We suggest that the time evolution of the collective response — is described by the Schrodinger wave equation ih d-x^l = _V2-(x, t) + y (x)-(x, t) , (9.23)
where 2nh is Planck's constant, -(x,t) is the wave function (probability amplitude) associated with the quantum object at space-time point (x,t), and m the mass of the quantum object. Further symbols such as i and V carry their usual meaning in the context of the Schrodinger wave equation. Another way to look at our proposed quantum brain is as follows. A neuronal lattice sets up a spatial potential field V(x). A quantum process described by a quantum state —, which mediates the collective response of a neuronal lattice, evolves in the spatial potential field V(x) according to (9.23). Thus the classical brain sets up a spatiotemporal potential field, while the quantum brain is excited by this potential field to provide a collective response.
9.4.2 An Eye-Tracking Model using RQNN with Nonlinear Modulation of Potential Field
In this section we present an extension of RQNN, briefly described in the previous section. Here, the potential field of the Schrodinger wave equation is modulated using a nonlinear neural circuit that results in a much pronounced soliton behavior of the wave packet. With this modification we now provide a plausible biological mechanism for eye tracking using the quantum brain model proposed in Sect. 9.4.1. The mechanism of eye movements, tracking a moving target consists of three stages as shown in Fig. 9.8: (i) stochastic-filtering of noisy data that impact the eye sensors; (ii) a predictor that predicts the next spatial position of the moving target; and (iii) a biological motor control system that aligns the eye pupil along the moving targets trajectory. The biological eye sensor fans out the input signal y to a specific neural lattice in the visual cortex. For clarity, Fig. 9.8 shows a one-dimensional array of neurons whose receptive fields are excited by the signal input y reaching each neuron through a synaptic connection described by a nonlinear MAP. The neural lattice responds to the stimulus by setting up a spatial potential field, V(x,t), which is a function of external stimulus y and estimated trajectory y of the moving target:
i=i where (.) is a Gaussian kernel function, n represents the number of such Gaussian functions describing the nonlinear MAP that represents the synaptic connections, v(t) represents the difference between y and y and W represents the synaptic weights as shown in Fig. 9.8. The Gaussian kernel function is taken as
where gi is the center of the i-th Gaussian function, fa. This center is chosen from input space described by the input signal, v(t), through uniform random sampling.
Our quantum-brain model proposes that a quantum process mediates the collective response of this neuronal lattice that sets up a spatial potential field V(x, t). This happens when the quantum state associated with this quantum
process evolves in this potential field. The spatiotemporal evolution follows as per (9.23). We hypothesize that this collective response is described by a wave packet, f (x,t) =| '^(x,t) |2, where the term tfi(x,t) represents a quantum state. In a generic sense, we assume that a classical stimulus in a brain triggers a wave packet in the counterpart "quantum brain". This subjective response, f (x, t), is quantified using the following estimate equation:
The estimate equation is motivated by the fact that the wave packet, f (x,t) =| tfi(x,t) |2 is interpreted as the probability density function. Although computation of (9.26) using the nonlinear Schrodinger wave equation is straightforward, we hypothesize that this computation can be done through an interaction between a quantum and a classical brain, using a suitable quantum measurement operator. At this point we will not speculate about the nature of such a quantum measurement operator that will estimate the ^ function necessary to compute (9.26). Based on this estimate, y, the predictor estimates the next spatial position of the moving target. To simplify our analysis, the predictor is made silent. Thus its output is the same as that of y. The biological motor control is commanded to fixate the eye pupil to align with the target position, which is predicted to be at y. Obviously, we have assumed that biological motor control is ideal.
After the above-mentioned simplification, the closed form dynamics of the model described by Fig. 9.8 becomes
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