#  Cortical Functional architectures
as contact and sub-Riemannian geometry

Giovanna Citti  and  Alessandro Sarti

###### Abstract.

In the paper [67], Jean Petitot together with the authors of the present paper
describe the functional geometry of the visual cortex as the symplectization
of a contact form to describe the family of cells sensitive to position,
orientation and scale. In the present paper, as a ”homage” to the enormous
contribution of Jean Petitot to neurogeometry, we will extend this approach to
more complex functional architectures built as a sequence of contactization or
a symplectization process, able to extend the dimension of the space. We will
also outline a few examples where a sub-Riemannian lifting is needed.

Giovanna Citti, Dipartimento di Matematica, University of Bologna
Alessandro Sarti, CNRS-CREA, Ecole Polytechnique, Paris

##  1\. Introduction

The mammalian visual cortex presents a modular structure since it is
constituted by many families of cells, each one sensitive to a specific
feature of the stimulus image: position, orientation, scale, color, curvature,
movement, stereo and many others. Each of these families presents a
topographical organization in such a way that over every point
$x=(x_{1},x_{2})$ of the retinal plane there is an entire set of cells, each
one sensible to a particular instance of the considered feature. The
topographical organization takes the form of a hypercolumnar structure where
every hypercolumn represents a point of the retinal plane respecting
retinotopy [36].

In his celebrated paper [33] W.C. Hoffmann introduced for the first time
differential fiber bundles for the description of the visual cortex. Here the
base of the fiber bundle represents the retinal plane while fibers represent
the engrafted variables. After his work, subsequent studies were mainly
focused on the set of simple cells responsible for the detection of position
and orientation of edges. A similar differential approach based on Elie Cartan
moving frames has been adopted by Steven Zucker (see [81] and the references
therein). He noted that differential geometry instruments can be used to
formalize the process of boundary completion.

Jean Petitot, together with Yannick Tondut [74] [56], described the set of
simple cells as a fiber bundle equipped with a contact structure and
introduced for the first time the notion of neurogeometry, that is the
intrinsic geometry of neural connectivity. They identified the connectivity
structure between simple cells with the Heisenberg group, and performed
contour completion in this structure minimizing a suitable length functional.
Co-axial long range connectivity found by neurophysiological experiments has
been modelled as integral curves of the contact structure. As well as the
boundary completion of celebrated Kanizsa triangle has been accomplished as
minimization of a functional constrained by the contact structure.

Based on the important work of Petitot and Tondut [74] [56], the authors of
the present paper proposed to represent the functional architecture in terms
of stratified Lie groups with a sub-Riemannian metric to take into account the
local symmetry of the cortex [21]. The hypercolumnar structure of simple cells
is constituted by the complete set of cells sensitive to all orientations and
each cell in the fiber is obtained from a fixed one by rotation. Since each
hypercolumn is obtained translating a fixed one, the structure of this layer
of cells is identified with the Lie group of translations and rotations. It is
well known that different geometries could be defined on this group. The sub-
Riemannian metric was chosen, since the integral curves of its generating
vector fields can also be considered as a mathematical representation of the
association fields of Field, Hayes and Hess [28]. In this model contour
completion is modelled as a front propagation in the sub-Riemannian geometry
[21], [13], [27]. In this structure it has been proved the relation between
neural mechanisms and perceptual completion in both cases of modal and a-modal
completion. See [23] for a review. Analytical properties of this model were
further studied by R. Hladky and Pauls [34], and a semidiscrete version was
proposed in [59]. Finally we recall the works of Duits, van Almsick, Franken,
ter Haar Romeny [30] [5] who proposed models of image processing in the same
Lie group.

As an extension of the models of Petitot [56] and Citti-Sarti [21], the three
authors together proposed a model of functional connectivity of simple cells
in a symplectic structure [67], taking into account translation, rotation and
scale. The approach has been particularly fecund, since it reveals the pattern
of co-axial and trans-axial connections found by neurophysiological
experiments as integral curves of the symplectic structure. The technique
adopted to extend the previous models consist to transform the one-form of the
contact structure in the two-form of the symplectic one, performing an
operation called symplectization. A similar procedure but considering
hyperbolic metrics has been adopted by the same authors in [68]. We also quote
the following more recent neurogeometric models of the functionality of the
cortex [2, 17, 18, 76, 47, 77, 37, 40]. For a more complete introduction to
these models, application to perceptual phenomena or image analysis, we refer
to [58], the entire volume [22] and the references therein.

In the present paper, that we are delighted to dedicate to Jean Petitot as a
homage for his 80th birthday, we will reconsider this approach and will extend
it to describe the modularity of the visual cortex. Particularly we will
consider subsequent contactization and symplectization procedures to model the
neurogeometry of different cortical layers and families of cells. Our approach
is compatible with the model proposed by Hubel and Wiesel in [36]. In fact
they introduced a type of wiring which could produce a simple-cell receptive
field starting from an alignment of retinal cells with radially symmetric
receptive field (see figure 1 left) . A similar type of wiring, produces a
complex cell, starting from an alignment of simple cells (see figure 1 right).

