Monday August 31, 2009

14:45
Opening session
15:30
Guy David (Université de Paris 11, France)
“Minor variants of Reifenberg's topological disk theorem”
Reifenberg's theorem gives a (local) biHolder parameterization of $E$ when $E$ is well approximated, in Hausdorff distance, by affine planes. We shall discuss variants where $E$ is allowed to have holes, or is only approximated by Lipschitz graphs, or we try to get a better parameterization. [Joint work with Tatiana Toro.]
16:15
Coffee break
16:30
Yves Fregnac (UNIC - CNRS, Gif-sur-Yvette, France)
“Multiscale functional imaging of dynamic association fields in V1”
Yves FREGNAC, Pedro CARELLI and Marc PANANCEAU Abstract : In vivo intracellular electrophysiology offers the unique possibility of listening to the "synaptic rumour" of the cortical network captured by the recording electrode in a single V1 cell. The analysis of synaptic echoes evoked during sensory processing is used to reconstruct the distribution of input sources in visual space and in time. It allows to infer, in cortical space, the dynamics of the effective input network afferent to the recorded cell. We have applied this method to demonstrate the propagation of visually evoked activity through lateral (and possibly feedback) connectivity in the primary cortex of higher mammals (Fregnac and Bringuier, 1996; Bringuier et al, 1999). This approach, based on functional synaptic imaging, is compared here with a real-time functional network imaging technique, based on the use of voltage-sensitive fluorescent dyes (Jancke et al, 2004; Roland et al, 2006; Benucci et al, 2007; Xu et al, 2007; Nauhaus et al, 2009). The former method gives access to microscopic convergence processes during synaptic integration in a single neuron, while the latter describes the macroscopic divergence process at the neuronal map level. The joint application of the two techniques which address two different scales of integration is used to elucidate the cortical origin of low-level (non-attentive) binding processes participating in the emergence of illusory motion percepts predicted by the psychological Gestalt theory (Chavane et al, submitted; Fr�gnac et al, 2009). Complementary studies, based on intracellular electrophysiology, network imaging (S�ries et al, 2002; 2003) and psychophysics (Georges et al, 2002), all point to the emergence of cooperative Gestalt-like interactions, when the stimulus carries a sufficient level of spatial and temporal coherence. Above a given activation threshold (yet to be quantitatively defined) a cooperative depolarizing or facilitatory wave becomes detectable in primary and secondary visual areas. This wave travels at low speed in the plane of the superficial layers (0.10 to 0.30 m/s) and becomes anisotropic for oriented inducer stimuli. The physiological features of the spatio-temporal propagation pattern recorded in V1 are highly correlated with the percept reported by the conscious observer and agree with predictions derived from the Gestalt theory. We will review two cases of motion illusion (apparent motion and line motion), which suggests the genesis of a wave of perceptual binding operating in V1, which modulates the integration of feed-forward inputs yet to come: this wave can be seen as the propagation of the V1 network belief of the possible presence of a global percept (the �whole�:) before the illusory percept becomes validated by the sequential presentation of the �parts� (signalled by direct focal feed-forward waves). This neuronal dynamics obeys closely the Gestalt prediction that the emergence of the �whole� should precede