Simulation of Long-Term Interaction of Spiral Waves in Excitable Media Modeled by Cellular Automata...
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![Page 1: Simulation of Long-Term Interaction of Spiral Waves in Excitable Media Modeled by Cellular Automata using a 20-CPU Linux Cluster Lukasz Grzegorz Maciak.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e665503460f94b612bd/html5/thumbnails/1.jpg)
Simulation of Long-Term Interaction of Spiral Waves in Excitable Media
Modeled by Cellular Automata using a 20-CPU Linux Cluster
Lukasz Grzegorz Maciak
Micheal Alexis
Faculty Supervisor: Dr. Roman ZaritskiCMPT 680-01 Parallel Architectures and Algorithms
![Page 2: Simulation of Long-Term Interaction of Spiral Waves in Excitable Media Modeled by Cellular Automata using a 20-CPU Linux Cluster Lukasz Grzegorz Maciak.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e665503460f94b612bd/html5/thumbnails/2.jpg)
Excitable Media Definition of Excitable Media:
An excitable medium is a nonlinear dynamical system which has the following properties: The capacity to propagate a wave of some description Inability to support the passing of another wave until a certain amount
of refractory time has passed Examples of Excitable Media in Nature:
Forest Fire Cardiac Muscle Tissue
Excited Wave Propagating Outwards
Refractory Area – cannot be excited
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Cellular Automaton
Excitable media can be modeled using the Cellular Automaton Theory: Cells on a grid are assigned discrete values Values are updated based on the state of the
neighboring cells according to some rules. Example:
Conway’s Game of Life:
Dead Cell
Live Cell
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Cellular Automata Rules for Excitable Media
The state of each cell is described by 2 values: Excitation (U) and Refraction (V)
The cell is Excited if U≥1 The cell stops being excited (U=0) if 0<V<0.6
Rules:1. If the value of U>0.3 then U=V=1
2. If the value of V<0.6 then U=0
3. While V>0 decrement V by 0.01
![Page 5: Simulation of Long-Term Interaction of Spiral Waves in Excitable Media Modeled by Cellular Automata using a 20-CPU Linux Cluster Lukasz Grzegorz Maciak.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e665503460f94b612bd/html5/thumbnails/5.jpg)
Cellular Automata Rules for Excitable Media
Rules Continued:4. The current value of U is determined based on
the state of either 4 or 8 neighboring cells using rules for diffusion (discretization of Laplace Operator)
1 2 3
4 CELL 5
6 7 8
Based on the U value ofU in the neighboring cellsU value of CELL is updated in small increments.
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Cellular Automata Rules for Excitable Media
Unexcited Cell Unexcited Cell(U diffuses)
Excited Cell
Excited Cell(V decreases)
Unexcited Cell(Refractory State)
Value of U
Value of V
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Implementing Computer Simulation
Implementation in C++ 20 CPU Linux Cluster
(located in RI 109) MPI library (parallel
communication) OpenGL library (data
visualization) LPThread library (the
graphical display functions)
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Implementation Details 1000 x 2000 Rectangular Grid Cells on the edges of the grid are always unexcited (u:0 v:0) Using Vertical Domain Slicing to divide workload among the
20 CPU’s Using MPI Send and Receive Functions to exchange cell
values on borders between each “chunk”
CPU 1
CPU 2
CPU 3
CPU 17
CPU 18
CPU 19
…
![Page 9: Simulation of Long-Term Interaction of Spiral Waves in Excitable Media Modeled by Cellular Automata using a 20-CPU Linux Cluster Lukasz Grzegorz Maciak.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e665503460f94b612bd/html5/thumbnails/9.jpg)
Parallelization Issues – Domain Slicing Problem
1 2 3 4
5 CELL 1 CELL 2 6
7 8 9 10
CELL1 depends on points 1,2,3,5,7,8, 9 and CELL2but 3, 9 and CELL2 values belong to CPUB
CPU A CPU B
