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The parameter Further, the extent of the cluster of cells participating in the neuronal transmission can be specified by the average range of interaction equal to <xij> and the size of the wave packet emulating the neuronal transmission can be decided as follows: Suppose (Δxij)in is the spatial spread of the incident wave at the site i. After the passage through the cell (local interaction site), the corresponding outgoing wave has the spread given by: where τij is the ith to jth site transit time. 7.9 Models of Peretto and Little versus Neuronal WaveBy way of analogy with statistical mechanics, Little [33] portrayed the existence of persistent states in a neural network under certain plausible assumptions. Existence of such states of persistent order has been shown directly analogous to the existence of long-range order in an Ising spin system, inasmuch as the relevant transition to the state of persistent order in the neurons mimics the transition to the ordered phase of the spin system. In the relevant analysis, Little recognizes the persistent states of the neural system being the property of the whole assembly rather than a localized entity. That is, the existence of a correlation or coherence between the neurons throughout the entire interconnected assembly of a large number of cells (such as brain cells) is implicitly assumed. Further, Little has observed that considering the enormous number of possible states in a large neural network (such as the brain) of the order of 2M (where M is the number of neurons of the order 1010) the number of states which determine the long-term behavior is, however, very much smaller in number. The third justifiable assumption of Little refers to the transformation from the uncorrelated to the correlated state in a portion of, or in the whole, neuronal assembly. Such transformation can occur by the variation of the mean biochemical concentrations in these regions, and these transformations are analogous to the phase transition in spin systems. On the basis of the above assumptions, Little has derived a (2M × 2M) matrix whose elements give the probability of a particular state |S1, S2, ..., SM> yielding after one cycle the new state It is well known that the correlation in the Ising model is a measure of the interaction(s) of the atomic spins. That is, the question of interaction refers to the correlation between a configuration in row q, say, and row r, for a given distance between q and r; and, when the correlation does exist, a long-range order is attributed to the lattice structure. For a spin system at a critical temperature (Curie temperature), the long-range order sets in and the system becomes ferromagnetic and exists at all temperatures below that. Analogously considering the neuronal assembly, existence of a correlation between two states which are separated by a long period of time is directly analogous to the occurrence of long-range order in the corresponding spin system. Referring to the lattice gas model of the neuronal assembly, the interaction refers to the incidence of neuronal wave(s) at the synaptic junction from the interconnected cells. The corresponding average range of interaction between say, the ith and jth cell, is given by <xij>. The passage of a wave-packet in the interaction zone with minimum spread can be considered as the neuronal transmission with a state-transition having taken place. This situation corresponds to the spread of an incident wave packet, namely, (Δxij)in equal to The presence of external stimuli (bias) from a kth (external) source at the synapse would alter the postsynaptic potential of the ith neuron. Little [33] observes that the external signals would cause a rapid series of nerve pulses at the synaptic junction; and the effect of such a barrage of signals would be to generate a constant average potential which can transform the effective threshold to a new value on a time-average basis. He has also demonstrated that this threshold shift could drive the network or parts of it across the phase boundary from the ordered to the disordered state or vice versa. That is, the external stimuli could play the role of initiating the onset of a persistent state representing the long-term memory. In terms of wave mechanics considerations, the effect of external stimuli corresponds to altering the neuronal transmission rate; and, in the presence of external bias, The above situation which concurs with Littles heuristic approach is justifiable since the active state proliferation or neural transmission is decided largely by the interneuronal connection and the strength of the synaptic junctions quantified by the intrinsic state-transition rate γij; and the external bias perturbs this value via an implicit change in the local threshold potential(s). The eigenstates which represent the neuronal information (or memory storage at the sites) warrant an extended state of the sites which is assured in the present analysis due to the translational invariancy of the neuronal assembly presumed earlier.
Copyright © CRC Press LLC
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