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In a later study, Hopfield [36] points out that real neurons have continuous input-output relations. Hence, he constructs another model based on continuous variables and responses which still retains all the significant, characteristics of the original model (based on two-state McCulloch-Pitts’ threshold devices having outputs of 0 or 1 only). Hopfield let the output variable σi for neuron i have a squashed range and considered it as a continuous and monotone-increasing function of the instantaneous input xi to neuron i. The typical input-output relation is then a S-shaped sigmoid with asymptotes and . (The sigmoidal aspects of a neural network will be discussed in detail in a later chapter.)

2.7 Neural Net: A Self-Organizing Finite Automaton

The general characteristic of a neural net is that it is essentially a finite automaton. In other words, its input-output behavior corresponds to that of a finite automaton.

A modular net (such as the neural net) being a finite automaton has the capability for memory and computation.

Further, the modular net emerges as a computer which has command over its input and output—it can postpone its input (delay) and refer back to earlier inputs (memory) by an effective procedure or by a set of rules (more often known as algorithms in reference to computers).

A neural net in its global operation achieves a formalized procedure in deciding its input-output relation. This effective or decision procedure is typically cybernetic in that a particular operation is amenable as a mathematical operation. Further, neural net operates on a logical basis (of precise or probabilistic form) which governs the basic aspect of cybernetic principles.

A neural net supports a progression of state transitions (on-off type in the simplest neuronal configuration) — channeling a flow of bit-by-bit information across it. Thus, it envisages an information or communication protocol, deliberating the cybernetic principle.

2.8 Concluding Remarks

A neural complex has a diversified complexity in its structure and functions but portrays a unity in its collective behavior. The anatomy and physiology of the nervous system facilitates this coorperative neural performance through the mediating biochemical processes manifesting as the informational flow across the interconnected neurons. The proliferation of neural information envisages the commands, control, and communication protocols among the neurons. The resulting automaton represents a self-organizing system—a neurocybernetic.

The mechanism of interaction between the neurons immensely mimics the various interactive phenomena of statistical physics; more specifically, it corresponds to the Ising spin interaction pertinent to the statistical mechanics of ferromagnetism. Further, the neural complex is essentially a stochastical system. Its random structural considerations and conjectural functional attributes fortify such stochastical attributes and dictate a probabilistic modus operandi in visualizing the complex behavior of the neural system.

Modeling the biological neural complex or an artificial neural network on the basic characteristics as listed above is, therefore, supplemented by the associated stochastical theory, principles of cybernetics, and physics of statistical mechanics. The government of these considerations in essence constitutes the contents of the ensuing chapters.


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