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Nonlinear Dyn. Author manuscript; available in PMC 2009 January 1. Published in final edited form as: | PMCID: PMC2605722 NIHMSID: NIHMS23453 |
Asymptotic approximation of an ionic model for cardiac restitution David G. Schaeffer,1,2* Wenjun Ying,1 and Xiaopeng Zhao2,3 1 Department of Mathematics, Duke University, Durham, North Carolina 27708, USA 2 Center for Nonlinear and Complex Systems, Duke University, Durham, North Carolina 27708, USA 3 Department of Biomedical Engineering, Duke University, Durham, North Carolina 27708, USA Cardiac restitution has been described both in terms of ionic models—systems of ODE’s—and in terms of mapping models. While the former provide a more fundamental description, the latter are more flexible in trying to fit experimental data. Recently we proposed a two-dimensional mapping that accurately reproduces restitution behavior of a paced cardiac patch, including rate dependence and accommodation. By contrast, with previous models only a qualitative, not a quantitative, fit had been possible. In this paper, a theoretical foundation for the new mapping is established by deriving it as an asymptotic limit of an idealized ionic model. 1.1 Background information When a small piece of cardiac muscle is subjected to a sequence of brief electrical stimuli whose strength exceeds a critical threshold, the myocytes respond by producing action potentials, see . The duration of action potential refers to the period when the voltage is elevated above its resting value. The interval between the time when the voltage returns to its resting value 1 and the next stimulus is called the diastolic interval. We use the acronyms APD for action potential duration; DI for diastolic interval; and, assuming periodic pacing, BCL for the interval between stimuli, also known as basic cycle length. There is great interest in cardiac restitution: i.e., determining how, under repeated stimulations, each APD depends on previous history. This is a key step in a program to understand how arrhythmias arise and sometimes progress to sudden cardiac death [ 1, 2, 12, 22]. Restitution information from experiments is often presented in a restitution curve: i.e., a graph of APD vs. DI under one of various protocols [ 11]. shows one of these, the so-called dynamic restitution curve. In this protocol, for each of many periods B, the tissue is paced periodically with this period until it reaches a steady-state 1:1 phase-locked response, and then the steady-state action potential duration Ass and diastolic interval Dss are recorded. The pairs of points ( Dss, Ass) resulting from various values of B form the dynamic restitution curve. Each APD depends most strongly on the previous DI. In their seminal paper [ 15], Nolasco and Dahlen abstracted this behavior in a phenomenological model where An denotes the duration of the nth action potential, Dn denotes the duration of the nth diastolic interval, and G( D) is a monotone increasing function of the diastolic interval. If B denotes the BCL with which the stimuli are applied, then Dn = B − An, see . Substituting into ( 1), we see that in this model the sequence An is determined recursively by iteration of a 1D mapping. If the data in were described by a model of the form ( 1), then the thin curve in the figure would be the graph of G. Despite its successes, the Nolasco-Dahlen model misses many important phenomena. In particular, it does not capture any memory effects [ 6, 10, 14]. To illustrate this, consider tissue that, after repeated pacing with period B0, has achieved a steady-state response. Then suppose the BCL is abruptly decreased to a new value and held there. According to ( 1), all pairs of points ( Dn, An+1) in the transient to the new steady state would lie on the graph of the function G( D) in the D, A-plane. However, in experiments (see for example Kalb et al. [ 11]), the approach to steady state occurs along a completely different curve, as illustrated in . Moreover, evolution towards the steady state is very slow, much slower than other time scales in the data. See Schaeffer et al. [ 17] and the references in that paper for a more thorough discussion of memory issues. 