Location of Repository

Spatiotemporal symmetries in the disynaptic canal-neck projection

By Martin Golubitsky, Liejune Shiau and Ian Stewart


The vestibular system in almost all vertebrates, and in particular in humans, controls\ud balance by employing a set of six semicircular canals, three in each inner ear, to detect angular\ud accelerations of the head in three mutually orthogonal coordinate planes. Signals from the canals are\ud transmitted to eight (groups of) neck motoneurons, which activate the eight corresponding muscle\ud groups. These signals may be either excitatory or inhibitory, depending on the direction of head\ud acceleration. McCollum and Boyle have observed that in the cat the relevant network of neurons\ud possesses octahedral symmetry, a structure that they deduce from the known innervation patterns\ud (connections) from canals to muscles. We rederive the octahedral symmetry from mathematical\ud features of the probable network architecture, and model the movement of the head in response to\ud the activation patterns of the muscles concerned. We assume that connections between neck muscles\ud can be modeled by a “coupled cell network,” a system of coupled ODEs whose variables correspond\ud to the eight muscles, and that this network also has octahedral symmetry. The network and its\ud symmetries imply that these ODEs must be equivariant under a suitable action of the octahedral\ud group. It is observed that muscle motoneurons form natural “push-pull pairs” in which, for given\ud movements of the head, one neuron produces an excitatory signal, whereas the other produces an\ud inhibitory signal. By incorporating this feature into the mathematics in a natural way, we are led\ud to a model in which the octahedral group acts by signed permutations on muscle motoneurons.\ud We show that with the appropriate group actions, there are six possible spatiotemporal patterns of\ud time-periodic states that can arise by Hopf bifurcation from an equilibrium representing an immobile\ud head. Here we use results of Ashwin and Podvigina. Counting conjugate states, whose physiological\ud interpretations can have significantly different features, there are 15 patterns of periodic oscillation,\ud not counting left-right reflections or time-reversals as being different. We interpret these patterns\ud as motions of the head, and note that all six types of pattern appear to correspond to natural head\ud motions

Topics: QA
Publisher: Society for Industrial and Applied Mathematics
Year: 2007
OAI identifier: oai:wrap.warwick.ac.uk:184

Suggested articles



  1. (1998). A modular network for legged locomotion, doi
  2. (1974). Connections between semicircular canals and neck motoneurons in the cat,
  3. (1993). Coupled nonlinear oscillators and the symmetries of animal gaits, doi
  4. (1996). Four convergent patterns of input from the six semicircular canals to motoneurons of different neck muscles in the upper cervical cord, doi
  5. (2001). Geometric visual hallucinations, Euclidean symmetry, and the functional architecture of striate cortex, doi
  6. (1993). Hexapodal gaits and coupled nonlinear oscillator models, doi
  7. (1985). Hopf bifurcation in the presence of symmetry, doi
  8. (2003). Hopf bifurcation with cubic symmetry and instability of ABC flow, doi
  9. (1994). Input patterns and pathways from six semicircular canals to motoneurons of neck muscles I: The multifidus muscle group, doi
  10. (1997). Input patterns and pathways from six semicircular canals to motoneurons of neck muscles II: The longissimus and semispinalis muscle groups, doi
  11. (2001). Models of central pattern generators for quadruped locomotion: I. Primary gaits, doi
  12. (1962). Representation Theory of Finite Groups and Associative Algebras, Wiley-Interscience, doi
  13. (2004). Rotations in a vertebrate setting: Evaluation of the symmetry group of the disynaptic canal-neck projection, doi
  14. (2007). Spatial symmetries in vestibular projections to the uvula-nodulus, doi
  15. (1999). Symmetry in locomotor central pattern generators and animal gaits, doi
  16. (2002). The Symmetry Perspective, doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.