Figure 1. The model of Hubel-Wiesel: the connectivity produces a simple cell
receptive profile starting from an alignment of retinal cells (left), and
produces a complex receptive profile starting from simple cells. Image taken
from [32].

Is an open problem to understand if this is just a mathematical procedure or
the natural process that has been implemented during evolution to introduce
new kind of cells from exiting ones.

##  2\. From functional geometry to differential structure via contactization,
symplectization and sub-Riermannian lifting

Contactization and symplectization are well known mathematical procedures,
which starting respectively from a symplectic or a contact space provide a
contact or a symplectic one, by adding new variables. This instrument was
first applied to model the functionality of the visual cortex by Jean Petitot
in [74],[56], and then extended in [67] and [68] (see also [40]). However they
are insufficient to generate the large variability of the families of cells in
V1, and we will also apply sub-Riemannian lifting.

###  2.1. The retinal plane

The retinal plane $M$ is often identified with the ${\mathbb{R}}^{2}$ space
with an Euclidean geometry. This representation is an idealization, and it is
a good approximation just close to the foveal point because it discards the
fact that the retina is a half sphere. A natural reference frame in this
setting is simply induced by a choice of coordinates and the classical choice
of vector fields

$X_{0;1}=\partial_{x_{1}},\;\;X_{0;2}=\partial_{x_{2}}.$

Here we have denoted $x=(x_{1},x_{2})$ the point of ${\mathbb{R}}^{2}.$ After
the first retinal process, the visual stimulus reaches the Lateral Geniculate
Nucleus without further processing and then the primary visual cortex V1.

###  2.2. Simple cells, modeled via contactization and symplectization

####  2.2.1. Odd Simple cells as a contactization of the retinal plane

Simple cells are the simplest family of cortical cells which process the
signal in V1. It is largely accepted that, in presence of a visual stimulus
$I:M\to R$, the main functionality of this family of cells is to extract the
orientation of level line of $I$ at every point $x=(x_{1},x_{2})$. We can
always assume that level lines of $I$ are parametrized by arch leth. This
means that, if we call $x(s)$ the level line, then $||x^{\prime}||=1,$ and
there exists $\theta$ such that

(1) $x^{\prime}=(\cos(\theta),\sin(\theta)).$

The angle $\theta=\theta(s)$ is called orientation at the point $x=x(s)$, and
the action of the cell is to detect it. In their celebrated paper [56] Petitot
and Tondut modelled the structure of this family of cells as a contact
structure, only based on their functionality. The model has been generalized
in many direction (see for example [21], [67], [68]), and in the present
context it can be expressed as follows. From condition (1), we deduce the
relations $dx_{1}=\cos(\theta)dt$ and $dx_{2}=\sin(\theta)dt$. Removing $dt$
we formally obtain $\sin(\theta)dx_{1}=\cos(\theta)dx_{2}$. Hence, the curve
$x=x(s)$ lying on the retinal plane is lifted to a new curve
$(x(s),\theta(s))$ in V1, which belongs to the kernel of the $1$-form

$\omega_{1}=-\sin(\theta)dx_{1}+\cos(\theta)dx_{2}.$

This is a contact form: a contact form is simply defined as an $1$-form such
that the volume form is non null:

(2) $\omega_{1}\wedge d\omega_{1}\not=0.$

As a result, the action of the simple cells can be considered as a
contactization process, which lift the 2-dimensional complex retinal plane
${\mathbb{C}}$ to a 3 dimensional contact structure: the space will be
obtained by adding the variable $\theta$ to the retinal space

$M_{1}=\\{(x,\theta):x\in{\mathbb{R}}^{2},\theta\in S^{1}\\}.$

The kernel of the form $\omega_{1}$ defines the so called horizontal
distribution 222recall that a distribution $\Delta$ is a sub-bundle of the
tangent space, so that for every $\eta\in M_{1}$, the space $\Delta_{p}$ is a
sub-space of the tangent space at the point $p$ $T_{p}M$., and a possible
choice of generators is defined by the following vector fields $X_{1;i}$, with
$i=1,2$ (as in footnote 1: the first pedix identifies the family of cells, the
second the vector field) :

(3) $X_{1;1}=\cos(\theta)\partial_{x_{1}}+\sin(\theta)\partial_{x_{2}},\quad
X_{1;2}=\partial_{\theta}.$

The fact that $\omega_{1}$ is a contact form, can be equivalently stated
saying that the vector fields satisfy the Hörmander condition 333a family of
vector fields satisfies the Hörmander condition if the generated Lie algebra
coincides with the tangent space at every point.. If we introduce the metric
which makes $X_{1;i}$, with $i=1,2$ orthonormal, we have defined a sub-
Riemannian geometry.

