1. Lukasz Grzegorz Maciak Micheal Alexis
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 Zaritski
CMPT Parallel Architectures and Algorithms

2. 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

3. 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

4. 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:
If the value of U>0.3 then U=V=1
If the value of V<0.6 then U=0
While V>0 decrement V by 0.01

5. Cellular Automata Rules for Excitable Media
Rules Continued:
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 of
U in the neighboring cells
U value of CELL is updated
in small increments.

6. Cellular Automata Rules for Excitable Media
Unexcited Cell
Unexcited Cell
(U diffuses)
Excited Cell
Value of U
Value of V
Excited Cell
(V decreases)
Unexcited Cell
(Refractory State)

7. 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)

8. 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 …

9. Parallelization Issues – Domain Slicing Problem
CPU A
CPU B 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 CELL2
but 3, 9 and CELL2 values belong to CPUB

10. OpenGL Graphical Thread
Work Distribution
Master Node (Control)
OpenGL Graphical Thread
1st Chunk
2nd Chunk …
19th Chunk

11. Cellular Automaton Rules - Implementation
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;
Diffusion Coefficient
(*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);
V decrement
Refraction Threshold
Ignition Threshold

12. 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);

13. Initiating Spiral Waves
Excited Cells
Intersecting squares of Excited and Refractory cells produce 2 Spiral Waves
Refractory Cells
Spiral Wave is a unique configuration of
Excited and Refractory cells – it will spin
forever, and it cannot be destroyed by
collisions with other waves

14. Rotation Core Depends on:
Spiral Wave Rotation
The Size of the
Rotation Core Depends on:
Ignition threshold
Refraction threshold
V decrement size
U diffusion coefficient
Rotation Core
Spiral Wave Tip
Spiral Wave rotates around a core – this makes it move around on the grid
and makes it collide with other waves – which alters the direction of the spin.

15. Computer Simulation
Initial Setup – Randomly placed intersecting squares.
This picture represents the state several generations old.

16. Computer Simulation
Spiral Waves are Forming (4 neighbor diffusion)

17. Computer Simulation
Runtime 5-10 minutes (4 neighbor diffusion)

18. Speedup Analysis
Using 8 neighbors we get much smoother waves.

19. Computer Simulation
Formation of Spiral Wave Pairs (8 neighbors)

20. Computer Simulation
Multiple Spiral Wave Clusters forming

21. Computer Simulation
Spiral Wave Pair becomes the dominant feature

22. Computer Simulation
Runtime 5-6 hours: single Spiral Wave dominates the grid

23. Speedup Analysis

24. 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?