Orientation Maps in V1 and NonEuclidean Geometry
 Alexandre Afgoustidis^{1}Email author
https://doi.org/10.1186/s1340801500247
© Afgoustidis 2015
Received: 13 November 2014
Accepted: 25 May 2015
Published: 17 June 2015
Abstract
In the primary visual cortex, the processing of information uses the distribution of orientations in the visual input: neurons react to some orientations in the stimulus more than to others. In many species, orientation preference is mapped in a remarkable way on the cortical surface, and this organization of the neural population seems to be important for visual processing. Now, existing models for the geometry and development of orientation preference maps in higher mammals make a crucial use of symmetry considerations. In this paper, we consider probabilistic models for V1 maps from the point of view of group theory; we focus on Gaussian random fields with symmetry properties and review the probabilistic arguments that allow one to estimate pinwheel densities and predict the observed value of π. Then, in order to test the relevance of general symmetry arguments and to introduce methods which could be of use in modeling curved regions, we reconsider this model in the light of group representation theory, the canonical mathematics of symmetry. We show that through the Plancherel decomposition of the space of complexvalued maps on the Euclidean plane, each infinitedimensional irreducible unitary representation of the special Euclidean group yields a unique V1like map, and we use representation theory as a symmetrybased toolbox to build orientation maps adapted to the most famous nonEuclidean geometries, viz. spherical and hyperbolic geometry. We find that most of the dominant traits of V1 maps are preserved in these; we also study the link between symmetry and the statistics of singularities in orientation maps, and show what the striking quantitative characteristics observed in animals become in our curved models.
Keywords
1 Introduction
In the primary visual cortex, neurons are sensitive to selected features of the visual input: each cell analyzes the properties of a small window in the visual field, its response depends on the local orientations and spatial frequencies in the visual scene [1, 2], on velocities or time frequencies [3, 4], it is subject to ocular dominance [1], etc. These receptive profiles are distributed among the neurons of area V1, and in many species they are distributed in a remarkably orderly way [5–7]. For several of these characteristics (position, orientation), the layout of feature preferences is twodimensional in nature: neurons form socalled microcolumns orthogonal to the cortical surface, in which the preferred simulus orientation or position does not change [1, 8]; across the cortical surface, however, the twodimensional pattern of receptive profiles is richly organized [1, 5, 7–10].
It is thus tempting to attribute a high perceptual significance to the geometry of orientation maps, but is a longstanding mystery that V1 should develop this way: there are species in which no orientation map is present, most notably rodents [19, 20], though some of them, like squirrels, have fine vision [21]; on the other hand, it is a fact that orientation maps are to be found in distantly related species whose common ancestor likely did not exhibit regular maps. This has led to an intense (and ongoing) debate on the functional advantage of these ordered maps for perception, on the conditions under which such maps develop, and on the part selforganization has to play in the individual (ontogenetic) development of V1like geometries [11, 20, 22, 23].
Our concern here is not with these general issues, but on geometrical principles that underlie some elements of the debate. We focus on models which have been quite successful in predicting precise quantitative properties of V1 maps from a restricted number of principles.
Our results will be based on methods set forth by Wolf, Geisel and others while discussing development models. They have shown that the properties of mature maps in a large region of V1 (that which is most easily accessible to optical imaging) are well reproduced by treating the mature map as a sample from a random variable with values in the set of possible orientation maps, and by imposing symmetry conditions on this random variable (see Sect. 2).
A remarkable chain of observations by Kaschube et al. [11, 24, 25] has shown that there are universal statistical regularities in V1 orientation maps, including an intriguing mean value of π for their density of topological defects (with respect to their typical length of quasiperiodicity; see Fig. 1 and Sect. 2). Wolf, Geisel and others [12, 26, 27] give a theoretical basis for understanding this; one of its salient features is the use of Euclidean symmetry.
In this discussion, the cortical surface is treated as a full Euclidean plane. Then conditions of homogeneity and isotropy of the cortical surface are enforced by asking for the probability distribution of the mentioned random variable to be invariant under translations and rotations of this plane. This is a condition of invariance under the action of the Euclidean group of rigid plane motions.