####  2.2.2. Scale dependent simple cells as a symplectization of odd cells

Starting with a contact structure induced by a $1-$form $\omega_{1}$ a
classical way to define a symplectic structure is to multiply $\omega_{1}$ by
a real variable and differentiating. This process is called symplectization.
Precisely the authors of [67] and [68] introduced a new variable $\sigma$ and
an enlarged space

$M_{2}=\\{(x,\eta_{2}):x\in{\mathbb{R}}^{2},\eta_{2}=(\theta,\sigma),\theta\in
S^{1},\sigma\in{\mathbb{R}}\\}.$

In this space they selected as fundamental 1-form:

$\tilde{\omega}_{1}=e^{-\sigma}\left(-\sin(\theta)dx_{1}+\cos(\theta)dx_{2}\right),$

and took as symplectic form on $T^{*}M$ the 2-form obtained by differentiating
$\tilde{\omega}_{1}$ with respect to all its variables. The resulting
symplectic 2-form writes:

$\displaystyle\omega_{2}=d\tilde{\omega}_{1}$ $\displaystyle=$
$\displaystyle\left(e^{-\sigma}\cos(\theta)dx_{1}+e^{-\sigma}\sin(\theta)dx_{2}\right)\wedge
d\theta+$
$\displaystyle+\left(-e^{-\sigma}\sin(\theta)dx_{1}+e^{-\sigma}\cos(\theta)dx_{2}\right)\wedge
d\sigma.$

A basis of the space is composed by the vector fields $X_{2;j}$, where:

(4) $\displaystyle
X_{2;1}=e^{\sigma}\Big{(}\cos(\theta)\partial_{x_{1}}+\sin(\theta)\partial_{x_{2}},\Big{)},\;\;$
$\displaystyle X_{2;2}=\partial_{\theta},$ $\displaystyle
X_{2;3}=e^{\sigma}\Big{(}-\sin(\theta)\partial_{x_{1}}+\cos(\theta)\partial_{x_{2}}\Big{)},\;\;$
$\displaystyle X_{2;4}=e^{\sigma}\partial_{\sigma}.$

###  2.3. Complex cells as extension of simple cells space as contact or
subriemannian structures

We have described a process of contactization and symplectization, which,
starting from the complex retinal plane, give rise to the space of odd and
even simple cells. It correctly models the increase of dimensionality of the
space, due to the selection of eingrafted variables. We can now go on, further
applying a contactization, a simplectization, or building a new sub-Riemannian
structure formally adding variables and differential constraints. We will see
that these extension can describe the functionality of complex cells.

####  2.3.1. Cells sensitive to movement

A large class of cells in V1 shows a very specific space-time behavior [26].
Many models have been proposed to reproduce these dynamics [26, 3], [24], and
we show that a contactization of the set of simple cells can describe the
behavior of these families of cells. Let us first recall that, due to the
aperture problem [35] , we can perceive only movement in the direction
orthogonal to level lines of the stimulus. Using the notation of the previous
section this means that the perceived velocity is defined as

$v=-\sin(\theta)\dot{x}+\cos(\theta)\dot{y}$

where $\dot{x}$ is the derivative with respect to time variable, (and do not
coincide with the derivative performed in (1)). However we are formally in a
similar situation, since the new variable $v$ is related to the spatial and
time variables via a differential constraint $vdt=\omega_{1}$. This condition
naturally suggests to start from $\omega_{1}$ and extend it by adding the
couple of new variables speed and time: $(v,t)$. The resulting $1$\- form is

(5) $\omega_{3}=\omega_{1}-vdt=-\sin(\theta)dx_{1}+\cos(\theta)dx_{2}-vdt.$

This form is defined on the spatio-temporal phase space with fixed frequency,
that is the 5-dimensional manifold

$M_{3}=\\{(x,t,\theta,v):x\in{\mathbb{R}}^{2},t\in{\mathbb{R}}^{+},\theta\in
S^{1}v\in{\mathbb{R}}\\}.$

We will also denote $\eta_{3}=(t,\theta,v),$ so that the general element of
the space, will be $(x,\eta_{3})$. We will call horizontal space the kernel of
$\omega_{3}$. A basis of this space is constituted of the so-called horizontal
or admissible vectors

(6) $\displaystyle
X_{3;1}=\cos\theta\partial_{x_{1}}+\sin\theta\partial_{x_{2}},\;\;X_{3;2}=\partial_{\theta},\;\;$
$\displaystyle
X_{3;3}=-v\sin(\theta)\partial_{x_{1}}+v\cos(\theta)\partial_{x_{2}}+\partial_{t},\;\;X_{3;4}=\partial_{v},$

that defines the horizontal tangent space. In particular we note that the
Euclidean metric on the horizontal planes makes the vector fields $X_{3;j}$
orthogonal. Let us explicitly note that
$[X_{3;1},X_{3;2}]=-\sin(\theta)\partial_{x_{1}}+\cos(\theta)\partial_{x_{2}}=[X_{3;3},X_{3;4}]$
so that the vector fields $X_{3;1},\cdots,X_{3;4}$ span the tangent space at
every point, and the Hörmander condition is satisfied. The same condition can
be expressed recognizing that $\omega_{3}$, is a contact form (see (2) for the
definition).

