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with {Si} denoting the state of the whole system at a given instant of time; and both Si and Sj are the elements of the same set {Si}.

The aforesaid mathematical depiction of simple neural activity thus presents a close analogy between the neural network and magnetic spin model(s). That is, the neurons can be regarded as analogs of the Ising spins (see Appendix A) and the strengths of the synaptic connections are analogous to the strengths of the exchange interactions in spin systems. The concept of interaction physics as applied to a system of atomic magnetic dipoles or spins refers to the phenomenon namely, atoms interact with each other by inducing a magnetic field at the location of other (neighboring) atoms which interact with its spin. The total local magnetic field at the location of an atom i is equal to ΣijWijSj where Wij is the dipole force, and diagonal term j = i (self-energy) is not included in the sum. Further, Newton’s third law, namely, action equates to reaction, ensures the coupling strengths Wij being symmetric. That is, Wij = Wji. If all Wij are positive, the material is ferromagnetic; if there is a regular change of sign between neighboring atoms, it refers to antiferromagnetism. If the signs and absolute values of the Wij are distributed randomly, the material is called spin-glass. The ferromagnetic case corresponds to a neural network that has stored a single pattern. The network which has been loaded with a large number of randomly composed patterns resembles a spin glass.

Synaptic activity in neuronal systems being excitatory or inhibitory, the competition between these two types of interactions can be considered as similar to the competition between the ferromagnetic and antiferromagnetic exchange interactions in spin-glass systems. That is, the dichotomous “all-or-none” variables of neurons correspond to Si = ±1 Ising spins where i labels the neurons, and range between I and N determines the size of the network. Further, the threshold condition stipulated for the neural complex can be regarded as the analog of the condition for metastability against single-spin flips in the Ising model (except that in a neural complex the symmetry relation, namely, Wij = Wji, does not necessarily hold). The evolution of the analogical considerations between interconnected neurons and the magnetic spins is discussed in detail in the following sections.

5.2 Cragg and Temperley Model

In view of the above analogical considerations between neurons and magnetic spins, it appears that the feasibility of applying quantum theory mathematics to neurobiology was implicitly portrayed by Gabor [10] as early as in 1946 even before Weiner’s [9] suggestion on cybernetic aspects of biological neurons. As indicated by Licklider [75], “the analogy … [to] the position-momentum and energy-time problems that led Heisenberg in 1927 to state his uncertainty principle … has led Gabor to suggest that one may find the solution [to the problems of sensory processing] in quantum mechanics.”

In 1954, Cragg and Temperley [32] were perhaps the first to elaborate and examine qualitatively the possible analogy between the organization of neurons and the kind of interaction among atoms which leads to the cooperative processes in physics. That is, the purported analogy stems from the fact that large assemblies of atoms which interact with each other correspond to the collective neural assembly exhibiting cooperative activities through interconnections.

As explained before, in the case of an assembly of atoms, there is an explicit degree of interaction manifesting as the phenomenon of ferromagnetism; and such an interaction between atomic magnets keeps them lined up (polarized) below a specific temperature (known as the Curie point). (Above this temperature, the increase in thermal agitation would, however, throw the atomic magnets out of alignment; or the material would abruptly cease to be ferromagnetic).


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