Section: New Results

Neural Fields

Modelling the dynamics of contextual motion integration in the primate

Participants : Heiko Neumann [Institute of Neural Information Processing, Ulm University, Ulm, Germany] , Pierre Kornprobst, Guillaume Masson [Institut de Neurosciences de la Timone, UMR 6193, CNRS, Marseille, France] , Emilien Tlapale.

The dynamics of motion integration show striking similarities when observed at neuronal, psychophysical, and oculomotor levels. Based on the inter-relation and complementary insights given by those dynamics, our goal is to investigate how basic mechanisms of dynamical cortical processing can be incorporated in a dynamical model to solve several aspects of 2D motion integration and segmentation.

Thanks to Emilien Tlapale PhD [13] (see also [16] ), we have obtained the following results:

• We proposed a recurrent model of motion integration. Proposing a simple readout mechanism, we reproduced not only motion perception but also the dynamics of smooth pursuit eye movements on various line figures and gratings viewed through different apertures. Our model can also solve various contextual problems where extrinsic junctions should be eliminated, without relying on complex junction detectors or depth computation  [71] . Finally, we have also shown how our model can be rewritten in the neural fields formalism (see [52] and the Software MotionLib), which has opened new perspectives as detailed in Section 6.4.2 .

• We confronted our results to artificial and biological vision. To formalize the comparison against visual performance, we proposed a new evaluation methodology based on human visual performance by establishing a database of image sequences taken from biology and psychophysics literature  [70] , [69] , [67] . We compared our results against the state of the art computer vision approaches and we found that our model also gives results comparable to recent computer vision approaches of motion estimation.

Neural fields models for motion integration: Characterising the dynamics of multi-stable visual motion stimuli

Participants : Olivier Faugeras, Pierre Kornprobst, Guillaume Masson [Institut de Neurosciences de la Timone, UMR 6193, CNRS, Marseille, France] , Andrew Meso [Institut de Neurosciences de la Timone, UMR 6193, CNRS, Marseille, France] , James Rankin, Emilien Tlapale, Romain Veltz.

In [57] we investigated the temporal dynamics of the neural processing of a multi-stable visual motion stimulus with two complementary approaches: psychophysical experiments and mathematical modelling. The so called “barber pole” stimulus is considered with an aperture configuration that supports horizontal (H), diagonal (D) or vertical (V) perceived directions for the same input. The phenomenon demonstrates an interesting variable and dynamic competition for perceptual dominance between underlying neural representations of the three directions. We probe the early processing from stimulus presentation to initial perceived direction (before perceptual reversals). Starting from a simplified neural fields model inspired from [13] , we constructed a model of the necessary motion integration that shows a shift in perceptual dominance from D to either H or V with increasing duration. Further, the timing of this shift is shown to be controlled by a stimulus gain parameter analogous to contrast. In psychophysics experiments with concurrent eye movement recordings, observers report their perceived direction of motion for presentation durations between 0.1s and 0.5s. There is a also consistent transition in perceptual dominance from D to H/V as duration is increased. This trend, seen in both perceived direction decisions and eye movement patterns, is consistent with previous experiments using similar stimuli with an aperture configured for two (D/H) rather than three (D/H/V) states. The basic dynamic properties of the early transition from D to H/V are well predicted by the model. The experimental work additionally reveals asymmetric data patterns that guide adjustments to the model's input equations. Observers have an H bias over V, which is also reflected in faster reaction times for H. In order to capture the bias between H and V a separate weighting is attributed to the local input corresponding to each state. The work presented forms a solid foundation for future experimental and modelling work investigating the longer term dynamics for which perceptual reversals are known to occur.

Analysis of a hyperbolic geometric model for visual texture perception

Participants : Pascal Chossat, Grégory Faye, Olivier Faugeras.

We study the neural field equations introduced by Chossat and Faugeras in [64] to model the representation and the processing of image edges and textures in the hypercolumns of the cortical area V1. The key entity, the structure tensor, intrinsically lives in a non-Euclidean, in effect hyperbolic, space. Its spatio-temporal behaviour is governed by nonlinear integro-differential equations defined on the Poincaré disc model of the two-dimensional hyperbolic space. Using methods from the theory of functional analysis we show the existence and uniqueness of a solution of these equations. In the case of stationary, i.e. time independent, solutions we perform a stability analysis which yields important results on their behavior. We also present an original study, based on non-Euclidean, hyperbolic, analysis, of a spatially localised bump solution in a limiting case. We illustrate our theoretical results with numerical simulations.

This work has been published in the Journal of Mathematical Neuroscience [21] .

Bifurcation of Hyperbolic Planforms

Participants : Pascal Chossat, Grégory Faye, Olivier Faugeras.

