Multi+CA+emergence

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=Multi-level CA: emergent entities=

So far we have looked at how CA models allow for mesoscale patterns in the sense that simple local interactions lead to [|complex] patterns. The main points that we made were:
 * patterns can be seen as a default expectation
 * large scale patterns form from local interactions (cf disturbances)
 * unpredictability arising despite deterministic rules, having to let it live its life

We have discussed how mean field approximations can be seen as a descriptor of a system. In this it is important that unique next state functions depend on the level of description of a system. Whether you want to predict the behaviour of single cells, the observed entities that arise as mesoscale patterns, or a global characteristic such as population size will determine whether there is a UNSF or not. For aggregate states (i.e. mesoscale patterns) the UNSF is usually **not** uniquely determined.

It is also interesting to note the contrast between space-based models and individual-based models (see later). On the one hand (space-based) one needs a predetermined number of states and variables to be in a dynamic system. In IBMs there can be a changing number of states and therefore this is an undefined system in term of dynamic systems.

//Mesoscale patterns//
The mesoscale entities that we have been discussing up to nowcan be referred to as "[|emergent properties]". However we should beware of the so called mystic feeling that we can get from such terms. There is nothing mystic about emergent properties, however it is important to discover them and to describe them. The question is then which ones to describe? The importance of such an approach can be illustrated by thinking of the universe as a CA, where we would be a mesoscale pattern based on simple local rules. We are not predefined in the transition table but only as an emergent pattern that can be recognized and described.

__Rule 54__
Crutchfield and Mitchell ([|1995]) conducted a groundbreaking study on mesoscale patterns in 1D CAs by using filtering techniques to discover complex behaviour in seemingly undynamic CAs. A typical example is the study of [|Elementary CA] 54. This 1D CA is named after the binary code for its next state function. In its space-time plot, rule 54 shows a seemingly uninteresting triangle pattern with a predominant pattern interspersed with larger triangles (see left figure). The dynamics of this system was studied by filtering out the predominant background pattern of small triangles, and instead showing the deviations of the pattern and marking them in black. What results is the emergence of a non-regular pattern showing mesoscale patterns that travel through space and time (see right figure, this is reminiscent of Class IV behaviour). So this is all very interesting, but how does it help us to understand the behaviour of the system?

Well, we know the rule (rule 54) at the rule level, however we may want more detail. We can achieve this by classifying different mesoscale patterns, and describing their interactions. In this case, we can classify the mesoscale patterns as 4 types of "particles" (alpha, beta, gamma+ and gamma-) which interact in particular ways. We can then derive a description of the system at this higher level using ODEs to describe the particle "concentrations", which is not much different to what is done in chemistry (using mass-action terms to describe reactions). Moreover, in time one can track particles and observe first remnants of initial conditions which die out quickly after which particles continue in time. At another higher level one can plot the particles themselves (a further abstraction) and view the CA as particle interactions over time.

Such analyses have been done for all 256 elementary CAs. Although it may not be biologically relevant, it is conceptually important. It demonstrates that given a bedrock fully defined deterministic universe, there can be mesoscale entities with behaviour of their own which are best described beyond the "full description" modeling frame-work. However this does not help predictability given random initial conditions.

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