ca_paragidm

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 * TODO List**
 * REF: Toffoli using CA to explore implication of discrete particles
 * REF Wolfram using 3D CA to only find transition table to gain understanding

=[|CA] as a paradigm system=

(**Concepts**: patterns as default expectation, (evolved) patterns as not necessarily optimal, models for baseline expectations)

Cellular automata have been used as paradigm systems in various study areas:
 * [|Experimental mathematics] ([|Ulam] 1940's, see Burks 1970)
 * [|Artificial Life] ([|von Neumann] 1966, [|Langton] 1988)
 * [|New Physics] ([|Wolfram] 1984, [|Toffoli] 1987)
 * Local interactions (Hogeweg 1988)

In all cases the main defining point is the discrete nature of the formalism in combination with local interactions and complexity arising from the emergence of mesoscale patterns. For instance, Toffoli used a CA to explore the implications of discrete particles (particle concept) and contrast that to field theory (continuous variables) (REF). Wolfram even went as far as to consider the entire universe as a 3D CA in which one only had to find the transition table in order to gain full understanding (REF).

Important to note is that CAs are **not bad PDEs** ([|partial differential equations]), but are instead really a different concept which allows us to focus on different aspects (such as discrete entities and local interactions). To illustrate this point we consider the following model of [|B-cell] nodule formation in [|lymph nodes] ([|Hogeweg 1989]). In lymph nodes, the B cells are clumped within a mass of T cells, forming B cell modules. Both cell types enter the lymph node randomly. Two obvious questions are **(i) how are these nodules formed,** and **(ii) why did the system evolve this pattern?** This was studied with the following model:

The model is a CA that represents a crosssection of a lymph node. The state of each gridpoint represents the absence/presence of a T or B cell, and the next state function is defined in terms of birth/death and influx/efflux of cells. We assume **random** influx of [|T] and B cells. T cells proliferate independently while B cells only proliferate with local help of T cells. Finally cells have a constant efflux from the system. This simple model leads to a striking observation. In the model B-cell nodules form, including higher density of B cells at the edge of nodules (i.e. where they are stimulated by T cells), as is observed in actual lymph nodes. This result demonstrates that such patterns need not be the result of an active process or mechanism (no such mechanism was included in the model), and does not require a functional explanation (i.e. how is the pattern "good" for the system?). Actually, in this case the question can be reversed: how "**bad"** is it for the system? Namely, patterns with clumped B cells tend to slow the B cell - T cell interactions needed for proliferation. Hence, proliferation would be optimized if the system would be well-mixed. It is however very hard to remain **well mixed** when proliferating locally!

This example illustrates two important points:
 * **patterns need not be products of optimization** or come about by some specific mechanism (here it is actually optimal to be well-mixed!)
 * **patterns as default expectation**: our default expectation should probably more often be **non-homogeneity** because interactions are local.

CAs provide a powerful modeling formalism to study the effects of local interactions in discrete entities while allowing for mesoscale patterns to arise and add to the dynamics in the system. As such they have been used as paradigm systems. Important is that such models can offer us **baseline expectations** for a given system.

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