# News

## Reentry and Ectopic Pacemakers Emerge in a Three-Dimensional Model for a Slab of Cardiac Tissue with Diffuse Microfibrosis near the Percolation Threshold

Arrhythmias in cardiac tissue are generally associated with irregular electrical wave propagation in the heart. Cardiac tissue is formed by a discrete cell network, which is often heterogeneous. Recently, it was shown in simulations of two-dimensional (2D) discrete models of cardiac tissue that a wave crossing a fibrotic, heterogeneous region may produce reentry and transient or persistent ectopic activity provided the fraction of conducting connections is just above the percolation threshold. Here, we investigate the occurrence of these phenomena in three-dimensions by simulations of a discrete model representing a thin slab of cardiac tissue. This is motivated (i) by the necessity to study the relevance and properties of the percolation-related mechanism for the emergence of microreentries in three dimensions and (ii) by the fact that atrial tissue is quite thin in comparison with ventricular tissue. Here, we simplify the model by neglecting details of tissue anatomy, e. g. geometries of atria or ventricles and the anisotropy in the conductivity. Hence, our modeling study is confined to the investigation of the effect of the tissue thickness as well as to the comparison of the dynamics of electrical excitation in a 2D layer with the one in a 3D slab. Our results indicate a strong and non-trivial effect of the thickness even for thin tissue slabs on the probability of microreentries and ectopic beat generation. The strong correlation of the occurrence of microreentry with the percolation threshold reported earlier in 2D layers persists in 3D slabs. Finally, a qualitative agreement of 3D simulated electrograms in the fibrotic region with the experimentally observed complex fractional atrial electrograms (CFAE) as well as strong difference between simulated electrograms in 2D and 3D were found for the cases where reentry and ectopic activity were triggered by the micro-fibrotic region.

## Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that is triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media with applications to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact for cardiac arrhythmias.

## New Book: Nonlinear Dynamics in Biological Systems

**Nonlinear Dynamics in Biological Systems**

This book presents recent research results relating to applications of nonlinear dynamics, focusing specifically on four topics of wide interest: heart dynamics, DNA/RNA, cell mobility, and proteins. The book derives from the First BCAM Workshop on Nonlinear Dynamics in Biological Systems, held in June 2014 at the Basque Center of Applied Mathematics (BCAM). At this international meeting, researchers from different but complementary backgrounds, including molecular dynamics, physical chemistry, bio-informatics and biophysics, presented their most recent results and discussed the future direction of their studies using theoretical, mathematical modeling and experimental approaches. Such was the level of interest stimulated that the decision was taken to produce this publication, with the organizers of the event acting as editors. All of the contributing authors are researchers working on diverse biological problems that can be approached using nonlinear dynamics. The book will appeal especially to applied mathematicians, biophysicists, and computational biologists.

**Chapter 4: Pattern Formation at Cellular Membranes by Phosphorylation and Dephosphorylation of Proteins**

**Chapter 6: Mechanisms Underlying Electro-Mechanical Cardiac Alternans**

## Miquel Marchena: Development of a computational model of calcium signaling in cardiac cells at the submicron scale

**Title:**

Development of a computational model of calcium signaling in cardiac cells at the submicron scale

**Abstract:**

Calcium is a key element of biological signaling. Cells have a calcium signaling toolkit with many components that can be mixed and matched to create a wide range of spatial and temporal signals. Inside the cell, the release of calcium occurs in a small space around the ryanodine receptors and L-type calcium channels. In cardiac cells, these micro domains are called dyadic clefts, where the control of calcium release takes place. In this project a whole cell model in the submicron scale has been developed in order to describe the dynamic in these micro domains. Specifically, this bidomain model of calcium concentration works with the cytoplasm and the sarcoplasmic reticulum. The homogenized macroscopic behavior is described in a two-concentration field model, using an effective diffusion coefficients of calcium in the SR and in the cytoplasm. The effects of buffering have also been taken into account. Results show the typical traces of calcium in the cytoplasm and the sarcoplasmic reticulum. Global concentrations fit with experimental data.

## Ferran Pla: Study of calcium sparks and calcium wave propagation in cardiac cells

**Title:**

Study of calcium sparks and calcium wave propagation in cardiac cells

**Abstract:**

Contraction of cardiac cells is initiated by an increase in the level of intracellular calcium concentration. The calcium response is the combination of the local stochastic release of tens of thousands of release sites. The possible responses range from sparks (local release) to a global calcium increase, passing from calcium waves that propagate along the cell. In this project we model the intracellular calcium dynamics as a network of excitable elements that fire stochastically, and study the occurrence of calcium waves and spark nucleation. In an initial part, we model the sparks of a homogeneous distribution of calcium nodes. We study the wave propagation of this model depending on properties accounting for the state of the heart. We develop a simplified mean-field theory which will shed light on various aspects of the dynamics of the model. In a second part, we proceed to add clustering onto our model, and with it, we study its new wave propagation and the dynamics of the model.

