a domain D that containss = 0. LetW(s) be a continuously differentiable function such that W is PD and$\dot{W}({{\mathbf{s}}})$is ND for alls ∈ D. Furthermore, suppose that$W({{\mathbf{s}}})=k_1\|{{\mathbf{s}}}\|^q$and$\dot{W}({{\mathbf{s}}}) \leq -k_2\|{{\mathbf{s}}}\|^q,$for some positive constantsk₁, k₂, and q. Thens = 0 is exponentially stable.

Theorem 2

Lets = 0 be an equilibrium point for the system$\dot{{{\mathbf{s}}}} = \phi({{\mathbf{s}}},t),$where$\phi\hbox{: }[t_0,\infty)\times D \rightarrow {\mathbb{R}}^N$is continuously differentiable, $D=\{{{\mathbf{s}}}\in{\mathbb{R}}^N | \|{{\mathbf{s}}}\|_2 < r\}, \, r>0,$and the Jacobian$\frac{\partial \phi}{\partial {{\mathbf{s}}}}$is bounded and Lipschitz on D, uniformly in t. Define$A(t)=\frac{\partial \phi}{\partial{{\mathbf{s}}}}({{\mathbf{s}}},t)|_{{{\mathbf{s}}}=0}.$Then, s = 0 is an exponentially stable equilibrium point for the non-linear system if it is an exponentially stable equilibrium point for$\dot{{{\mathbf{s}}}}=A(t){{\mathbf{s}}}.$

The version of Theorem 1 in Khalil15 is more general in that it covers cases where W either has explicit time dependence or takes forms other than $k_1\|{{\mathbf{s}}}\|^q.$ In Theorem 2, note that the region over which stability holds may be smaller than r. See Khalil15 for proofs of the theorems.

Let D be a ball of finite radius that includes the origin. To apply the first theorem, let ϕ₁(s, t) = A(t)s, where A(t) is the Jacobian of ϕ(s, t) (where ϕ is the right-hand side of the closed-loop single-cell error dynamics), evaluated at s = 0. Specifically,

$$ \dot{{{\mathbf{s}}}} = \phi_1({{\mathbf{s}}},t) = A(t){{\mathbf{s}}} = {\frac{d}{dt}}\left[\begin{array}{l} \tilde{V}_1\\ \tilde{n}_1\\ \end{array}\right] (t) = \left[\begin{array}{ll} \alpha_{11}(t) + cK & \alpha_{12}(t)\\ \alpha_{21}(t) & \alpha_{22}\\ \end{array}\right] \left[\begin{array}{l} \tilde{V}_1\\ \tilde{n}_1\\ \end{array}\right] (t), $$

where

$$ \begin{aligned} \alpha_{11} &=\frac{1}{\tau_V}\left(-1+\left(V^*-\left(\frac{n^d}{n_B}\right) ^M\right)h'(V^d)\right),\quad \alpha_{12} =\frac{1}{\tau_V}\left(-M\left(\frac{n^d}{n_B}\right)^{M-1} \left(\frac{1}{n_B}\right)h(V^d)\right)\\ \alpha_{21} &=\frac{1}{\tau_n}\left(\frac{1}{b}\theta_{c}^{\prime}(V^d-V_n)\right), \quad \alpha_{22} =-\frac{1}{\tau_n}. \\ \end{aligned} $$

ϕ₁(s, t) is at least piecewise continuous in t (in fact, it is fully continuous in t), since the elements of A(t) are continuous functions of the continuous functions V ^d(t) and n ^d(t). Additionally, ϕ₁(s, t) is Lipschitz in D because all the elements of A(t) are bounded for all t ≥ t ₀. Each matrix element is bounded for bounded values of V ^d and n ^d, which are bounded by assumption.

Consider the function

$$ W({{\mathbf{s}}})= {\frac{1}{2}}\|{{\mathbf{s}}}\|_2^2 = {\frac{1} {2}}\tilde{V}^{2} + {\frac{1}{2}}\tilde{n}^{2}. $$

(8)

W as defined in Eq. (8) is continuously differentiable and PD by inspection. Applying the chain rule and substituting in the linearized two-variable Karma equations for a single-cell,

$$ \dot{W} = \tilde{V}\dot{\tilde{V}} + \tilde{n}\dot{\tilde{n}} = (\alpha_{11} + cK)\tilde{V}^{2} + (\alpha_{12} + \alpha_{21})\tilde{V}\tilde{n} + \alpha_{22}\tilde{n}^2 $$

is obtained. To satisfy the remainder of Theorem 1, it would be sufficient to find k ₂ > 0 for which

$$ \dot{W} = (\alpha_{11} + cK)\tilde{V}^2 + (\alpha_{12} + \alpha_{21})\tilde{V}\tilde{n} + \alpha_{22}\tilde{n}^2 \leq -k_2\|{{\mathbf{s}}}\|_{2}^{2}. $$

