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

Andrew Adamatzky; Jordi Valverdu

Barcelona_Raval_phi_0_050.mov: Raval. Initial perturbation site is at the beginning of  Les Rambles. $\phi=0.050$

Barcelona_Raval_phi_0_065.mov: Raval. Initial perturbation site is at the beginning of  Les Rambles. $\phi=0.065$

Barcelona_Raval_phi_0_074.mov: Raval. Initial perturbation site is at the beginning of  Les Rambles. $\phi=0.074$

Barcelona_Gracia_phi_0_0666.mov: Gracia. Initial perturbation site is Sagrada Familia.  $\phi=0.0666$

====== Description of the model ====

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 
Belousov-Zhabotinsky (BZ) reaction with applied illumination~\cite{beato2003pulse}:

\begin{eqnarray}
  \frac{\partial u}{\partial t} & = & \frac{1}{\epsilon} (u - u^2 - (f v + \phi)\frac{u-q}{u+q}) + D_u \nabla^2 u \nonumber \\
  \frac{\partial v}{\partial t} & = & u - v 
\label{equ:oregonator}
\end{eqnarray}

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. 

 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 $\Phi=[0.05,0.08]$.

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 Barcelona_Gracia.png  and Barcelona_Raval point.png. Time-lapse snapshots provided in the paper were recorded at every 150\textsuperscript{th} time step, we display sites with $u >0.04$; 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.  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.

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Barcelona_Gracia.png
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Barcelona_Gracia_phi_0_0666.mov
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Barcelona_Raval point.png
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Barcelona_Raval_phi_0_050.mov
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Barcelona_Raval_phi_0_065.mov
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Barcelona_Raval_phi_0_074.mov
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