Cellular Automata Models of Cardiac Conduction. In: Glass L, Hunter P, Mcculloch A, editors. Theory of Heart. New York: Springer; 1991. pp. 437-76. [DOI: 10.1007/978-1-4612-3118-9_18]

Conductance heterogeneities induced by multistability in the dynamics of coupled cardiac gap junctions

In this paper, we study the propagation of the cardiac action potential in a one-dimensional fiber, where cells are electrically coupled through gap junctions (GJs). We consider gap junctional gate dynamics that depend on the intercellular potential. We find that different GJs in the tissue can end up in two different states: a low conducting state and a high conducting state. We first present evidence of the dynamical multistability that occurs by setting specific parameters of the GJ dynamics. Subsequently, we explain how the multistability is a direct consequence of the GJ stability problem by reducing the dynamical system's dimensions. The conductance dispersion usually occurs on a large time scale, i.e., thousands of heartbeats. The full cardiac model simulations are computationally demanding, and we derive a simplified model that allows for a reduction in the computational cost of four orders of magnitude. This simplified model reproduces nearly quantitatively the results provided by the original full model. We explain the discrepancies between the two models due to the simplified model's lack of spatial correlations. This simplified model provides a valuable tool to explore cardiac dynamics over very long time scales. That is highly relevant in studying diseases that develop on a large time scale compared to the basic heartbeat. As in the brain, plasticity and tissue remodeling are crucial parameters in determining the action potential wave propagation's stability.

Actin droplet machine

The actin droplet machine is a computer model of a three-dimensional network of actin bundles developed in a droplet of a physiological solution, which implements mappings of sets of binary strings. The actin bundle network is conductive to travelling excitations, i.e. impulses. The machine is interfaced with an arbitrary selected set of k electrodes through which stimuli, binary strings of length k represented by impulses generated on the electrodes, are applied and responses are recorded. The responses are recorded in a form of impulses and then converted to binary strings. The machine's state is a binary string of length k: if there is an impulse recorded on the ith electrode, there is a '1' in the ith position of the string, and '0' otherwise. We present a design of the machine and analyse its state transition graphs. We envisage that actin droplet machines could form an elementary processor of future massive parallel computers made from biopolymers.

Towards fungal computer

We propose that fungi Basidiomycetes can be used as computing devices: information is represented by spikes of electrical activity, a computation is implemented in a mycelium network and an interface is realized via fruit bodies. In a series of scoping experiments, we demonstrate that electrical activity recorded on fruits might act as a reliable indicator of the fungi's response to thermal and chemical stimulation. A stimulation of a fruit is reflected in changes of electrical activity of other fruits of a cluster, i.e. there is distant information transfer between fungal fruit bodies. In an automaton model of a fungal computer, we show how to implement computation with fungi and demonstrate that a structure of logical functions computed is determined by mycelium geometry.

Mechanisms of stochastic onset and termination of atrial fibrillation studied with a cellular automaton model

Mathematical models of cardiac electrical excitation are increasingly complex, with multiscale models seeking to represent and bridge physiological behaviours across temporal and spatial scales. The increasing complexity of these models makes it computationally expensive to both evaluate long term (more than 60 s) behaviour and determine sensitivity of model outputs to inputs. This is particularly relevant in models of atrial fibrillation (AF), where individual episodes last from seconds to days, and interepisode waiting times can be minutes to months. Potential mechanisms of transition between sinus rhythm and AF have been identified but are not well understood, and it is difficult to simulate AF for long periods of time using state-of-the-art models. In this study, we implemented a Moe-type cellular automaton on a novel, topologically equivalent surface geometry of the left atrium. We used the model to simulate stochastic initiation and spontaneous termination of AF, arising from bursts of spontaneous activation near pulmonary veins. The simplified representation of atrial electrical activity reduced computational cost, and so permitted us to investigate AF mechanisms in a probabilistic setting. We computed large numbers (approx. 10⁵) of sample paths of the model, to infer stochastic initiation and termination rates of AF episodes using different model parameters. By generating statistical distributions of model outputs, we demonstrated how to propagate uncertainties of inputs within our microscopic level model up to a macroscopic level. Lastly, we investigated spontaneous termination in the model and found a complex dependence on its past AF trajectory, the mechanism of which merits future investigation.

Emergent behaviors in a deterministic model of the human uterus

The human birth process is powered by uterine contractions that have observable patterns that depend on the physiology of muscular activity. We explored a previously designed model(1) simulating the uterus to assess global contractile patterns. The model is a cellular automaton that simulates the complexities of uterine activity from a few simple rules of cellular interaction and uterine geometry. Multiple experiments using the cellular automaton involved different uterine shapes, cell numbers, and initial distributions of active and resting cells. Results demonstrate complex contraction patterns similar to those observed in human labor. At least 2 modes of behavior appear in the simulations, one consistent with effective labor and one not. Experiments with cellular automata provide insights into stereotypic and disordered labor patterns that produce patient discomfort without progress in labor. We hypothesize that complex uterine contraction patterns may have other roles in the preparation for labor and birth.

