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Spatiotemporal symmetries in the disynaptic canal-neck projection

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Golubitsky, Martin, Shiau, Liejune and Stewart, Ian, 1945-. (2007) Spatiotemporal symmetries in the disynaptic canal-neck projection. SIAM Journal on Applied Mathematics, Vol.67 (No.5). pp. 1396-1417. ISSN 0036-1399

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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 “push-pull 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 time-periodic 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 left-right reflections or time-reversals 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:	0036-1399
Date:	20 July 2007
Volume:	Vol.67
Number:	No.5
Page Range:	pp. 1396-1417
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 DMS-0604429 (NSF)
References:	[1] P. Ashwin and O. Podvigina, Hopf bifurcation with cubic symmetry and instability of ABC flow, Proc. R. Soc. Lond. A Math. Phys. Eng. Sci., 459 (2003), pp. 1801–1827. [2] P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas, and M. C. Wiener, Geometric visual hallucinations, Euclidean symmetry, and the functional architecture of striate cortex, Phil. Trans. Royal Soc. London B, 356 (2001), pp. 299–330. [3] P. L. Buono and M. Golubitsky, Models of central pattern generators for quadruped locomotion: I. Primary gaits, J. Math. Biol., 42 (2001), pp. 291–326. [4] J. J. Collins and I. Stewart, Hexapodal gaits and coupled nonlinear oscillator models, Biol. Cybern., 68 (1993), pp. 287–298. [5] J. J. Collins and I. Stewart, Coupled nonlinear oscillators and the symmetries of animal gaits, J. Nonlinear Sci., 3 (1993), pp. 349–392. [6] C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley-Interscience, New York, 1962. [7] I. Z. Foster, D. A. Hanes, N. H. Barmack, and G. McCollum, Spatial symmetries in vestibular projections to the uvula-nodulus, Biol. Cybernet., 96 (2007), pp. 439–453. [8] M. Golubitsky and I. N. Stewart, Hopf bifurcation in the presence of symmetry, Arch. Ration. Mech. Anal., 87 (1985), pp. 107–165. [9] M. Golubitsky and I. Stewart, The Symmetry Perspective, Progr. Math. 200, Birkh¨auser- Verlag, Basel, 2002. [10] M. Golubitsky, I. Stewart, P.-L. Buono, and J. J. Collins, A modular network for legged locomotion, Phys. D, 115 (1998), pp. 56–72. [11] M. Golubitsky, I. Stewart, P.-L. Buono, and J. J. Collins, Symmetry in locomotor central pattern generators and animal gaits, Nature, 401 (1999), pp. 693–695. [12] G. McCollum and R. Boyle, Rotations in a vertebrate setting: Evaluation of the symmetry group of the disynaptic canal-neck projection, Biol. Cybern., 90 (2004), pp. 203–217. [13] Y. Shinoda, Y. Sugiuchi, T. Futami, N. Ando, and T. Kawasaki, Input patterns and pathways from six semicircular canals to motoneurons of neck muscles I: The multifidus muscle group, J. Neurophysiol., 72 (1994), pp. 2691–2702. [14] Y. Shinoda, Y. Sugiuchi, T. Futami, N. Ando, and J. Yagi, Input patterns and pathways from six semicircular canals to motoneurons of neck muscles II: The longissimus and semispinalis muscle groups, J. Neurophysiol. 72 (1997), pp. 2691–2702. [15] Y. Shinoda, Y. Sugiuchi, T. Futami, S. Kakei, Y. Izawa, and J. Na, Four convergent patterns of input from the six semicircular canals to motoneurons of different neck muscles in the upper cervical cord, Ann. New York Acad. Sci., 781 (1996), pp. 264–275. [16] V. J. Wilson and M. Maeda, Connections between semicircular canals and neck motoneurons in the cat, J. Neurophysiol., 37 (1974), pp. 346–357.
URI:	http://wrap.warwick.ac.uk/id/eprint/184