Learning Process in a Neural Network Model

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The firing and the learning processes are the dynamics in a neural system on fast and slow time scales. In this paper, we study a neural network model that describes the two different dynamics in a unified manner. The model makes it possible to predict the firing probability or the firing correlation in a biological neural system much exactly via a path integral formulation. Also, an ideal learning process in the model system is defined to minimize the path-integral free energy. The learning principle explains how the typical form of plasticity in a biological synapse is determined. Further, the model makes it possible to understand or predict the emergent structures in a biological neural system through the basic principles in (statistical) physics. We investigate how the thermodynamic potentials in the free energy, such as the internal energy, the Gibbs potential, and the entropy, exert effects on the general learning behavior of the model system. In addition, we demonstrate that the derivative of an effective free energy leads to a learning equation for feature map formation.