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As stated earlier, Thompson and Gibson [37] indicate that the spin-system definition of long-range order (fixing the spin at one lattice site causes the spins at sites far away from it to show a preference for one orientation) is not applicable to the neural problem. Contrary to Little [33], Thompson and Gibson state that the existence of order (a correlation between the probability distribution Df the network at some initial time, and the probability distribution after in (m ≥ 1) time-steps does not mean that the network has a persistent state; and rather, order should only be considered over a moderate number of time-steps. However, inasmuch as order does imply a correlation between states of the network separated by time-steps, it seems reasonable to assume that order is associated with a memory mechanism.
Clearly, Littles model which is derived assuming a close similarity between it and the problem of an Ising system does not provide a comprehensive model of neural-firing behavior. However, it is advantageous in that the model neuron is both mathematically simple and able to produce a remarkably wide range of output patterns which are similar to the discharge patterns of many real neurons.
Further, considering Hopfields model, Hopfield [31] states that for Wij being symmetric and having a random character (analogous to the spin glass), state changes will continue until a local minimum is reached. That is, the equations of motions for a network with symmetric connections (Wij = Wji) always lead to a convergence to stable states in which the outputs of all neurons remain constant. Again, the symmetry of the network is essential to the mathematical description of the network. Hopfield notes that real neurons need not make synapses both of i → j and j → i; and, without this symmetry, the probability of errors would increase in the input-output neural network simulation, and there is a possibility that the minimum reached via algorithmic search would only be metastable and could be replaced in time by another minimum. The question of symmetry and the symmetry condition can, however, be omitted without destroying the associative memory. Such simplification is justifiable via the principle of universality in physics which permits study of the collective aspects of a systems behavior by introducing separate (and more than one) simplifications without essentially altering the conclusions being reached.
The concept of memory or storage and retrival of information pertinent to the Little and Hopfield models differ in the manner in which the state of the system is updated. In Littles model all neurons (spins) are updated synchronously as per the linear condition of output values, namely, oi(t) = Σj Wijxi(t), where the neurons are updated sequentially one at a time (either in a fixed order or randomly) in the Hopfield model. (Though sequential updating can be more easily simulated by conventional digital logic, real neurons do not operate sequentially.)
5.9 Ising Spin System versus Interacting Neurons
In view of the various models as discussed above, the considerations in the analogous representation of interacting neurons vis-a-vis the Ising magnetic spins and the contradictions or inconsistencies observed in such an analogy are summarized in Tables 5.1 and 5.2.
5.10 Liquid-Crystal Model
Basically, the analogy between Ising spins system and the neural complex stems from the fact that the organization of neurons is a collective enterprise in which the neuronal activity of interactive cells represents a cooperative process similar to that of spin interactions in a magnetic system. As summarized in Table 5.1, the strengths of synaptic connections between the cells representing the extent of interactive dynamics in the cellular automata are considered analogous to the strengths of exchange interactions in magnetic spin systems. Further, the synaptic activity, manifesting as the competition between the excitatory and inhibitory processes is regarded as equitable to the competition between the ferromagnetic and antiferromagnetic exchange interactions in spin-glass systems. Also, the threshold condition stipulated for the neuronal network is considered as the analog of the condition of metastability against single spin flips in the Ising spin-glass model.
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