## Talk: Roman Grigoriev

Title:

Spiral wave chaos: Tiling, local symmetries, and asymptotic freedom

Abstract:

Excitable systems can generate dynamics ranging from solitary waves in 1D to spiral/scroll wave chaos in 2D/3D. Complex spatiotemporally chaotic dynamics featuring spiral waves are associated with phenomena such as cardiac arrhythmias (e.g., fibrillation) and seizures (epilepsy). Understanding the nature of spatiotemporal chaos in excitable systems therefore is not only of fundamental interest, but also of high practical importance. This talk will give an overview of recent progress in understanding the dynamical mechanisms that initiate and maintain spiral wave chaos featuring multiple interacting spiral waves that repeatedly break up and merge.

Periodic orbit theory, which aims to describe chaotic dynamics using the properties of unstable periodic solutions embedded in the chaotic attractor, produced a lot of insight into the dynamics of low-dimensional systems, starting with the work of Poincare on celestial mechanics. Recently, a similar approach has been applied rather successfully to spatiotemporal chaos in a range of systems (complex GinzburgLandau, Kuramoto-Sivashinsky, and Navier-Stokes equation). In excitable systems, however, it fails rather spectacularly due to a special property of spiral waves: they have extremely short spatial correlations.

Although it is tempting to associate the relevant length scale with the wavelength of a spiral wave, the former is instead defined by the width of the adjoint eigenfunctions associated with the dominant modes of the linearization. For typical models of excitable dynamics these eigenfunctions are exponentially localized around the spiral core, with the width much smaller than the wavelength. Hence, interaction between two spiral waves falls off exponentially, and the dynamics of individual spirals become effectively independent once the separation between the spiral cores exceeds this length scale (spiral waves become asymptotically free).

As a result, typical multi-spiral states break the global Euclidean symmetry of the problem, but respect local symmetries (translations and rotations in 2D). Local symmetries imply that time-periodic solutions are extremely rare due to the slow relative drift in the position and orientation of individual spirals. This drift can be understood by partitioning the domain into tiles, each of which supports a single spiral wave. The dynamics of each spiral can then be understood completely based on the shape of the corresponding tile and the position of the spiral core. This formalism produces a number of specific predictions that are fully supported by numerical simulations and offers a novel way to understand and describe spiral wave chaos.

## Guillem Sanchis: Simulation of calcium release units in cardiomyocytes to study pulsus alternans

**Title:**

Simulation of calcium release units in cardiomyocytes to study pulsus alternans

**Abstract:**

Cardiac alternans, characterized by a beat-to-beat alternation in the strength of the heart contraction, is a phenomenon that currently drives a very active body of research due to its link to life-threatening heart conditions such as arrhythmia and fibrillation. At the moment, the mechanism through which alternans arises is not well understood. It is known to be associated by an alternation in the levels of intracellular calcium, although it has also found to be related to an instability in the electrical propagation through the tissue; these two mechanisms have been observed both separately and together in several experimental setups. In particular, the role of RyR in alternans is a current topic of debate.

In cardiac cells, the contraction is driven by the release of Ca2+ ions from the sarcoplasmic reticulum. This happens across the cell in thousands of calcium release units, which are sets of coupled protein clusters that release calcium in response to a signal in the form of an action potential, through a positivefeedback mechanism known as calcium-induced calcium release.

The objective of this work is to implement a stochastic model of a calcium release unit to compute how the probability of a calcium spark to occur depends on a variety of biophysical parameters related with the dynamics of intracellular calcium and the stochastic configuration transitions of the protein channels. Then, the obtained probability functions are used in a simplified coupled return maps model of a cardiac cell to explore the parameter ranges that allow the appearance of calcium alternans. The fact that the homeostasis is simplified in this model allows for the study of the alternans as a consequence mainly of the behavior of local units.

