Nonlinear dynamics in cardiology

Abstract

The dynamics of many cardiac arrhythmias, as well as the nature of transitions between different heart rhythms, have long been considered evidence of nonlinear phenomena playing a direct role in cardiac arrhythmogenesis. In most types of cardiac disease, the pathology develops slowly and gradually, often over many years. In contrast, arrhythmias often occur suddenly. In nonlinear systems, sudden changes in qualitative dynamics can, counterintuitively, result from a gradual change in a system parameter-this is known as a bifurcation. Here, we review how nonlinearities in cardiac electrophysiology influence normal and abnormal rhythms and how bifurcations change the dynamics. In particular, we focus on the many recent developments in computational modeling at the cellular level that are focused on intracellular calcium dynamics. We discuss two areas where recent experimental and modeling work has suggested the importance of nonlinearities in calcium dynamics: repolarization alternans and pacemaker cell automaticity.

Fig. 1

Schematic illustration of ionic currents and action potential in a ventricular myocyte. a: The major channels and transporters involved in generating the cardiac action potential. Also shown is a rudimentary schematic of the intracellular calcium dynamics, which will be discussed more in Chapter 3. b: The fast upstroke of the action potential is generated by a large inward (negative) total current carried by sodium ions (off scale; goes to about −200 pA/pF in this model (158)). The slowly changing plateau phase is caused by a near-balance between I_Ca,L (inward) and I_Ks and I_Kr (outward). The rapid relaxation back to the resting membrane potential is due to I_K1. Abbreviations: I_Na: sodium current; I_b,Na: background sodium current; I_Ca,L: L-type calcium current; I_b,Ca: background calcium current; I_K,p: plateau potassium current; I_to: transient outward current; I_Ks: slow delayed rectifier current; I_Kr: rapid delayed rectifier current; I_K1: inward rectifier current; I_NaK: sodium/potassium pump current; I_NaCa: sodium/calcium exchange current; I_pCa: calcium pump current.

Fig. 2

Alternans, higher-order periodic rhythms, and irregular dynamics. a: Action potential duration restitution curve obtained by applying premature stimuli in the Ten Tusscher and Panfilov model (158). b: Cobweb iterations of restitution map given by APD_i+1=220–180exp(–DI_i/60) (blue curve) and DI_i=PCL–APD_i (red line)(13), where APD is the action potential duration, DI is the diastolic interval, PCL is the constant pacing cycle length. Left: period-1 rhythm for PCL=400 ms. Right: alternans for PCL=220 ms; dashed line indicates the value of DI for which the slope of the restitution curve equals one. c: Bifurcation diagram of the restitution map (similar to that in Ref. (13)) with the condition that action potentials occur only for DI>5 ms. Numbers give stimuli:response values; ID indicate irregular dynamics, which is preceded by period-doubling bifurcations.

Fig. 3

Early afterdepolarization (EAD) dynamics. a: EADs in a mathematical model of a rabbit ventricular myocyte modified to simulate oxidative stress. The occurrence of EADs depends on a fine balance between inward and outward currents during the plateau of the action potential and is therefore sensitive to changes in stimulus timing. Here, a certain stimulus timing leads to a normal action potential (blue trace), while a stimulus applied 5 ms later causes an EAD (red trace). b: Iteration of a one-dimensional map based on the myocyte model leads to irregular (chaotic) dynamics, with aperiodic switching between short (no EAD) and long APD values (with EAD). c: APD values from iteration in (B) plotted vs. beat number. Modified from Ref. (15) with permission.

Fig. 4

Intracellular calcium handling. A ventricular myocyte consists of approximately 75 sarcomeres, each of length ~2 μm. Sarcomeres span the distance between adjacent t-tubules that contain sodium-calcium exchangers (NCX; blue) as well as L-type calcium-channels (LCC; cyan), which are colocated in functional calcium release units with Ryanodine receptors (RyR; green) in the junctional sarcoplasmic reticulum (JSR). A cell contains about 10,000–20,000 release units, or couplons, each of which in turn consists of 10–25 LCCs and 100–200 RyRs. The total calcium concentration in the cytosol, arising as a global contribution from all couplons, gives the calcium transient (CaT). Expanded from Ref. (114) with permission.

Fig. 5

Subcellular calcium alternans. a: Subcellular calcium alternans during alternans control pacing in an isolated guinea pig ventricular cell. Modified from Ref. (115) with permission. b: Turing instability can occur with unstable calcium transient dynamics causing local growth of calcium transient alternans in combination with positive calcium-to-voltage coupling and negative voltage-to-calcium coupling to cause out-of-phase calcium transient alternans down the myocyte (C).

Fig. 6

Abolishment of spontaneous oscillations. Simulations of the Irisawa-Noma model of a rabbit sinoatrial node cell (159) illustrating three ways in which spontaneous activity can stop (118). Left: Gradual decline in amplitude due to a supercritical Hopf bifurcation for decreasing calcium current conductance. Right: Skipped beats and increased period due to a homoclinic bifurcation occurring with increasing positive bias current injection. Top: Annihilation by injection of a brief stimulus current (arrow) due to bistability between a stable limit cycle and a stable fixed point occurring with zero pacemaker current and injected positive bias current. Such bistability can arise in different ways, e.g., a subcritical Hopf bifurcation followed by a saddle node bifurcation of limit cycles. Numbers on the ordinates give transmembrane potential in mV. See Ref. (118) for details on simulations.