Memory of a Nonlinear Process

The previous experiment depicted a stationary oscillating CA-1 with a period of 46 days, which is called here the parent.  One of its border cells was injured (set to zero = white), and the parent  was transformed into an oscillating  CA-2 with a period of 29, called here the offspring. The offspring was later injured in the same way as its parent before. The experiment was repeated 29 times. Each time a different border cell was injured,  and the response of the CA monitored. 30% perturbations ended in CA-2 death.  20% led to the creation of  a new attractor with the same period and structure (phenotype) .30% became chaotic, and 20% assumed the parent phenotype with a period of 46 days.

The  image depicts the transition of the CA-2 offspring back to its parent CA-1 attractor. The left CA-2 was injured and the outcome is depicted in the right CA-2. The image in between is the absolute difference between the two CA. It shows that the perturbation progressed from the left CA-2 border to the right. Then offspring assumed its parent phenotype.

Not all transitions were so fast (short). Some perturbations initiated a prolonged chaotic behavior which occasionally settled on the parent attractor like in the image below, The experiment lasted 300 days. It is possible that later on, also other chaotic attractors may have transformed into to their parent phenotype.


The experiment illustrates an interesting phenomenon. Offspring remembers its parent phenotype, and occasionally assumes it. Where is this memory stored?  The fate of the CA is controlled by a triplet : {state, rule#, max age}. Here max age was set to infinite and had no effect on CA behavior. Parent memory was stored in the structure and rule. The latter is inherited by each cell, and may be regarded as its gene, which controls also CA structure, or phenotype.

This simple relationship holds only in an isolated CA which does not interact with the environment. When perturbed (injured), phenotype is determined by gene, state, and perturbation.  The memory of this  interaction is stored in the CA structure, and was described in the chapter: Regeneration during chronic Injury.


Although perturbation caused  the change, it did not fully control its outcome, which was determined also by  the triplet {state, rule#, max age}.  While in Mechanics, perturbation fully determines the outcome, in Life (and CA) it does not.  Here the outcome has four causes :

1. Material cause:  CA structure, and age
2. Formal cause:  Its rule
3. Efficient cause: Perturbation (injury)
4. Final cause: To stay alive

They were defined by Aristotle (v. Chapter on CA Creativity ), and are essential for understanding life phenomena. While Physics can manage with efficient cause, Medicine, cannot. Its rejection of the other causes leads to treatment induced diseases, called Iatrogenesis.

Different kinds of Chaos

It is striking that  even after prolonged periods of chaotic change, the offspring may occasionally assume its parent phenotype. Even in the state of chaos it remembers its ancestor. Apparently the CA has a small repertory of oscillatory attractors (phenotypes)  which it can establish. Two are depicted above. There may by more, but not many, since each CA rule excludes most of them. The chapter on Injury and Repair , depicts an outcome of a chaotic change induced by rule #2058.

A chaotic change caused by the rule #600, will produce different oscillators than that caused by #2058. The outcome of a chaotic phase will thus depend on the CA rule, and since each rule is associated with a distinct oscillator set, each CA rule generates a different chaotic trajectory.   Wolfram's rule numbering (p. 54), may therefore be applied also to number the chaotic changes following  a perturbation. We start with a zygote ( = 1), let it grow, and set one of its progeny to zero. Each CA type will create different chaotic trajectories , which may be distinguished from each other by their oscillator sets. One may regard this chaos set as a number system.

A CA may respond to perturbation in four ways:

1. No effect,
2. Oscillation
3. Chaotic change
4. Death

Actually there are two kinds of oscillation, stationary and non stationary . The latter is depicted in the chapter on Recollection. We shall be concerned with stationary processes, like the two described above. While  a stationary oscillation is stable, chaos is not. Generally, injury will drive the CA from one stationary oscillating attractor to another. Chaotic response is extremely dangerous and unless it settles in an stationary attractor, CA dies.

effect[1,500]; effect[2,500];  injurystate[1, j, 47, f[[1,1]], 1, 0]; injurystate[2, j, 47, f[[2,1]], 1, 0]; injurystate[2, j, 127, f[[2,1]], 1, 0];

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