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5.5 Thompson and Gibson Model
Thompson and Gibson in 1981 [37] advocated in favor of Littles model governing the probabilistic aspects of neuronal-firing behavior with the exception that the concept of long-range order introduced by Little is considered rather inappropriate for the neural network; and they suggested alternatively a more general definition of the order. The relevant synopsis of the studies due to Thompson and Gibson follows.
Considering a spin system, if fixing the spin at one lattice site causes spins at sites far away from it to show a preference for one orientation, it refers to the long-range order of the spin system. To extend this concept to the neural assembly, it is necessary first to consider the Ising model of the two-dimensional ferromagnet in detail. In the Ising spin system, a regular lattice of spins Si = ±1 with an isotropic nearest-neighbor interaction is built by successive addition of rows, each consisting of M spins, where M is finite. The probability distribution of spins in the (m + 1)th row depends only on the distribution in the mth row, depicting a Markov process with a transition matrix TM. In this respect, the neural network and spin structure are formally analogous; and the time-steps for the neural network correspond to the spatial dimension of the spin lattice, as discussed earlier.
In the spin problem, the transition matrix TM is strictly a positive stochastic matrix for all positive values of the temperature T such that the long-range order for any finite spin system with T > 0 is not feasible. However, in the limit as M → ∞, the largest eigenvalue of TM is asymptotically degenerate provided T < TC, (TC being the Curie point). In this case, no longer approaches a matrix with equal components when m becomes arbitrarily large. This infinite two-dimensional spin system undergoes a sharp phase transition at TC. For T > TC, there is no long range order and each spin is equally likely to be up or down, whereas for T < TC, there is a long-range order and the spins are not randomly oriented (see Appendix A).
The nearest-neighbor spin-spin interactions in a ferromagnetic system are symmetric as discussed earlier, and the effect that one spin has on the orientation of any other spin depends only on their spatial separation in the lattice. Hence, the successive rows of the spin system can be added in any direction; however, considering the neural system, the analogous time development steps have only a specific forward direction. That is, the neuronal interaction is inherently anisotropic. The state of the neuron at any time is determined by the suite of all the neurons at the previous time. This interaction for a given neuron can be distinctly different and unique when considered with other neurons. It also depends on the synaptic connectivity of the particular network in question. Generally, the interaction of the jth neuron with the ith neuron is not the same as that of the ith neuron with the jth; and the transition matrix TM is, therefore, nonsymmetric. That is, the synaptic connections are generally not symmetric and are often maximally asymmetrical or unidirectional. (Müller and Reinhardt [1] refers to such networks as cybernetic networks and indicate the feed-forward, layered neural networks as the best-studied class of cybernetic networks. They provide an optimal reaction or answer to an external stimulus as dictated by a supervising element [such as the brain] in the self-control endeavors. Further, as a result of being asymmetric the theories of thermodynamic equilibrium systems have no direct bearing on such cybernetic networks.)
Thompson and Gibson hence declared that the spin system definition of long-range order is rather inapplicable to neural assembly due to the following reasons: (1) Inasmuch as the interaction between different neurons have different forms, any single neuron would not influence the state of any other single neuron (including itself) at a later time; and (2) because the transition matrix is asymmetric (not necessarily diagonalizable) in a neuronal system, the long-range order does not necessarily imply a tendency for the system to be in a particular or persistent state. (On the contrary, in a spin system, the order is strictly a measure of the tendency of the spins aligned in one direction in preference to random orientation.)
As a result of the inapplicability of the spin-system based definition of long-range order to a neural system, Thompson and Gibson proposed an alternative definition of long-range order which is applicable to both the spin system as well as the neuronal system. Their definition refers to the order of the system applied to a moderate time scale and not for the long-range epoch. In this moderate time-frame order, plastic changes in synaptic parameters would be absent; and by considering the neural network as a finite (and not as an arbitrarily large or infinite system), the phase transition process (akin to that of the spin system) from a disordered to an ordered state would take place in a continuous graded fashion rather than as a sharp transition. Thus, the spin system analogy is still applicable to the neural system provided a finite system assumption and moderate time-scale order are attributed explicitly to the neuronal state transition process.
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