Quantum activation function (Schrodinger wave equation)

Quantum activation function (Schrodinger wave equation)

Fig. 9.2. A stochastic filter using RQNN with linear modulation embedded in noise (|(t)), i.e. y(t) = ya(t) + |(t). The signal excites N neurons spatially located along the x-axis after being preprocessed by synapses. In the model the synapses are represented by time-varying synaptic weights K(x, t). The unified dynamics of the one-dimensional neural lattice consisting of N neurons is described by the Schrodinger wave equation given as

where i,h, '^(x,t) and V carry their usual meaning in the context of Schrodinger wave equation. The tfi(x,t) function represents the solution of (9.1). The potential field of the Schrodinger wave equation given in (9.1) consists of two terms:

U(x,t) = -K(x,t)y(t) , G(\ ^ \2) = K(x,t) J xf (x, t)dx , where f (x,t)=\ ^(x,t) \2

Since the potential field term in (9.1) is a function of ^(x,t), the Schrodinger wave equation that describes the stochastic filter is nonlinear. In contrast to artificial neural networks studied in the literature, in our model the neural lattice consisting of N neurons is described by the state tfi(x,t) which is the solution of (9.1). Simultaneously, the model is recurrent as the dynamics consists of a feedback term G(.). The information about the signal is thus transferred to the potential field of the Schroodinger wave equation and the dynamics is evolved accordingly. Here we have used a linear neural circuit to set up the potential field where K(x,t)s are the associated linear synaptic weights. The signal is then estimated using a maximum-likelihood estimator y(t) = J xf (x,t)dx.

y as

When the estimate y(t) is the actual signal, then the signal that generates the potential field for the Schrodinger wave equation, V(t), is simply the noise that is embedded in the signal. If the statistical mean of the noise is zero, then this error-correcting signal z>(t) has little effect on the movement of the wave packet. Precisely, it is the actual signal content in the input y(t) that moves the wave packet along the desired direction that, in effect, achieves the goal of the stochastic-filtering. It is expected that the synaptic weights evolve in such a manner so as to drive the ^ function to carry the exact information of the pdf of the observed stochastic variable y(t).

Learning and Estimation

The nonlinear Schrodinger wave equation given by (9.1) exhibits a soliton property, i. e. the square of | ^(x, t) | is a wave packet that moves like a particle. The importance of this property is as follows. Let the stochastic variable y(t) be described by a Gaussian probability density function f (x,t) with mean k and standard deviation a. Let the initial state of (9.1) correspond to zero mean Gaussian probability density function f '(x, t) with standard deviation a'. As the dynamics evolves with online update of the synaptic weights K(x,t), the probability density function f '(x,t) should ideally move toward the pdf, f (x) of the signal y(t). Thus the filtering problem in this new framework can be seen as the ability of the nonlinear Schrodinger wave equation to produce a wave packet solution that glides along with the time-varying pdf corresponding to the signal y(t).

The synaptic weights K(x,t), which is a N x 1-dimensional vector, is updated using the Hebbian learning algorithm

where v(t) = y(t) — y(t). y(t) is the filtered estimate of the actual signal ya(t). We compute the filtered estimate according to (9.5). We will show later that the wave packet moves in the required direction in our new model.

9.2.2 Integration of the Schrodinger Wave Equation

The nonlinear Schrodinger wave equation is - from the mathematical point of view - a partial differential equation with two variables: x and t. In an abstract sense, receptive fields of N neurons span the entire distance along the x-axis. (9.1) is converted into the finite difference form by dividing the x-axis into N mesh points so that x and t are represented as follows:

where j varies from -N/2 to +N/2. The finite-difference form of (9.1) is expressed as

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