####  2.3.2. Cells sensible to frequency and phase

Let us explicitly note that the same mathematical procedure of starting from
the set of simple cells and introducing the time and velocity variables, can
be repeated with a couple of different variables satisfying the same
differential relation. In particular it has been experimentally proved that
frequency and phase are selected in the cortex (see [41, 39, 45, 61, 75]). The
resulting mathematical structure which codes for time and velocity will be the
same as coding for frequency and phase: in [9] (see also [40]) a
contactization procedure has been used to extend the form $\omega_{4}$ to the
form

$\omega_{4}=-fsin(\theta)dx_{1}+fcos(\theta)dx_{2}-ds.$

where $f$ is the frequency and $s$ the phase. As before this is a contact form
and the horizontal vector fields are formally defined as the generators of the
kernel of the one form, and satisfy the Hörmander condition.

Let us also mention that the symplectization of this structure give rise to a
multiscale phase and frequency model $M_{5}$ as in [7].

####  2.3.3. Cells selective to curvature

Finally let us explicitly recall that simplectisation and contactization are
not the only way to define a new sub-Riemannian structure. Cells sensible to
curvature were studied by [6], [52] and then by Petitot in [54]. Models of the
space have been proposed in [1, 14]: the associated set of variables is

$M_{6}=\\{(x,\theta,k):x\in{\mathbb{R}}^{2},\theta\in
S^{1},k\in{\mathbb{R}}\\}.$

As before we will call $(x,\eta_{6})$ the general element of the space, where
$\eta_{6}=(\theta,k).$ The curvature is the derivative of $\theta$ with
respect to arch length, so that the associated one form reads

$\omega_{6}=d_{\theta}-k(\cos(\theta)dx_{1}+\sin(\theta)dx_{2})$

If we onsider the structure $(M_{6},\omega_{1},\omega_{6})$, we have obtained
a 4 dimensional space, associated to two $1$ forms. Clearly it is neither a
contact nor a symplectic space, but it can describe the functionality of cells
able to select position, orientation and curvature, by endowing the space with
a subriemannian structure, by defining horizontal tangent plane the kernel of
the forms $\omega_{1}$ and $\omega_{6}$. A set of generators will be:

(7)
$X_{6;1}=\cos(\theta)\partial_{x_{1}}+\sin(\theta)\partial_{x_{2}}+k\partial_{\theta},\;\;X_{6;2}=\partial_{\theta}.$

Note that also these vector fields satisfy the Hörmander condition.

###  2.4. Further extension: hierarchically higher families of cells

From a geometric point of view we can further extend any of these structures,
adding new differntial constraints, new $1$ or 2 forms and choosing as
horizontal space their kernel. Let us mention for example spaces of jet set
[16], stereo [12] or structure in the motor cortex [44]. The element of these
spaces will be denoted $(x,\eta_{i})$, where $x\in{\mathbb{R}}^{2}$ denote the
retinal point and $\eta_{i}$ represent the eingrafted variable selected by the
considered family of cells. We will obtain a family of sub-Riemannian
structures $\\{M_{i}:i\in{\mathbb{N}}\\}$, which is not ordered by the
inclusion. The families of cells we have described in detail are organized as
follows

Retina Simple cells Complex cells $\;\;velocity\;\;\atop{time}\;$
$orientation$ $frequency\atop{phase}$ $\;contrast\;$
$orientation\atop{curvature}$ $orientation\atop{scale}$
$orientation\atop{curvature,scale}$
$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$$\subset$other
cellsother cellsother cellsother cells Figure 2. Structure of families of
cells. The left column represent the retina, whose cells are sensible to
contrast. In the middle column we represent the simple cells, sensible to
orientation and scale. Right column: the complex cells process the output of
simple cells and select velocity and time, frequency and phase, orientation
and curvature.

##  3\. From differential structure to Lie groups representation
as models of receptive profiles

By now we have shown that the formal process of extension of sub-Riemannian
structure can model families of cortical cells, sensible to different
geometric features. We have used only the functionality of cells: the
selectivity of a cell to a differential feature introduces a differential
constraint in the space, which is naturally endowed with a contact, symplectic
or sub-Riemannian structure. We would like to show that this formal process is
consistent with the shape of cells receptive profiles, and the structure of
their set, as they are measured experimentally.

The receptive field (RF) of a visual neuron is typically modeled as functions,
which are positive in the excitatory region, and negative in the inhibitory
one. In the examples considered in the previous sections the set of profiles
is parametrized by the point in the n-space (for example the 3-space
$(x,y,\theta)$ for simple cells): there is a mother profile and the same
profile is transported over every point.

This property will be interpreted as an invariance and described as a Lie
group condition. In order to recover these profiles, we will apply the
uncertainty principle induced by the non commutative properties of the vector
fields $X_{h;i}$ to identify the mother profile. The profiles will be the
minimizers of the uncertainty principle. Then we consider the Lie group
associated to the Lie algebra and use the unitary representation of the group
to define all the other profiles. In other words the group law can be used to
generate the whole family of cells starting from the mother profile. We have
adopted this procedure for example in [8].

