title: Exponentially Long Orbits in Hopfield Neural Networks.
abstract: We show that Hopfield neural networks with synchronous dynamics and asymmetric weights admit stable orbits that form sequences of maximal length. For [Formula: see text] units, these sequences have length [Formula: see text]; that is, they cover the full state space. We present a mathematical proof that maximal-length orbits exist for all [Formula: see text], and we provide a method to construct both the sequence and the weight matrix that allow its production. The orbit is relatively robust to dynamical noise, and perturbations of the optimal weights reveal other periodic orbits that are not maximal but typically still very long. We discuss how the resulting dynamics on slow time-scales can be used to generate desired output sequences.
authors.affiliation: School of Computer and Communication Sciences and Brain Mind Institute, School of Life Sciences, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland samuel.muscinelli@epfl.ch.
authors.forename: Samuel P
authors.initials: SP
authors.lastname: Muscinelli
authors.affiliation: School of Computer and Communication Sciences and Brain Mind Institute, School of Life Sciences, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland wulfram.gerstner@epfl.ch.
authors.forename: Wulfram
authors.initials: W
authors.lastname: Gerstner
authors.affiliation: School of Computer and Communication Sciences and Brain Mind Institute, School of Life Sciences, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland johanni.brea@epfl.ch.
authors.forename: Johanni
authors.initials: J
authors.lastname: Brea
mesh_terms: D000818:Animals; D001921:Brain / Q000502:physiology; D006801:Humans; D007858:Learning / Q000502:physiology; D008959:Models, Neurological; D016571:Neural Networks, Computer; D009434:Neural Pathways / Q000502:physiology; D009474:Neurons / Q000502:physiology