Greenberg-Hastings Model

step

12

b

3

n

7

random seed

123

The spatio-temporal patterns generated by means of the Greenberg–Hastings model, which is given by

t+1

c

ij

=

mod t c ij ,n	if t c ij ≥1
1	if t c ij =0∧# t c ij =1>b
t c ij	otherwise

,

where

t

c

ij

represents the state of the

i

th

j

cell at the

th

t

time step,

n

is the number of different states distinguished, the cardinality

#

t

c

ij

=1

is determined within an

r=2

Moore neighborhood, and

b

is a threshold value representing the number of excited cells (

#

t

c

ij

=1>b

) that have to enclose an excitable cell in order for it to become excited itself. A cell is in an excitable, excited, or refractory state if

t

c

ij

=0

,

t

c

ij

=1

, or

1<

t

c

ij

≤n-1

, respectively. This cyclic cellular automaton (CA) generates patterns that resemble the spiral waves generated by chemical clock reactions such as the Belousov–Zhabotinsky reaction. The evolution starts from random initial conditions, so the outcome will be different every time the model is run and might be homogeneously blank if there were not enough excited cells at the beginning.