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where N is the number of cellular units and υ is their (associated) variety discussed earlier. The function measures implicitly the uncertainty or entropy due to the complexity and therefore is logarithmic as specified by Hartley’s law.

The uncertainty indicator of the complexity should therefore be limited to the disorderliness function by a criterion that, in the event of the system tending to be well-ordered (minimization of disorderliness) with the condition Yj→ 0, it should be accommodated by an appropriate functional relation between Yj and Cs.

Further, by considering the geometrical space representing the disorderliness, the variate y defines a measure of uncertainty region (deterministic or probabilistic) and it also refers implicitly to the a priori probability of a sample within the region.

The collective influence of disorderliness and the complexity, therefore, determine the extent of entropy (or uncertainty) associated with the self-organizational efforts in the neural system in achieving a specific control goal. That is, the net complexity and disorderliness function bear the information on the degree of disorganization or the self-organization deficiency defined earlier.

If self-organizational deficiency is attributed to every synaptic coupling, across N cellular units, the overall self-organization deficiency can be stipulated by:

where pj refers to the probability of encountering the jth cell (in the neural spatial complex) wherein the disorganization is observed, do is a function to be specified, and ΘY is a disorderliness parameter defined by the relation:

where WYj is a weighting function measuring the deviation of Yj of jth realization from those of other realizations of the state variable; and CY is a conditional coefficient which sets doYj) = 0 when Yj = 0. In writing the above relation, it is presumed that the system is ergodic with the entire ensemble of the parameter space having a common functional relation do.

Thus, the disorganization is an ensemble-generalized characteristic of disorder in the state of the neural system, weighted primarily by the probability of encountering a cell with a disorderly behavior and secondarily by its relevance. The relevance of the disorder to the jth situation is decided both by the functional relation do common to the entire ensemble and the additional weight WYj accounting for the deviation of Yj with the corresponding disorderliness of situations other than j.

The functional relation do pertinent to the self-organizing control endeavors of a neural complex refers to the sensitivity of the high-level goal towards the degree of failure to attain the subgoals considered. Such a failure or deviatory response arises due to the entropy of the system. Therefore, more explicitly OD can be written as an entropy function in terms of Hartley’s law as:

Considering both temporal and spatial disorganizations associated with interconnected neurons, a superposition leads to the following general expression for the summation of the effects:

From the foregoing discussions it is evident that the disorganization of a neural complex is the consequence of:

  Spatial factors conceived in terms of the random locations of the neuronal cells.
  Temporal characteristics as decided by the random occurrence of state transitions across the interconnected cells.
  Stochastical attributes of the neural complexity.
  Combinatorial aspects due to the number and variety of the participating subsets in the neural activity.

Referring to spatiotemporal disorderliness as indicated earlier, let the a priori probability pj denote the probability of occurrence (in time and space) of the jth neural event. When pj → 1, it amounts to a total disorderliness with Yj → 0. Likewise, pj → 0 sets a total disorderliness with Yj → ∞. The above conditions are met simultaneously by the following coupled relations:

Therefore, the Hartley measure of the extent of disorganization can be written as:

which is again in the standard form as Shannon's statistical measure of entropy.

The structural or the combinatorial aspect of disorganization pertains to the number of alternatives (such as the paths of state-transition proliferation or traces) in the interconnected network of cellular automata. Usually, only n out of such N alternatives are warranted to confirm a total orderliness. Therefore, the disorderliness is written as:

so that Yj → ∞ for Nj → ∞ with n being a constant and Yj → 0 for Nj → n. The corresponding measure of disorganization written in conformity with Hartley’s measure of entropy simplifies to:


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