Pioneered by Ulam and von Neumann, a cellular automaton (CA) is a formalism with the following characteristics.

there is a grid or lattice

each gridcell is a finite state machine (FSM), but as simple as possible. The entire grid is an autonomous FSM again.

there is speed of "light", the velocity at which information can travel across the grid

"the whole is less than the sum of its parts", since the input of the small FSMs is not arbitrary, but constrained to its neighbours

CAs have been used mainly as experimental mathematics (Ulam), artificial physics (Ulam), "new" physics (Wolfram), artificial life (von Neumann, Langton) or as a modelling tool (Toffoli). In biology the last case is the most interesting one. The features that make a CA a (good) model are:

structural stability: stable, but not too stable. When changing any parameter doesn't result in a structural effect, the model has no meaning (is not "saying" anything)

multiple stable points: starting in a certain initial state, a lot of stable states are possible

local optima: f.i. additive voting rules can be seen as minimizing the zero-one borderline

variable/FSM identification: the correspondence between the real world and the features in the model. The active components in CAs are not the FSMs but the spaces

dynamical systems constraints: fixed set of FSMs. When one relaxes these constraints, one gets individual oriented models

People started studying 1D elementary CAs and attempted to classify them by the temporal-spatial pattern they generated. The well-known classification by Wolfram is listed in the table below.

Class

Spatial pattern

Non-spatial equivalent

I

to uniform pattern

fixed point

IIa

domains, localized: patches

limit cycles

IIb

domains, non-stationary: waves

limit cycles

III

non-periodic, non-localized

chaos

IV

localized, long transient

universal computation

Particle conservation, biological models vs physical models
Space time scaling
behavior scaling
Synchronous vs asynchronous
critically slowing down phenomena
travelling patterns

## Cellular Automata

Pioneered by Ulam and von Neumann, a cellular automaton (CA) is a formalism with the following characteristics.

- there is a grid or lattice
- each gridcell is a finite state machine (FSM), but as simple as possible. The entire grid is an autonomous FSM again.
- there is speed of "light", the velocity at which information can travel across the grid
- "the whole is

CAs have been used mainly as experimental mathematics (Ulam), artificial physics (Ulam), "new" physics (Wolfram), artificial life (von Neumann, Langton) or as a modelling tool (Toffoli). In biology the last case is the most interesting one. The features that make a CA a (good) model are:lessthan the sum of its parts", since the input of the small FSMs is not arbitrary, but constrained to its neighbours- structural stability: stable, but not too stable. When changing any parameter doesn't result in a structural effect, the model has no meaning (is not "saying" anything)
- multiple stable points: starting in a certain initial state, a lot of stable states are possible
- local optima: f.i. additive voting rules can be seen as minimizing the zero-one borderline
- variable/FSM identification: the correspondence between the real world and the features in the model. The active components in CAs are not the FSMs but the spaces
- dynamical systems constraints: fixed set of FSMs. When one relaxes these constraints, one gets individual oriented models

People started studying 1D elementary CAs and attempted to classify them by the temporal-spatial pattern they generated. The well-known classification by Wolfram is listed in the table below.ClassSpatial patternNon-spatial equivalentParticle conservation, biological models vs physical models

Space time scaling

behavior scaling

Synchronous vs asynchronous

critically slowing down phenomena

travelling patterns

Crutchfield's rule 54 (mesoscale pattern algebra)

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