Symmetry and bifurcation in vestibular system

Abstract

AbstractThe vestibular system in almost all vertebrates, humans included, controls balance by employing a set of six semicircular canals, three in each inner ear, to detect angular accelerations of the head. Signals from the canals are transmitted to neck motoneurons and activate eight corresponding muscle groups. These signals may be either excitatory or inhibitory, depending on the direction of acceleration. McCollum and Boyle have observed that in the cat the network of neurons concerned possesses octahedral symmetry, a structure deduced from the known innervation patterns (connections) from canals to muscles.We re‐derive the octahedral symmetry from mathematical features of the probable network architecture, and model the movement of the head in response to the activation patterns of the muscles concerned. We assume that connections among neck muscles can be modeled by a ‘coupled cell network’, a system of coupled ODEs whose variables correspond to the eight muscles, and that network also has octahedral symmetry. The network and its symmetries imply that these ODEs must be equivariant under a suitable action of the octahedral group.Using results of Ashwin and Podvigina, we show that with the appropriate group actions, there are six possible spatiotemporal patterns of time‐periodic states that can arise by Hopf bifurcation from an equilibrium corresponding to natural head motions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)