in time the detection of the �parts�. It remains to be determined whether the correlations we report between perception and horizontal propagation are the result solely of neural processes intrinsic to V1, or whether they reflect the reverberation in V1 of a collective feedback originating from multiple secondary cortical areas, each encoding for a distinct functional representation of the visual field. It may be indeed envisioned that the primary visual cortex plays the role of a generalized echo chamber fed by other cortical areas (visual or not) which participate in the coding of shape and motion in space: accordingly, the waves travelling across V1 would signal the emergence of perceptual coherence when a synergy is reached between the different cortical analyzers. References: (1) ANGELUCCI, A., LEVITTE, J.B., WALTON, E.J.S., HUP�, J.M., BULLIER, J. AND LUND, J.S. (2002). Circuits for local and global signal integration in primary visual cortex. J Neurosci., 22, 8633-46. (2) BENUCCI, A., FRAZOR, RA, CARANDINI, M. (2007). Standing waves and traveling waves distinguish two circuits in visual cortex. Neuron, 55(1), 103-117 (3) BRINGUIER, V., CHAVANE, F., GLAESER, L. and FR�GNAC, Y. (1999). Horizontal propagation of visual activity in the synaptic integration field of area 17 neurons. Science, 283, 695-699. (4) CHAVANE, F., MONIER, C., BRINGUIER, V., BAUDOT, P., BORG-GRAHAM, L., LORENCEAU, J. AND FR�GNAC, Y. (2000). The visual cortical association field: a Gestalt concept or a physiological entity? Journal of Physiology (Paris), 94, 333-342 (5) CHAVANE, F., SHARON, D., JANCKE, D., MARRE, O., FR�GNAC, Y. AND GRINVALD, A. (submitted). Horizontal spread of orientation selectivity in V1 requires intracortical cooperativity (6) FR�GNAC, Y. and BRINGUIER, V. (1996). Spatio-temporal dynamics of synaptic integration in cat visual cortical receptive fields. In Brain Theory: Biological Basis and Computational Theory of Vision , eds. Aertsen, A. and Braitenberg, V., pp. 143-199. Springer-Verlag, Amsterdam. (7) FREGNAC, Y., BAUDOT, P., CHAVANE, F., MONIER, C., LORENCEAU, J., MARRE, O., PANANCEAU, O., CARELLI, P.V. AND SADOC, G. Multiscale functional imaging in V1 and cortical correlates of apparent motion. In �Dynamics of Visual Motion Processing� Ed. G. masson and U. Ilig, Springer (in press). (8) GEORGES, S., S�RIES, P., FR�GNAC, Y., AND LORENCEAU, J. (2002). Orientation dependent modulation of apparent speed: psychophysical evidence. Vision Res., 42, 2757-2772. (9) JANCKE, D., CHAVANE, F., NAAMAN, S. and GRINVALD, A. (2004). Imaging cortical correlates of illusion in early visual cortex. Nature 428, 423-426. (10) NAUHAUS, I., BUSSE, L., CARANDINI, M. AND RINGACH, D.L. (2009). Stimulus contrast modulates functional connectivity in visual cortex. Nature Neurosci, 12, 70-76. (11) ROLAND, PE., HANAZAWA, A., UNDEMAN, C., ERIKSSON, D., TOMPA, T., NAKAMURA, H., VALENTINIENE, S., AHMED, B. (2006). Cortical feedback depolarization waves: a mechanism of top-down influence on early visual areas. Proc Natl Acad Sci USA, 103(33),12586-12591. (12) S�RIES, P., GEORGES, S., LORENCEAU, J., AND FR�GNAC, Y. (2002). Orientation dependent modulation of apparent speed: a model based on the dynamics of feed-forward and horizontal connectivity in V1 cortex. Vision Res., 42, 2781-2797. (13) SERIES, P., LORENCEAU, J. and FREGNAC, Y. (2003). The silent surround of V1 receptive fields : theory and experiments. Journal of Physiology (Paris), 97(4-6): 453-474. (14) XU, W., HUANG, X., TAKAGAKI, K., WU, JY. (2007). Compression and reflection of visually evoked cortical waves. Neuron, 55(1), 119-129

