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

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

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

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

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.

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

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)

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

…

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

Work Distribution

Master Node (Control)

2nd Chunk

OpenGL Graphical Thread

1st Chunk

19th Chunk

…

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);

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);

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

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

Computer Simulation

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

Computer Simulation

Spiral Waves are Forming (4 neighbor diffusion)

Computer Simulation

Runtime 5-10 minutes (4 neighbor diffusion)

Speedup Analysis

Using 8 neighbors we get much smoother waves.

Computer Simulation

Formation of Spiral Wave Pairs (8 neighbors)

Computer Simulation

Multiple Spiral Wave Clusters forming

Computer Simulation

Spiral Wave Pair becomes the dominant feature

Computer Simulation

Runtime 5-6 hours: single Spiral Wave dominates the grid

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

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?