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## Mathematical model of signal propagation in excitable media

 Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Prague, 12000, Czech Republic

* Corresponding author: Michal Beneš

Received  February 2020 Revised  January 2019 Published  June 2020

This article deals with a model of signal propagation in excitable media based on a system of reaction-diffusion equations. Such media have the ability to exhibit a large response in reaction to a small deviation from the rest state. An example of such media is the nerve tissue or the heart tissue. The first part of the article briefly describes the origin and the propagation of the cardiac action potential in the heart. In the second part, the mathematical properties of the model are discussed. Next, the numerical algorithm based on the finite difference method is used to obtain computational studies in both a homogeneous and heterogeneous medium with an emphasis on interactions of the propagating signals with obstacles in the medium.

Citation: Jakub Kantner, Michal Beneš. Mathematical model of signal propagation in excitable media. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020382
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An example of nullclines for problem (1). The green curve represents cubic function $v_1$ and the dashed red line is the graph of the linear function $v_2$. The values of the parameters are $\varepsilon = 0.008, \alpha = 0.139, \beta = 2.54, g_1 = 20, g_2 = 1.5, \text{ and } g_3 = -5.5$
The time evolution of the component $u_1$ of the solution for the set parameter values and the initial conditions for Example 1. Each subfigure represents one selected time level $t$
The time evolution of the component $u_1$ of the solution for the set parameter values and the initial conditions for Example 2. Each subfigure represents one selected time level $t$. The obstacle is marked in orange in Figures Figures 3a and 3b
The time evolution of the component $u_1$ of the solution for the set parameter values and the initial conditions for Experiment 3. Each subfigure represents one selected time level $t$. The obstacle is marked in orange
The common parameters for all Examples
 Model characteristics $u$ right hand side coefficients $\epsilon = 0.008, \alpha = 0.139 , \beta = 2.54$ $v$ right hand side coefficients $g_1 = 20, g_2 = 1.5, g_3 = -7.5$ Fixed point location $u_1^0 = 0.198598, u_2^0 = -2.352028$ Diffusion coefficients $\qquad D_1 = 8\cdot 10^{-4} \quad \text{ in }\Omega\setminus\Omega_{obs}$ $\qquad D_2 = 4\cdot 10^{-4} \quad \text{ in }\Omega\setminus\Omega_{obs}$ $\qquad D_1 = 0 \quad \text{ in } \Omega_{obs}$ $\qquad D_2 = 0 \quad \text{ in } \Omega_{obs},$ where $\Omega$ is a domain identical with the one in the initial conditions and $\Omega_{obs}$ is an obstacle, further described in Table 4 for each example. Initial state and boundary conditions $\qquad u_{ini, 1} = u_1^0 + \sin \left( \frac{\pi (x-a_x^0)}{b^0_x-a^0_x}\right)\sin \left( \frac{\pi (y-a^0_y)}{b^0_y-a^0_y}\right) \quad \text{ in }\Omega_0$ $\qquad u_{ini, 1} = u_1^0 \quad \text{ in }\Omega\setminus\Omega_0$ $\qquad u_{ini, 2} = u_2^0 \quad \text{ in }\Omega$ $\qquad u_1|_{\partial\Omega} = u_1^0$ $\qquad u_2|_{\partial\Omega} = u_2^0$ $\quad \Omega \ = \left( a_x, b_x\right) \times\left( a_y, b_y\right) \ = \left( 0, 1\right)\times\left( 0, 1\right)$ $\quad$ The parameters for $\Omega_0 = (a_x^0, b_x^0)\times (a_y^0, b_y^0)$ are described in Table 4 for each experiment. Numerical characteristics Total length of simulation $. \dots\dots\dots\dots\dots . \ T$ $15$ Internal time step $. \dots\dots\dots\dots\dots\dots\dots\dots \ \tau$ $6.1\cdot10^{-5}$ Mesh $. \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \ \omega_h$ $128\times 128$
