Lyapunov functions for rotor neural networks

The state of a rotor neuron is constrained to live on the surface of a sphere in ℜⁿ. A rotor neural network is used to minimize an arbitrary cost function with respect to these "spherical" states. One practical example of such a situation is optimal charge distribution on a sphere in electromagnetism. In this paper, I show that if the cost function is quadratic in the neuron states, the synchronous, iterated-map algorithm used to find the fixed-points of the network has a Lyapunov function. I also propose a continuous-time dynamical system for finding the fixed-points, that is valid for any cost function. Moreover, I show that this continuous-time dynamics has a Lyapunov function. Finally, I show that a similar continuous-time algorithm and a similar Lyapunov function can be used for solving fixed-point equations more general than those of rotor neural networks.