1.2 The goal of this paper In Schaeffer et al. [ 17], we introduced a 2D mapping and chose parameters that gave a quantitatively accurate description of the full restitution behavior, including memory, measured from a bullfrog ventricle. The present paper is concerned with this model, but we study its theoretical foundations rather than apply it to fitting experimental data. A mapping provides a flexible way to fit restitution data from experiments, but it suffers from the limitations of a phenomenological model. For example, a more complete description of propagation of action potentials in extended tissue is provided by a more fundamental type of model, an ionic model. For a single cell or a small piece of tissue, an ionic model consists of a system of ODEs that specifies how the voltage across the cell membrane changes in response to currents of ions that flow through the cell walls. Both idealized ionic models [ 14, 18]—low-dimensional systems aimed at qualitative understanding—and realistic models [ 8, 13]—complicated systems intended to describe all currents in the cell—have been proposed. As in the Hodgkin-Huxley equations, given these ODEs, one may introduce an appropriate diffusive term in the voltage equation (or equations, if the bidomain model is used) to obtain PDEs that describe propagation in extended tissue. In this paper we complement the modeling of Schaeffer et al. [ 17] by showing that the mapping of that paper arises as an asymptotic limit of an idealized ionic model. We begin in Section 2 by recalling from Schaeffer et al. [ 17] both the ionic model and the mapping. In Section 3 we derive the mapping from the ionic model as the leading term in an asymptotic expansion. A concluding discussion is presented in Section 4. 2 The ionic and mapping models 2.1 The idealized ionic model The present model builds on the two-current ionic model of Karma [ 12] and of Mitchell and Schaeffer [ 14]. The two-current model contains two functions of time, the transmembrane potential v( t) and a gating variable h( t), both of which are dimensionless and scaled to lie in the interval (0, 1). We augment the two-current model by adding a third variable, a (dimensionless) generalized concentration c, and modifying the equations as given in equations (2, 4, 6) below. These equations involve ten positive parameters, values of which can be obtained by fitting the model with experimental data. For example, the values listed in were obtained in Schaeffer et al. [ 17] from experiments with a bullfrog ventricle. | Table 1Parameters for the ionic model (2), (4) and (6) |
(i) The equation for the transmembrane potential reads where the outward current in ( 2) is linear in the voltage, Jout( v) = v/ τout, and the nonlinear inward current is the sum of concentration- independent and concentration- dependent parts where β > 0 is a constant. It may be seen from ( 3) that the build-up of charge in the cell weakens the inward current, thereby shortening action potentials. The behavior of the model is not very sensitive to the exact form of the functions ![[var phi]](/corehtml/pmc/pmcents/x03C6.gif) ci( v) and cd( v). In the present work, to facilitate the calculations, we set these functions equal to piecewise linear functions of v, as follows: and (ii) Depending on the voltage, the gating variable h opens or closes according to the equation The voltage-dependent closing rate is taken as piecewise linear in v, Note that two different time-scale parameters, τfclose and τsclose, derive from the closing of the gate. (Remark: The subscripts fclose and sclose are mnemonic for “fast close” and “slow close”, respectively; sldn, for “slow down”.) (iii) The concentration is determined by a balance between I(t), the current which leads to the build-up of charge in the cell, and constant linear pumping, which removes charge from the cell: The current I(t) should satisfy two key properties: - I(t) is nonzero only during the upstroke of an action potential, and
- A fixed charge ε enters the cell during each action potential; in symbols
The precise form of I( t) is not important; to achieve the properties above we choose 2Note that the time constants in satisfy the following property: Below in deriving a mapping to describe the behavior of ( 2), ( 4) and ( 6), we shall assume that ( 9) holds, as well as Although it is not critical, we shall also assume that 1 − vsldn ≤ vcrit ![[double less-than sign]](/corehtml/pmc/pmcents/x226A.gif) 1. The ionic model ( 2), ( 4) and ( 6) can be used to model action potentials produced by a cardiac patch under repeated stimulation. For example, shows two time traces of solutions at a basic cycle length B = 650ms, with model parameters chosen as in . The solid curve represents the steady-state response following many stimuli at this basic cycle length, while the dashed curve represents the response to the first stimulus with B = 650ms, following many stimuli at B = 750ms. 2.2 Approximation of the ionic model by a mapping Complicated evolution of the ionic model, such as in , can be described approximately, with far less computation, in terms of the 2D mapping introduced in [ 17]. The variables in the mapping are An and Cn. Here An denotes the duration of the nth action potential as illustrated in , and Cn specifies the ion concentration c at the start of the ( n + 1) st action potential. Intuitively, one may think of Cn as a memory variable 3: i.e., a slowly evolving, auxiliary quantity that modifies the electrical properties of the cell. Provided the diastolic interval