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indicating no correlation. However, if the maximum eigenvalue λmax is degenerate, the factorization of Γ(α1, α2) is not possible; and there will be correlation in time in the neuronal-firing behavior. This type of degeneracy occurs in the spin system for some regions of a β - (φij - φpBi) plane and refers to the transition from the paramagnetic to the ferromagnetic phase. Relevant to the neural complex, Little suggests that such time ordering is related to short-term memory. Since time correlations of the order less than or equal to a second are of interest in the neural dynamics, a practical degeneracy will result if the two largest λs are degenerate to ~1 %.
In the above treatment, the parameter β assumed is arbitrary. However, this β could represent all the spread in the uncertainty of the firing of the neuron. This has been demonstrated by Shaw and Vasudevan [76] who suggested that the ad hoc parameter β in reality relates to the fluctuations governing the total (summed up) potential gathered by the neuron in a time-step (which eventually decides the state of the neuron at the end of the time-step as well). The relevant analysis was based on the probabilistic aspects of synaptic transmission, and the temperature-like factor or the pseudo-temperature universe β (= 1/kBT in the Ising model) was termed as a smearing parameter.
Explicitly, this smearing parameter (β) has been shown equal to , where Δ is a factor decided by the gaussian statistics of the action potentials and the poissonian process governing the occurrence rate of the quanta of chemical transmitter (ACh) reaching the postsynaptic membrane (and hence causing the postsynaptic potential). The relevant statistics indicated refer to the variations in size and the probability of release of these quanta manifesting (and experimentally observed) as fluctuations in the postsynaptic potentials.
In a continued study on the statistical mechanics aspects of neural activity, Little and Shaw [77] developed a model of a large neural complex (such as the brain) to depict the nature of short- and long-term memory. They presumed that memory results from a form of synaptic strength modification which is dependent on the correlation of pre- and postsynaptic neuronal firing; and deduced that a reliable, well-defined behavior of the assembly would prevail despite of noisy (and hence random) characteristics of the membrane potentials due to the fluctuations (in the number and size) of neurochemical transmitter molecules (ACh quanta) released at the synapses. The underlying basis for their inference is that the neuronal collection represents an extensive assembly comprised of a (statistically) large number of cells with complex synaptic interconnections, permitting a stochastically viable proliferation of state changes through all the possible neuronal configurations (or patterns of neural conduction).
In the relevant study, the pertinent assumption on modifiable synapses refers to the Hebbian learning process. In neurophysiological terms it is explicitly postulated as: When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that As efficiency as one of the cells firing B is enhanced [19].
In further studies concerning the analogy of neural activity versus Ising spin model, Little and Shaw [78] developed an analytical model to elucidate the memory storage capacity of a neural network. They showed thereof that the memory capacity is decided by the (large) number of synapses rather than by the (much smaller) number of neurons themselves; and by virtue of this large memory capacity, there is a storage of information generated via patterns of state-transition proliferation across the neural assembly which evolves with time. That is, considering the long-term memory model, the synaptic strengths cannot be assumed as time-invariant. With the result, a modified Hebbs hypothesis, namely, that the synaptic changes do occur as a result of correlated pre- and postneuronal firing behavior of the linear combinations of the (spatial) firing pattern, was suggested in [78]. Thus, the relevant study portrayed the existence of possible spatial correlation (that is, firing correlation of neighboring neurons, as evinced in experimental studies) in a neural assembly. Also, such correlations resulting from the linear combination of firing patterns corresponds to M2 transitions, where M is the number of neurons; and with every neuron connected to every other neuron, there are M2 number of synapses wherein the transitions would take place.
The aforesaid results and conclusions of Little and Shaw were again based mainly on the Ising spin analogy with the neural system. However, the extent of their study on the linear combination firing patterns from the statistical mechanics point of view is a more rigorous, statistically involved task warranting an analogy with the three-dimensional Ising problem unfortunately, this remains unsolved to date. Nevertheless, the results of Little and Shaw based on the elementary Ising spin model, indicate the possibility of spatial firing correlations of neighboring neurons which have been confirmed henceforth via experiments using two or more closely spaced microelectrodes.
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