Differentiability implies continuity in neuronal dynamics

Recent work has identified nonlinear deterministic structure in neuronal dy namics using periodic orbit theory. Troublesome in this work were the signi ficant periods of time where no periodic orbits were extracted - ''dynamica lly dark" regions. Tests for periodic orbit structure typically require tha t the underlying dynamics are differentiable. Since continuity of a mathema tical function is a necessary but insufficient condition for differentiabil ity, regions of observed differentiability should be fully contained within regions of continuity. We here verify that this fundamental mathematical p rinciple is reflected in observations from mammalian neuronal activity. Fir st, we introduce a null Jacobian transformation to verify the observation o f differentiable dynamics when periodic orbits are extracted. Second, we sh ow that a less restrictive test for deterministic structure requiring only continuity demonstrates widespread nonlinear deterministic structure only p artially appreciated with previous approaches. (C) 2001 Published by Elsevi er Science B.V.