Nonlinear Wave Processes in Excitable Media

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Then we can divide the right hand side of Equation 15 into two terms. The term which would cause an increase of u r with time is named as the source term. It reads as. The other term which would cause a decrease of u r with time is named the sink term. From the above expressions of two terms, we find that larger D would enhance the source term Equation 16 but enhances the sink term Equation 17 even more. Larger A would not affect the source term but enhance the sink term.

Larger r , i. Therefore, the conclusion is that the larger D , A and R are, the stronger the sink term would be, and the later u r reaches the excitation threshold. This is the cause of the delay of the start time of the excitation penetration near the corner of the rounded rectangular FPR, i. As shown in Figure 9 , the numerical simulation results prove our explanation of the delay effect by plotting the value of u at the corner of the rounded rectangular FPR with time. Figure 9. Temporal dynamics of the value of u at the junction point between the flat boundary and the corner of a rectangularly shaped FPR.

Inside the rounded rectangular FPR, the accelerating effect on the propagation velocity c v occurs near its vertical flat boundary. When the initial plane wave is blocked at the flat boundary, although it does not penetrate inside the FPR, it yet increases the value of u in a vicinity of the FPR boundary. This is quite similar to a preheating effect in the flame propagation when the fuel temperature ahead of the flame front is increased [ 39 , 40 ]. The mechanism of the accelerated wave front could be analytically understood from Equation 5 for the bistable distributed system.

Thus, Equation 5 would be simplified as. Based on the theory described in Keener and Sneyd [ 36 ], the propagation velocity of the excitation wave front could be expressed as. The analytical expressions of Equation 19 would be obtained for two limiting cases. The first one is the unpreheated case at which u p is equal to the resting state. That gives. This gives.

Project Area C - Biological Systems: Self-organization and nonlinear waves in active media.

We also investigate the preheated propagation velocity in the numerical simulations of Equation 5. As shown in Figure 10 , the numerical results elucidate the acceleration of the propagation velocity c u p as a function of the preheated state u p.

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The analytical results from Equations 20 and 21 perfectly describe both limiting cases following from these numerical data. Figure The solid line represents the numerical simulation results in a one-dimensional medium described by Equation 5. The value of u ahead of the wave front is set to be u p. Our results demonstrate that a self-sustained rotor could be initiated from the spatial heterogeneity, i.

We use a generic model to parameterize the heterogeneity with three parameters D , A , and R. The two boundaries of the rotor initiation region could be estimated by the analytical equation for the bistable distributed system and the simulations in a one-dimensional medium for a circular FPR, respectively.

We also show that to initiate the self-sustained rotor the length of the rounded rectangular FPR should be larger than the critical L c. Our findings in the generic model might be applicable to describe the electrophysiological dynamics of cardiac tissue. Indeed, the distribution of transmembrane potential V in a two-dimensional tissue could be described by the reaction-diffusion equation as follows [ 41 ].

Equations 22 and 23 can be generalized into a two-component reaction-diffusion system as follows. Thus, the reduced system which describes electrophysiological properties of the cardiac tissue looks similar to the reaction-diffusion model we use. Nowadays many detailed models of human atria incorporate both structural and electrophysiological heterogeneities leading to differences in conduction velocity between the neighboring regions [ 42 — 44 ].

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It is also well known that atrial fibrosis in the aging heart can result in spatial variations in the electrical conductivity of a part of the cardiac muscle [ 45 ]. If some regions within this part remain unchanged, they can resemble fast propagation regions introduced in our model. Note, that a similar nonhomogeneity in the electrical conductivity can appear, for instance, when fresh stem cells aggregates implanted in strongly remodeled cardiac tissue form gap junctions with adult cardiac myocytes [ 46 ].

Moreover, some cardiac diseases cause ion channel remodeling [ 47 ]. This remodeling can be represented as a variation of the term I ion in Equation This is to some extent equivalent to a variation of the parameter A in our model. If this remodeling occurs non-uniformly in space, one can expect the creation of some spots with a negligible variation of this parameter in comparison to its strong decrease in the surrounding regions.

Thus, nonhomogeneous remodeling can results in a creation of fast propagation regions considered in this study. Of course, the model used above is aimed to reproduce only most generic features of electrical activity in myocardial tissue. Investigation of specific dynamical features can be done by application of more detailed models widely used in the literature [ 48 — 50 ].

It is important to note that our recent results based on the Fenton-Karma model [ 48 ] indicate that all scenarios of rotor initiation obtained with the Barkley model are perfectly reproducible [ 33 ]. An obvious reason for this is that the restitution of action potential duration in detail reproduced in the Fenton-Karma model plays only a restricted role in the described scenarios, where spiral waves are generated after application of a single excitation stimulus. Of course, the following dynamics of the initiated rotors is strongly influenced by many other factors and specific features of cardiac tissue, which are not reproduced in the framework of the generic model used in the study.

Therefore, computer simulations of a real tissue in the framework of much more detailed models and most importantly experimental investigations definitely can help to verify the role of the observed scenario for generation of cardiac arrhythmias. All authors conceived of the presented idea.

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All authors discussed the results and contributed to the final manuscript. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Super-spiral structures in an excitable medium

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