# π Qu 2014 | Nonlinear and stochastic dynamics in the heart

Nonlinear and stochastic dynamics in the heart

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## Introduction

- 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

### Arrhythmias and anti-arrhythmic therapies

### 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

- 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

### Alternans

- 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

### 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

#### Action potential models

- There have been over 100 cardiac action potential models developed [137], most of them are available at CellML (http://models.cellml.org/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 β3/2 exponent

By solving the Master equation, they were able to obtain a closed-time distribution with a β3/2-power law, p(Ο) β Οβ3/2, 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

- 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
- 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
- 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 $$ slope =\left.\frac{d f}{d d_{n}}\right|_{d=d_{s}}=-\left.\frac{d f}{d a_{n}}\right|_{a=a_{s}}>1 $$
- memory can both inhibit and promote APD instabilities

#### Hysteresis

- 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

#### 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

- 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
- 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

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

### 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

### 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

### Dynamics of reentry in cardiac tissue and organ

- a: INa; b: iCaL

### Formation of focal excitations in cardiac tissue

- 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
- 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

## Nonlinear dynamics of the pacemaking system

- But SAN oscillation is regulated by two coupled voltage and Ca2+ oscillators, not just one

## Control and termination of arrhythmias

- waves run into refractory tissue, and drift off the tissue border
- Domination of fast waves
- Depolarizations as obstacles

## 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 βrareβ 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.

## Reference

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