Memory-Induced Chaos in Cardiac Excitation

Intracellular ion accumulation in the genesis of complex action potential dynamics under cardiac diseases

Intracellular ions, including sodium (Na^{+}), calcium (Ca^{2+}), and potassium (K^{+}), etc., accumulate slowly after a change of the state of the heart, such as a change of the heart rate. The goal of this study is to understand the roles of slow ion accumulation in the genesis of cardiac memory and complex action-potential duration (APD) dynamics that can lead to lethal cardiac arrhythmias. We carry out numerical simulations of a detailed action potential model of ventricular myocytes under normal and diseased conditions, which exhibit memory effects and complex APD dynamics. We develop a low-dimensional iterated map (IM) model to describe the dynamics of Na^{+}, Ca^{2+}, and APD and use it to uncover the underlying dynamical mechanisms. The development of the IM model is informed by simulation results under the normal condition. We then use the IM model to perform linear stability analyses and computer simulations to investigate the bifurcations and complex APD dynamics, which depend on the feedback loops between APD and intracellular Ca^{2+} and Na^{+} concentrations and the steepness of the APD response to the ion concentrations. When the feedback between APD and Ca^{2+} concentration is positive, a Hopf bifurcation leading to periodic oscillatory behavior occurs as the steepness of the APD response to the ion concentrations increases. The negative feedback loop between APD and Na^{+} concentration is required for the Hopf bifurcation. When the feedback between APD and Ca^{2+} concentration is negative, period-doubling bifurcations leading to high periodicity and chaos occurs. In this case, Na^{+} accumulation plays little role in the dynamics. Finally, we carry out simulations of the detailed action potential model under two diseased conditions, which exhibit steep APD responses to ion concentrations. Under both conditions, Hopf bifurcations leading to slow oscillations or period-doubling bifurcations leading to high periodicity and chaotic APD dynamics occur, depending on the strength of the ion pump-Na^{+}-Ca^{2+} exchanger. Using functions reconstructed from the simulation data, the IM model accurately captures the bifurcations and dynamics under the two diseased conditions. In conclusion, besides using computer simulations of a detailed high-dimensional action-potential model to investigate the effects of slow ion accumulation and short-term memory on bifurcations and genesis of complex APD dynamics in cardiac myocytes under diseased conditions, this study also provides a low-dimensional mathematical tool, i.e., the IM model, to allow stability analyses for uncovering the underlying mechanisms.

Electromechanical model for object roughness perception during finger sliding

Touch allows us to gather abundant information in the world around us. However, how sensory cells embedded in the fingers convey texture information into their firing patterns is still poorly understood. Here, we develop an electromechanical model for roughness perception by incorporating main ingredients such as voltage-gated ion channels, active ion pumps, mechanosensitive channels, and cell deformation. The model reveals that sensory cells can convey texture wavelengths into the period of their firing patterns as the finger slides across object surfaces, but they can only convey a limited range of texture wavelengths. We also show that an increase in sliding speed broadens the decoding wavelength range at the cost of reduction of lower perception limits. Thus, a smaller sliding speed and a bigger contact force may be needed to successfully discern a smooth surface, consistent with previous psychophysical observations. Moreover, we show that cells with slowly adapting mechanosensitive channels can still fire action potentials under static loadings, indicating that slowly adapting mechanosensitive channels may contribute to the perception of coarse textures under static touch. Our work thus provides a new theoretical framework to study roughness perception and may have important implications for the design of electronic skin, artificial touch, and haptic interfaces.

Why Is Only Type 1 Electrocardiogram Diagnostic of Brugada Syndrome? Mechanistic Insights From Computer Modeling

BACKGROUND

Three types of characteristic ST-segment elevation are associated with Brugada syndrome but only type 1 is diagnostic. Why only type 1 ECG is diagnostic remains unanswered.

METHODS

Computer simulations were performed in single cells, 1-dimensional cables, and 2-dimensional tissues to investigate the effects of the peak and late components of the transient outward potassium current (Ito), sodium current, and L-type calcium current (ICa,L) as well as other potassium currents on the genesis of ECG morphologies and phase 2 reentry (P2R).

RESULTS

Although a sufficiently large peak Ito was required to result in the type 1 ECG pattern and P2R, increasing the late component of Ito converted type 1 ECG to type 2 ECG and suppressed P2R. Increasing the peak Ito promoted spiral wave breakup, potentiating the transition from tachycardia to fibrillation, but increasing the late Ito prevented spiral wave breakup by flattening the action potential duration restitution and preventing P2R. A sufficiently large ICa,L conductance was needed for P2R to occur, but once above the critical conductance, blocking ICa,L promoted P2R. However, selectively blocking the window and late components of ICa,L suppressed P2R, countering the effect of the late Ito. Blocking either the peak or late components of sodium current promoted P2R, with the late sodium current blockade having the larger effect. As expected, increasing other potassium currents potentiated P2R, with ATP-sensitive potassium current exhibiting a larger effect than rapid and slow component of the delayed rectifier potassium current.