There are several reasons for wondering why the cortical surface should be treated as a Euclidean plane, and not as a curved surface like the ones supporting nonEuclidean geometries.

the geometry of the visual field,

the geometry of the cortical surface, that of the actual biological tissue,

or an intermediate functional geometry (e.g. treating motions of solid objects against a fixed background).
It is true that the part of V1 which is accessible to optical imaging is mostly flat, and that we may imagine an affine visual field to be flat as well.
However, we feel these three “planes” should be carefully distinguished, and that using Euclidean geometry simultaneously at all levels is not without significance.
A first, casual remark is that the way the (spherical) retina records the visual field uses its projective properties; it is on a rather functional level that we wan think of “the” affine visual field related to it by central projection from the retina (eye movements are amazingly well adapted to this reconstruction: see [28] for a discussion of motor computation in the Listing plane).
But more importantly, for almost all (if not all) animals which have been investigated, the correspondence between the accessible cortical region and the visual field (the retinotopic map) strongly departs from a central projection: it is logarithmic in nature, with a large magnification factor. For instance, even for the Tree Shrew which is known to have cortical V1 mostly flat, the observed region does not correspond to the center of the retina and the representation of the central field covers the major part of V1. A consequence is that Euclidean plane motions on the cortical surface and rigid motions in the visual field are very different. This is even more strikingly true for cats, primates and humans, whose calcarine sulcus has a more intricate 3D structure [29]. With this in mind, it seems very striking that the functional architecture of V1 should rely on a structure, the “association field” (see [13], Chap. 4) and its condition of “coaxial alignment” of orientation preferences, which simultaneously uses Euclidean geometry at several of these levels. It is also very interesting to note the successful use of shifttwist symmetry (see Sect. 3.2.4), a geometrical transformation which relates rotations on the cortical plane and rotations in the visual plane, in the study of hallucinatory patterns with contours [30] and of fine geometrical properties of V1 maps [31].
Thus when discussing plane motions, we feel that one should carefully keep track of the level (anatomical, functional, “external”) to which they refer. On the other hand, it is quite clear in Wolf and Geisel’s development models that the Euclidean invariance conditions are imposed at the cortical level, independently of the retinotopic map [11, 31].
Is then using flat Euclidean geometry at the cortical level indispensable? A closer look at the literature reveals that, when it appears, Euclidean geometry is endorsed only as a way to enforce conditions of homogeneity and isotropy on the twodimensional surface of cortical V1. This makes it reasonable to look at the conditions of homogeneity and isotropy in nonEuclidean cases.
Now, these two notions are not at all incompatible with curvature; they are central in studying twodimensional geometries with nonzero curvature, discovered and made famous by Gauss, Bolyai, Lobatchevski, Riemann and others. Extending the notions from geometry, analysis and probability to these spaces has been a source of great mathematical achievements in the late 19th and throughout the 20th century (Lie, Cartan, Weyl, HarishChandra, Yaglom). The central concept is that of transformation group, and the corresponding mathematical tools are those of noncommutative harmonic analysis, grounded on Lie group representations. In fact, Wolf and Geisel’s ingredients precisely match the basic objects of invariant harmonic analysis.
Our aim in this paper is to use these tools to define natural V1like patterns on nonEuclidean spaces. Because symmetry considerations are central to the whole discussion, we need our nonEuclidean spaces to admit enough symmetries for the conditions of homogeneity and isotropy to make sense, and we thus consider the twodimensional symmetric spaces. Aside from the Euclidean plane there are but two continuous families of models for such spaces, isomorphic to the sphere and the hyperbolic plane, so these two spaces will be the nonEuclidean settings for our constructions.
