Daniel Bennequin

Movement Timing and Invariance Arise from Several Geometries

Human movements show several prominent features; movement duration is nearly independent of movement size (the isochrony principle), instantaneous speed depends on movement curvature (captured by the 2/3 power law), and complex movements are composed of simpler elements (movement compositionality). No existing theory can successfully account for all of these features, and the nature of the underlying motion primitives is still unknown. Also unknown is how the brain selects movement duration. Here we present a new theory of movement timing based on geometrical invariance. We propose that movement duration and compositionality arise from cooperation among Euclidian, equi-affine and full affine geometries. Each geometry posses a canonical measure of distance along curves, an invariant arc-length parameter. We suggest that for continuous movements, the actual movement duration reflects a particular tensorial mixture of these canonical parameters. Near geometrical singularities, specific combinations are selected to compensate for time expansion or compression in individual parameters. The theory was mathematically formulated using Cartans moving frame method. Its predictions were tested on three data sets: drawings of elliptical curves, locomotion and drawing trajectories of complex figural forms (cloverleaves, lemniscates and limaçons, with varying ratios between the sizes of the large versus the small loops). Our theory accounted well for the kinematic and temporal features of these movements, in most cases better than the constrained Minimum Jerk model, even when taking into account the number of estimated free parameters. During both drawing and locomotion equi-affine geometry was the most dominant geometry, with affine geometry second most important during drawing; Euclidian geometry was second most important during locomotion. We further discuss the implications of this theory: the origin of the dominance of equi-affine geometry, the possibility that the brain uses different mixtures of these geometries to encode movement duration and speed, and the ontogeny of such representations.

Geometry of the superior colliculus mapping and efficient oculomotor computation

Numerous brain regions encode variables using spatial distribution of activity in neuronal maps. Their specific geometry is usually explained by sensory considerations only. We provide here, for the first time, a theory involving the motor function of the superior colliculus to explain the geometry of its maps. We use six hypotheses in accordance with neurobiology to show that linear and logarithmic mappings are the only ones compatible with the generation of saccadic motor command. This mathematical proof gives a global coherence to the neurobiological studies on which it is based. Moreover, a new solution to the problem of saccades involving both colliculi is proposed. Comparative simulations show that it is more precise than the classical one.

Gaze anticipation during human locomotion

During locomotion, a top-down organization has been previously demonstrated with the head as a stabilized platform and gaze anticipating the horizontal direction of the trajectory. However, the quantitative assessment of the anticipatory sequence from gaze to trajectory and body segments has not been documented. The present paper provides a detailed investigation into the spatial and temporal anticipatory relationships among the direction of gaze and body segments during locomotion. Participants had to walk along several mentally simulated complex trajectories, without any visual cues indicating the trajectory to follow. The trajectory shapes were presented to the participants on a sheet of paper. Our study includes an analysis of the relationships between horizontal gaze anticipatory behavior direction and the upcoming changes in the trajectory. Our findings confirm the following: 1) The hierarchical ordered organization of gaze and body segment orientations during complex trajectories and free locomotion. Gaze direction anticipates the head orientation, and head orientation anticipates reorientation of the other body segments. 2) The influence of the curvature of the trajectory and constraints of the tasks on the temporal and spatial relationships between gaze and the body segments: Increased curvature resulted in increased time and spatial anticipation. 3) A different sequence of gaze movements at inflection points where gaze plans a much later segment of the trajectory.

Assessing morphology and function of the semicircular duct system: introducing new in-situ visualization and software toolbox

The semicircular duct system is part of the sensory organ of balance and essential for navigation and spatial awareness in vertebrates. Its function in detecting head rotations has been modelled with increasing sophistication, but the biomechanics of actual semicircular duct systems has rarely been analyzed, foremost because the fragile membranous structures in the inner ear are hard to visualize undistorted and in full. Here we present a new, easy-to-apply and non-invasive method for three-dimensional in-situ visualization and quantification of the semicircular duct system, using X-ray micro tomography and tissue staining with phosphotungstic acid. Moreover, we introduce Ariadne, a software toolbox which provides comprehensive and improved morphological and functional analysis of any visualized duct system. We demonstrate the potential of these methods by presenting results for the duct system of humans, the squirrel monkey and the rhesus macaque, making comparisons with past results from neurophysiological, oculometric and biomechanical studies. Ariadne is freely available at http://www.earbank.org.

Geometrical Invariance and Smoothness Maximization for Task-Space Movement Generation

In human motor control studies, end-effector (e.g., hand) trajectories have been successfully modeled using optimization principles. Yet, it remains unclear how such trajectories are updated when the end-effector or task goals are perturbed. Here, we present an approach to human and robotic task-level trajectory planning and modification using geometrical invariance and optimization, allowing to adapt learned movements to a priori unknown boundary conditions. The optimization criterion represents a tradeoff between smoothness (minimum jerk) and accuracy (jerk-accuracy model). We show that planning maximally smooth movements allows recovery from perturbations by superimposing specific affine orbits on maximally smooth preplanned trajectories. The generated trajectories are compared with those resulting from other recent approaches used in robotics. Finally, we discuss conditions for affine invariance of maximally smooth task-space trajectories. Possible applications of this study to both human motor control and robotics research studies are discussed.