Dn is not too short, the mapping is given by the formula Where and The new constants Amax and α are expressed in terms of the parameters of in equations (28) and (29) below, respectively. As we shall see, Amax is the longest possible APD. The evolution of APD and the concentration in the simulation behind is illustrated in , which graph An and Cn as functions of the beat number n. If, as in , all BCL’s are equal to some constant B and if the first stimulus occurs at t = 0, then Cn = c(nB). The first fifty beats in show the steady values for these variables following many stimuli at BCL = 750ms (assuming parameters as given in ). At n = 51, the BCL is abruptly decreased to 650ms. This results in an immediate decrease in An, followed by a slow evolution over 250 beats during which Cn increases and An decreases. show blow-ups of the evolution during the first few beats after the change in BCL; note that Cn changes only slightly over this short time. Regarding short DI’s, the above formulas hold provided For the parameters in , we have Dsldn = 46ms. See Section 3.2(b) for treatment of DI’s shorter than this. 3 Leading order approximation of the ionic model 3.1 Overview of the derivation As sketched in , an action potential has four distinct phases. In each phase there are different balances between the equations (2, 4, 6) and their associated time scales. Note from ( 9) that the fastest time scales are associated with the voltage equation (2). Thus, the nullcline of this equation, plays a central role in the asymptotics. By contrast, equation (6) for the concentration contains only an extremely slow time scale, so c is nearly constant over one action potential; thus, in ( 16), c is regarded as a constant. Apart from the trivial case v = 0, equation (16) expresses the condition that the inward and outward currents are exactly balanced. Solving this equation for h as a function of v yields The nullcline is the dashed curve graphed in . Let hmin( c) be the minimum value for h on this curve. Since β > 0, it is easy to find from ( 17) that Equation (16) may also be solved for v as a function of h, but one encounters multivaluedness: i.e., as may be seen in , for a given value of h, besides v = 0 there typically are two nonzero solutions of ( 16). The dominant behavior in each phase of an action potential may be described as follows and as summarized in . The fact that c is approximately constant over each phase is not repeated in the description. (See Mitchell and Schaeffer [ 14] for a more detailed discussion of the asymptotics.) | Table 2Summary of asymptotics during the four phases of an action potential |
(1) Upstroke phase Following a successful stimulus 4, the inward current Jin dominates the outward current Jout. In a time on the order of τin, during which the change in the gating variable h is negligible, the voltage rises quickly to the right branch of the nullcline ( 16). (2) Plateau phase As the gate closes according to equation (4), the voltage follows the nullcline, keeping the inward and outward currents balanced. In the present model, the plateau phase may be subdivided into a fast-closing subphase ( v > vsldn) and a slow-closing subphase ( v < vsldn), which have time scales τfclose and τsclose, respectively. (3) Repolarization phase When the gating variable reaches hmin(c) on the nullcline, the solution trajectory “falls off the nullcline”: i.e., the outward current Jout dominates the inward current Jin and the voltage drops toward v = 0 (see the solid line in ). This occurs on a time scale of order τout. (4) Resting phase, or diastolic interval The voltage stays small and the gate reopens with a time constant τopen. This continues until the next stimulus is applied. An APD consists of all of phase 2 plus parts of phases 1 and 3. According to ( 9), phases 1 and 3 are much shorter, so to a first approximation 5, the APD is the duration of phase 2. 3.2 Derivation of the mapping (a) Preliminaries Assuming that ( 2, 4, 6) is stimulated repeatedly, we wish to estimate An+1—the duration of the action potential produced by the ( n + 1) st stimulus (assumed successful)—and Cn+1—the concentration when the ( n + 2) nd stimulus arrives. In our approximation, these quantities depend only on Dn, the diastolic interval preceding the ( n + 1) st stimulus; Cn, the concentration when the ( n + 1) st stimulus arrives; and B, the interval between the ( n + 1) st and the ( n + 2) nd stimuli. In our principal application, periodic stimulation, every two consecutive stimuli are separated by the same interval, so in our notation we do not include a subscript on B. The estimate for Cn+1 is easily obtained. Given ( 9) and the assumptions on I( t) in the ODE ( 6), we see that following phase 1 of the ( n + 1) st action potential, the concentration evolves by where t = 0 corresponds to the arrival time of the ( n + 1) st stimulus. Thus ( 