CONCLUSIONS

The peak Ito promotes type 1 ECG and P2R, whereas the late Ito converts type 1 ECG to type 2 ECG and suppresses P2R. Blocking the peak ICa,L and either the peak or the late sodium current promotes P2R, whereas blocking the window and late ICa,L suppresses P2R. These results provide important insights into the mechanisms of arrhythmogenesis and potential therapeutic targets for treatment of Brugada syndrome. Graphic Abstract: A graphic abstract is available for this article.

Life and death saddles in the heart

Saddle points are responsible for threshold phenomena of many biological systems. In the heart, saddle points determine the normal excitability and conduction, but are also responsible for certain abnormal action potential behaviors associated with lethal arrhythmias. We investigate the dynamical mechanisms for the genesis of lethal extra heartbeats in heterogeneous cardiac tissue under two diseased conditions. For both conditions, the lethal events occur when the system is close to the saddle point, implying the pivotal role of the saddle point in cardiac arrhythmogenesis.

The transient outward potassium current plays a key role in spiral wave breakup in ventricular tissue

Spiral wave reentry as a mechanism of lethal ventricular arrhythmias has been widely demonstrated in animal experiments and recordings from human hearts. It has been shown that in structurally normal hearts spiral waves are unstable, breaking up into multiple wavelets via dynamical instabilities. However, many of the second-generation action potential models give rise only to stable spiral waves, raising issues regarding the underlying mechanisms of spiral wave breakup. In this study, we carried out computer simulations of two-dimensional homogeneous tissues using five ventricular action potential models. We show that the transient outward potassium current (I_to), although it is not required, plays a key role in promoting spiral wave breakup in all five models. As the maximum conductance of I_to increases, it first promotes spiral wave breakup and then stabilizes the spiral waves. In the absence of I_to, speeding up the L-type calcium kinetics can prevent spiral wave breakup, however, with the same speedup kinetics, spiral wave breakup can be promoted by increasing I_to. Increasing I_to promotes single-cell dynamical instabilities, including action potential duration alternans and chaos, and increasing I_to further suppresses these action potential dynamics. These cellular properties agree with the observation that increasing I_to first promotes spiral wave breakup and then stabilizes spiral waves in tissue. Implications of our observations to spiral wave dynamics in the real hearts and action potential model improvements are discussed.NEW & NOTEWORTHY Spiral wave breakup manifesting as multiple wavelets is a mechanism of ventricular fibrillation. It has been known that spiral wave breakup in cardiac tissue can be caused by a steeply sloped action potential duration restitution curve, a property mainly determined by the recovery of L-type calcium current. Here, we show that the transient outward potassium current (I_to) is another current that plays a key role in spiral wave breakup, that is, spiral waves can be stable for low and high maximum I_to conductance but breakup occurs for intermediate maximum I_to conductance. Since I_to is present in normal hearts of many species and required for Brugada syndrome, it may play an important role in the spiral wave stability and arrhythmogenesis under both normal condition and Brugada syndrome.

Key aspects for effective mathematical modelling of fractional-diffusion in cardiac electrophysiology: a quantitative study

Microscopic structural features of cardiac tissue play a fundamental role in determining complex spatio-temporal excitation dynamics at the macroscopic level. Recent efforts have been devoted to the development of mathematical models accounting for non-local spatio-temporal coupling able to capture these complex dynamics without the need of resolving tissue heterogeneities down to the micro-scale. In this work, we analyse in detail several important aspects affecting the overall predictive power of these modelling tools and provide some guidelines for an effective use of space-fractional models of cardiac electrophysiology in practical applications. Through an extensive computational study in simplified computational domains, we highlight the robustness of models belonging to different categories, i.e., physiological and phenomenological descriptions, against the introduction of non-locality, and lay down the foundations for future research and model validation against experimental data. A modern genetic algorithm framework is used to investigate proper parameterisations of the considered models, and the crucial role played by the boundary assumptions in the considered settings is discussed. Several numerical results are provided to support our claims.

