πŸ“’ Qu 2014 | Nonlinear and stochastic dynamics in the heart


Nonlinear and stochastic dynamics in the heart1



  • Sudden cardiac death (SCD) is not only a problem of biology and medicine, but also a problem of physics and mathematics

The normal heart rhythm and arrhythmias
The normal heart structure and function

Arrhythmias and anti-arrhythmic therapies
Transitions in cardiac arrhythmias

Multi-scale regulation of the heart

  • Time spans from milliseconds to years and length scales from nanometers to centimeters
  • complex action potential and Ca2+ dynamics emerge via bifurcations and dynamical instabilities

Nonlinear and stochastic dynamics in the heart

Nonlinear dynamics of heart rhythms and heart rate variability
Complex ECG dynamics during sinus rhythm

  • the RR interval in a normal heart is not truly periodic, but shows variation called heart rate variability (HRV). In diseased conditions such as heart failure, the variability may be diminished. => higher risk of SCD


  • electrical alternans (ECG) and APD alternans
  • when the slope of the APD restitution curve (see Section 6.1.1 for definition) is greater than one, a bifurcation from regular beating (period-1) to alternans (period-2) occurs
  • Ca2+ alternans can alternate in-phase, or out-of-phase from instability caused by bi-directional coupling of voltage and Ca2+

Regular and irregular premature excitations

  • premature ventricular contractions (PVCs)
  • automaticity, early afterdepolarizations (EADs), and delayed afterdepolarizations (DADs)

Ventricular arrhythmias and electrical wave dynamics

  • normal excitations of the heart are target waves originating from the SAN
  • a PVC is a focal excitation from the Purkinje fiber or the ventricles
  • circulating repetitively around an obstacle (anatomical reentry)
  • spiral wave: functional reentry

Nonlinear dynamics at sub-cellular scales

  • Ca2+ sparks and Ca2+ waves
  • can alter action potential dynamics, such as EADs and DADs
  • this transition is similar to a second-order phase transition in which criticality occurs
    Intracellular Ca2+ cycling dynamics in ventricular myocytes

Transitions in the heart

  • Bifurcations via dynamical instabilities: APD alternans or Ca2+ alternans
  • Thermodynamic phase transitions and criticality: Ca2+ sparks / waves
  • Transitions between multiple solutions: sinus rythm, reentry, spiral wave
  • Synchronization in SAN
  • Wave competition or entrainment: slower oscillation frequencies will always be taken over by the faster waves
  • Dynamical transients: a sudden change in heart rate can induce transient alternans

Biophysics of excitable cells and mathematical modeling of the heart

Electrical circuit theory of excitable cells

Nernst potential

$$ \frac{c_{i}}{c_{o}}=\frac{p_{i}}{p_{o}}=\frac{e^{-e \phi_{i} / k T}}{e^{-e \phi_{o} / k T}} $$ $$ E=\phi_{i}-\phi_{o}=-\frac{k T}{e} \ln \frac{c_{i}}{c_{o}}=-\frac{R T}{F} \ln \frac{c_{i}}{c_{o}} $$ $$ E=-\frac{R T}{z F} \ln \frac{c_{i}}{c_{o}} $$

Hodgkin-Huxley model

$$ C_{m} \frac{d V}{d t}=-\left(I_{i o n}+I_{s t i}\right) $$

FitzHugh-Nagumo model

  • simplified excitable medium $$ \begin{aligned} \frac{d u}{d t}&=\varepsilon\left(u-\frac{1}{3} u^{3}-v\right) \cr \frac{d v}{d t}&=u-\beta v+a \end{aligned} $$
  • The stabilities of the fixed points are determined by the eigenvalues of the Jacobian: $$ J=\left( \begin{array}{cc}{\varepsilon f_{u}^{\prime}} & {-\varepsilon} \cr {1} & {-\beta}\end{array}\right) $$

The cardiac action potential and modeling
The cardiac action potential and excitation-contraction coupling

Action potential models

  • There have been over 100 cardiac action potential models developed [137], most of them are available at CellML (
  • Variety of cell types, species
  • From 1994 Luo-Rudy model to recent ones
  • In most cases, it is not possible to reproduce or code the model into a computer program from the original publication due to typographical errors in parameters values, units, and even equations
  • Since most of the models were fit to one set of experimental data and/or at a certain pacing cycle length (PCL), the robustness and parameter sensitivity have not been well-tested
  • The first and second generation models are low dimensional models developed phenomenologically based on whole-cell data. Not useful for transitions near cirticality like Ca2+ sparks to Ca2+ wave.
  • Some of the models may be very sensitive to initial conditions and parameter changes, and some may require very small time steps
  • a model with more physiological details is better at relating dynamical features to specific molecular properties, but not necessarily better at illuminating the mechanisms underlying the dynamical features (harder to fit)