Tuesday September 1, 2009

09:15
Joan Verdera (Universitat Autònoma de Barcelona, Spain)
“Singular Integrals in action: Painlevé's problem and vortex patches”
Singular Integrals of Calderón-Zygmund type have been widely used in PDE and different branches of analysis. The purpose of this lecture is to present an overview of their role in the solution of two problems, one in complex analysis and the other in PDE. First we will deal with Painlevé's problem, which consists in describing metrically the removable sets for bounded analytic functions. We will show how singular integrals enter the scene and we will explain the need for a more sophisticated Calderón-Zygmund Theory dealing with non-doubling underlying measures. Then we will discuss the weak solutions of Euler's equation called vortex patches. We will explain the role of singular integrals in Chemin's Theorem on the persistence of boundary.
10:00
Andrea Malchiodi (SISSA, Trieste, Italy)
“Minimal surfaces in the Heisenberg group”
We study the minimal surface equation for graphs in the Heisenberg group: this is a highly degenerate equation which has both elliptic and hyperbolic features. We discuss existence-regularity of solutions, and in particular the structure of the singular set, where the tangent plane of the graph coincides with the contact distribution.
10:45
Coffee break
11:00
Xiao Zhong (Jyväskylän yliopisto, Finland)
“Quasilinear elliptic equations in the Heisenberg group”
I will talk about the regularity of solutions to quasilinear elliptic equations of p-Laplacian type in the Heisenberg group. Some open problems in this topic will be mentioned.
14:45
Zoltan Balogh (Universität Bern, Switzerland)
“Fractals in Carnot groups”
I will present a summary of joint results with R. Berger, R. Monti, J. Tyson, B. Warhurst. We consider Iterated Function Sytems in the sub-Riemannian setting of Carnot groups. Almost sure dimension formulae for the underlying invariant sets are derived and the size of the exceptional sets of parameters for which these formulae do not hold is estimated.
15:30
Luca Capogna (University of Arkansas, USA)
“A sub-Riemannian analogue of the mean curvature flow”
I will discuss some analytic and geometric aspects of the evolution PDE that describes the motion of smooth hypersurfaces in Carnot groups along the gradient flow of the (sub-Riemannian) perimeter.
16:15
Coffee break
16:30
Poster Session

Wednesday September 2, 2009

09:15
Pertti Mattila (Helsingin yliopisto, Finland)
“Singular integrals on Ahlfors regular subsets of Heisenberg groups”
We investigate certain singular integral operators with Riesz-type kernels on s-dimensional Ahlfors regular subsets of Heisenberg groups. We show that L^2-boudedness implies that s must be an integer and the set can be approximated at some arrbitrary small scales by homogeneous subgroups. It follows that the operators cannot be bounded on many self-similar fractal subsets of Heisenberg groups.
10:00
Ermanno Lanconelli (Università di Bologna, Italy)
“Global L^p estimates for Ornstein-Uhlenbeck operators: a general approach”
We first present a recent result proved in collaboration with M.Bramanti, G. Cupini and E. Priola concerning global L^p estimates for degenerate Ornstein-Uhlenbeck operators L. The estimates were obtained with a new approach based on: (i) The left translation invariance, with respect to a Lie group structure, of the evolution counterpart H of L; (ii) The existence of a global fundamental soltion for H, with a suitable fast decay at infinity. (iii) A Calderon-Zygmund type theorem for singular integrals in non-doubling spaces. Then, we present some other results obtained in collaboration with A. Bonfiglioli, aimed to pave the way to apply the previous thecniques to a more general class of H\"ormander operators: "sum of squares" plus drift. We present sufficient conditions for these operators to be left translation
10:45
Coffee break
11:00
Brynjulf Owren (NTNU, Trondheim, Norway)
“Numerical computation of motion in Lie groups and subriemannian geometries”
Traditional numerical methods for solving ordinary differential equations were designed with the objectives of rendering accurate, efficient and robust solution to any ODE problem. Errors were usually measured on finite time intervals and black-box software of high quality was developed. Many of the most popular integrators used today were developed in this early period of computing and are excellent choices for a large variety of problems. In recent years however, more attention has been given to developing approximation methods which somehow respect or inherit the geometric structure of the particular differential equation or class of equations at hand. A famous examples of this is the class of Hamiltonian problems. Since the flow of such problems belong to the class of symplectic maps, it is natural to insist that also the numerical approximation method takes the form of a symplectic mapping. Such numerical methods show excellent behaviour over long integration intervals. Another important example is the Lie group integrators. When the ODE is defined on a Lie group or via a Lie group acting on a manifold, the Lie group integrators work by using the action itself as a tool for moving on the manifold. In this talk we shall give a general introduction to integrators on Lie groups and homogeneous manifolds. Although relatively little is known about the behaviour of such methods in the subriemannian setting, there actually exist classes of Lie group integrators which seem well suited for such geometries.
14:30
Special session: Neuromathematics of vision
20:00
Social dinner