Pinwheel-dipole configuration in cat early visual cortex.

In the early visual cortex, information is processed within functional maps whose layouts are thought to underlie visual perception. However, the precise organization of these functional maps as well as their interrelationships remain unsettled. Here, we show that spatial frequency representation in cat early visual cortex exhibits singularities around which the map organizes like an electric dipole potential. These singularities are precisely co-located with singularities of the orientation map: the pinwheel centers. To show this, we used high resolution intrinsic optical imaging in cat areas 17 and 18. First, we show that a majority of pinwheel centers exhibit in their neighborhood both semi-global maximum and minimum in the spatial frequency map (i.e. extreme values of the spatial frequency in a hypercolumn). This contradicts pioneering studies suggesting that pinwheel centers are placed at the locus of a single spatial frequency extremum. Based on an analogy with electromagnetism, we proposed a mathematical model for a dipolar structure, accurately fitting optical imaging data. We conclude that a majority of orientation pinwheel centers form spatial frequency dipoles in cat early visual cortex. Given the functional specificities of neurons at singularities in the visual cortex, it is argued that the dipolar organization of spatial frequency around pinwheel centers could be fundamental for visual processing.

Differences in gaze anticipation for locomotion with and without vision

Previous experimental studies have shown a spontaneous anticipation of locomotor trajectory by the head and gaze direction during human locomotion. This anticipatory behavior could serve several functions: an optimal selection of visual information, for instance through landmarks and optic flow, as well as trajectory planning and motor control. This would imply that anticipation remains in darkness but with different characteristics. We asked 10 participants to walk along two predefined complex trajectories (limaçon and figure eight) without any cue on the trajectory to follow. Two visual conditions were used: (i) in light and (ii) in complete darkness with eyes open. The whole body kinematics were recorded by motion capture, along with the participants right eye movements. We showed that in darkness and in light, horizontal gaze anticipates the orientation of the head which itself anticipates the trajectory direction. However, the horizontal angular anticipation decreases by a half in darkness for both gaze and head. In both visual conditions we observed an eye nystagmus with similar properties (frequency and amplitude). The main difference comes from the fact that in light, there is a shift of the orientations of the eye nystagmus and the head in the direction of the trajectory. These results suggest that a fundamental function of gaze is to represent self motion, stabilize the perception of space during locomotion, and to simulate the future trajectory, regardless of the vision condition.

Striola magica. A functional explanation of otolith geometry

Otolith end organs of vertebrates sense linear accelerations of the head and gravitation. The hair cells on their epithelia are responsible for transduction. In mammals, the striola, parallel to the line where hair cells reverse their polarization, is a narrow region centered on a curve with curvature and torsion. It has been shown that the striolar region is functionally different from the rest, being involved in a phasic vestibular pathway. We propose a mathematical and computational model that explains the necessity of this amazing geometry for the striola to be able to carry out its function. Our hypothesis, related to the biophysics of the hair cells and to the physiology of their afferent neurons, is that striolar afferents collect information from several type I hair cells to detect the jerk in a large domain of acceleration directions. This predicts a mean number of two calyces for afferent neurons, as measured in rodents. The domain of acceleration directions sensed by our striolar model is compatible with the experimental results obtained on monkeys considering all afferents. Therefore, the main result of our study is that phasic and tonic vestibular afferents cover the same geometrical fields, but at different dynamical and frequency domains.

How the Learning Path and the Very Structure of a Multifloored Environment Influence Human Spatial Memory.

Few studies have explored how humans memorize landmarks in complex multifloored buildings. They have observed that participants memorize an environment either by floors or by vertical columns, influenced by the learning path. However, the influence of the building’s actual structure is not yet known. In order to investigate this influence, we conducted an experiment using an object-in-place protocol in a cylindrical building to contrast with previous experiments which used rectilinear environments. Two groups of 15 participants were taken on a tour with a first person perspective through a virtual cylindrical three-floored building. They followed either a route discovering floors one at a time, or a route discovering columns (by simulated lifts across floors). They then underwent a series of trials, in which they viewed a camera movement reproducing either a segment of the learning path (familiar trials), or performing a shortcut relative to the learning trajectory (novel trials). We observed that regardless of the learning path, participants better memorized the building by floors, and only participants who had discovered the building by columns also memorized it by columns. This expands on previous results obtained in a rectilinear building, where the learning path favoured the memory of its horizontal and vertical layout. Taken together, these results suggest that both learning mode and an environment’s structure influence the spatial memory of complex multifloored buildings.

Non-linear Galilean vestibular receptive fields

We present a set of formulas for the receptive fields of the vestibular neurons that are motivated by Galilean invariance. We show that these formulas explain non-trivial data in neurophysiology, and suggest new hypothesis to be tested in dynamical 3D conditions. Moreover our model offers a way for neuronal computing with 3D displacements, which is reputed to be hard, underlying the vestibular reflexes. This computation is presented in a Bayesian framework. The basis of the model is the necessity of living bodies to work invariantly in space-time, allied to the necessary discreteness of neuronal transmission.