Role of the Cerebellum in the Construction of Functional and Geometrical Spaces

Abstract

The perceptual and motor systems appear to have a set of movement primitives that exhibit certain geometric and kinematic invariances. Complex patterns and mental representations can be produced by (re)combining some simple motor elements in various ways using basic operations, transformations, and respecting a set of laws referred to as kinematic laws of motion. For example, point-to-point hand movements are characterized by straight hand paths with single-peaked-bell-shaped velocity profiles, whereas hand speed profiles for curved trajectories are often irregular and more variable, with speed valleys and inflections extrema occurring at the peak curvature. Curvature and speed are generically related by the 2/3 power law. Mathematically, such laws can be deduced from a combination of Euclidean, affine, and equi-affine geometries, whose neural correlates have been partially detected in various brain areas including the cerebellum and the basal ganglia. The cerebellum has been found to play an important role in the control of coordination, balance, posture, and timing over the past years. It is also assumed that the cerebellum computes forward internal models in relationship with specific cortical and subcortical brain regions but its motor relationship with the perceptual space is unclear. A renewed interest in the geometrical and spatial role of the cerebellum may enable a better understanding of its specific contribution to the action-perception loop and behavior’s adaptation. In this sense, we complete this overview with an innovative theoretical framework that describes a possible implementation and selection by the cerebellum of geometries adhering to different mathematical laws.

Appendix

“Definitions”: a group, noted G for instance, is a collection of entities, that we think of as transformations of something, not necessarily well determined. This collection contains a privileged element, say e, named the neutral element, or the identity of G; and G is equipped with an operation, named its law of composition, or its product, which associates an element to every pair of elements of G, satisfying three axioms: associativity, neutrality of e, and existence of the inverse. These axioms express the usual properties of displacements of objects in space: the succession of two compositions is well defined, without memory of the intermediate step, the identity, being the immobility, when nothing changes, has no effect in composition, and finally, that every motion can be reversed to come back at the original state.

The spatial rigid displacements form the main example of the concept of a group. Another important example is given by the group of permutations of a set of objects: it describes all the manners to exchange the elements of this set between themselves; the product law being the composition of successive permutations; the neutral element being the identity; the inverse of a permutation consisting of replacing things in the former order.

A subgroup H of G is a subset (i.e. a part of G), containing e, which is preserved by-products and inversions.

For instance, if the objects are colored, we can consider the subgroup of permutations that preserve the color.

In general, we define an action (or representation) of G on a set X, as an operation by permutations. More precisely, this is an action to the left, expressed by: (g, x) gives g.x.

The group GA(E) of all the affine transformations of the Euclidian space E, or the Euclidian plane, contains the displacements, but it is strictly richer, it is made by all permutations of E, which sends parallel lines to parallel lines, for instance all dilatation, stretching, squeezing are affine transformations.

For computation, it is practical here to use linear algebra, because every affine transformation can be written as the composition of a translation with a linear map, representable by a matrix.

Generally, the affine transformations do not respect the distance, or the volume. The subgroup that preserves the volume is named equi-affine.

In 3D space, the number of dimensions (degrees of liberty) of displacements is $6$, but for affine transformations, it is 12. For the equi-affine group, we get 11.

Larger than GA(E), we can consider the 3D projective group PGL(P), acting on the projective space P, which is obtained from the affine space E by adding a set of points at infinity. Projective transformations depend on 15 dimensions.

Much larger is the diffeomorphism group, which preserves the continuity and differential regularity of figures; it has an infinite number of dimensions. Without differentials, we get the homeomorphism group.

When we restrict ourselves to the Euclidean plane, of dimension two, the Euclide group of displacements has dimension 3, the equi-affine group dimension 5, and the full affine group dimension 6. More generally, Felix Klein suggested that Geometry is associated with a group G and a subgroup H, in such a manner that the points of the space that G transforms (i.e. permutes) are the subsets (i.e. parts) of G of the special form xH, obtained by applying element x to any element of H, when x describes all G; two elements x, y of G are said equivalent modulo H when xH coincides with yH. The set of equivalence classes is denoted by G/H and named a Klein model of space.

However, observe that if we adopt this definition, a specially marked point appears in the space, which is H itself. A manner to “forget about this point” as in usual geometry, is to consider an action of G on a set X, having two properties: given any pair of points x, y, there exists at least one transformation sending x to y. In some sense this consists in looking at the points through the subgroups that stabilize them, for instance in the Euclidian case, points are replaced conceptually by the rotations around them. The model of space X is called a homogeneous space.

For all the above examples of groups, there exist natural families of sub-groups that define convenient geometries, which are useful for Mathematics, Physics, and Biology.