The success of Euclideansymmetrybased arguments for describing flat parts of V1 makes it quite natural, from a neural point of view, to wonder whether in curved regions of V1, the layout of orientation preferences develops according to the same principles, and what could be the importance of the metric induced by cortical folding or of “coordinates” which would be induced by flattening the surface (and with respect to which the notion of curvature loses its meaning). It is a matter of current debate whether the threedimensional structure induced by cortical folding has functional benefits; present understanding seems to be that its structure is the result of anatomical constraints (like the tension along corticocortical connections, or the repartition of blood flow; see [32]), but several hypotheses have been put forward to assess its functional meaning (see for instance [32, 33]). In a study trying to assess the importance of cortical folding for orientation maps it would be natural of course to consider variable curvature, but it is difficult to see how symmetry arguments could generalize and even make sense, whereas in regions having large (local) symmetry groups we shall see that it is very natural to adapt the successful arguments for flat V1 after a suitable interpretation of the latter. As we shall point out in the upcoming Discussion, there might also be benefits (in terms of information processing) in having symmetry groups as large as possible in rather extended regions.
Here is an outline of the paper. In Sect. 2, we first proceed to describe some aspects of Wolf and Geisel’s models with the words of representation theory; in this situation the relevant group is the Euclidean group of rigid plane motions. We introduce the probabilistic setting to be used in this paper, that of Gaussian random fields, in Sect. 2.1, and discuss the crucial Euclidean symmetry arguments in Sect. 2.2. We bring group theory into the picture in Sect. 2.3, and irreducible representations in Sect. 2.4.
To pass over to nonEuclidean geometries, we then examine what happens if the Euclidean group is replaced by the isometry groups of other symmetric spaces; we thus define “orientation maps” on surfaces of negative or positive curvature. For symmetric spaces the curvature is a numerical constant, and after a renormalization the twodimensional symmetric spaces turn out to be isomorphic with the Euclidean plane, the hyperbolic plane or the round sphere. We begin Sect. 3 with the hyperbolic, negatively curved setting rather than the spherical, positivelycurved one, because there are closer links with flat harmonic analysis in that case. After introducing our orientationpreferencelike maps on these spaces, we emphasize the important part symmetry plays in the existence of the universal value for defect (pinwheel) densities in V1 maps by discussing the density of topological defects in nonEuclidean orientation maps.
As we shall see, in the Euclidean case, irreducible representations enter the picture through the existence of a dominant wavelength in the correlation spectrum; our recent paper in this journal [34] focuses on the role of this monochromaticity condition in getting a precise pinwheel density and quasiperiodicity. Although some of our results can find motivation from a few remarks in that paper, the present study is independent from [34].
2 Methods
2.1 Gaussian Random Fields
Let us add that if there is a pinwheel center at \(x_{0}\), by definition^{1} \(\mathbf{z}_{\mathrm{exp}}\) takes all values of the argument in a neighbourhood of \(x_{0}\), so \(\mathbf{z}_{\mathrm{exp}}(x_{0})\) must be zero. On the other hand, the modulus of \(\mathbf {z}_{\exp}\) may loosely be interpreted as a measure of orientation selectivity: when orientation tuning at \(x_{0}\) is poor, all of the \(a_{\vartheta_{k}}(x_{0})\) will be approximately the same, so \(x_{0}\) will be close to a zero of \(\mathbf{z}_{\mathrm{exp}}\), while if orientation selectivity at \(x_{0}\) is sharp, the numbers \(a_{\vartheta_{k}}(x_{0})\) for which \(\vartheta_{k}\) is close to the preferred orientation will be much larger than the others, and the modulus of z will be rather high at \(x_{0}\).
With this interpretation, we may discuss any complexvalued smooth function z on a surface X as if its argument were an orientation map, and its modulus were a measure of orientation selectivity. Orientation selectivity near pinwheel centers is being actively researched and debated, see [9, 35] and the references in [36], so interpreting the modulus of the vector sum \(z_{\mathrm{exp}}\) in this way might be questioned, but this tradition dates back to 1982 [18].
If mathematical models yielding plausible maps z are to be furnished, then certainly they should be compared to the multitude of maps observed in different individuals. Let us neglect, for a given species, the slight differences in cortical shape and assume that each test subject comes with a coordinate system on the surface of its V1, so that we may compare a given map from \(\mathbb {R}^{2}\) to \(\mathbb {C}\) to the orientation map observed in this individual.