 Model characteristics $u$ right hand side coefficients $\epsilon = 0.008, \alpha = 0.139 , \beta = 2.54$ $v$ right hand side coefficients $g_1 = 20, g_2 = 1.5, g_3 = -7.5$ Fixed point location $u_1^0 = 0.198598, u_2^0 = -2.352028$ Diffusion coefficients $\qquad D_1 = 8\cdot 10^{-4} \quad \text{ in }\Omega\setminus\Omega_{obs}$ $\qquad D_2 = 4\cdot 10^{-4} \quad \text{ in }\Omega\setminus\Omega_{obs}$ $\qquad D_1 = 0 \quad \text{ in } \Omega_{obs}$ $\qquad D_2 = 0 \quad \text{ in } \Omega_{obs},$ where $\Omega$ is a domain identical with the one in the initial conditions and $\Omega_{obs}$ is an obstacle, further described in Table 4 for each example. Initial state and boundary conditions $\qquad u_{ini, 1} = u_1^0 + \sin \left( \frac{\pi (x-a_x^0)}{b^0_x-a^0_x}\right)\sin \left( \frac{\pi (y-a^0_y)}{b^0_y-a^0_y}\right) \quad \text{ in }\Omega_0$ $\qquad u_{ini, 1} = u_1^0 \quad \text{ in }\Omega\setminus\Omega_0$ $\qquad u_{ini, 2} = u_2^0 \quad \text{ in }\Omega$ $\qquad u_1|_{\partial\Omega} = u_1^0$ $\qquad u_2|_{\partial\Omega} = u_2^0$ $\quad \Omega \ = \left( a_x, b_x\right) \times\left( a_y, b_y\right) \ = \left( 0, 1\right)\times\left( 0, 1\right)$ $\quad$ The parameters for $\Omega_0 = (a_x^0, b_x^0)\times (a_y^0, b_y^0)$ are described in Table 4 for each experiment. Numerical characteristics Total length of simulation $. \dots\dots\dots\dots\dots . \ T$ $15$ Internal time step $. \dots\dots\dots\dots\dots\dots\dots\dots \ \tau$ $6.1\cdot10^{-5}$ Mesh $. \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \ \omega_h$ $128\times 128$
Table of parameter values in which the Examples differ
 Example $\Omega_{obs}$ $\Omega_0$ 1 no obstacle $(0.02, 0.12)\times(0.02, 0.92)$ 2 triangle obstacle in orange in Figure 3 $(0.1, 0.3)\times(0.3, 0.5)$ 3 triangle obstacle in orange in Figure 4 $(0.1, 0.3)\times(0.3, 0.5)$
 Example $\Omega_{obs}$ $\Omega_0$ 1 no obstacle $(0.02, 0.12)\times(0.02, 0.92)$ 2 triangle obstacle in orange in Figure 3 $(0.1, 0.3)\times(0.3, 0.5)$ 3 triangle obstacle in orange in Figure 4 $(0.1, 0.3)\times(0.3, 0.5)$
Table of the numerical parameters and the maximal $L_1, L_2$ and $L_\infty$ errors at 20 time levels for an excitation in a medium with heterogeneous diffusion. Measured against the reference mesh with spatial step $h = 1.95 \cdot 10^{-3}$
 Mesh time step $L_1$ $L_1$ $L_2$ $L_2$ $L_\infty$ $L_\infty$ $h$ $\tau$ error of $u$ error of $v$ error of $u$ error of $v$ error of $u$ error of $v$ 0.0625 0.001953 0.056240 0.161228 0.108336 0.265209 0.363029 0.779687 0.0313 0.000488 0.021852 0.049984 0.043336 0.094244 0.235573 0.651494 0.0156 0.000122 0.004384 0.008584 0.008694 0.014913 0.053264 0.107053 0.0078 0.000031 0.000884 0.001813 0.001997 0.002953 0.013961 0.021748 0.0039 0.000008 0.000217 0.000472 0.000505 0.000698 0.003454 0.004255
 Mesh time step $L_1$ $L_1$ $L_2$ $L_2$ $L_\infty$ $L_\infty$ $h$ $\tau$ error of $u$ error of $v$ error of $u$ error of $v$ error of $u$ error of $v$ 0.0625 0.001953 0.056240 0.161228 0.108336 0.265209 0.363029 0.779687 0.0313 0.000488 0.021852 0.049984 0.043336 0.094244 0.235573 0.651494 0.0156 0.000122 0.004384 0.008584 0.008694 0.014913 0.053264 0.107053 0.0078 0.000031 0.000884 0.001813 0.001997 0.002953 0.013961 0.021748 0.0039 0.000008 0.000217 0.000472 0.000505 0.000698 0.003454 0.004255
Table of the EOC coefficients for an excitation in a medium with the heterogeneous diffusion
 Mesh Mesh EOC u EOC v EOC $u$ EOC $v$ EOC $u$ EOC $v$ $h_1$ $h_2$ $L_1$ $L_1$ $L_2$ $L_2$ $L_\infty$ $L_\infty$ 0.0625 0.0313 1.363831 1.689564 1.321875 1.492657 0.623911 0.259143 0.0313 0.0156 2.317446 2.541744 2.317474 2.659830 2.144942 2.605427 0.0156 0.0078 2.310130 2.243271 2.122186 2.336317 1.931758 2.299371 0.0078 0.0039 2.026351 1.941520 1.983479 2.080882 2.015062 2.353652
 Mesh Mesh EOC u EOC v EOC $u$ EOC $v$ EOC $u$ EOC $v$ $h_1$ $h_2$ $L_1$ $L_1$ $L_2$ $L_2$ $L_\infty$ $L_\infty$ 0.0625 0.0313 1.363831 1.689564 1.321875 1.492657 0.623911 0.259143 0.0313 0.0156 2.317446 2.541744 2.317474 2.659830 2.144942 2.605427 0.0156 0.0078 2.310130 2.243271 2.122186 2.336317 1.931758 2.299371 0.0078 0.0039 2.026351 1.941520 1.983479 2.080882 2.015062 2.353652
The parameters for the second wave in Example 1 – the functional reentry
 Initial state of the second wave at time $t=3$ $\qquad u_{ini, 1} = u_1^0 + \sin \left( \frac{\pi (x-a^2_x)}{b^2_x-a^2_x}\right)\sin \left( \frac{\pi (y-a^2_y)}{b^2_y-a^2_y}\right) \quad \text{ in }\Omega_2$ $\qquad u_{ini, 2} = u_2^0 \quad \text{ in }\Omega_2$ $\quad \Omega_2 = \left( a^2_x, b^2_x\right)\times\left( a^2_y, b^2_y\right) = \left( 0.3, 0.4\right)\times\left( 0.4, 0.6\right)$
 Initial state of the second wave at time $t=3$ $\qquad u_{ini, 1} = u_1^0 + \sin \left( \frac{\pi (x-a^2_x)}{b^2_x-a^2_x}\right)\sin \left( \frac{\pi (y-a^2_y)}{b^2_y-a^2_y}\right) \quad \text{ in }\Omega_2$ $\qquad u_{ini, 2} = u_2^0 \quad \text{ in }\Omega_2$ $\quad \Omega_2 = \left( a^2_x, b^2_x\right)\times\left( a^2_y, b^2_y\right) = \left( 0.3, 0.4\right)\times\left( 0.4, 0.6\right)$
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