## Pol Canal: Measurement and modeling of atrial cell electrodynamics and atrial arrhythmias

**Title:**

Measurement and modeling of atrial cell electrodynamics and atrial arrhythmias

**Abstract:**

Here I present the study of instabilities in human atrial cardiac tissue using a complex non-linear set of differential equations. More specifically, the topic I will focus on is atrial calcium alternans driven by SR Ca content fluctuations and SR refractoriness when extending the model from the cell level to tissue. This thesis will be structured in four blocks, each of which will be separated in subsections. The first one will introduce the reasons why studying instabilities in cardiac tissue is a hot topic as well as a research motor that helps other disciplines like computer science advance faster. Then I will briefly talk about the state of art experimental setup (optical mapping) and not so briefly explain the bibliographic research about the biology behind this model for a better comprehension of the connection between the model and reality. To end with this second block I will summarize the GPU programming concept and compare CPU vs GPU to highlight the parallel programming benefits. The third block will summarize the main concepts of the model, providing the capability of understanding figures and results in the forth block. The forth block will focus on simulating the 2D model and analyzing its results in terms of instabilities and performance in order to understand why those instabilities appear when changing some parameter values in the equations. Finally, I will discuss the obtained results and conclude with limitations and future work. The main concepts this thesis wants you to learn are calcium cycling, and more specifically what's the role of the Ryanodine Receptors in the SR, calcium and action potential sustained alternans and the mechanisms behind it and the differences between CPU and GPU and why is it important.

## David Casas: Gpu-accelerated simulations of the chemotactic response of amoeba Dictyostelium discoideum.

**Title:**

Gpu-accelerated simulations of the chemotactic response of amoeba Dictyostelium discoideum.

**Abstract:**

Having computer simulations of biophysical systems is an important step for checking the validity of our models, improve our understanding of their behaviour and serve as a first testing ground for further applications. The aim of this project was twofold: rstly, the design and implementation of a generic framework that easies the creation of gpu-accelerated physical simulations. Secondly, it tests a model for the chemotactic response and locomotion of the amoeba Dictyostelium discoideum, using the aforementioned framework. The framework has proven itself very useful during the testing of the model, although some improvements could be done. The model used in this project yielded satisfactory results, but should be completed in order to provide more interesting insights.

## Oscillations and uniaxial mechanochemical waves in a model of an active poroelastic medium: Application to deformation patterns in protoplasmic droplets of Physarum polycephalum

Self-organization in cells often manifests itself in oscillations and waves. Here, we address deformation waves in protoplasmic droplets of the plasmodial slime mould Physarum polycephalum by modelling and experiments. In particular, we extend a one-dimensional model that considered the cell as a poroelastic medium, where active tension caused mechanochemical waves that were regulated by an inhibitor (Radszuweit et al., 2013). Our extension consists of a simple, qualitative chemical reaction\u2013diffusion model (Brusselator) that describes the regulation of the inhibitor by another biochemical species. The biochemical reaction enhances the formation of mechanochemical waves if the reaction rates and input concentrations are near or inside an oscillatory regime. The period of the waves is found to be controlled by the characteristic oscillation period, whereas their wavelength is set by mechanical parameters. The model also allows for a systematic study of the chemical activity at the onset of mechanochemical waves. We also present examples for pattern formation in protoplasmic droplets of Physarum polycephalum including global oscillations where the central region of the droplets is in antiphase to the boundary zone, as well as travelling and standing wave-like uniaxial patterns. Finally, we apply our model to reproduce these experimental results by identifying the active tension inhibitor with the intracellular calcium concentration in the Physarum droplets and by using parameter values from mechanical experiments, respectively knowledge about the properties of calcium oscillations in Physarum. The simulation results are then found to be in good agreement with the experimental observations.

Link: doi:10.1016/j.physd.2015.09.017

## Symposium: Nonlinear Processes in Cardiac Electrophysiology: from cardiac tissue modeling to single cell experiments

Symposium announcement (PDF file)

11:30 **Two different discrete models of cardiac electromechanics**

Dr. Rodrigo Weber dos Santos (Federal University of Juiz de Fora, Brazil)

12:30 **High Performance Computing Cardiovascular Modelling**

Dr. Jazmin Aguado Sierra (Barcelona Supercomputing Center)

13:00 **Altered calcium homeostasis in atrial myocytes from patients with 4q25 risk variants for atrial fibrillation**

Dr. Leif Hove (Catalan Cardiovascular Research Institute)

Location:

Sala de Graus (Room 022)

Escola Politècnica Superior d'Edificació de Barcelona (EPSEB)

Av Doctor Marañón 44-50, 08028, Barcelona

## Simulations in Heart Function: Special issue on Cardiac modeling in BioMed Res. Int.

The special issue Simulations in Heart Function focuses on cardiac modeling and simulations that can contribute to improving the understanding of this multifaceted system under normal conditions and different cardiac pathologies. Multiple types of computational and mathematical models are used to describe heart function at different levels of details. For instance, relatively simple models have been employed to characterize the main properties of action potential propagation and wave dynamics in cardiac tissue, and detailed physiological models have been employed to improve our understanding of arrhythmia generation, fibrillation, and defibrillation. Coupled models of cardiac electromechanics that involve multiple scales, from intracellular to whole-organ, have been developed to describe the relation between electric signals and heart contraction.