We can then compare the different individual maps, leading to map statistics; if orientation maps are to be described mathematically, it seems fair to hope for a mathematical object that produces, rather than a single complexvalued function with the desired features, statistical ensembles of realisticlooking maps [12]. This approach might not be the best way to account for the finer properties of mature maps as experimentally observed, and it is certainly a rough approximation that needs to be confronted with the output of more biologically plausible development models. However, it does have the advantage of mathematical simplicity, and as we shall see, it is particularly well suited to discussing the part symmetry arguments have to play in producing realistic maps.
So what we need is a random field, that is, a random variable with values in the set of smooth maps from \(\mathbb {R}^{2}\) to \(\mathbb {C}\). Since the set of smooth maps is infinitedimensional, we cannot expect to find interesting “probability distributions” from closed formulae [37, 38]; but in the case of V1, the general theory of random fields and the available biological information make it possible to describe special fields whose “typical realizations” yield rather realistic maps [26, 38]. When we go over to nonEuclidean settings in this paper, we shall see that the mathematical description can be adapted to provide special random fields defined on nonEuclidean spaces; their typical realizations will yield V1like maps adapted to the considered nonEuclidean geometry.
But let us now make our way toward the special fields on Euclidean space whose typical realizations look like orientation maps.
Measured statistical properties of real orientation maps include correlation functions [31]: it turns out that the structures of correlation measured in different individuals look very much alike. This is important: many discussions take the architecture of correlations to be essential to the horizontal wiring of V1, and to be at the heart of its perceptual function [13, 39]; it is also at the heart of striking results on the distribution of singularities in OPMs [11, 12, 26]. So using models that reproduce this correlation structure seems to be a good idea, and there is a way to associate special random fields to correlation structures.
Definition
A complexvalued random function z on a smooth manifold M (a collection \(\mathbf{z}(x)\), \(x \in M\) of complexvalued random variables) is a complexvalued centered Gaussian random field (GRF) if, for every integer n and every ntuple \((x_{1}, \ldots ,x_{n}) \in M^{n}\), the \(\mathbb {C}^{n}\)valued random variable \((\mathbf{z}(x_{1}), \ldots, \mathbf{z}(x_{n}))\) is Gaussian with zero mean. Its correlation function \(C: M^{2} \rightarrow \mathbb {C}\) is the (deterministic) map \((x, y) \mapsto\mathbb{E} [ \mathbf {z}(x) \bar{\mathbf{z}}(y) ]\).
Just as a Gaussian probability distribution on \(\mathbb {R}\) is available when a value for expectation and a value for variance are given (and is the “best bet”, that is, the minimum entropy distribution, given these data [40]), a continuous twopoint correlation function \(C: M^{2} \mapsto \mathbb {C}\) (together with the zeromean requirement in the definition we use here) determines a unique GRF thanks to an existence theorem by Kolmogorov: see [41], Theorem 12.1.3, [38], and [37], p. 4.^{2}
In what follows, we shall always require that \(C: M^{2} \rightarrow \mathbb {C}\) be smooth enough; in fact we will only meet fields with realanalytic correlation functions. Maps drawn from such fields are almost surely smooth, so there is no regularity problem ahead.
Before we add symmetry constraints on our Gaussian fields, note that \(C(x, x)\) is the variance of \(\mathbf{z}(x)\); this depends on a choice of unit for measuring orientation selectivity. We shall proceed to a convenient one in the next subsection.
2.2 Euclidean Symmetry in V1
Let us for the moment deal with the cortical surface as if it were an Euclidean plane \(\mathbb {R}^{2}\). In a grown individual, different points on this plane correspond to neurons that usually do not have the same orientation preference, whose connectivity reaches out to different subsets of the cortex [13, 30, 39]; at some points we find sharp orientation tuning and under others (pinwheels) a less clear behaviour. In short, two different points on the cortical surface usually have different parts to play in the processing of visual information. But experimental evidence [7, 42] suggests very clearly that no particular point on this plane should have any distinguished part to play in the general design of the orientation map (e.g. be an organizational center for the development of the map, or have a systematic tendency to exhibit a particular orientation preference in the end).
The above assumption is then that the probability distribution of z is invariant under the action of E(2) on the set of maps [26].
