International Workshop "Nonsmooth Phenomena in Cardiac Dynamics"

Date: Tue, Apr 23 - Wed, Apr 24 2013

Hour: 11:00

Speakers: Luca Gerardo Giorda, Blas Echebarria, Ruediger Thul, David Schaeffer

The main objective of the International Workshop "Nonsmooth phenomena in cardiac dynamics" is to enhance the dialog between applied mathematicians and engineers within the rapidly emerging field of nonsmooth dynamical systems. During this meeting we will discuss the intrinsic value of nonsmooth models (versus smooth ones) towards understanding the occurrence of cardiac alternans.

Supported by Basque Government 


10:30-11:00 Waking up coffee

11:00-11:30 Luca GERARDO-GIORDA
11:30-12:00 Blas ECHEBARRIA

12:00-12:30 Coffee break

12:30-13:00 Ruediger THUL
13:00-13:30 David SCHAEFFER

from 13:30 on: Pintxos lunch and discussions

Luca GERARDO-GIORDA, BCAM - Basque Center for Applied Mathematics, Spain

"Some aspects of the spatial modeling in heart electrophysiology and its numerical approximation"

Abstract: The numerical simulation of the dynamical system describing the propagation of the electrical signal in the heart chambers is a difficult task. The most commonly used model, the Bidomain one, is describing the dynamics of both intracellular and extracellular potential. A simpler model, called Monodomain, is often used to reduce computational cost. The latter describes the dynamics of the membrane potential, which is the difference between the intra and extracellular potential. Both models feature a coupled PDE-ODE system to cope with both space and time modeling. In this talk, starting from the model for a single excitable cell, I will give an overview on the derivation of the spatial models and discuss the computational difficulties arising in their numerical simulation.

Blas ECHEBARRIA, Universitat Politecnica de Catalunya, Spain

"Origin and dynamics of cardiac alternans"

Abstract: In cardiac muscle, a change in transmembrane cellular potential produces a response known as action potential, that propagates along tissue. Many cardiac malfunctions are associated to problems in propagation, sometimes inducing the formation of rotors. When unstable, they can give rise to ventricular fibrillation, in which synchronous excitation is lost among different parts of the ventricle, impeding contraction, and causing death in a few minutes. Among the known precursors of life-threatening ventricular arrhythmias and sudden cardiac death are T wave alternans, defined as a periodic beat to beat change in the amplitude or shape of the ECG T wave. Although T wave alternans provide a global measure of the propagation at the whole heart level, they have been related to alternations in the duration of the excited phase (or action potential duration APD) at the single cell level, thereby establishing a causal link between electrical alternans and the initiation of ventricular fibrillation. In this talk, I will review recent results regarding the origin and dynamics of cardiac alternans. At the cellular level, this rhythm often appears due to a steep relationship between the APD and the time elapsed from the end of the previous action potential. In other situations, it has been observed to be due to an instability in intracellular calcium dynamics. In tissue, the oscillations at the single cell level can be spatially in or out of phase. This latter case is termed discordant alternates (DA) and is known to be related with the induction of reentry. Its origin can be related to the dispersion properties of the tissue. In this regard, recent developments will be presented, including the effect of anomalous dispersion, and tissue contraction.

Ruediger THUL, Nottingham University, UK

"Understanding cardiac alternans: a piecewise linear modelling framework"

Abstract: Cardiac alternans is a beat-to-beat alternation in action potential duration and intracellular calcium cycling seen in cardiac myocytes under rapid pacing that is believed to be a precursor to fibrillation. Here we establish that the key dynamical behaviours of a model developed by Shiferaw and Karma for the intracellular calcium dynamics in paced cardiac myocytes are arranged around a set of switches. These are shown to be the main elements for organising the nonlinear behaviour of the model. Exploiting this observation we show that a piecewise linear caricature of the Shiferaw-Karma model, with a set of appropriate switching manifolds, can be constructed that preserves the physiological interpretation of the original model whilst being amenable to a systematic mathematical analysis. In illustration of this point we formulate the dynamics of calcium cycling (in response to pacing) and compute the properties of periodic orbits in terms of a stroboscopic map that can be constructed without approximation. Using this we show that alternans emerge via a period-doubling instability and track this bifurcation in terms of physiologically important parameters. We also show that when coupled to a spatially extended model for calcium transport the model supports spatially varying patterns of alternans. We analyse the onset of this instability with a generalisation of the master stability approach to accommodate the non-smooth nature of our system.

David SCHAEFFER, Duke University, USA

"Smooth and border-collision characteristics of the bifurcation to alternans in cardiac dynamics"

Abstract: The period-doubling bifurcation to alternans in heart tissue has been mod- eled with both smooth and border-collision dynamics. In this talk we show that experimental data is most accurately described by hybrid behavior: very close to the bifurcation point the dynamics is smooth, whereas further away it is more simply described by a border-collision model. The essence of this behavior is captured by what we call an unfolded border-collision bifurcation. This new description elucidates that, in experiments (where only a limited number of data points can be measured) smooth behavior of the bifurcation can easily be missed.


Blas ECHEBARRIA (Universitat Politecnica de Catalunya, Spain)
Francesco FANELLI (BCAM)
David SCHAEFFER (Duke University, USA)
Ruediger THUL (Nottingham University, UK)

Organiser: Oleg MAKARENKOV ( - registration and other enquiries

Venue: BCAM, Mazarredo, 14, 48009 Bilbao, Basque Country - Spain

Poster is available here.




BCAM; Universitat Politecnica de Catalunya, Spain; Nottingham University, UK; Duke University, USA

Confirmed speakers:

Luca Gerardo Giorda, Blas Echebarria, Ruediger Thul, David Schaeffer