Differentiability implies continuity in neuronal dynamics |
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| Institution: | 1. REC “Artificial Intelligence Systems and Neurotechnology”, Yuri Gagarin State Technical University of Saratov, 77 Politechnicheskaya str., Saratov 410054, Russia;2. Department of Automation, Control, Mechatronics, Yuri Gagarin State Technical University of Saratov, 77 Politechnicheskaya str., Saratov 410054, Russia;3. Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India;4. Department of Mathematics, Jadavpur University, Kolkata 700032, India;5. Saratov State University, 83 Astrakhanskaya str., Saratov 410012, Russia;1. Laboratory of Electronics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Cameroon;2. Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Cameroon;3. Le2i FRE2005, CNRS, Arts et Métiers, Univ. Bourgogne Franche-Comté, Dijon F-21000, France;1. Department of Mathematics, University of Exeter, Exeter, EX4 4QF, UK;2. Living Systems Institute, University of Exeter, Exeter, EX4 4QD, UK;3. EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, EX4 4QJ, UK;4. Department of Mathematics, Cankiri Karatekin University, Cankiri, 18100, Turkey;5. VU University Amsterdam, Department of Mathematics Faculteit der Exacte Wetenschappen De Boelelaan 1081a 1081 HV Amsterdam, The Netherlands;6. Inria Sophia Antipolis Méditerranée Research Centre, MathNeuro Team, 2004 route des Lucioles-Boîte Postale 93 06902 Sophia Antipolis, Cedex, France |
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| Abstract: | Recent work has identified nonlinear deterministic structure in neuronal dynamics using periodic orbit theory. Troublesome in this work were the significant periods of time where no periodic orbits were extracted — “dynamically dark” regions. Tests for periodic orbit structure typically require that the underlying dynamics are differentiable. Since continuity of a mathematical function is a necessary but insufficient condition for differentiability, regions of observed differentiability should be fully contained within regions of continuity. We here verify that this fundamental mathematical principle is reflected in observations from mammalian neuronal activity. First, we introduce a null Jacobian transformation to verify the observation of differentiable dynamics when periodic orbits are extracted. Second, we show that a less restrictive test for deterministic structure requiring only continuity demonstrates widespread nonlinear deterministic structure only partially appreciated with previous approaches. |
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