Spatially Discordant Repolarization Alternans in the Absence of Conduction Velocity Restitution

Spatially discordant alternans (SDA) of action potential duration (APD) has been widely observed in cardiac tissue and is linked to cardiac arrhythmogenesis. Theoretical studies have shown that conduction velocity restitution (CVR) is required for the formation of SDA. However, this theory is not completely supported by experiments, indicating that other mechanisms may exist. In this study, we carried out computer simulations using mathematical models of action potentials to investigate the mechanisms of SDA in cardiac tissue. We show that when CVR is present and engaged, such as fast pacing from one side of the tissue, the spatial pattern of APD in the tissue undergoes either spatially concordant alternans or SDA, independent of initial conditions or tissue heterogeneities. When CVR is not engaged, such as simultaneous pacing of the whole tissue or under normal/slow heart rates, the spatial pattern of APD in the tissue can have multiple solutions, including spatially concordant alternans and different SDA patterns, depending on heterogeneous initial conditions or pre-existing repolarization heterogeneities. In homogeneous tissue, curved nodal lines are not stable, which either evolve into straight lines or disappear. However, in heterogeneous itssue, curved nodal lines can be stable, depending on their initial locations and shapes relative to the structure of the heterogeneity. Therefore, CVR-induced SDA and non-CVR-induced SDA exhibit different dynamical properties, which may be responsible for the different SDA properties observed in experimental studies and arrhythmogenesis in different clinical settings.

Bifurcations Caused by Feedback between Voltage and Intracellular Ion Concentrations in Ventricular Myocytes

We develop an iterated map model to describe the bifurcations and complex dynamics caused by the feedback between voltage and intracellular Ca^{2+} and Na^{+} concentrations in paced ventricular myocytes. Voltage and Ca^{2+} can form either a positive or a negative feedback loop, while voltage and Na^{+} form a negative feedback loop. Under certain diseased conditions, when the feedback between voltage and Ca^{2+} is positive, Hopf bifurcations occur, leading to periodic oscillatory behaviors. When this feedback is negative, period-doubling bifurcation routes to alternans and chaos occur.

Spatiotemporal correlation uncovers characteristic lengths in cardiac tissue

Complex spatiotemporal patterns of action potential duration have been shown to occur in many mammalian hearts due to period-doubling bifurcations that develop with increasing frequency of stimulation. Here, through high-resolution optical mapping experiments and mathematical modeling, we introduce a characteristic spatial length of cardiac activity in canine ventricular wedges via a spatiotemporal correlation analysis, at different stimulation frequencies and during fibrillation. We show that the characteristic length ranges from 40 to 20 cm during one-to-one responses and it decreases to a specific value of about 3 cm at the transition from period-doubling bifurcation to fibrillation. We further show that during fibrillation, the characteristic length is about 1 cm. Another significant outcome of our analysis is the finding of a constitutive phenomenological law obtained from a nonlinear fitting of experimental data which relates the conduction velocity restitution curve with the characteristic length of the system. The fractional exponent of 3/2 in our phenomenological law is in agreement with the domain size remapping required to reproduce experimental fibrillation dynamics within a realistic cardiac domain via accurate mathematical models.

Control of voltage-driven instabilities in cardiac myocytes with memory

Sudden cardiac death is known to be associated with dynamical instabilities in the heart, and thus control of dynamical instabilities is considered a potential therapeutic strategy. Different control methods were developed previously, including time-delayed feedback pacing control and constant diastolic interval pacing control. Experimental, theoretical, and simulation studies have examined the efficacy of these control methods in stabilizing action potential dynamics. In this study, we apply these control methods to control complex action potential (AP) dynamics under two diseased conditions: early repolarization syndrome and long QT syndrome, in which voltage-driven instabilities occur in the presence of short-term cardiac memory. In addition, we also develop a feedback pacing method to stabilize these instabilities. We perform theoretical analyses using iterated map models and carry out numerical simulations of AP models. We show that under the normal condition where the memory effect is minimal, all three methods can effectively control the action potential duration (APD) dynamics. Under the two diseased conditions where the memory effect is exacerbated, constant diastolic pacing control is least effective, while the feedback pacing control is most effective. Under a very strong memory effect, all three methods fail to stabilize the voltage-driven instabilities. The failure of effective control is due to memory and the all-or-none AP dynamics which results in very steep changes in APD.

