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Spatiotemporal symmetries in the disynaptic canalneck projection
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Golubitsky, Martin, Shiau, Liejune and Stewart, Ian, 1945. (2007) Spatiotemporal symmetries in the disynaptic canalneck projection. SIAM Journal on Applied Mathematics, Vol.67 (No.5). pp. 13961417. ISSN 00361399

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Official URL: http://dx.doi.org/10.1137/060667773
Abstract
The vestibular system in almost all vertebrates, and in particular in humans, controls
balance by employing a set of six semicircular canals, three in each inner ear, to detect angular
accelerations of the head in three mutually orthogonal coordinate planes. Signals from the canals are
transmitted to eight (groups of) neck motoneurons, which activate the eight corresponding muscle
groups. These signals may be either excitatory or inhibitory, depending on the direction of head
acceleration. McCollum and Boyle have observed that in the cat the relevant network of neurons
possesses octahedral symmetry, a structure that they deduce from the known innervation patterns
(connections) from canals to muscles. We rederive 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 between neck muscles
can be modeled by a “coupled cell network,” a system of coupled ODEs whose variables correspond
to the eight muscles, and that this 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. It is observed that muscle motoneurons form natural “pushpull pairs” in which, for given
movements of the head, one neuron produces an excitatory signal, whereas the other produces an
inhibitory signal. By incorporating this feature into the mathematics in a natural way, we are led
to a model in which the octahedral group acts by signed permutations on muscle motoneurons.
We show that with the appropriate group actions, there are six possible spatiotemporal patterns of
timeperiodic states that can arise by Hopf bifurcation from an equilibrium representing an immobile
head. Here we use results of Ashwin and Podvigina. Counting conjugate states, whose physiological
interpretations can have significantly different features, there are 15 patterns of periodic oscillation,
not counting leftright reflections or timereversals as being different. We interpret these patterns
as motions of the head, and note that all six types of pattern appear to correspond to natural head
motions.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Vestibular apparatus, Bifurcation theory 
Journal or Publication Title:  SIAM Journal on Applied Mathematics 
Publisher:  Society for Industrial and Applied Mathematics 
ISSN:  00361399 
Official Date:  20 July 2007 
Volume:  Vol.67 
Number:  No.5 
Page Range:  pp. 13961417 
Identification Number:  10.1137/060667773 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
Funder:  National Science Foundation (U.S.) (NSF), Engineering and Physical Sciences Research Council (EPSRC) 
Grant number:  DMS 0244529 and DMS0604429 (NSF) 
References:  [1] P. Ashwin and O. Podvigina, Hopf bifurcation with cubic symmetry and instability of ABC 
URI:  http://wrap.warwick.ac.uk/id/eprint/184 
Data sourced from Thomson Reuters' Web of Knowledge
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