###  3.1. The retinal plane: a commutative group

In Section 2.1 we recognized that the Lie algebra acting on the retinal plane
is generated by $\partial_{x_{1}}$ and $\partial_{x_{2}}$. The associated Lie
group is the Euclidean group ${\mathbb{R}}^{2}$. This is in good agreement
with the well known fact that the RPs of the retinal ganglion cells are
usually modeled by Laplacians of Gaussians [42],

$\psi_{0}(y)=\Delta G(y),$

where $G(y)=e^{-|y|^{2}}$ is the Gaussian bell and where $\Delta$ is the
standard Laplacian. This symmetric profile clearly respect the rotational
symmetry of the space. The same receptive profile is replicated over each
point in the retina, i.e. of the ${\mathbb{R}}^{2}$ space and this point
allows to parametrize the receptive profiles with the point where it is
centered.

###  3.2. Simple cells Receptive profiles

####  3.2.1. The set of Simple cells Receptive Profiles as a group

We already recognized that the functionality of simple cells, can be modeled
by a contact structure and that the generators of the horizonal distribution
are the vector fields $X_{1;1},X_{1;2}$, defined in (3). The commutation rules
satisfied by the vector fields $X_{1;i}$ univocally identify the group $SE(2)$
of rotation and translation: it coincides with the experimental observation
that the set of simple cells RPs is obtained via translations and rotations
from a mother profile $\psi_{1}$ (see [38]).

####  3.2.2. The mother profile as the minimum of an uncertaintly principle

The shape of the mother profile has been identified by Daugmann (see [25]) as
a Gabor filter which is minimum of the uncertainty principle in the Heisenberg
group. Folland proved in [29], that it is possible to state an uncertainty
principle any time we have non commuting vector fields, and in [64] we showed
that Gabor filters can be identified as the minima of uncertainty principle
associated to the vector fields $X_{1;i}$, and identifies receptive profiles
of odd and simple cells, for fixed scale as solution of the following PDE:

$X_{1;1}\psi_{1}=iX_{1;2}\psi_{1}.$

The mother profile turn out to be the following one:

(8) $\psi_{1}(y)=e^{-|y|^{2}}e^{2iy_{2}}.$

Note that this profile can be considered the product of the receptive profile
acting on the retinal plane and the complex exponential. This shape is in good
agreement with the model proposed by Hubel and Wiesel in [36], since the
profile is obtained by composing profiles of a hierarchically lower family of
receptive profile.

####  3.2.3. The whole set of profiles and the unitary representation of the
group

The action of the group on ${\mathbb{R}}^{2}$ is expressed as follows,

(11) $\displaystyle A_{x,\theta}(y)$ $\displaystyle=$ $\displaystyle
x+R_{\theta}y,\;\;\text{where
}\;\;R_{\theta}=\left(\begin{array}[]{cc}\cos(\theta)&-\sin(\theta)\\\
\sin(\theta)&\cos(\theta)\end{array}\right)$

Since $\theta$ is the first feature extracted we will call $\eta_{1}=\theta$
and the action of the group on the mother profile, which generates all the
family of filters is expressed as

$A_{x,\eta_{1}}\psi_{1}(y)=\psi_{1}(A^{-1}_{x,\eta_{1}}(y)).$

This can be expressed as a unitary representation of the group. If $G_{1}$
denotes the $SE(2)$ group, $H$ the Hilbert space $L^{2}({\mathbb{R}}^{2}),$ a
unitary representation of the group reads

$\Phi_{1}:G_{1}\to
H\;\quad\Phi_{1}(x,\eta_{1})\psi_{1}(y)=\psi_{1}(A^{-1}_{(x,\eta_{1})}(y)).$

As a result the complete set of filters is obtained as the image of the
unitary representation of the group.

####  3.2.4. cell sensible to scale

In perfect analogy, when we consider the commutation rules associated to the
vector fields defined in (4), identify the group $G_{2}$ of translation,
rotation, and dilation, which indeed is the one experimentally associated to
the receptive profiles able to select position, orientation, and scale (see
[38]).

It has been proved in [4], that it is not possible to impose the uncertainty
principle introduced by Folland and used here in section 3.2.2 to find a
minimum. Other uncertaintly criteria which can provide a minimum, have been
introduced in [4] and [63], but the problem is still partially open. Here we
choose as a mother profile the function introduced in (8):
$\psi_{2}=\psi_{1}$. Then we consider the action of the group $G_{2}$ on the
retinal plane

$\displaystyle A_{x,\eta_{2}}(y)=A_{x,\theta,\sigma}(y)$ $\displaystyle=$
$\displaystyle x+e^{\sigma}R_{\theta}y$

As noted in [67], the group action on the mother filter provides a family of
receptive profiles, $A_{x,\eta_{2}}\psi_{2}$ which well describes cells
sensible to scale.

