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8.6 Entropy of Neurocybernetic Self-Regulation

In the self-organization endeavor, the neurocybernetic system attempts to (self) regulate (via feedback techniques), the state of the system (corresponding to say the ith realization of cellular state variable being nondeviatory with respect to a target state). That is, designating the generalized state of the system vector by yi in the ith realization from the target vector yT, the corresponding diversion is represented by (yi - yT) and |yi - yT = Qi. A bounded two-dimensional spread-space, ΩS includes all such likely diversions as depicted in Figure 8.2. It can be decomposed into elementary subspaces ΔΩ which can be set equal to δ2, where δ represents the one-dimensional quantizing level of the space.

Suppose the a priori probability (pi) of finding the vector yi at the ith elementary subspace is known, along with the distance of this ith realization from the goal (target), namely, Qi. The corresponding diversion ensemble of the entire spread-space {ΩS} can be written as:

where Σκ i=1 pi = 1, and the number of form realizations in each subspace over the domain ΩS is κ = ΩS/ΔΩ. Assuming equal weights, namely, W1 = W2 = ... = Wκ = 1, a goal-associated positional entropy can be defined as follows:

The function defines a new entropy space () as distinct from the parameter space ΩS. That is, each value of in the entropy space could be mapped onto the parameter spread-space and denoted on a one-to-one basis by . This value of represents the expected mean diversion from the goal in the spread-space ΩS. Further, the spaces ΩS and are affinely similar and the goal-related entropy satisfies the conditions specified in the following lemmas [111]:

Lemma 1:

Lemma 2:

Lemma 3:

where

and

Lemma 4: Sum of two entropies satisfying the conditions of independence and summation in the spread-space of the state vector leads to:

where

Lemmas 1 and 2 represent the intuitive concept of neural disorganization in the state of being controlled towards the goal. If an ideal control is perceived, it strikes the well-ordered target domain in all realizations specified by . Diversions from the ideality of the ensemble with an increasing or decreasing trend enable to increase or decrease, respectively.

Lemma 3 stipulates that, in the event of equiprobable diversions, the relation between the spread-space of the state vector and the entropy space is logarithmic with an error εκ → 0 for |yi - yT| >> 1.

Entropy associated with target-seeking endeavor is not additive. That is, goal-associated entropies cannot be added or subtracted directly in the entropy space. However, these superposition operations can be performed in the spread-space of the state vector. After the necessary extent of such operations are executed in the spread-space, the consequent effects can be translated to the entropy space.

Lemma 4 specifies the rule of additivity in the spread space, pertaining to independent goal-associated position entropies with an accuracy set by ε(1,2) → 0 with .

8.7 Subjective Neural Disorganization

Shown in Figure 8.3 is an arrangement depicting the ensemble of the state-vector realizations grouped with a center which does not coincide with the center of order or the goal-vector. That is, the goal center may lie outside the spread space. This happens when the neural system has self-organizing characteristics controlled subjectively. The subjectiveness refers to the neural intellect that has an information (or entropy) from the source, classifies the situations in it, predicts its behavior, and makes decisions leading the processed information to a desired functional attribute. While the ensemble of subjective realizations of the state-vectors tend to lie in a spread domain, the goal-dictated (objective) state-variables may normally cluster outside this region as shown in Figure 8.3.


Figure 8.3  Objective and subjective neural disorganizations

The question of subjectiveness refers to the situation when the system prescribes inherently its own goal or subjective function regardless of the training-dictated objective function. This condition in the nervous system is, for example, due to external influences such as drug, alcohol etc. Such external factors may cause the control strategy of neural self-regulation to seek the subjective goal rather than the objective goal learned through experience.

Correspondingly, there are two possible disorganizations with respect to subjective and objective goals designated as yTS and yTO, respectively. The mutual positional disorganization defined between yTS and yTO is given by:

where gso is the mutual diversion between the subjective and objective goals; and the corresponding fractional position entropy defined relative to the subjective goal is given by:

where |yj - yTS| is the spread about the subjective target. It should be noted that as gso → 0, and the fractional positional entropy satisfies all the four conditions imposed earlier.

While yTO is a steady-state (time-invariant) factor, yTS may change subjectively with time. Therefore, H(o/s)y represents a dynamic parameter as decided by the dynamic wandering of the subjective goal center. In general, gso is a stochastical quantity the extent of which is dictated by the neural (random) triggering induced by external influences. The concept of fractional positional entropy may find application in the analysis of a pathogenic neural complex and is a measure of fault-tolerancy in the network operation.


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