Nicolas Tabareau

A Contraction Theory Approach to Stochastic Incremental Stability

We investigate the incremental stability properties of Ito stochastic dynamical systems. Specifically, we derive a stochastic version of nonlinear contraction theory that provides a bound on the mean square distance between any two trajectories of a stochastically contracting system. This bound can be expressed as a function of the noise intensity and the contraction rate of the noise-free system. We illustrate these results in the contexts of nonlinear observers design and stochastic synchronization.

2008 Special Issue: Where neuroscience and dynamic system theory meet autonomous robotics: A contracting basal ganglia model for action selection

Action selection, the problem of choosing what to do next, is central to any autonomous agent architecture. We use here a multi-disciplinary approach at the convergence of neuroscience, dynamical system theory and autonomous robotics, in order to propose an efficient action selection mechanism based on a new model of the basal ganglia. We first describe new developments of contraction theory regarding locally projected dynamical systems. We exploit these results to design a stable computational model of the cortico-baso-thalamo-cortical loops. Based on recent anatomical data, we include usually neglected neural projections, which participate in performing accurate selection. Finally, the efficiency of this model as an autonomous robot action selection mechanism is assessed in a standard survival task. The model exhibits valuable dithering avoidance and energy-saving properties, when compared with a simple if-then-else decision rule.

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.

Implementation of a neurophysiological model of saccadic eye movements on an anthropomorphic robotic head

In this paper we investigated the relevance of a robotic implementation in the development and validation of a neurophysiological model of the generation of saccadic eye movements. To this aim, a well-characterized model of the brainstem saccadic circuitry was implemented on a humanoid robot head with 7 degrees of freedom (DOFs), which was designed to mimic the human head in terms of the physical dimensions (i.e. geometry and masses), the kinematics (i.e. number of DOFs and ranges of motion), the dynamics (i.e. velocities and accelerations), and the functionality (i.e. the ocular movements of vergence, smooth pursuit and saccades). Our implementation makes the robot head execute saccadic eye movements upon a visual stimulus appearing in the periphery of the robot visual field, by reproducing the following steps: projection or the camera images onto collicular images, according to the modeled mapping between the retina and the superior colliculus (SC); transformation of the retinotopic coordinates of the stimulus obtained in the camera reference frame into their corresponding projections on the SC; spatio-temporal transformation of these coordinates according to what is known to happen in the brainstem saccade burst generator of primates; and execution of the eye movement by controlling one eye motor of the robot, in velocity. The capabilities of the robot head to execute saccadic movements have been tested with respect to the neurophysiological model implemented, in view of the use of this robotic implementation for validating and tuning the model itself, in further focused experimental trials