Combination of computer simulations and experimental measurements as the training dataset for statistical estimation of epicardial activation maps from venous catheter recordings

One of the epicardial mapping techniques requires the insertion of multiple multi-electrode catheters into the coronary vessels. The recordings from the intracoronary catheters reflect the electrical activity on the nearby epicardial sites; however, most of epicardial surface is still inaccessible. In order to overcome this limited access problem, a method called the linear least squares estimation was proposed for the reconstruction of high-resolution maps using sparse measurements. In this technique, the relationship between catheter measurements and the remaining sites on the epicardium is created from previously obtained high-resolution maps (training dataset). Even though open-chest surgery is still a relatively frequent occurrence, an additional burden on the patient to obtain epicardial maps might impose an important risk on the patient. In this study, we hypothesize that epicardial maps created from computer simulations might be used in combination with the experimental data. In order to test this hypothesis, we used high-resolution epicardial activation maps acquired from 13 experiments performed on canine hearts that were stimulated via unipolar pacing from sites distributed all over the epicardium. We investigated the feasibility of the Aliev-Panfilov model that generated focal epicardial arrhythmias on Auckland heart. We started the simulations from the sites that corresponded to the pacing sites on the experimental geometry after a registration procedure between the experimental and simulation geometries. We then compared the simulation results with the corresponding experimental activation maps. Finally, we included simulated activation maps alone (100%) and in combination (simulated maps constituted 90%, 75%, 50%, 25%, 10%, and 0% of the training dataset) with experimental maps in the training set, performed the statistical estimation, and obtained the error statistics. The mean correlation coefficient (CC) between the simulated epicardial activation maps with the experimental maps was 0.88. The results of the estimation indicated that only 2.6 mm localization error and 0.05 CC value degradation occurred on average for the replacement of 75% of the experimental maps with the simulated counterparts. This indicated that including an important percentage from simulations may lead to decreased need for open-chest procedures and less burden on the patients.

A computer model of human atria with reasonable computation load and realistic anatomical properties

Atrial fibrillation is the most frequent arrhythmia, provoking discomfort, heart failure and arterial embolisms. The aim of this work is to develop a simplified anatomical computer model of human atria for the study of atrial arrhythmias and the understanding of electrical propagation mechanisms. With the model we propose, up to 40 s of real-time propagation have been simulated on a single-processor computer. The size and the electrophysiological properties of the simulated atria are within realistic values and information about anatomy has been taken into account in a three-dimensional structure. Besides normal sinus beat, pathological phenomena such as flutter and fibrillation have been induced using a programmed stimulation protocol. One important observation in our model is that atrial arrhythmias are a combination of functional and anatomical reentries and that the geometry plays an important role. This virtual atrium can reproduce electrophysiological observations made in humans but with the advantage of showing in great detail how arrhythmias are initiated and sustained. Such details are difficult or impossible to study in humans. This model will serve us as a tool to evaluate the impact of new therapeutic strategies and to improve them.

Strength-interval curves in canine myocardium at very short cycle lengths

While ventricular electrophysiological properties have been intensively studied at normal heart rates, little is known about these properties at the very short cycle lengths (approximately 100 msec), which are present in ventricular fibrillation. We examined refractoriness in the right ventricles of six dogs at stimulation intervals of 80 to 300 msec. Starting at 300 msec, the basic (S1) cycle length was decremented by 10 msec each beat to 200, 150, or 125 msec. A 1-msec premature (S2) stimulus of 1, 5, 10, or 20 mA was then introduced. The S1-S2 interval was decremented until capture was lost. The refractory period was considered to be the shortest interval that captured the heart for each S2 strength. Only pacing episodes that did not induce fibrillation were included. Strength-interval curves maintained the same hyperbolic shape but shifted to very short refractory periods as the S1-S1 interval was decreased. At the shortest S1-S1 intervals, premature stimuli were capable of capturing the heart without inducing ventricular fibrillation for S1-S2 intervals as short as 83 +/- 3 msec. Thus, decremental rapid pacing can produce refractory periods shorter than the cycle length during ventricular fibrillation. This finding suggests that there is no need to postulate a discontinuous jump to new electrophysiological properties or relationships at the onset of fibrillation, but that the capability for fibrillation is an integral part of normal electrophysiological parameters when they are pushed to values that do not occur normally. The results of this study should be useful in the further development of active membrane models and cellular automata models of cellular electrical behavior.