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

PDF
WRAP_Stewart_GetPDFServlet.pdf  Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (233Kb) 
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  
Dates: 


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
Request changes or add full text files to a record
Actions (login required)
View Item 
Downloads
Downloads per month over past year