Thursday September 3, 2009

09:15
Jean Petitot (CREA, Paris, France)
10:00
Steven Zucker (Johns Hopkins University, USA)
10:45
Coffee break
11:00
Walter Schempp (Universität Siegen, Germany)
“Third-Generation Nuclear Magnetic Resonance: Diffusion Tensor Magnetic Resonance Tomography and Cerebral White Matter Tractography”
14:45
Abbas Bahri (Rutgers, The State University of New Jersey, USA)
15:30
William Hamilton Meeks III (University of Massachusetts, Amherst, USA)
“The local and global geometry of embedded minimal and CMC surfaces in 3-manifolds”
16:15
Coffee break
16:30
Poster Session

Friday September 4, 2009

09:15
Bart M. ter Haar Romeny (Technische Universiteit, Eindhoven, Netherlands)
10:00
Tai Sing Lee (Carnegie Mellon University, USA)
“Natural scene statistics and neural basis of 3D inference”
From a computational perspective, visual perception is the inference of the underlying physical causes of the images we observed. It has been long hypothesized that the brain adopts a Bayesian inference approach to solve this problem by learning and exploiting the statistical regularities in the natural environment. Understanding these statistical regularities, particularly the correlation structures between 3D scene structures and observed 2D images, is therefore important for understanding the neural basis of 3D scene inference. In this talk, I will discuss a number of empirical correlation structures that we have found in a set of natural color images with precisely co-registered range images: correlations between depth and brightness, correlations between shape and shading and the scaling laws governing them. I will show that these statistical regularities can be useful in 3D inference in computer vision. Finally, I will describe some of our multi-electrode recording results in the early visual cortex of awake monkeys that begin to reveal how these statistical regularities are encoded at the neuronal level to constrain perceptual computation.
10:45
Coffee break
11:00
Ennio Mingolla (Boston University, USA)
“Neural dynamics of perceptual 3-D surface formation: Multi-scale boundary webs and featural filling-in”
A real-time visual processing theory provides a unified approach to the analysis of important aspects of surface perception through mechanisms closely analogous to those of the primate early visual pathways. Neural network interactions within a multiple-scale boundary contour and feature contour system generate a representation of curved 3D surfaces with variations of surface lightness. Each spatial scale of the system contains a hierarchy of orientationally-tuned processes, which include spatially short-range competitive interactions and spatially long-range cooperative interactions. Feedback between the competitive and cooperative stages synthesizes a coherent multiple-scale structural representation of a 3D surface, called a boundary web, which regulates multiple-scale filling-in reactions of featural quality, such as color or lightness, to generate a percept of form-and-color-in-depth. Feedback between boundary and surface featural representations regulates figure-ground separation and also determination of border ownership, amodal completion of occluded surfaces, and perception of transparency. Simulations of shape-from-texture illustrate key properties of boundary webs and related model mechanisms. Two basic challenges are: (1) Patterns of spatially discrete 2D texture elements must be transformed into a spatially smooth surface representation of 3D shape. (2) Changes in the statistical properties of texture elements across space must induce the perceived 3D shape of the resulting surface representation. This is achieved in the model through multiple-scale filtering of a 2D image, followed by a cooperative-competitive grouping network that coherently binds texture elements into boundary webs at the appropriate depths using a scale-to-depth map and a subsequent depth competition stage. These boundary webs then gate filling-in of surface lightness signals in order to form a smooth 3D surface percept. The model quantitatively simulates challenging psychophysical data about perception of surface curvature-in-depth (Grossberg, Kuhlmann, and Mingolla; Vision Res. 2007, 47(5):634-72).