12) follows for stimuli separated by period B. In phase 2, v is determined to leading order as a function of h and c by ( 16). On substitution of the resulting formula for v into ( 4), we obtain an ODE for h. In this equation, c, which may be approximated as constant over one APD, appears as a parameter. We will solve this ODE subject to the initial value for h given in the following lemma. Lemma 3.1 At the start of phase 2 of the (n + 1)st action potential and at the end of this phase Proof As noted above, hterm ≈ hmin( Cn) defines the end of phase 2: i.e., the point at which h has decayed so much that the inward current can no longer balance the outward current. This verifies ( 20). Equation (19) may be verified by analyzing the preceding DI. The initial condition for the gate h at the start of this DI, say h(0) where we have redefined the time origin, is approximately hmin( Cn), which is the value of h at the end of phase 2 of the previous action potential. More accurately, because h continues to decay during phase 3 of the previous action potential, we have However, hmin( Cn) ≤ τin/ τout, which by ( 9) is a small quantity. Thus we may take h(0) ≈ 0. By solving the initial-value problem for dh/ dt = (1 − h)/ τopen with h (0) = 0 over the interval 0 < t < Dn, we see that the value of h at the end of the nth DI is given by ( 19). Since h does not change appreciably during phase 1 of the ( n + 1) st action potential, the lemma is proved. ■ (b) Short DI’s Except for very fast pacing, the diastolic interval Dn is larger than Dsldn, where If Dn < Dsldn, then at the start of the ( n + 1) st action potential, hinit < hsldn as can be seen from ( 19) and ( 21). According to ( 16), at this time v < vsldn; thus, in solving ( 4) only the simpler alternative occurs: i.e., dh/ dt = − h/ τsclose. Note that v does not appear in this equation. Thus, regardless of the behavior of v, the gate h has simple exponential decay. Hence, if Dn < Dsldn, then An+1, the time required for h to decay from hinit to hmin( Cn), is given by (c) General DI’s If Dn > Dsldn, then both fast-closing and slow-closing subphases are present in phase 2. The slow-closing phase begins at h = hsldn and ends at h = hmin(Cn), so it has duration To determine the duration of the fast-closing subphase, we note from ( 16) that if v > vsldn, then Substituting into ( 4), we obtain the linear ordinary differential equation for h( t) Resetting (without loss of generality) t = 0 in the initial condition for ( 24), we find the formula for the gating variable in the fast closing sub-phase The duration of this subphase, Afclose, is determined by solving for the time when h(t) = hsldn: where we have substituted ( 17) for ( 19). Of course An+1 is the sum of ( 26) and ( 23), (d) A convenient rewriting At slow pacing Dn is large, and under repeated slow pacing Cn converges to approximately zero. Thus, recalling ( 18) and ( 21), we see that under repeated slow pacing where Adding and subtracting Amax to ( 27) and rearranging, we obtain equations (11, 13, 14). Incidentally, for the parameter values in , we have Amax = 840ms and α = 1.1. 3.3 Threshold for stimulation Up to now we have been assuming that each stimulus was successful in producing an action potential. Let us examine the stimulation process more carefully. When a stimulus current is applied, an extra term must be added to ( 2), Assume that Jstim in nonzero only for an interval of length τstim that is short compared to all other time scales in the equations. Then at the end of the stimulus, v ≈ vstim, where Let hstim( Cn) be the corresponding value of h on the nullcline ( 17): i.e., Lemma 3.2 The (n + 1)st stimulus will produce an action potential if and only if Proof Immediately following the (n + 1)st stimulus, where hinit is given by ( 19). If ( 30) holds, then the point ( 31) lies inside the nullcline ( 16) where Jin dominates Jout, and the system will begin a normal action potential. If ( 30) does not hold, then Jout dominates Jin, and the voltage will quickly decay back to zero with no lasting change in the evolution. ■ 3.4 Comparison of the mapping and the ionic model shows the dynamic restitution curves produced by both the mapping ( 11– 12) and the ionic model ( 2, 4, 6). As the figure shows, the errors are larger than one would like, especially for faster pacing rates. Cain and Schaeffer [ 3] have shown that the asymptotic mapping of the two-current model in [ 14] can be greatly improved by including higher-order corrections. Following a similar line of approach, one can obtain an improved mapping for the ionic model ( 2, 4, 6). A preliminary version of the improved mapping significantly reduces the errors. This will be discussed elsewhere. Based on asymptotic approximation of a system of nonlinear ODEs, we have derived a two-dimensional mapping, which is able to accurately describe restitution in paced cardiac