Memory-induced nonlinear dynamics of excitation in cardiac diseases

Excitable cells, such as cardiac myocytes, exhibit short-term memory, i.e., the state of the cell depends on its history of excitation. Memory can originate from slow recovery of membrane ion channels or from accumulation of intracellular ion concentrations, such as calcium ion or sodium ion concentration accumulation. Here we examine the effects of memory on excitation dynamics in cardiac myocytes under two diseased conditions, early repolarization and reduced repolarization reserve, each with memory from two different sources: slow recovery of a potassium ion channel and slow accumulation of the intracellular calcium ion concentration. We first carry out computer simulations of action potential models described by differential equations to demonstrate complex excitation dynamics, such as chaos. We then develop iterated map models that incorporate memory, which accurately capture the complex excitation dynamics and bifurcations of the action potential models. Finally, we carry out theoretical analyses of the iterated map models to reveal the underlying mechanisms of memory-induced nonlinear dynamics. Our study demonstrates that the memory effect can be unmasked or greatly exacerbated under certain diseased conditions, which promotes complex excitation dynamics, such as chaos. The iterated map models reveal that memory converts a monotonic iterated map function into a nonmonotonic one to promote the bifurcations leading to high periodicity and chaos.

Mechanisms linking T-wave alternans to spontaneous initiation of ventricular arrhythmias in rabbit models of long QT syndrome

KEY POINTS

T-wave alternans (TWA) and T-wave lability (TWL) are precursors of ventricular arrhythmias in long QT syndrome; however, the mechanistic link remains to be clarified. Computer simulations show that action potential duration (APD) prolongation and slowed heart rates promote APD alternans and chaos, manifesting as TWA and TWL, respectively. Regional APD alternans and chaos can exacerbate pre-existing or induce de novo APD dispersion, which combines with enhanced I_Ca,L to result in premature ventricular complexes (PVCs) originating from the APD gradient region. These PVCs can directly degenerate into re-entrant arrhythmias without the need for an additional tissue substrate or further exacerbate the APD dispersion to cause spontaneous initiation of ventricular arrhythmias. Experiments conducted in transgenic long QT rabbits show that PVC alternans occurs at slow heart rates, preceding spontaneous intuition of ventricular arrhythmias.

ABSTRACT

T-wave alternans (TWA) and irregular beat-to-beat T-wave variability or T-wave lability (TWL), the ECG manifestations of action potential duration (APD) alternans and variability, are precursors of ventricular arrhythmias in long QT syndromes. TWA and TWL in patients tend to occur at normal heart rates and are usually potentiated by bradycardia. Whether or how TWA and TWL at normal or slow heart rates are causally linked to arrhythmogenesis remains unknown. In the present study, we used computer simulations and experiments of a transgenic rabbit model of long QT syndrome to investigate the underlying mechanisms. Computer simulations showed that APD prolongation and slowed heart rates caused early afterdepolarization-mediated APD alternans and chaos, manifesting as TWA and TWL, respectively. Regional APD alternans and chaos exacerbated pre-existing APD dispersion and, in addition, APD chaos could also induce APD dispersion de novo via chaos desynchronization. Increased APD dispersion, combined with substantially enhanced I_Ca,L , resulted in a tissue-scale dynamical instability that gave rise to the spontaneous occurrence of unidirectionally propagating premature ventricular complexes (PVCs) originating from the APD gradient region. These PVCs could directly degenerate into re-entrant arrhythmias without the need for an additional tissue substrate or could block the following sinus beat to result in a longer RR interval, which further exacerbated the APD dispersion giving rise to the spontaneous occurrence of ventricular arrhythmias. Slow heart rate-induced PVC alternans was observed in experiments of transgenic LQT2 rabbits under isoproterenol, which was associated with increased APD dispersion and spontaneous occurrence of ventricular arrhythmias, in agreement with the theoretical predictions.

Bursting dynamics in the normal and failing hearts

A failing heart differs from healthy hearts by an array of symptomatic characteristics, including impaired Ca²⁺ transients, upregulation of Na⁺/Ca²⁺ exchanger function, reduction of Ca²⁺ uptake to sarcoplasmic reticulum, reduced K⁺ currents, and increased propensity to arrhythmias. While significant efforts have been made in both experimental studies and model development to display the causes of heart failure, the full process of deterioration from a healthy to a failing heart yet remains deficiently understood. In this paper, we analyze a highly detailed mathematical model of mouse ventricular myocytes to disclose the key mechanisms underlying the continual transition towards a state of heart failure. We argue that such a transition can be described in mathematical terms as a sequence of bifurcations that the healthy cells undergo while transforming into failing cells. They include normal action potentials and [Ca²⁺]_i transients, action potential and [Ca²⁺]_i alternans, and bursting behaviors. These behaviors where supported by experimental studies of heart failure. The analysis of this model allowed us to identify that the slow component of the fast Na⁺ current is a key determining factor for the onset of bursting activity in mouse ventricular myocytes.