**Pr. Dmitri Alekseevsky**

Institute for Information Transmission Problems RAS and Hull University, Russia

**Talk Title**

**Conformal model of hypercolumns in V1 cortex and Mobius group. Application to the visual stability problem**

**Talk Abstract**

We propose a conformal spherical model of hypercolumns of primary visual cortex V1, which is a modifcation of the Bressloff- Cowan Riemannian spherical model and is closely related to the Sarti-Citti-Petit symplectic model of V1 cortex. Application to visual stability problem will be considered. D. Hubel and T. Wiesel put forward the idea that the visual cortex should be viewed as a fiber bundle over the retina R. Fiber of the bundle corresponds to different internal parameters ( orientation, spatial frequency, ocular dominance, direction of motion, curvature, three parameters of color space etc) that affect the excitation of visual neurons. N.V. Swindale (2000) estimated the dimension of the fibers ( = the number of internal parameters) as 6-7 or 9-10. Following the idea by Hubel and Wiesel, J. Petitot proposed a contact model of V1 cortex as the contact bundle ∏ : F -> R of orientations (directions) over the retina R (identified with the Euclidean plane R = R2). The space E has coordinates (x; y; Ɵ) where (x; y) 2 R2 and the orientation Ɵ is the angle between the tangent line to a contour in retina and the axis 0x. The space F is identified with the bundle of ( oriented) orthonormal frames and with the group SE(2) = SO(2) . R2 of (unimodular) Euclidean isometries.

The main assumption is that simple neurons are parametrised by the points of F = SE(2). More precisely , the simple neuron, associated to a frame f, is working as the mother Gabor filter in Euclidean coordinates defined by the frame f. Note that in this model, "points" of retina correspond to pinwheels, that is, singular columns of cortex, which contains simple neurons of any orientation. Recall that all simple neurons of a regular column act as identical Gabor filters with (almost) the same receptive field D and they fire when the contour on the retia, which cross D has an appropriate orientation Ɵ.

Recently, this model ( with an appropriate sub-Riemannian metric ) had been successfully applied by B. Franceschiello, A. Mashtakov, G. Citti, and A. Sarti for explanation of some optical illusions.

The contact model had been extended by Sarti, Citti and Petitot to symplectic model, with two-dimensional fiber, associated with the orientation Ɵ and the scaling σ. In this model, simple cells are parametrised by conformal frames or points of the conformal group Sim(E2) = R+ . SE(2). Again, the simple neuron, associated with a frame f acts as Gabot filter, written w.r.t. coordinates associated with f. Hubel and Wiesel introduced a very fruitful notion of hypercolumn of V1 cortex as a minimal collection of regular columns, which contains simple neuron, detecting any possible values of given internal parameters. P. Bressloff and J. Cowan proposed a Riemannian spherical model of a hypercolumn H associated with the orientation Ɵ and spatial frequency p. They assume that a hypercolumn H is associated with two pinwheels S;N , which correspond to minimum and maximum of the spatial frequency. Simple neurons are parametrised by Ɵ and normalised spatial frequency σ ϵ[-∏/2; ∏/2]. More precisely, this means that the simple neuron n(Ɵ;σ) is fired if a stimulus has the orientation Ɵ and the normalised spatial frequency σ. The exception are simple neurons from singular columns, which corresponds to South and North poles S;N and have spatial frequency σ = - ∏/2 and , respectively, ∏/2. They contain simple neurons of any orientation and the longitude coordinate Ɵ is not defined for them. We present a conformal modification of the Bressloff-Cowan model. We assume that a hypercolumn associated with two pinwheels N; S is a conformal sphere with spherical coordinates Ɵ; σ. Simple neurons are parametrised by the conformal Mobius group G ' SL(2;C), hence depends of 6 parameters. More precisely, each simple neuron acts as the mother Gabot filtder w.r.t. conformal coordinated, obtained from the standard coordinates by a conformal transformation from G. We show that in a small neighborhood DS of the South pole , responsible for perception of low frequency stimuli, the model reduces to the symplectic model by Sarti, Citti, Petitot. Similarly, a small neighborhood DN of the north pole N, responsible for perception of higher frequency stimuli, is identify with another copy of the symplectic model. The identification is realised by the stereographic projections from North and respectively , South pole.

