Stochastic Processes and Neuronal Modelling: Quantum Harmonic Oscillator Dynamics in Neural Structures

This paper studies neural structures with weights that follow the model of the quantum harmonic oscillator (Q.H.O.). The proposed neural networks have stochastic weights which are calculated from the solution of Schrödinger’s equation under the assumption of a parabolic (harmonic) potential. These weights correspond to diffusing particles, which interact to each other as the theory of Brownian motion (Wiener process) predicts. The learning of the stochastic weights (convergence of the diffusing particles to an equilibrium) is analyzed. In the case of associative memories the proposed neural model results in an exponential increase of patterns storage capacity (number of attractors). Finally, it is shown that conventional neural networks and learning algorithms based on error gradient can be conceived as a subset of the proposed quantum neural structures. Thus, the complementarity between classical and quantum physics is also validated in the field of neural computation.