Modeling electrical wave conduction in tissue and organ

  • Modeled as aniostropic RD-type PDE

Stochastic modeling

  • Stochastic ion channel openings and closings are modeled using Markov transitions
  • ion channels have been modeled using Markov models with deterministic simulations to generate the whole cell current

Sub-cellular modeling

  • SR
  • CaRU
  • mitochondria

Some notes on experimental technologies

  • voltage clamp, patch clamp, and optical mapping

Dynamics at the molecular and sub-cellular scales

Power-law distribution of ion channel closed times

  • the closed time distribution (but not the open time distribution) exhibits a power-law with a unique βˆ’3/2 exponent

  • By solving the Master equation, they were able to obtain a closed-time distribution with a βˆ’3/2-power law, p(Ο„) ∝ Ο„βˆ’3/2, under the conditions that the number of closed states is very large

  • The relationship is also in On-off intermittency, a noise or chaos induced bursting behavior in nonlinear systems $$ \frac{d x}{d t}=[a+\xi(t)] x-x^{3} $$ The distribution of the dwell time Ο„ around x=0 (the off state) can be analytically derived as $$ p(\tau) \propto \tau^{-3 / 2} e^{-\left(a^{2} / 4 D\right) \tau} $$

Dynamics of Ca2+ sparks

  • Ca2+ sparks are considered as the elementary Ca2+ release events for Ca2+ signaling in heart cells, and collective behavior of CaRU
  • A spark is not a purely random event, but since the number of RyRs in a CRU is limited (~100 RyRs), stochasticity still has a very important influence on Ca2+ sparks

Criticality in the transition from Ca2+ sparks to waves and oscillations
Phase transition and criticality in Ca2+ signaling

  • when Ca2+ in the cell is low or the CRU coupling is weak, the system is dominated by quarks and sparks, and the spark cluster size distribution is exponential
  • As Ca2+ level increases, the distribution changes toward a power-law
    Transition to whole-cell Ca2+ oscillations
  • at low Ca2+ levels, the system is dominated by random quarks and sparks
  • Close to the critical state, Ca2+ waves occur occasionally which result in Ca2+ spikes at the whole-cell level and irregular Ca2+ oscillations
  • Close to criticality, the distribution of the inter-spike interval exhibits a fat-tail while at the high Ca2+, the distribution becomes Gaussian
    Power-law clustering in an agent-based model
  • the CRU network in the real cell is not homogeneous [229], and thus self-organization of spark clusters is also not uniform in space

Nonlinear dynamics in single myocytes

APD alternans and chaos

  • TW(T-wave)A is a manifestation of APD alternans in the ECG
  • Source: EP & Ca cycling

APD restitution

  • In most cases, APD is shorter at faster heart rates

APD alternans and chaos induced by steep APD restitution

  • The steady state becomes unstable when $$ slope =\left.\frac{d f}{d d_{n}}\right|_{d=d_{s}}=-\left.\frac{d f}{d a_{n}}\right|_{a=a_{s}}>1 $$
    Effects of cardiac memory on APD alternans
  • memory can both inhibit and promote APD instabilities


  • one observes an overlap region in which two action potential behaviors co-exist for the same PCL

Intracellular Ca2+ alternans

  • Ca2+ and voltage are bi-directionally coupled
  • the regulation of Ca2+ cycling system is also very complex, and includes positive feedback loops (CICR) => alternans
    Steep fractional SR Ca2+ release as a mechanism of Ca2+ alternans

A mean-field theory of Ca2+ alternans

$$ f\left(\alpha, \beta, \gamma, N_{k}\right)=1-\left[1-\alpha \gamma\left(1-\beta N_{k} / N_{0}\right)\right]^{M} $$

  • The steep fractional Ca2+ release mechanism was challenged by experimental studies
  • three critical properties of a CRU or a Ca spark
    • randomness
    • refractoriness
    • recruitment
      From microscopic to macroscopic alternans (from disorder to order)
  • the mean-field theory of Ca2+ alternans links the microscopic (spark) behaviors to the macroscopic (alternans) behaviors

Alternans dynamics due to voltage and Ca2+ coupling

  • APD and Ca2+ alternate in-phase: electromechanically concordant alternans
  • One of the important tasks of computational modeling is to identify characteristics of alternans that can be used to infer the origins of instabilities, which is not a trivial task in real cells

Mechanisms of sub-cellular discordant Ca2+ alternans

  • Obs: Ca2+ alternates out of phase at the two end of the cells
    Mechanisms of EAD chaos
    Turing instability induced sub-cellular discordant Ca2+ alternans
  • In most realistic conditions, both Ca2+-to-APD and APD-to-Ca2+ couplings are positive, and thus the Turing instability may not occur naturally in real cells. Sub-cellular discordant Ca2+ alternans could develop after they changed APD-to-Ca2+ coupling from positive to negative
  • Other mechanisms: Ca2+ wave occurring during pacing can reverse the phase in one region