###  3.3. Complex cells Receptive profiles

####  3.3.1. Spatio-temporal Receptive profiles: the Galilean group

When applying this construction to the vector fields $X_{3;i}$ defined in (6)
we obtain a smooth composition law on $M_{3}$ which reads as follows

(12)
$\begin{array}[]{l}(x,t,\theta,v)\odot(x^{\prime},t^{\prime},\theta^{\prime},v^{\prime})=(R_{\theta}(x^{\prime}+\textstyle{\binom{v}{0}}t^{\prime})+x,t^{\prime}+t,\theta^{\prime}+\theta,v^{\prime}+v).\end{array}$

The manifold $M_{3}$. together with this composition law can also be
identified with a subset $G_{3}$ of the Galileian group. $G_{3}$ acts on the
space of variables $(y,t)\in{\mathbb{R}}^{2}\times{\mathbb{R}}^{+}$ and the
action of the group will be denoted $A_{(}x,t,\eta)$.

An ad hoc version of the uncertainty principle leads to identify as minimum
the function

(13)
$\psi_{3}(y,s)=e^{iy_{2}}e^{-\frac{|y|^{2}}{\sigma_{y}^{2}}-\frac{s^{2}}{\sigma_{s}^{2}}}$

and we will identify all the other filters via the action of $G_{3}$ so that
the generic receptive profile will be denoted

$A_{x,t,\eta_{3}}\psi_{3}(y,s),$

where we recall that $(x,\eta_{3})=(x,\theta,v,t)$. Then the whole set of
filter will be obtained using the unitary representation

$\Phi_{3}:G_{3}\to
H_{3}\;\quad\Phi_{3}(x,\eta_{3})\psi_{3}(y,s)=\psi_{3}(A^{-1}_{x,\eta_{3}}(y,s)),$

where $H_{3}=L^{2}({\mathbb{R}}^{2}\times{\mathbb{R}})$. The family of filters
obtained in this way is the family of Gaussian Gabor filters centered at
position $x\in{\mathbb{R}}^{2}$ on the image plane, activated around time
$t\in{\mathbb{R}}$, with spatial frequency
$\cos(\theta),\sin(\theta),\in{\mathbb{R}}^{2}$, temporal frequency
$\nu\in{\mathbb{R}}$, spatial width $\sigma_{x}\in{\mathbb{R}}^{+}$ (circular
gaussians) and temporal width $\sigma_{t}\in{\mathbb{R}}^{+}$

(14)
$\Phi_{3}(x,\eta_{3})\psi_{3}(y,s)=e^{i(R_{\theta}\cdot(y-x)-\nu(t-s))}e^{-\frac{|x-y|^{2}}{\sigma_{x}^{2}}-\frac{(t-s)^{2}}{\sigma_{t}^{2}}}$

The structure has been obtained as a complectization of the family of simple
cells. Correspondingly the mother filter $\psi_{3}$ is obtained by the by
multiplying the mother profile by a function of time: this seems to be
compatible with the model proposed by Hubel and Wiesel [36], since receptive
profiles are obtained by composition of receptive profiles of hierarchically
lower level cells. In addition these RP are in good agreement with the
experimental evidence provided in [26, 3] and the model proposed in [24],
where it has been shown that a 3-dimensional sum of Gabor filters can fit very
well experimental data of both separable and inseparable RPs.

####  3.3.2. Receptive profiles of cells selective for curvature

The two vector fields introduced in (7), associated to cells sensitive to
curvature, do not commute, and the uncertainty principle stated by Folland,
can be expressed as

$X_{6,1}\psi_{6}=iX_{6,1}\psi_{6}$

The solution will be used to model the mother filter for this family of cells.

From the commutator rules between the vector fields, we can deduce that the
associated Lie group is the Engel one, and that the complete set of receptive
fields can be obtained from a fixed one, by the action of the Engel group:

$RP_{(}x,\eta_{6})(y)=\Phi(x,\eta_{6})\psi_{6}(y).$

###  3.4. Receptive profiles of hierarchically higher cells

The procedure we have described here can be repeated for hierarchically higher
families of cells, identified via differential constraints only if the Lie
algebra generated by the associated vector fields induces a Lie group
structure on the space.

However we can not hope to apply in general Lie groups the Folland uncertainty
principle to identify the associated mother filter. There are indeed groups,
in which the system obtained imposing all uncertainty principle has no
solutions. It is still an open problem to understand which optimization
principles lead to the definition of the mother filter in these groups.

##  4\. Cortical Connectivity and horizontal curves

###  4.1. Fiber bundle structure and the hypercolumnar structure

The differential structure introduced in section 2 describes the direction of
propagation, but all the horizontal directions have the same role.

This does not happen in the spatial organization of the receptive profiles and
in the structure of the connectivity. Experimentally it has been shown that
over each retinal point $x$ there is a whole fiber of variables $\eta_{i}$,
extracted by the functionality of the different families of cells and called
engrafted variables. The roles of these variables is different, and this
organization is called hypercolumnar organization.

Hoffmann [34], and Petitot [56], proposed to use the structure of fiber bundle
to describe the hypercolumnar. The basis of the space are the variables $x$,
and the fiber is constituted by the other variables $\eta_{i}$. However, even
the fiber bundle structure alone is not able to capture the invariance
properties of neural space, which are coded in the Lie group.