tissue. Unlike ad hoc mappings, the mapping developed here clearly relates to physiological variables through the underlying ODEs, also known as an ionic model. The developed mapping provides a tool to understand cardiac instabilities that may lead to fatal arrhythmias. Since the underlying ionic model is piecewise defined, the resulting mapping also exhibits piecewise smoothness. Piecewise smooth dynamical systems may exhibit various discontinuity-induced bifurcations, such as grazing bifurcations in systems with discontinuous changes in states [ 7, 19, 20] and border-collision bifurcations in piecewise continuous maps [ 5, 16, 21]. To explore the possibility for discontinuous bifurcations, we first examine the type of discontinuities in the mapping. As established in Section 3, An+1 relates to Dn by ( 22) when Dn < Dsldn and by ( 28) when Dn > Dsldn. Thus, a discontinuity boundary of the mapping is associated with Dn = Dsldn. It follows from ( 21) that hsldn = 1 − e−Dsldn/τopen. Therefore, values of the mapping are continuous at Dn = Dsldn, as can be seen from ( 22) and ( 28). Moreover, one can verify that first derivatives of the mapping are continuous at the discontinuity boundary, although second derivatives jump. Thus, the mapping satisfies the usual
 hypothesis of smooth bifurcations. In any event, in almost the entire range of the experiment of [ 17], the system is responding in the range Dn > Dsldn above the discontinuity boundary. Support of the National Institutes of Health under grant 1R01-HL-72831 and the National Science Foundation under grants DMS-9983320 is gratefully acknowledged. 1. Banville I, Gray RA. Effect of action potential duration and conduction velocity restitution and their spatial dispersion on alternans and the stability of arrhythmias. J Cardiovasc Electrophysiol. 2002;13:1141–1149. [PubMed] 2. Cherry EM, Fenton FH. Suppression of alternans and conduction blocks despite steep APD restitution: electrotonic, memory, and conduction velocity restitution effects. Am J Physiol. 2004;286:H2332–H2341. 3. Cain JW, Schaeffer DG. Two-term asymptotic approximation of a cardiac restitution curve. 2005. Preprint. 4. Chialvo DR, Michaels DC, Jalife J. Supernormal excitability as a mechanism of chaotic dynamics of activation in cardiac Purkinje fibers. Circ Res. 1990;66:525–545. [PubMed] 5. di Bernardo M, Budd C, Champneys A, Kowalczyk P. Bifurcation and Chaos in Piecewise-smooth Dynamical Systems: Theory and Applications. Springer-Verlag; in process. 6. Elharrar V, Surawicz B. Cycle length effect on restitution of action potential duration in dog cardiac fibers. Am J Physiol. 1983;244:H782–H792. [PubMed] 7. Foale S, Bishop R. Bifurcations in impacting systems. Nonlinear Dynamics. 1994;6:285–299. 8. Fox JJ, McHarg JL, Gilmour RF., Jr Ionic mechanism of electrical alternans. Am J Physiol. 2002c;282:H516–H530. 9. Fox JJ, Bodenschatz E, Gilmour RF., Jr Period-doubling instability and memory in cardiac tissue. Phys Rev Lett. 2002a;89:138101-1–138101-4. [PubMed] 10. Hall GM, Bahar S, Gauthier DJ. Prevalence of rate-dependent behaviors in cardiac muscle. Phys Rev Lett. 1999;82:2995–2998. 11. Kalb SS, Dobrovolny HM, Tolkacheva EG, Idriss SF, Krassowska W, Gauthier DJ. The restitution portrait: a new method for investigating rate-dependent restitution. J Cardiovasc Electrophysiol. 2004;15:698–709. [PubMed] 12. Karma A. Spiral breakup in model equations of action potential propagation in cardiac tissue. Phys Rev Lett. 1993;71:1103–1107. [PubMed] 13. Luo C, Rudy Y. A dynamic model of the cardiac ventricular action potential. Circ Res. 1994;74:1071–1096. [PubMed] 14. Mitchell CC, Schaeffer DG. A two-current model for the dynamics of cardiac membrane. Bull Math Bio. 2003;65:767–793. [PubMed] 15. Nolasco JB, Dahlen RW. A graphic method for the study of alternation in cardiac action potentials. J Appl Physiol. 1968;25:191–196. [PubMed] 16. Nusse HE, Yorke JA. ‘Border-collision bifurcations including period two to period three for piecewise smooth systems,’ Physica D. 1992;57:39–57. 17. Schaeffer DG, Cain JW, Gauthier DJ, Kalb SS, Oliver RA, Tolkacheva EG, Ying W, Krassowska W. An ionically based mapping model with memory for cardiac restituion. Submitted. 18. Shiferaw Y, Watanabe MA, Garfinkel A, Weiss JN, Karma A. Model of intracellular calcium cycling in ventricular myocytes. Biophys J. 2003;85:3666–3686. [PMC free article] [PubMed] 19. Zhao X, Dankowicz H, Reddy CK, Nayfeh AH. Modeling and Simulation Methodology for Impact Microactuators. Journal of Micromechanics and Microengineering. 2004;14:775–784. 20. Zhao X, Reddy CK, Nayfeh AH. Nonlinear Dynamics of an Electrically Driven Impact Microactuator. Nonlinear Dynamics. 2005;40:227–239. 21. Zhusubaliyev ZT, Mosekilde E. Bifurcations and chaos in piecewise-smooth dynamical systems. World Scientific; Singapore: 2003.
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