Multiple-valued energy function in neural networks with asymmetric connections.

Abstract

We apply the graphic transformation method [K. Mogi, J. Theor. Biol. 162, 337 (1993)] to obtain the steady-state distribution of asymmetric Boltzmann machines as an extension of the symmetric equilibrium case. We give the magnitude of deviation from the equilibrium explicitly as a function of asymmetry in the connections between the neurons. We show that the steady state of asymmetric Boltzmann machines is characterized by multiple energy values, rather than by a single energy value as in the equilibrium state of symmetric Boltzmann machines. The equilibrium scalar energy function is generalized to a multiple-valued energy function in the case of asymmetric Boltzmann machines.