Lyapunov theory demonstrating a fundamental limit on the speed of systems consolidation

The nervous system reorganizes memories from an early site to a late site, a commonly observed feature of learning and memory systems known as systems consolidation. Previous work has suggested learning rules by which consolidation may occur. Here, we provide conditions under which such rules are guaranteed to lead to stable convergence of learning and consolidation. We use the theory of Lyapunov functions, which enforces stability by requiring learning rules to decrease an energy-like (Lyapunov) function. We present the theory in the context of a simple circuit architecture motivated by classic models of cerebellum-mediated learning and consolidation. Stability is only guaranteed if the learning rate in the late stage is not faster than the learning rate in the early stage. Further, the slower the learning rate at the late stage, the larger the perturbation the system can tolerate with a guarantee of stability. We provide intuition for this result by mapping a simple example consolidation model to a damped driven oscillator system and showing that the ratio of early- to late-stage learning rates in the consolidation model can be directly identified with the oscillator's damping ratio. We then apply the theory to modeling the tuning by the cerebellum of a well-characterized analog short-term memory system, the oculomotor neural integrator, and find similar stability conditions. This work suggests the power of the Lyapunov approach to provide constraints on nervous system function.