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Work Distribution
Master Node (Control)
2nd Chunk
OpenGL Graphical Thread
1st Chunk
19th Chunk
…
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Diffusion of U:
(*uu)(i,j) = U+2.76*D*dt/dx/dx*((*u)(i+1,j)+(*u)(i-1,j)+(*u)(i,j+1)+(*u)(i,j-1)-4*U);
Change of V:
if (V>0) {
V=V-0.01; if (V<0.6) (*uu)(i,j)=0;}else {
if(((*uu)(i,j)>0.3)&&((*uu)(i,j)<1)){
(*uu)(i,j)=1.0; V=1.0; }
}
Cellular Automaton Rules - Implementation
Diffusion Coefficient
V decrement Refraction Threshold
Ignition Threshold
(*uu)(i,j) = U+2.76*D*dt/dx/dx*((*u)(i+1,j)+(*u)(i-1,j)+(*u)(i,j+1)+(*u)(i,j-1) +(*u)(i-1,j-1) +(*u)(i-1,j+1) +(*u)(i+1,j-1) +(*u)(i+1,j+1)-8*U);
![Page 12: Simulation of Long-Term Interaction of Spiral Waves in Excitable Media Modeled by Cellular Automata using a 20-CPU Linux Cluster Lukasz Grzegorz Maciak.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e665503460f94b612bd/html5/thumbnails/12.jpg)
Parallel Communication
MPI Library functions MPI_Send and MPI_Recv are used for communication
The U and V values on the “chunk” border need to be swapped among CPU’s
MPI Code Samples:
MPI_Send((void*)uu->GetColumnPtr(L_min_2), N_plus_1, MPIFPTYPE, right, 11+DiscrTime, MPI::COMM_WORLD);
MPI_Recv((void*)u->GetColumnPtr((k-1)*(L-2)), L*(N+1), MPIFPTYPE, ltor[k], k, MPI::COMM_WORLD, status);
![Page 13: Simulation of Long-Term Interaction of Spiral Waves in Excitable Media Modeled by Cellular Automata using a 20-CPU Linux Cluster Lukasz Grzegorz Maciak.](https://reader030.fdocuments.in/reader030/viewer/2022032604/56649e665503460f94b612bd/html5/thumbnails/13.jpg)
Initiating Spiral Waves
Excited Cells
Refractory Cells
Intersecting squares of Excited and Refractory cells produce 2 Spiral Waves
Spiral Wave is a unique configuration ofExcited and Refractory cells – it will spinforever, and it cannot be destroyed by collisions with other waves
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Spiral Wave Rotation
Spiral Wave Tip
Rotation Core
Spiral Wave rotates around a core – this makes it move around on the gridand makes it collide with other waves – which alters the direction of the spin.
The Size of the Rotation Core Depends on:
• Ignition threshold • Refraction threshold• V decrement size• U diffusion coefficient
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Computer Simulation
Initial Setup – Randomly placed intersecting squares.This picture represents the state several generations old.
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Computer Simulation
Spiral Waves are Forming (4 neighbor diffusion)
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Computer Simulation
Runtime 5-10 minutes (4 neighbor diffusion)
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Speedup Analysis
Using 8 neighbors we get much smoother waves.
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Computer Simulation
Formation of Spiral Wave Pairs (8 neighbors)
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Computer Simulation
Multiple Spiral Wave Clusters forming
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Computer Simulation
Spiral Wave Pair becomes the dominant feature
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Computer Simulation
Runtime 5-6 hours: single Spiral Wave dominates the grid
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Speedup Analysis
Speed-Up Curve
0
2
4
6
8
10
12
14
16
18
0 5 10 15 20
# CPU
Sp
ee
d-U
p
DT=100
DT=500
DT=1000
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FUTURE DIRECTIONS
Using different boundary conditions Continuous Domain (Cylinder, Taurus) Reflective Boundaries
Discovering effective ways to eliminate spiral ways Long term statistical analysis of Spiral Wave
behavior: What are the optimal conditions for forming pairs and
triplets? What is the least time needed for a triplet to form? Is there a time period after which we can safely say, that if
no pairs have formed, it is unlikely that they will form at all?