Motivated by a model for the perception of textures by the visual cortex in primates, we analyze the bifurcation of periodic patterns for nonlinear equations describing the state of a system defined on the space of structure tensors, when these equations are further invariant with respect to the isometries of this space. We show that the problem reduces to a bifurcation problem in the hyperbolic plane D (Poincaré disc).We make use of the concept of a periodic lattice in D to further reduce the problem to one on a compact Riemann surface $D/\Gamma$ , where $\Gamma$ is a cocompact, torsion-free Fuchsian group. The knowledge of the symmetry group of this surface allows us to use the machinery of equivariant bifurcation theory. Solutions which generically bifurcate are called “H-planforms”, by analogy with the “planforms” introduced for pattern formation in Euclidean space. This concept is applied to the case of an octagonal periodic pattern, where we are able to classify all possible H-planforms satisfying the hypotheses of the Equivariant Branching Lemma. These patterns are, however, not straightforward to compute, even numerically, and in the last section we describe a method for computation illustrated with a selection of images of octagonal H-planforms.

This work has been published in the Journal of Nonlinear Science [19] .

Bifurcation diagrams and heteroclinic networks of octagonal H-planforms

Participants : Grégory Faye, Pascal Chossat [correspondent] .

This paper completes the classification of bifurcation diagrams for H-planforms in the Poincaré disc $𝔻$ whose fundamental domain is a regular octagon. An H-planform is a steady solution of a PDE or integro-differential equation in $𝔻$, which is invariant under the action of a lattice subgroup $\Gamma$ of $U(1,1)$, the group of isometries of $𝔻$. In our case $\Gamma$ generates a tiling of $𝔻$ with regular octagons. This problem was introduced in [19] as an example of spontaneous pattern formation in a model of image feature detection by the visual cortex where the features are assumed to be represented in the space of structure tensors. Under "generic" assumptions the bifurcation problem reduces to an ODE which is invariant by an irreducible representation of the group of automorphisms $𝒢$ of the compact Riemann surface $𝔻/\Gamma$. The irreducible representations of $𝒢$ have dimension one, two, three and four. The bifurcation diagrams for the representations of dimension less than four have already been described and correspond to already well known goup actions. In the present work we compute the bifurcation diagrams for the remaining three irreducible representations of dimension four, thus completing the classification. In one of these cases, there is generic bifurcation of a heteroclinic network connecting equilibria with two different orbit types.

This work has been accepted for publication in the Journal of Nonlinear Science [22] .

Hopf bifurcation curves in neural field networks with space-dependent delays

Participant : Romain Veltz.

We give an analytical parametrization of the curves of purely imaginary eigenvalues in the delay-parameter plane of the linearized neural field network equations with spacedependent delays. In order to determine if the rightmost eigenvalue is purely imaginary, we have to compute a finite number of such curves; the number of curves is bounded by a constant for which we give an expression. The Hopf bifurcation curve lies on these curves.

This work has appeared in the Comptes Rendus Mathématiques de l'Académie des Sciences [30] .

Stability of the stationary solutions of neural field equations with propagation delays

Participants : Olivier Faugeras, Romain Veltz.

We consider neural field equations with space-dependent delays. Neural fields are continuous assemblies of mesoscopic models arising when modeling macroscopic parts of the brain. They are modeled by nonlinear integro-differential equations. We rigorously prove, for the first time to our knowledge, suffcient conditions for the stability of their stationary solutions. We use two methods 1) the computation of the eigenvalues of the linear operator defined by the linearized equations and 2) the formulation of the problem as a fixed point problem. The first method involves tools of functional analysis and yields a new estimate of the semigroup of the previous linear operator using the eigenvalues of its infinitesimal generator. It yields a sufficient condition for stability which is independent of the characteristics of the delays. The second method allows us to find new sufficient conditions for the stability of stationary solutions which depend upon the values of the delays. These conditions are very easy to evaluate numerically. We illustrate the conservativeness of the bounds with a comparison with numerical simulation.

This work has appeared in the Journal of Mathematical Neuroscience [29] .

Neural Mass Activity, Bifurcations and Epilepsy

Participants : Patrick Chauvel [INSERM U751, Marseille, Assistance Publique-Hopitaux de Marseille Timone, and Universite Aix-Marseille, Marseille] , Olivier Faugeras, Jonathan Touboul, Fabrice Wendling [INSERM, U642, Rennes] .

We propose a general framework for studying neural mass models defined by ordinary differential equations. By studying the bifurcations of the solutions to these equations and their sensitivity to noise we establish an important relation, similar to a dictionary, between their behaviors and normal and pathological, especially epileptic, cortical patterns of activity. We then apply this framework to the analysis of two models that feature most phenomena of interest, the Jansen and Rit model, and the slightly more complex model recently proposed by Wendling and Chauvel. This model-based approach allows to test various neurophysiological hypotheses on the origin of pathological cortical behaviors and to investigate the effect of medication. We also study the effects of the stochastic nature of the inputs which gives us clues about the origins of such important phenomena as interictal spikes, inter-ical bursts and fast onset activity, that are of particular relevance in epilepsy.

This work has appeared in Neural Computation [27] .