Energy approach to rivalry dynamics, soliton stability, and pattern formation in neuronal networks

Hopfield's Lyapunov function is used to view the stability and topology of equilibria in neuronal networks for visual rivalry and pattern formation. For two neural populations with reciprocal inhibition and slow adaptation, the dynamics of neural activity is found to include a pair of limit cycles: one for oscillations between states where one population has high activity and the other has low activity, as in rivalry, and one for oscillations between states where both populations have the same activity. Hopfield's Lyapunov function is used to find the dynamical mechanism for oscillations and the basin of attraction of each limit cycle. For a spatially continuous population with lateral inhibition, stable equilibria are found for local regions of high activity (solitons) and for bound states of two or more solitons. Bound states become stable when moving two solitons together minimizes the Lyapunov function, a result of decreasing activity in regions between peaks of high activity when the firing rate is described by a sigmoid function. Lowering the barrier to soliton formation leads to a pattern-forming instability, and a nonlinear solution to the dynamical equations is found to be given by a soliton lattice, which is completely characterized by the soliton width and the spacing between neighboring solitons. Fluctuations due to noise create lattice vacancies analogous to point defects in crystals, leading to activity which is spatially inhomogeneous.