###  4.2. Cortical connectivity of simple cells

####  4.2.1. odd simple cells which select orientation

We have modelled the set of simple cells as the group $SE(2)$, and denoted
$(x,\theta)$ its variables, and we selected a horizontal tangent plane spanned
by $X_{1,i}$. The space can also be considered a fiber bundle with basis
$B_{1}={\mathbb{R}}^{2}$, and fiber $F_{1}=S^{1}$. In order to take into
account both structures: the fiber bundle and the sub-Riemannian structure, we
remark that the intersection of the basis $B_{1}$ of the fiber bundle and the
horizonal tangent plane $HM_{1}$ is spanned by the vector $X_{1;1}$: it has a
non trivial projection on the retinal plane. This means that only integral
curves of the vector fields with non vanishing component along $X_{1;1}$ can
be considered lifting of curves defined on the retinal planes, and describe
meaningfull cortical curves.

For this reason the structure of connectivity was modelled in [21] as a curve
$\gamma$ satisfying the following condition:

(15)
$\gamma^{\prime}=X_{1;1}(x,\theta)+kX_{1;2}(x,\theta),\quad\gamma(0)=(x_{0},\theta_{0}).$

Here the parameter $k$ varies arbitrarily in ${\mathbb{R}}$ while the
coefficient of $X_{1}$ is always 1. This means that integral curves of
$X_{1;1}$ will have non trivial projection on the basis of the fiber bundle.
These vector fields provide a good model of the local association field
introduced in [28] and [31]).

The model has been modified in [27] and [13], allowing the coefficient of
$X_{1;1}$ to vanish, considering geodesics of the subriemannian space and
discarding the fiber bundle structure. The curves better fit the association
fields. However the authors proved that, even if they start with this
apparently more general condition, they are obliged to stop the geodesics when
the coefficient of $X_{1;1}$ vanish, since the curves produces cusps. A more
natural approach has been introduced in [43] where the authors go back to the
original model and look for a curve of minimal lenght directly in the set of
curves expressed in the form (15). the curves coded in the cortex, and whose
projection does not produce cusps.

####  4.2.2. Simple cells sensible to scale

Simple cells sensible to scale have been modelled as a symplectic structure in
Section 3.2.4. A symplectic space is a quasi complex space, where the
multiplication for $i$ is replaced by the linear transform $J$ such that

$J(X_{2;1})=X_{2;2},\;J(X_{2;2})=-X_{2;1},\;J(X_{2;3})=X_{2;4},\;J(X_{2;4})=-X_{2;3}.$

Hence in [68] together with J. Petitot, the authors first considered
separately integral curves of vector fields $X_{2;1}$ and $X_{2;2}$ and of the
vector fields $X_{2;3}$ and $X_{2;4}$ . Their pattern generated by the first
one is in good agreement with the pattern of long range connections found both
in neurophysiological and psychophys- ical experiments. Integral curves of the
vector fields $X_{2;3}$ and $X_{2;4}$ model the trans-axial connections, which
extends orthogonally from the orientation axis of the cell.

###  4.3. Cortical connectivity of complex cells

####  4.3.1. cells sensible to movement

The set of cells sensible to movement has been modelled as the sub-Riemannian
structure defined by the manifold $M_{3}=\\{(x,t,\eta_{3})$, where
$\eta_{3}=(\theta,v)$, and the distribution generated by the vector fields
$X_{3,i}$ defined in (6). The space can also be considered a fiber bundle with
basis $B_{3}={\mathbb{R}}^{2}\times{\mathbb{R}}^{+}$, of variables $(x,t)$ and
fiber $F_{3}=S^{1}\times{\mathbb{R}}$, which contains the eingrafted variables
$\eta_{3}=(\theta,v)$. We select the vector fields with non trivial projection
on the retinal plane: $X_{3;1}$ and $X_{3;3}$. As a result, the connectivity
will be modelled by the integral cuves of the vector fields
$X_{3;1}+kX_{3;2}$, which describe long range connectivity for boundary
completion as in the odd simple cells. In addition integral curves of the
vector fields $X_{3;3}+kX_{3;2}$ related to the perception of movement.

###  4.4. Cortical connectivity of hyerarchically higer cells

All families of cells considered up to now have been described as manifolds
$M_{h}$ parametrized as $(x,\eta_{h})$, where $x$ is the retinal variables and
$\eta_{h}$ describes the eingrafted variables. Hence they have a natural
structure of fiber bundle, where $x$ belong to the basis and $\eta$ to the
fiber. For fixed $h$ we selected vector fields $X_{h;1},X_{h;2},...$ which
define a subriemannian metric. The connectivity is modelled by integral curves
of linear combination of these vector fields, with coefficient 1 for the
vector field with non trivial projection on the retinal plane.