Guest Editors: Rodrigo Weber dos Santos, Sergio Alonso, Elizabeth M. Cherry, and Joakim Sundnes.

## New project funded by "La Marató de TV3"

During the ceremony on November 5^{th} the principal investigator Blas Echebarria was awarded with the grant diploma for the project:Development and application of atrial myocyte models to investigate mechanisms that confer patients a high risk of atrial fibrillation.

Here a complete list of funded projects can be found.

## Simulation of Ectopic Pacemakers in the Heart: Multiple Ectopic Beats Generated by Reentry inside Fibrotic Regions

The inclusion of nonconducting media, mimicking cardiac fibrosis, in two models of cardiac tissue produces the formation of ectopic beats. The fraction of nonconducting media in comparison with the fraction of healthy myocytes and the topological distribution of cells determines the probability of ectopic beat generation. First, a detailed subcellular microscopic model that accounts for the microstructure of the cardiac tissue is constructed and employed for the numerical simulation of action potential propagation. Next, an equivalent discrete model is implemented, which permits a faster integration of the equations. This discrete model is a simplified version of the microscopic model that maintains the distribution of connections between cells. Both models produce similar results when describing action potential propagation in homogeneous tissue; however, they slightly differ in the generation of ectopic beats in heterogeneous tissue. Nevertheless, both models present the generation of reentry inside fibrotic tissues. This kind of reentry restricted to microfibrosis regions can result in the formation of ectopic pacemakers, that is, regions that will generate a series of ectopic stimulus at a fast pacing rate. In turn, such activity has been related to trigger fibrillation in the atria and in the ventricles in clinical and animal studies.

## Turbulent Bubble Jets in Microgravity. Spatial Dispersion and Velocity Fluctuations

We perform a detailed statistical analysis of bubble dispersion in turbulent jets based on data from drop tower experiments. A stochastic model is also introduced in order to capture these statistics to a large extent, treating bubbles as passive tracers with a local diffusivity given by a k-ε description of the turbulence. It is found that, although interactions cannot be neglected very close to the inlet, the model predictions for the overall spatial distribution of the bubble ensemble are compatible with data within experimental uncertainty, and within the limited statistics of the experiments. In addition, the velocity fluctuations from the same experiments are analyzed, obtaining the local standard deviation of bubble velocities. We also find good agreement between experimental data and the effective model. Slight deviations between the model predictions and the experimental data are found at the jet margins, concerning the dependence on Reynolds number of jet angle and the relative velocity fluctuations. Consequently, significant bubble-flow interactions seem to be confined at the boundaries of the jets.

## Transitions Between Symmetric and Nonsymmetric Regimes in Binary-Mixture Convection

We have studied the different bifurcations found for small to moderate Rayleigh number in binary-mixture convection with lateral heating and negative separation ratio (S). The present work connects the symmetric regime found for pure fluid (S=0) with the fundamentally nonsymmetric regime found for S=-1. We give a global context as well as an interpretation for the different associations of bifurcations found, and in particular we interpret an association of codimension-two bifurcations in terms of a higher codimension bifurcation never found, to our knowledge, in the study of an extended system.

## The new page of the group is already public

From today April 30th our webpage is public. Please enter and take a look in nolin.upc.edu

## Calcium Alternans is Due to an Order-Disorder Phase Transition in Cardiac Cells

Electromechanical alternans is a beat-to-beat alternation in the strength of contraction of a cardiac cell, which can be caused by an instability of calcium cycling. Using a distributed model of subcellular calcium we show that alternans occurs via an order-disorder phase transition which exhibits critical slowing down and a diverging correlation length. We apply finite size scaling along with a mapping to a stochastic coupled map model, to show that this transition in two dimensions is characterized by critical exponents consistent with the Ising universality class. These findings highlight the important role of cooperativity in biological cells, and suggest novel approaches to investigate the onset of the alternans instability in the heart.

## Molecular Na-channel excitability from statistical physics

The excitable properties of the neural cell membrane is the driving mechanism of the neural pulses. Coordinated ionic fluxes across Na and K channels are the devices responsible of this function. Here we present a simple microscopic physical scenario which accounts for this phenomenology. The main elements are ions and channel doors that obey the standard formulation of statistical physics (overdamped Langevin equations) with appropriate nonlinear interacting potentials. From these equations we obtain the ionic flux and the dynamics of the membrane potential. We show that the excitable properties of the membrane are present in a single and simple Na channel. From this framework, additional microscopic information can be obtained, such as statistics of single-channels dynamics or the energetics of action potential events.

## New web soon

Under construction