Early afterdepolarizations (EADs)

Early afterdepolarizations and irregular dynamics

  • EADs arise from a dual Hopf-homoclinic bifurcation

A bifurcation theory of EADs

$$ \begin{aligned} I_{Q S S}(V, x) &=\overline{G}_{N a} m_{\infty}^{3} h_{\infty} j_{\infty}\left(V-E_{N a}\right)+\overline{G}_{C a, L} d_{\infty} f_{\infty}(V \cr &-E_{C a} )+I_{0}(V)+\overline{G}_{K} x x_{1}\left(V-E_{K}\right) \end{aligned} $$
Chaotic dynamics of EADs

Delayed afterdepolarizations (DADs)

  • During a Ca2+ wave, Ca2+ concentration is elevated, which increases INCX and other Ca2+-sensitive currents (electrogenic) => triggers rested Na channel => DADs

Electrical wave dynamics in tissue and organ
Conduction in cardiac tissue

Dynamical repolarization pattern formation in tissue

Spatially discordant alternans

  • all cells in the tissue may alternate in the same phase (in-phase) or alternate in the opposite phase (or antiphase) to the cells in neighboring regions

Chaos synchronization

Initiation of reentry
Induction of reentry around an obstacle
Induction of reentry by a strong stimulus
Unidirectional conduction block in heterogeneous tissue
Induction of reentry due to discordant alternans
Induction of PVC and reentry by EADs in a 2D heterogeneous tissue
Reentry initiation via spatiotemporal chaotic dynamics

Dynamics of reentry in cardiac tissue and organ
Quasi-periodicity in a ring of cardiac tissue
Spiral wave dynamics in generic excitable medium
Characteristics of a stable spiral wave in cardiac tissue
Spiral wave dynamics in a homogeneous 2D tissue model
Spiral wave drift and dynamics in heterogeneous 2D tissue models
Dynamics of scroll waves in 3D tissue models
Effects of tissue thickness on scroll wave stability
Fiber rotation induced scroll wave twist and breakup
Computer simulation of scroll wave dynamics in a dog heart
Negative filament tension
Effect of APD restitution on spiral wave stability in a rabbit heart
Frequency distribution during fibrillation in real hearts
Bistable spiral wave conduction in the FHN model

  • a: INa; b: iCaL

Formation of focal excitations in cardiac tissue
PVC formation in tissue with DAD cells

  • 79 DADs cells in the center of a 400-cell cable results in no PVC. b. 80 DADs cells in the center of the same cable result in a PVC
    Oscillations induced via coupling of normal excitable cells and non-excitable cells
  • Fibroblasts are small and non-excitable cells which play an important role in cardiac mechanics and wound repair. They also develop gap junction coupling with myocytes causing myocyte depolarization
    Chaos synchronization induced focal excitations

Nonlinear dynamics of the pacemaking system
SAN oscillations and abolishment

  • But SAN oscillation is regulated by two coupled voltage and Ca2+ oscillators, not just one
    Voltage and Ca2+ oscillation in SAN cells
    Phase locking and chaos in periodically paced pacemaker cells
    Fractal heart rate variability
    Heart rate variability in cultured monolayers

Control and termination of arrhythmias
Controlling APD alternans
Spontaneous termination of spiral turbulence

  • waves run into refractory tissue, and drift off the tissue border
    Fibrillation duration versus tissue size and stability
    Fast pacing induced termination of arrhythmias
  • Domination of fast waves
    Low energy defibrillation in a computer model
  • Depolarizations as obstacles
    Low energy defibrillation in the real heart

Conclusions and perspective

  • Unlike many other human diseases, the transition from normal sinus rhythm to arrhythmias, particularly atrial and ventricular arrhythmias, is a change in the dynamical state of the same heart which does not necessarily require a change in the properties of the heart itself

  • Lethal ventricular arrhythmias, such as VF, are β€œrare” and random events
  • Even if computational power were not limiting, high-dimensional cardiac models are too complex to systematically analyze the dynamics arising from the complex nonlinear interactions
    • Use Nonlinear dynamics and statistical physics
  • The data available for model parameters: 1) are limited and 2) vary from lab to lab and from experiment to experiment
  • Finally, the ultimate goal of understanding the underlying mechanisms of arrhythmias is to use them for arrhythmia risk stratification and development of effective therapeutics.


  1. Qu Z, Hu G, Garfinkel A, Weiss JN. Nonlinear and Stochastic Dynamics in the Heart. Phys Rep. 2014;543(2):61-162. PMC4175480 ↩︎