Moving all terms to the left side of the inequality, and completing the square (which involves adding a net zero quantity to both sides) yields

$$ \dot{W} + k_2\|{{\mathbf{s}}}\|_2^2 = (\bar{\alpha}_{11} + cK)\tilde{V}^2 + \bar{\alpha}_{22}\left({\frac{(\alpha_{12} + \alpha_{21})}{2\bar{\alpha}_{22}}}\tilde{V} + \tilde{n}\right)^2 - {\frac{(\alpha_{12} + \alpha_{21})^2} {4\bar{\alpha}_{22}}}\tilde{V}^2 \leq 0,$$

where $\bar{\alpha}_{11}=\alpha_{11} + k_2$ and $\bar{\alpha}_{22}=\alpha_{22} + k_2.$ Suppose that $0<k_2<\frac{1}{\tau_n}$ (τ _n  = 250 in the default units). Then, $\bar{\alpha}_{22}=-\frac{1}{\tau_n}+ k_2 <0,$ which renders the second term, $\bar{\alpha}_{22}\left(\frac{(\alpha_{12} + \alpha_{21})}{2\bar{\alpha}_{22}}\tilde{V} + \tilde{n}\right)^{2},$ on the left side of the inequality nonpositive; hence,

$$ \dot{W} + k_2\|{{\mathbf{s}}}\|_{2}^{2} \leq (\bar{\alpha}_{11} + cK)\tilde{V}^{2} - {\frac{(\alpha_{12} + \alpha_{21})^2} {4\bar{\alpha}_{22}}}\tilde{V}^2.$$

The quantity on the right side of the inequality is non-positive as long as

$$ K \leq -{\frac{1}{c}}\left(\bar{\alpha}_{11}(t) - {\frac{(\alpha_{12}(t) + \alpha_{21}(t))^2} {4\bar{\alpha}_{22}}}\right), $$

for all t ≥ t ₀. The expression on the right side of the inequality is bounded, since the elements of A(t) are bounded, and $\bar{\alpha}_{22}\neq 0.$ For values of K sufficiently far to the left (possibly negative) on the real axis, the conditions of Theorem 1 are satisfied for $k_1=\frac{1}{2},\quad 0<k_2<\frac{1}{\tau_n},$ and q = 2. Thus, the origin is exponentially stable for $\dot{{{\mathbf{s}}}} = \phi_1({{\mathbf{s}}},t) = A(t){{\mathbf{s}}}.$

The first condition of Theorem 2 concerns the non-linear system $\dot{{{\mathbf{s}}}} = \phi({{\mathbf{s}}},t).$ Note that ϕ _t (s, t) exists and is continuous, since $\dot{V}^d$ and $\dot{n}^d$ are continuous by assumption. Writing out the Jacobian matrix, $J({{\mathbf{s}}},t)=\frac{\partial \phi}{\partial {{\mathbf{s}}}},$ reveals that its elements are continuous functions of $V^d-\tilde{V}$ and $n^d-\tilde{n}.$ Hence, ϕ(s, t) is continuously differentiable. The Jacobian is bounded for all t ≥ t ₀, since its elements are functions that are bounded for all bounded values of $V^d-\tilde{V}$ and $n^d-\tilde{n}.$ V ^d and n ^d are bounded, for all time, by assumption, and $\tilde{V}$ and $\tilde{n}$ are bounded because s ∈ D. For the product of the Jacobian and the state, ζ(s, t) = J(s, t)s, to be Lipschitz in D, with the same Lipschitz bound for all t ≥ t ₀, it is sufficient to establish that (1) ζ is continuous in its arguments, (2) $\frac{\partial \zeta}{\partial {{\mathbf{s}}}}$ exists and is continuous, and (3) $\frac{\partial \zeta}{\partial {{\mathbf{s}}}}$ is bounded, for all t ≥ t ₀ and points in D. ζ(s, t) is continuous since J(s, t) is continuous in its arguments. Assuming M ≥ 2, and noting that the elements of the Jacobian are continuously differentiable functions of $V^d-\tilde{V}$ and $n^d-\tilde{n},$ the elements of $\frac{\partial \zeta}{\partial {{\mathbf{s}}}}$ exist and are continuous functions of s and t, defined for all bounded values of s and all t ≥ t ₀. Finally, $\frac{\partial \zeta}{\partial {{\mathbf{s}}}}$ is bounded for all t ≥ t ₀, since its elements are functions that are bounded for all bounded values of $V^d-\tilde{V}$ and $n^d-\tilde{n}.$ Therefore, the conditions of Theorem 2 are satisfied, which means that s = 0 is an exponentially stable equilibrium point for the non-linear system $\dot{{{\mathbf{s}}}} = \phi({{\mathbf{s}}},t),$ because it is an exponentially stable equilibrium point for $\dot{{{\mathbf{s}}}}=A(t){{\mathbf{s}}}.$