Geometry of the visuo-vestibular information

07/11/2017
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Geometry of the visuo-vestibular information

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Geometry of the visuo-vestibular information GSI17 07/10/2017 Daniel Bennequin IMJ (Paris 7) The labyrinth designs the group of displacement in 3D euclidian space, isomorpphic to the linear part of the Galilée’s group: 6 dimensions: 3 parameters of rotation (2 pour the axis and 1 for the angle) detected by the gradient of pressure in the three semi-circular canals, et 3 parameters of translation, detected by cristals (otoliths) beyond two (or three) epithelial surfaces Let us go back in the past, just before the vertebrates: An exemplar animal: the tunicate larva, with one eye, one otolith, a chord and muscles To control movements, to perceive space, is the origin of the brain. A nice local minimum of motor control (500000000 years) Two experimental main points support a galilean structure of vestibular information: 1) In the vestibular cerebellum, neurons have separated data for the gravitation vector and the linear acceleration vector, giving the galilean frame for the head (Yakusheva et al. 2007). 2) Otoliths and canals send informations at the same order with respect to galilean tensors, linear acceleration and rotation velocity, or linear jerk (i.e. third derivative w.t. time) and rotation acceleration, that are mainly separated in two systems (that overlap): regular and tonic afferent neurons versus irregular and phasic afferent neurons. More precisely, the « vestibular space » is the Lie algebra L(G) of six dimensions, made by infinitesimal rotations (velocity) and linear acceleration. This follows from the arguments of Poincaré, telling that « mutual information » between animals and their environment are first coded by the structure of the compensations between voluntary actions and passive activations, giving rize to the euclidian group G, then the ambient space itself is given by the indifferences of sensors (visual or tactile), E=G/H. When the same arguments are applied to motions themselves, G must be replaced by the tangent group T(G)=G×L(G), and the indifference of the vestibular end sensors is measured by G (the set of inertial frames), thus the quotient space is the Lie algebra L(G). Note an important difference with Physics: in T(G) we find the extension of G by the abelian group T of spatial translations, but not by the group R of temporal translations, confirming that the brain contains no absolute representation of time as a line. Time is active and multiple. But this makes no important changes for the invariance itself. Manifold of reflexes and sensations: VOR, Nystagmus, VCR, locomotor, postural, respiratory, visceral,… The CLONS* Project * CLOsed-loop Neural prosthesis for vestibular disorderS Implanted electrodes & electronics Artificial vestibular system Decoding Learning algorithms Feedback algorithms Neurophysiology Most of what I present now is joint work with Alain Berthoz, CdF, LPPA With the software Ariadne IEE Boston 2011 It is the proper labyrinths Hodge De Rham homology Navier-Stokes in the ducts fluid, coupled with Reissner-Mindlin for the elastic cupula Once projected along the Kelvin currents, this gives three approximate o.d.e. s Note that animals prefer to use an infinite dimensional system for registering a finite dimensional information Crocodylus cataphractus Labyrinth information 2010. C. R. Palevol. 9 (7-8). P 397-410 Microtomography and 3D Scans Humans sensibility Cupula Geometry Set of contacted type I and type II hair cells At least two types of hair cells and afferent neurons (I,II) Parallel with vision: Hair cells correspond to rods and cones (for transduction); Vestibular afferents to bipolar cells; Central nuclei to ganglion cells (except vestibular loops); Then many subcortical areas, thalamus and so on Means Standard deviations Frequencies CV is the ratio of the standard deviation by the mean interspike, measured experimentally by Goldberg and Fernandez (70, 80’s) = Determination of the mean and the variance of the IFR 1) Regular neuron 2) Irregular neuron 3) Cupula deflection Both results accord with Sadeghi et al. 2007 More general hypothesis: With a good approximation, Linear transfer Information spectrum Probability dependent on velocity, i.e. on L(G) Fisher Information metric Irregular case Regular case General case Model of the crista (with Prisca Marianelli) Crista egregia principles Minimal surfaces Catenoids Maculae and striolae Optimal information striola the circular helices Translation surfaces of Lie and Poincaré Jacobian to be optimized (M.Dimiccoli, B.Girard, A.Berthoz, D.Bennequin, 2012) (a natural restriction for the tonic channel) cnl is the inverse of the lemniscate integral: Striola magica, principles Virtual phasic macula A mathematical coincidence? • For the crista: a minimal surface that is formed by the real points of the translation surface of an isotropic circular helix in C3 • For the sriola, the real translation surface of a real circular helix in R3 • Dualitity of compact and non-compact symmetric space (Lobatchevski: i versus 1) • Duality of angular velocities and linear accelerations Central visuo-vestibular receptive fields In the brain, space and time are not given first by position and date but they are accessible through impulsion and frequency Receptive Fields Or in 3D+1, with disparity: 2D+1 frequency in vision With Abelian characters: More generally, for many groups, a duality holds between functions on G and functions on its unitary dual. For instance, for the Galilée group. Which permits unitary receptive fields for vestibular information: 2D simplest case: Approximate: Galilean Invariant NLRFs