Cell neighborhoods

Von Neumann was able to construct a self-reproducing UTM embedded within a 29-state/5-cell neighborhood two-dimensional cellular automaton, composed of several tens of thousands of cells. It was, to say the least, an enormously complex machine . Its set of 29 states consist largely of various logical building blocks (AND and OR gates, for example), several types of transmission lines, data encoders and recorders, clocks, etc. Von Neumann was unfortunately unable to finish the proof that his machine was a UTM before his death, but the proof was later completed and published by Arthur Burks [vonN66]. 

Figure 2.5. Cell neighborhoods (a) the von Neumann neighborhood, (b) the Moore neighborhood, and (c) the extended von Neumann neighborhood of cell A...

Local interactions each cell interacts only with cells that are in its local neighborhood. 

Discrete dynamics at each discrete unit time, each cell updates its current state according to a transition rule taking into account the states of cells in its neighborhood. 

Since each cell takes on one of k possible valnes-that is, Cj(t) G 0,1, 2,..., k -the rnle F is completely defined by specifying the valne assigned to each of the possible (2r -f l)-tnple confignrations for a given range-r neighborhood ... 

Dynamical Rule d> E x E x x E —> E, where n specifies the number of cells needed to define the neighborhood of a given cell. Defining M i] to be the neighborhood about cell i, the transition rule is most generally written as... 

Figure 2.7, for example, shows how a 2" -order radius r = 1 rule 4> may be converted into a F -order radius r = 2 rule Effective values d 0,1,..., define the states of the single remaining cell of a two-state collapse in time, and extra sites are added on to either side of the original neighborhood. 

Infected Cells Infected cells are assumed to gradually, and linearly, approach the ill state as a function of the average degree of infection in their neighborhood. Define Sij t) as the average degree of infection of sites neighboring site (bi) at time ... 

Some or all of the vertices in each fragment may be representative of a water molecule. The trace of each fragment may be mapped onto a two-dimensional grid (Figure 3.1c). This trace is equated with the mapping of a cellular automaton von Neumann neighborhood. The cellular automata transition rules operate randomly and asynchronously on the central cell, i, in each von... 

In a similar spirit, Inoue et al. [120] and Hashimoto et al. [121] generalized MPC dynamics so that the collision operator reflects the species compositions in the neighborhood of a chosen cell. More specifically, consider a binary mixture of particles with different colors. The color of particle i is denoted by c,-. The color flux of particles with color c in cell E, is defined as... 

Precellular solute ionization dictates membrane permeability dependence on mucosal pH. Therefore, lumenal or cellular events that affect mucosal microclimate pH may alter the membrane transport of ionizable solutes. The mucosal microclimate pH is defined by a region in the neighborhood of the mucosal membrane in which pH is lower than in the lumenal fluid. This is the result of proton secretion by the enterocytes, for which outward diffusion is slowed by intestinal mucus. (In fact, mucosal secretion of any ion coupled with mucus-restricted diffusion will provide an ionic microclimate.) Important differences in solute transport between experimental systems may be due to differences in intestinal ions and mucus secretion. It might be anticipated that microclimate pH effects would be less pronounced in epithelial cell culture (devoid of goblet cells) transport studies than in whole intestinal tissue. 

A function that defines the neighborhood around each cell. A cell can interact only with other cells that are within its neighborhood. 

The cells of the CA lie at the vertices of a regular lattice that homogenously covers a region of space. Although it is possible in principle to use an irregular lattice, we need to be able to define unambiguously the "neighborhood"... 

The state of every cell is updated once each cycle, according to transition rules, which take into account the cell s current state and the state of cells in the neighborhood. Transition rules may be deterministic, so that the next state of a cell is determined unambiguously by the current state of the CA, or partly or wholly stochastic. 

The neighborhood in a one-dimensional cellular automaton. Usually this includes only the immediate neighbors, but it can extend farther out to include more distant cells. 

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