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It would be possible for the neurons not to interact with each neighbor, only if there were a specific circuit arrangement of fiber processes preventing such interactions. That is, an extracellular current from an active neuron would not pass through the membranes of its neighbors, if specifically dictated by an inherent arrangement. However, evidence suggests [32] that such ramification is not governed by any arrangements in some definite scheme, but rather by some random mechanical principles of growth. Therefore, each neuron seems to be forced to interact with all of its immediate neighbors as well as some more distant neurons.

The theory of a cooperative organization of neurons does not require a definite arrangement of neural processes. What is required for an assembly to be cooperative is that each of its units should interact with more than two other units, and that the degree of interaction should exceed a certain critical level. Also, each unit should be capable of existing in two or more states of different energies. This theory applies to a (statistically) large assembly of units. Considering such a cooperative assembly, a small change in the external constraints may cause a finite transition in the average properties of the whole assembly. In other words, neural interaction is an extensive phenomenon.

In short, neural networks are explicitly cooperative. The presence or absence of an action potential contributes at least two different states (all-or-none response), and the propagation process pertaining to dichotomous state transitions provides the mode of relevant interaction(s).

Assuming that the organization of neurons is cooperative, as mentioned earlier, there is a possible analogy between the neuronal organization and the kind of interaction that exists among atoms which leads to interatomic cooperative processes. Little [33] developed his neural network model based on this analogy. He considered the correspondence between the magnetic Ising spin system and the neural network. For a magnetic spin system which becomes ferromagnetic, the long-range order (defined as fixing the spin at one lattice site causes the spins at sites far away from it to show a preference for one orientation) sets in at the Curie point and exists at all temperatures below that critical point (see Appendix A). The onset of this long-range order is associated with the occurrence of a degeneracy of the maximum eigenvalue of a certain matrix describing the mathematics of the interactive process.

Considering the largest natural neural network, namely, the brain, the following mode of temporal configuration determines the long-range interaction of state-transitions: The existence of correlation between two states of the brain which are separated by a long period of time is directly analogous to the occurrence of long-range order in the corresponding spin problem; and the occurrence of these persistent states is also related to the occurrence of a degeneracy of the maximum eigenvalue of a certain matrix.

In view of the analogy between the neural network and the two-dimensional Ising problem as conceived by Little, there are two main reasons or justifications posed for such an analogy: One is that already there exists a massive theory and experimental data on the properties of ferromagnetic materials; and, therefore, it might be possible to take the relevant results and apply them to the nervous system in order to predict certain properties. The other reason is that, on account of unfamiliarity with biological aspects of the neural system, it is simple and logical to relate the relevant considerations to those with which one is already familiar.

Continuing with the same connotations, Hopfield and Tank [34] stated “that the biological system operates in a collective analog mode, with each neuron summing the inputs of hundreds or thousands of others in order to determine its graded output.” Accordingly, they demonstrated the computational power and speed of collective analog networks of neurons in solving optimization problems using the principle of collective interactions.

In Hopfield’s original model [31], each neuron i has two states: σi = 0 (“not firing”) and σi = 1 (“firing at maximum rate”). That is, he uses essentially the McCulloch-Pitts neuron [7]. This “mathematical neuron” as deliberated earlier is capable of being excited by its inputs and of giving an output when a threshold VTi is exceeded. This neuron can only change its state on one of the discrete series of equally spaced times. If Wij is the strength of the connection from neuron i to neuron j, a binary word of M bits consisting of the M values of σi represents the instantaneous state of the system; and the state progresses in time or the dynamic evolution of Hopfield’s network can be specified according to the following algorithm:

Here, each neuron evaluates randomly and asynchronously whether it is above or below the threshold and readjusts accordingly; and the times of interrogation of each neuron are independent of the times at which other neurons are interrogated. These considerations distinguish Hopfield’s net from that of McCulloch and Pitts.

Hopfield’s model has stable limit points. Considering the special case of symmetric connection weights (that is, Wij = Wji), an energy functional E can be defined by a Hamiltonian (HN) as:

and ΔE due to Δσj is given by:

Therefore, E is a monotonically decreasing function, with the result the state-changes continue until a least local E is reached. This process is isomorphic with the Ising spin model [35]. When Wij is symmetric, but has a random character (analogous to spin-glass systems where atomic spins on a row of atoms in a crystal with each atom having a spin of one half interact with the spins on the next row so that the probability of obtaining a particular configuration in the mth row is ascertained), there are known to be many (locally) stable states present as well.


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