In this article, we compare quantitatively the efficiency of three different protocols commonly used in commercial defibrillators. These are based on monophasic and both symmetric and asymmetric biphasic shocks. A numerical one-dimensional model of cardiac tissue using the bidomain formulation is used in order to test the different protocols. In particular, we performed a total of 4.8 Â 10 6 simulations by varying shock waveform, shock energy, initial conditions, and heterogeneity in internal electrical conductivity. Whenever the shock successfully removed the reentrant dynamics in the tissue, we classified the mechanism. The analysis of the numerical data shows that biphasic shocks are significantly more efficient (by about 25%) than the corresponding monophasic ones. We determine that the increase in efficiency of the biphasic shocks can be explained by the higher proportion of newly excited tissue through the mechanism of direct activation. In the present paper, we show how numerical simulations can be used to understand the efficiency of different defibrillation protocols. Fibrillation is a rapid, irregular electrical activity of the heart. This fatal medical condition is usually treated by the application of an external electric shock to the patient chest through external paddle electrodes. The shape of the electric waveforms that are usually applied are either monophasic or biphasic. This means that in the latter the polarity is switched at some point in the course of the application of the shock. Empirical observations suggest that biphasic shocks are more efficient than monophasic shocks in terminating fibrillation. In this paper, by using a simplified mathematical model of cardiac tissue, which, however, includes a realistic response of the cells to large electric fields, we confirm and explain this experimental observation. The model developed here could be used in subsequent studies in order to design and test more complex waveforms, which could be done systematically because the model is simple and not very computationally costly. The next goal is to find the optimal waveform that reduces the energy needed for defibrillatory shocks. This would be of great benefit for patients undergoing defibrillation by limiting the damage to the heart tissue caused by such a strong electric shock.
Connexins are specialized ionic channels that control the action potential propagation between cardiac myocytes. In this paper, we study the connexin dynamics in a one-dimensional model of cardiac tissue. We show that the connexin dynamics may lead to a spatial organization of the gap junction conductance. In the numerical simulations presented in this paper we have found two different regimes for the spatial organization of the conductances: (a) a spatially uniform conductance; (b) a spatially complex pattern of local values of high and low conductances. In addition, we have observed that, locally, the two final states are limit cycles with a period equal to the period associated with the external excitation of the tissue strand. The conductance dispersion usually takes place on a very large time scale, i.e. thousands of heart beats, and on a very short spatial scale. Due to its simplicity, the one-dimensional setting allows a detailed study of the emerging structure and in particular very long simulations. We have studied the transition between the two aforementioned states as a function of the gap junction conductance characteristics. Furthermore, we have studied the effect of initially added noises on the outcome of the system. Finally, using spatial autocorrelation functions we have characterized the spatial dispersion in conductance values.
The purpose of this paper is to analyze the operators induced by relations and conversely the relations induced by operators in fuzzy logic. Given a t-norm * and given a non-empty universal set X, it is well known that if R is a fuzzy *-preorder on X then the operator induced by R, [Formula: see text], is a fuzzy consequence operator (FCO). In fact, [Formula: see text] is a *-coherent FCO. It is also known that if C is a *-coherent FCO then the relation induced by C, RC, is a fuzzy *-preorder. We explore the *-coherence axiom because we do not know in the literature any example of non-coherent operator. Then, several families of these operators will be shown. Moreover we prove that the equivalence between fuzzy preorders and fuzzy consequence operators is held in only one way. As a result, a characterization of the *-preorder concept using the induced operator is given. Also some characterizations which show when an operator induces a *-preorder are proved. Finally, we will show that the characterization of the operators induced by relations given for finite universes cannot be generalized for infinite universes.
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.
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