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Using the terminology of a quantum mechanical spin system, the state of the brain (as determined by the set of neurons that have fired most recently and those that have not) is the configuration represented by Ψ at the (discrete) time t.

Si = +SU( = +1), if the ith neuron fires at time t or Si = -SL ( = -1) and if it is silent corresponding to the up and down states of a spin system.* Let φij be the postsynaptic potential of the ith neuron due to the firing of the jth neuron. Thus the total potential of the ith neuron is given by ΣMj=1 Φij (Sj + 1)/2. If the total potential exceeds a threshold value φpBi (the barrier potential, possibly independent of i), the neuron will probably fire; and the probability of firing is assumed by Little, in analogy with the spin system, to be:


*For a particular configuration of spins, say {S1, S2, …}, Si = +SU (or +1) refers to the spin being up; and, when Si = -SL (or -1), the spin is labeled as down.

for time t’ = (t + τ).

The temperature factor β = 1/kBT in the spin system (with kB denoting the [pseudo] Boltzmann constant) is related to the uncertainty in the firing of the neuron. The probability of not firing is 1- ρi(+SU). Thus, the probability of obtaining at time (t + τ) the state Ψ’ = |S1’S2’ … SM’>, given the state Ψ’ = |S1’S2’ … SM’>, …, SM> at one unit of time τ preceding it, is given by:

where Φ(Sj) = ΣMj=1ij (Sj + 1)/2] - φpBi.

It may be noted that this expression is very similar to that occurring in the study of propagation of order in crystals with rows of atomic spins. Further, the interaction is between spins with dashed indices and undashed indices, that is, for spins in adjacent rows and not in the same row (or different time steps in the neural network). Long-range order exists in the crystal whenever there is a correlation between distant rows. Ferromagnetism sets in when long range order gets established below the Curie temperature of the spin system.* There are 2M possible spin states as specified by Equation (5.3); and likewise 2M × 2M matrix elements as specified by Equation (5.5) which constitute a 2M × 2M transfer matrix TM. This long range order is associated with the degeneracy of the maximum eigenvalue of TM. In the neural problem, the firing pattern of the neuron at time t corresponds to the up-down state of spin in a row and the time steps are the different rows of the crystal problem. Since the neuronal φij is not equal to φji, the matrix TM from Equation (5.5) is not symmetric as in the spin problem and thus cannot always be diagonalized. This problem can, however, be handled in terms of principal vectors rather than eigenvector expansions in the spin system without any loss of generality.


*A crystal undergoes a phase transition from the paramagnetic state to the ferromagnetic state at a sharply defined temperature (Curie point). At this transition temperature, the properties of the matrix change so that the ferromagnetic state information contained in the first row of the crystal propagates throughout the crystal. In a neural network, this represents analogously the capability of the network to sustain a persistent firing pattern.

Let Ψ(αi) represent the 2M possible states Equation (53). Then the probability of obtaining state Ψ(α’) having started with Ψ(α) (m time intervals [τ] earlier) can be written in terms of the transfer matrix of Equation (5.5) as:

As is familiar in quantum mechanics, Ψ(α) can be expressed in terms of the 2M (orthonormal) eigenvectors (with eigenvalues λr) of the operator TM. Each has 2M components, one for each configuration α:

Hence

Little’s approach is concerned with the probability Γ(α1) of finding a particular state α1 (after in time-steps) starting with an arbitrary initial state. Analogous to the method of using cyclic boundary conditions in the spin problem (in order to simplify derivations, yet with no loss of generality), it is assumed that the neural system returns to the initial conditions after Mo (>>m) steps. Hence, it follows that:

(which is independent of m). Now, with the condition that the maximum eigenvalue λmax is nondegenerate, Equation (5.9) reduces to:

Hence, it follows that the probability of obtaining the state α2 after time steps, given α1 after m steps, is given by:


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