##  5\. Action and functionality of receptive profiles, as Bargmann transform

We started from an abstract notion of functionality as selection of
differential features to express our model. We obtained contact, symplectic
and subriemannian structures, directly from the differential properties of the
features. Up to now, the model seems expressed in terms of universal and
geometrical founding principles. The fit between the properties of Lie group
and the shape of minima of uncertainty principle from one side and the
structure and the shape of receptive profiles, provides a first validation of
the project. We verified that integral curves compatibles with the horizontal
distribution and the fiber bundle structure are a good model for the cortical
connectivity. Finally we provide an other validation of the model, proving
that receptive profiles obtained in this way, are able to select the desired
features.

###  5.1. Action of receptive profiles as Bargmann transform

The overall output $O$ of the parallel filtering performed by receptive
profiles is given by the integral of the signal $I$ times the bank of filters:

(16) $\displaystyle O(x,t,\eta)=\int_{M}RP_{x,t,\eta}(y)I(y)dy.$

We present here an unitary description which applies to all the considered
families of cells considered before. In particular for $t=0$ we recover family
of cells independent of time. As remarked in [64] the whole output will be
identified with a Bargmann transform:

(17) $B(I)(x,t,\eta)=\int_{M}\Phi(x,t,\eta)\psi_{h}(y)I(y)dy.$

This transform maps the image $I$ defined on the 2D space, to a function which
depends on all the variables of the total space of the cortical layer. This
integral has the natural meaning of projection of $I$ in the direction of the
specific element $\Phi(x,t,\eta)$ of the frame.

###  5.2. Action of intracortical connectivity and feature selection

Several models have been presented to explain the selectivity of features in
the primary visual cortex. Even if the basic mechanism producing strong
orientation selectivity is controversial (see for example [46, 51, 72]),
nevertheless it is evident that the intra-cortical circuitry is able to filter
out all the spurious directions and to strictly keep the direction of maximum
response of the simple cells.

####  5.2.1. Non maximal suppression and orientation selectivity in simple
cells

While considering simple cells sensible to orientation, the output $O$ in
formula (16) is a function of the variables $x$ and $\theta$ alone. Since the
output $O$ is a regular function, its maximum at a value
$\theta^{*}=\theta^{*}(x)$

$\max_{\theta}O(x,\theta)=O(x,\theta^{*})$

This mechanism was introduced by [21], who proved that the vector
$(-\sin\theta^{*},\cos\theta^{*})$ is orthogonal to the level lines of $I(x)$.
As a consequence, the level lines of $I$ are lifted to curves admissible in
the sense that their tangent vector lies in the kernel of the form
$\omega_{1}$. And this provides a validation of the model since it ensures
that the functionality of these filters is orientation selection.

####  5.2.2. Non maximal suppression in complex cells and selectivity of time
and movement

While taking into account time, the output $O$ defined in formula (16) will be
a function of variables $(x,t,\eta_{3})$ defined on $M_{3}$. The output $O$
takes its maximum at the values $\theta^{*}$ and $v^{*}$

$\max_{\theta,v}O(x,t,\theta,v)=O(x,t,\theta^{*},v^{*}).$

In perfect analogy with the lower dimensional case, the vector
$(-\sin\theta^{*},\cos\theta^{*},v^{*})$ is orthogonal to the level set of
$I$. The vector $(-\sin\theta^{*},\cos\theta^{*})$ is orthogonal to the
spatial level line, and the scalar $v^{*}$ represents the apparent velocity in
this direction (see [24] for the proof).

In this way the receptive profiles are able to select the features of
orientation and velocity.

####  5.2.3. Hierarchically higher level cells

For general families of cells, independent of time and selective to a feature
$\eta$, the output $O$ will be a function of the variables $(x,\eta)$ and the
non maxima suppression mechanism will be expressed as the existence of a value
$\eta^{*}$

$\max_{\eta}O(x,\eta)=O(x,\eta^{*})$

The value of $\eta^{*}$ represents the selection of the value of the engrafted
variables performed by the layer of cells.

##  6\. Discussion

We presented a model of the visual cortex able to describe different families
of cells, and based on three genearal principles. The functionality of each
family of cells defines a one form. The invariance properties of the set of
cells are captured by a Lie group structure. The hypercolumnar structure is
modelled by a fiber bundle structure.

Further models of contact/symplectic structures in neurogeometry have been
proposed recently to model the functional architecture of M1 cells encoding
hand movement direction [44] and to model the functional architecture of
stereo vision [12]. Metric models of the visual cortex have been presented in
[48][49].

Other models have been proposed to face the problem of the sparseness of
orientation hypercolumns in the visual cortex. In fact, being the dimension of
a hypercolumn about $1mm^{2}$ and being a typical dimension of V1 in primates
around $12-16cm^{2}$, the number of hypercolumns will be respectively of
$30\times 40-40\times 40$, that is absolutely insufficient to sample a high
resolution stimulus image, even taking into account the anisotropy of
sampling. To overcome the problem, it has been proposed that all the cells of
a hypercolumn are actively sampling the image stimulus and contribute to its
reconstruction. A heterogeneous Poisson problem has been posed and solved in
[70][71], starting from the concept of heterogenous differential operator
proposed in [69].

Finally a new line of research emerged from the integration of geometric
technics and learning procedures [10] [11].

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