Pulse propagation in discrete systems of coupled excitable cells

Propagation of pulses in myelinated fibers may be described by appropriate solutions of spatially discrete FitzHugh-Nagumo systems. In these systems, propagation failure may occur if either the coupling between nodes is not strong enough or the recovery is too fast. We give an asymptotic construction of pulses for spatially discrete FitzHugh-Nagumo systems which agrees well with numerical simulations and discuss evolution of initial data into pulses and pulse generation at a boundary. Formulas for the speed and length of pulses are also obtained.Comment: 16 pages, 10 figures, to appear in SIAM J. Appl. Mat

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 are 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 and their application 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 in cardiac arrhythmias.Peer ReviewedPreprin

A reaction-diffusion model of cholinergic retinal waves

Prior to receiving visual stimuli, spontaneous, correlated activity called retinal waves drives activity-dependent developmental programs. Early-stage waves mediated by acetylcholine (ACh) manifest as slow, spreading bursts of action potentials. They are believed to be initiated by the spontaneous firing of Starburst Amacrine Cells (SACs), whose dense, recurrent connectivity then propagates this activity laterally. Their extended inter-wave intervals and shifting wave boundaries are the result of the slow after-hyperpolarization of the SACs creating an evolving mosaic of recruitable and refractory cells, which can and cannot participate in waves, respectively. Recent evidence suggests that cholinergic waves may be modulated by the extracellular concentration of ACh. Here, we construct a simplified, biophysically consistent, reaction-diffusion model of cholinergic retinal waves capable of recapitulating wave dynamics observed in mice retina recordings. The dense, recurrent connectivity of SACs is modeled through local, excitatory coupling occurring via the volume release and diffusion of ACh. In contrast with previous, simulation-based models, we are able to use non-linear wave theory to connect wave features to underlying physiological parameters, making the model useful in determining appropriate pharmacological manipulations to experimentally produce waves of a prescribed spatiotemporal character. The model is used to determine how ACh mediated connectivity may modulate wave activity, and how the noise rate and sAHP refractory period contributes to critical wave size variability.Comment: 38 pages, 10 figure

Excitable Media Seminar

The simulation data presented here, and the conceptual framework developed for their interpretation are, both, in need of substantial refinement and extension. However, granting that they are initial pointers of some merit, and elementary indicators of general principles, several implications follow: the activity patterns of neurons and their assemblies are\ud interdependent with the extracellular milieu in which they are embedded, and to whose time varying composition they contribute. The complexity of this interdependence in the temporal dimension forecloses any time and context invariant relation between what the experimenter may consider stimulus input and its representation in neural activity. Hence, ideas of coding by (quasi)-digital neurons are called in question by the mutual interdependence of neurons and their\ud humoral milieu. Instead, concepts of 'mass action' in the Nervous system gain a new perspective: this time augmented by including the chemical medium surrounding neurons as part of the dynamics of the system as a whole. Accordingly, a meaningful way to describe activity in a neuron assembly would be in terms of a state space in which it can move along an infinite number of trajectories.\u

Sparks and waves in a stochastic fire-diffuse-fire model of Ca2+

Calcium ions are an important second messenger in living cells. Indeed calcium signals in the form of waves have been the subject of much recent experimental interest. It is now well established that these waves are composed of elementary stochastic release events (calcium puffs) from spatially localized calcium stores. Here we develop a computationally inexpensive model of calcium release based upon a stochastic generalization of the Fire-Diffuse-Fire (FDF) threshold model. Our model retains the discrete nature of calcium stores, but also incorporates a notion of release probability via the introduction of threshold noise. Numerical simulations of the model illustrate that stochastic calcium release leads to the spontaneous production of calcium sparks that may merge to form saltatory waves. In the parameter regime where deterministic waves exist it is possible to identify a critical level of noise defining a non-equilibrium phase-transition between propagating and abortive structures. A statistical analysis shows that this transition is the same as for models in the directed percolation universality class. Moreover, in the regime where no initial structure can survive deterministically, threshold noise is shown to generate a form of array enhanced coherence resonance whereby all calcium stores release periodically and simultaneously

Calcium puffs are generic InsP₃-activated elementary calcium signals and are downregulated by prolonged hormonal stimulation to inhibit cellular calcium responses

Elementary Ca2+ signals, such as "Ca2+ puffs", which arise from the activation of inositol 1,4,5-trisphosphate receptors, are building blocks for local and global Ca2+ signalling. We characterized Ca2+ puffs in six cell types that expressed differing ratios of the three inositol 1,4,5-trisphosphate receptor isoforms. The amplitudes, spatial spreads and kinetics of the events were similar in each of the cell types. The resemblance of Ca2+ puffs in these cell types suggests that they are a generic elementary Ca2+ signal and, furthermore, that the different inositol 1,4,5-trisphosphate isoforms are functionally redundant at the level of subcellular Ca2+ signalling. Hormonal stimulation of SH-SY5Y neuroblastoma cells and HeLa cells for several hours downregulated inositol 1,4,5-trisphosphate expression and concomitantly altered the properties of the Ca2+ puffs. The amplitude and duration of Ca2+ puffs were substantially reduced. In addition, the number of Ca2+ puff sites active during the onset of a Ca2+ wave declined. The consequence of the changes in Ca2+ puff properties was that cells displayed a lower propensity to trigger regenerative Ca2+ waves. Therefore, Ca2+ puffs underlie inositol 1,4,5-trisphosphate signalling in diverse cell types and are focal points for regulation of cellular responses

Evolution of spiral and scroll waves of excitation in a mathematical model of ischaemic border zone

Abnormal electrical activity from the boundaries of ischemic cardiac tissue is recognized as one of the major causes in generation of ischemia-reperfusion arrhythmias. Here we present theoretical analysis of the waves of electrical activity that can rise on the boundary of cardiac cell network upon its recovery from ischaemia-like conditions. The main factors included in our analysis are macroscopic gradients of the cell-to-cell coupling and cell excitability and microscopic heterogeneity of individual cells. The interplay between these factors allows one to explain how spirals form, drift together with the moving boundary, get transiently pinned to local inhomogeneities, and finally penetrate into the bulk of the well-coupled tissue where they reach macroscopic scale. The asymptotic theory of the drift of spiral and scroll waves based on response functions provides explanation of the drifts involved in this mechanism, with the exception of effects due to the discreteness of cardiac tissue. In particular, this asymptotic theory allows an extrapolation of 2D events into 3D, which has shown that cells within the border zone can give rise to 3D analogues of spirals, the scroll waves. When and if such scroll waves escape into a better coupled tissue, they are likely to collapse due to the positive filament tension. However, our simulations have shown that such collapse of newly generated scrolls is not inevitable and that under certain conditions filament tension becomes negative, leading to scroll filaments to expand and multiply leading to a fibrillation-like state within small areas of cardiac tissue.Comment: 26 pages, 13 figures, appendix and 2 movies, as accepted to PLoS ONE 2011/08/0