Bifurcation and Chaos in a Model of Cardiac Early Afterdepolarizations

Abstract

Excitable cells can exhibit complex patterns of oscillations, such as spiking and bursting. In cardiac cells, pathological voltage oscillations, called early afterdepolarizations (EADs), have been widely observed under disease conditions, yet their dynamical mechanisms remain unknown. Here, we show that EADs are caused by Hopf and homoclinic bifurcations. During period pacing, chaos always occurs at the transition from no EAD to EADs as the stimulation frequency decreases, providing a novel explanation for the irregular EAD behavior frequently observed in experiments.Complex oscillatory behaviors, such as spiking and bursting dynamics in pancreatic β-cells [1], neurons [1][2][3][4][5], and optical lasers [6], are common phenomena in excitable systems. These complex dynamics are generally described by systems with fast and slow time scales, where the full system behavior can be described by slow dynamics evolving the fast subsystem through a series of bifurcations [1,2]. Cardiac myocytes can exhibit pathological excitations called early afterdepolarizations (EADs), which are voltage oscillations during the repolarizing phase of the action potential (AP). They have been implicated as a cause of lethal cardiac arrhythmias [7][8][9] and have been widely investigated in experiments [8,[10][11][12] and also in simulations [13][14][15][16]. It is commonly agreed that EADs occur when inward (depolarizing) currents are increased and/or outward (repolarizing) currents are decreased. But many such changes do not produce EADs, and the general underlying dynamical mechanism still remains unknown. In single myocytes, EADs typically occur irregularly [10][11][12], which is generally attributed to random fluctuations of the underlying ion channels [13]. In a recent study [16], we presented evidence from isolated myocyte experiments and computational simulations that irregular EAD behavior is not random, but rather dynamical chaos, and gives rise to novel tissue scale dynamics.EADs have typically been studied in computational simulations using highly detailed AP models making dynamical analysis difficult [13][14][15][16] Dynamical origin of EADsThere are typically three time scales in a normal cardiac AP. The sodium (Na) current activates very rapidly, causing the fast upstroke of the AP, and then rapidly inactivates. The L-type calcium (Ca) current activates and inactivates more slowly than the Na current, playing a key role in maintaining the long AP plateau. Time-dependent potassium (K) currents activate even more slowly and eventually overcome the inward currents, repolarizing the cell back to the resting potential. EADs have typically been studied using highly detailed AP models [13][14][15][16] where C m = 1 µF/cm 2 ; I Na is the Na current; We set E si = 80 mV and E K = −77 mV. To study the effects of the time constants for the gating variables d, f, and x on the AP dynamics, we change these time constants by multiplying them by a scalar factor, i.e.,, and τ x (V) → γτ x (V). We refer to this modified LR1 model ...

Calcium-Voltage Coupling in the Genesis of Early and Delayed Afterdepolarizations in Cardiac Myocytes

“…Depending on the stability of the quasi-equilibrium state, the decay rate of voltage, and the activation speed of the slowly increasing net outward current, as well as Cavoltage coupling, different EAD patterns can occur. Based on the bifurcation theory and results in previous simulations (21,44,(55)(56)(57) and the observations in this study, we summarize below the mechanisms underlying six characteristic patterns of EAD behaviors:…”

“…Computer modeling and simulations are a complementary means to reveal the underlying ionic mechanisms. Many computer modeling studies (41)(42)(43)(44)(45)(46) have been carried out to investigate the mechanisms of EADs, but for the most part these have focused on EADs caused by voltage-driven oscillations, because the AP models lacked detailed spatiotemporal Ca cycling required to simulate Ca waves and oscillations. In this study, we investigated the effects of bidirectional Ca-voltage coupling on the genesis of EADs and DADs in ventricular myocytes, using an AP model with a detailed spatiotemporal Ca cycling regulation incorporating stochastic LCC and RyR openings and experiments of mouse ventricular myocytes.…”

“…6, 7, 8, and S1-S3, EADs can exhibit very complex patterns and behaviors. From this study and the nonlinear dynamics analysis (21,44), we can provide a unified theory for these complex EAD patterns based on the interactions of Ca dynamics with the Hopf-homoclinic bifurcation mechanism of voltage-driven EADs. Fig.…”

Bifurcation Analysis of a Certain Hodgkin-Huxley Model Depending on Multiple Bifurcation Parameters

“…Here, we want to remark that for instance in [1,26] the occurrence of EADs via the reduction of the potassium current is studied. The authors used a time scale separation argument (not explicit) to identify the gating variable x as the slowest variable and then, they argued in principle that EADs are Hopf-induced, cf.…”

“…Nevertheless, this is a suitable starting point for the study of EADs, since EADs appear either in the plateau or the repolarisation phase. Here, the potassium (K + ) and the calcium (Ca 2+ ) currents are important, see [1,26]. The model is a three-dimensional ODE system, which contains the potassium current and the corresponding mathematical model read as follows:…”