The main assumption is that simple neurons are parametrised by the points of F = SE(2). More precisely , the simple neuron, associated to a frame f, is working as the mother Gabor filter in Euclidean coordinates defined by the frame f. Note that in this model, "points" of retina correspond to pinwheels, that is, singular columns of cortex, which contains simple neurons of any orientation. Recall that all simple neurons of a regular column act as identical Gabor filters with (almost) the same receptive field D and they fire when the contour on the retia, which cross D has an appropriate orientation Ɵ.

Recently, this model ( with an appropriate sub-Riemannian metric ) had been successfully applied by B. Franceschiello, A. Mashtakov, G. Citti, and A. Sarti for explanation of some optical illusions.

The contact model had been extended by Sarti, Citti and Petitot to symplectic model, with two-dimensional fiber, associated with the orientation Ɵ and the scaling σ. In this model, simple cells are parametrised by conformal frames or points of the conformal group Sim(E2) = R+ . SE(2). Again, the simple neuron, associated with a frame f acts as Gabot filter, written w.r.t. coordinates associated with f. Hubel and Wiesel introduced a very fruitful notion of hypercolumn of V1 cortex as a minimal collection of regular columns, which contains simple neuron, detecting any possible values of given internal parameters. P. Bressloff and J. Cowan proposed a Riemannian spherical model of a hypercolumn H associated with the orientation Ɵ and spatial frequency p. They assume that a hypercolumn H is associated with two pinwheels S;N , which correspond to minimum and maximum of the spatial frequency. Simple neurons are parametrised by Ɵ and normalised spatial frequency σ ϵ[-∏/2; ∏/2]. More precisely, this means that the simple neuron n(Ɵ;σ) is fired if a stimulus has the orientation Ɵ and the normalised spatial frequency σ. The exception are simple neurons from singular columns, which corresponds to South and North poles S;N and have spatial frequency σ = - ∏/2 and , respectively, ∏/2. They contain simple neurons of any orientation and the longitude coordinate Ɵ is not defined for them. We present a conformal modification of the Bressloff-Cowan model. We assume that a hypercolumn associated with two pinwheels N; S is a conformal sphere with spherical coordinates Ɵ; σ. Simple neurons are parametrised by the conformal Mobius group G ' SL(2;C), hence depends of 6 parameters. More precisely, each simple neuron acts as the mother Gabot filtder w.r.t. conformal coordinated, obtained from the standard coordinates by a conformal transformation from G. We show that in a small neighborhood DS of the South pole , responsible for perception of low frequency stimuli, the model reduces to the symplectic model by Sarti, Citti, Petitot. Similarly, a small neighborhood DN of the north pole N, responsible for perception of higher frequency stimuli, is identify with another copy of the symplectic model. The identification is realised by the stereographic projections from North and respectively , South pole.

We apply this conformal model to visual stability problem which consists in explanation how brain perceives stable external visual objects as stable in spite of transformation of their retina images caused by eyes rotations

**Short Biography**

Name: Dmitri V. Alekseevsky

Birth: 20.08.1940, Moscow, Russia

Citizenship: Russia, UK

Current position: Leading Researcher, Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia,

Emeritus Professor of Hull University..

**Talk Keywords**

primary visual cortex, model of hypercolumns, simple cells and Gabor filter, conformal geometry of sphere and Moebius group, fixational eye movements, saccades, self-avoiding random walk, visual stability problem, diffusion geometry.

**Target Audience**

Students, Post doctoral, Industry, Doctors and professors

**Speaker-intro video**

TBA