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It can be observed that this smearing action is seen implicitly in terms of wave function attributes specified as the spreads (Δx, Δp). Thus Λ = βGβ = βG/kBT is controlled by the smearing action. That is, in the total absence of fluctuations in the postsynaptic potentials, kBT → 0; and Λ → ∞ depicts McCulloch-Pitts regime. In the real neurons, the inevitable fluctuations in the summed up potential would always lead to a finite smearing with corresponding finite spreads in the wave function parameters leading to nonspontaneous state-transitions with a finite value Λ. 7.4 Wave Mechanics Representation of Neural ActivityAs discussed earlier, the biological neural assembly refers to a complex linkage of an extremely large number of cells having synaptic junctions across which the neuronal transmission takes place. Such a transmission is governed by a complex biochemical (metabolic) activity and manifests as a train of spiky impulses (action potentials). That is, the operation of neural networks, in essence, refers to the all-or-none response of the basic units (neurons) to incoming stimuli. A neuron is activated by the flow of chemicals across synaptic junctions from the axons leading from other neurons. These electrical effects which reach a neuron may be excitatory or inhibitory postsynaptic potentials. If the potential gathered from all the synaptic connections exceeds a threshold value, the neuron fires and an action potential propagates down its one output axon which communicates with other neurons via tributary synaptic connections. The neuronal topology, as a model, includes therefore a huge collection of almost identical (cell) elements; and each such element is characterized by an internal state (of activity) with transitional dynamics and connected to each other via synaptic interactions. In general, the interconnected set of neurons (Figure 7.1) represents a disordered system with the following attributes: (1) In the assembly of M neurons, the ith neuron is characterized by the ordered pair (Si, Wij) representing the short-term (memory) state and the long-term (memory) state respectively; (2) The short-term (memory) state Si refers to the state of the neurons local (biochemical) activity; (3) The long-term (memory) state Wij is a multidimensional vector modeling the synaptic weights associated between the ith and the other interconnected neurons; (4) With a feedback that exists between the output to the input, the neural assembly represents a recurrent network wherein the input is updated or modified dynamically by the feedback from the output. That is, in response to an applied input vector, each neuron delivers an output; and the ith neurons output is taken commonly as a sigmoid function, F(Si), of the short-term state. The state equation for a neuron can be written as: where θi is a constant external bias; and (5) the neuronal inputs represented by a set of vectors The activity of a biological neuron as summarized above in an organization of a large neuronal assembly represents a cooperative process in which each neuron interacts with many of its neighbors; and, as dictated by a threshold, the input-output response of a neuron was represented as a simple dichotomous state system active and inactive by McCulloch and Pitts. This formal or logical neuron model has been developed into a more comprehensive model since then with the state equation given by Equation (7.27); and the corresponding Hamiltonian associated with the neuronal interaction is written as:
In many respects, the aforesaid mathematical depiction of the neuron is analogous to the magnetic spin model in which a large ensemble of magnetic dipoles/domains interact with each other in varying weights as detailed in Chapter 5. In addition to the statistical mechanics characterization of neuronal activity, the input-output relation depicting the collective response of the neuronal aggregates has also been treated as due to a flow of flux spreading across the interacting units as described earlier in this chapter. In view of these considerations which display the analogies between statistical mechanics and neuronal dynamics indicating the feasibility of modeling the neuronal activity by a flow system, an alternative approach based on wave mechanical perspectives in conjunction with the associated wave function(s) depicting the neural transmission can be considered as discussed in the following sections. 7.5 Characteristics of Neuronal Wave FunctionThe neural activity as modeled here refers to the motion of the neuronal wave or the collective movement of state-transitions through a set of interconnected sites (neuronal cells) in the domain Ω. These cells are at random dichotomous potentials as a result of short-term biochemical ionic activity across the cellular membrane. Each time when the neuronal assembly is stimulated by an input, it corresponds to a set of inputs causing a collective movement of neurons through a domain which can be designated as the long-term (memory) space (Ω). This weight-space, in which both the moving state vector {Wij} and the applied input vector exist (Figure 7.2), constitutes a continuum wherein the evolution of neuronal distribution and a dynamic activity persist.
The proliferation of neuronal wave through this system can be described by a time-dependent wave function Ψ(x, t) namely: where E(x, t) refers to the random site potential with a given statistics,
Copyright © CRC Press LLC
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