Excitation waves propagating on the streets of Barcelona. Numerical integration of Oregonator equations.

<p><a href="https://zenodo.org/api/files/38cdb802-1f50-4990-afbe-073fefbb46fe/Barcelona_Raval_phi_0_050.mov?versionId=285ba608-3f27-422f-b9f2-72137a5525a7">Barcelona_Raval_phi_0_050.mov</a>:&nbsp;Raval.&nbsp;Initial perturbation site is at the beginning of &nbsp;Les Rambles. $\phi=0.050$<br> <br> <a href="https://zenodo.org/api/files/38cdb802-1f50-4990-afbe-073fefbb46fe/Barcelona_Raval_phi_0_065.mov?versionId=30d2c271-9058-4587-a1ec-b5fdda78482b">Barcelona_Raval_phi_0_065.mov</a>:&nbsp;Raval.&nbsp;Initial perturbation site is at the beginning of &nbsp;Les Rambles. $\phi=0.065$<br> <br> <a href="https://zenodo.org/api/files/38cdb802-1f50-4990-afbe-073fefbb46fe/Barcelona_Raval_phi_0_074.mov?versionId=d0a4c922-b29c-46da-b8fd-618a404b482f">Barcelona_Raval_phi_0_074.mov</a>:&nbsp;Raval.&nbsp;Initial perturbation site is at the beginning of &nbsp;Les Rambles. $\phi=0.074$</p> <p>Barcelona_Gracia_phi_0_0666.mov:&nbsp;Gracia. Initial perturbation site is Sagrada Familia.&nbsp; $\phi=0.0666$</p> <p>====== Description of the model ====</p> <p>Two fragments of Barcelona street map --- Gracia and Raval, were mapped onto a grid of 2500 by 2500 nodes. Nodes of the grid corresponding to streets are considered to be filled with a Belousov-Zhabotinsky medium, i.e. excitable nodes, other nodes are non-excitable. We use two-variable Oregonator equations~\cite{field1974oscillations} adapted to a light-sensitive&nbsp;<br> Belousov-Zhabotinsky (BZ) reaction with applied illumination~\cite{beato2003pulse}:</p> <p>\begin{eqnarray}<br> &nbsp; \frac{\partial u}{\partial t} &amp; = &amp; \frac{1}{\epsilon} (u - u^2 - (f v + \phi)\frac{u-q}{u+q}) + D_u \nabla^2 u \nonumber \\<br> &nbsp; \frac{\partial v}{\partial t} &amp; = &amp; u - v&nbsp;<br> \label{equ:oregonator}<br> \end{eqnarray}</p> <p>The variables $u$ and $v$ represent local concentrations of an activator, or an excitatory component of BZ system, and an inhibitor, or a refractory component. Parameter $\epsilon$ sets up a ratio of the time scale of variables $u$ and $v$, $q$ is a scaling parameter depending on rates of activation/propagation and inhibition, $f$ is a stoichiometric coefficient.&nbsp;</p> <p>&nbsp;We integrated the system using Euler method with five-node Laplace operator, time step $\Delta t=0.001$ and grid point spacing $\Delta x = 0.25$, $\epsilon=0.02$, $f=1.4$, $q=0.002$. We varied value of $\phi$ from the interval $\Phi=[0.05,0.08]$.</p> <p>To generate excitation waves we perturb the medium by square solid domains of excitation, $20 \times 20$ sites in state $u=1.0$, site of the perturbation is shown by red discs in <a href="https://zenodo.org/api/files/38cdb802-1f50-4990-afbe-073fefbb46fe/Barcelona_Gracia.png?versionId=cfeb3a2a-1f8a-427d-a2b1-41f9893a4d66">Barcelona_Gracia.png&nbsp;</a>&nbsp;and <a href="https://zenodo.org/api/files/38cdb802-1f50-4990-afbe-073fefbb46fe/Barcelona_Raval%20point.png?versionId=c0ad4d24-ff59-4287-a817-2c7bf72f509c">Barcelona_Raval point.png</a>. Time-lapse snapshots provided in the paper were recorded at every 150\textsuperscript{th} time step, we display sites with $u &gt;0.04$; videos supplementing figures were produced by saving a frame of the simulation every 50\textsuperscript{th} step of numerical integration and assembling them in the video with play rate 30 fps. &nbsp;All figures in this paper show time lapsed snapshots of waves, initiated just once from a single source of stimulation; these are not trains of waves following each other.</p>