This implies that \(C(x,y)\) depends only on \(\xy\\), and in the case of a Gaussian field this apparently weaker form of invariance is actually equivalent to the invariance of the full probability distribution. Let us write \(\varGamma: {\mathbb {R}^{2}} \rightarrow \mathbb {C}\) for the radial function such that \(C(x,y) = \varGamma(xy)\) for all x and y, and note that up to a global rescaling of the modulus used to measure orientation selectivity, we may (and will) assume \(\varGamma(0) = 1\).
Further discussion of correlations may be conducted using Γ, and there is an important remark to be made here: the highfrequency components of its Fourier transform record local correlations, while lowfrequency components in \(\widehat{\varGamma}\) (the Fourier transform of Γ) point to longrange correlations. If z is to produce a quasiperiodic layout of orientation preferences with characteristic distance Λ, this seems to leave no room for systematic correlations at a much longer or much shorter distance than Λ. So it seems reasonable to expect that Gaussian fields generating plausible maps have \(\widehat{\varGamma}\) supported on the neighbourhood of a circle with radius \(\frac{2\pi}{\varLambda}\). Following Niebur and Worgotter, Wolf and Geisel and others, we note that this further hypothesis on \(\widehat{\varGamma}\) is all that is needed to generate realisticlooking maps.
The simplest way to test this claim is to use what we shall call a monochromatic invariant random field, a field in which \(\widehat {\varGamma}\) actually has support in a single circle, and consequently is the Dirac distribution on this circle:^{3} \(\varGamma(\vec{r}) = \int_{\mathbb {S}^{1}} e^{i \frac{2\pi}{\varLambda} \vec{u} \cdot\vec{r}} \,d\vec{u}\).
This looks realistic enough. Now, what is truly remarkable is that it is not only on a first, qualitative look that this map, which has been computergenerated from simple principles, has the right features: it also exhibits a pinwheel density of π, which Kaschube et al. have observed in real maps with 2 % precision [11].
This result appeared in physics [12, 26, 47] and is now supported by full mathematical rigor (see [48], Chap. 6, for the general setting and [49], Sect. 4 for the full proof); we shall use the same methods to derive nonEuclidean pinwheel densities in Sect. 3. We should note here that as translationinvariant random fields of our type have ergodicity properties (see [38], Sect. 6.5), it is quite reasonable to compare ensemble expectations for Gaussian fields and pinwheel densities which, in experiments, are measured on individual orientation maps.
Of course the correlation spectra measured in real V1 maps are not concentrated on an infinitely thin annulus (for precise measurements, see Schnabel [31], p. 103). But upon closer examination (see for instance [34]), one can see that maps sampled from invariant Gaussian fields whose spectra are not quite monochromatic, but concentrated on thin annuli, do not only look the same as that of Fig. 2, but that many interesting quantitative properties (such as pinwheel density or a low variance for the spacing between isoorientation domains) have vanishing firstorder terms as functions of spectral thickness. In other words, it is reasonable to say that monochromatic invariant random fields provide as good a description for the layout of orientation preferences as invariant fields with more realistic spectra do (perhaps even better; see [34]). As we shall see presently, neglecting details in the power spectrum and going for maximum simplicity allows for a generalization that will lead us to pinwheellike arrangements in nonEuclidean settings. We shall take this step now and start looking for nonEuclidean analogues of monochromatic fields.
 (1)
a randomness structure, that of a smooth Gaussian field;
 (2)
an assumption of Euclidean invariance;
 (3)
and a monochromaticity, or neartomonochromaticity condition out of which quasiperiodicity in the map arose.
When we go over to nonEuclidean settings, these are the three properties that we shall look for. The first only needs the surface on which we draw orientation maps to be smooth. For analogues of the last two conditions in nonEuclidean geometries we need group actions, of course, and a nonEuclidean notion of monochromatic random field. In the next two subsections, we shall describe the appropriate tool.
Before we embark on our program, let us note that in spite of the close resemblance between maps sampled from monochromatic Gaussian fields and real mature maps, there are notable differences. As we remarked above, real correlation spectra are not infinitely thin, and the precise measurements by Schnabel make it possible to give quantitative arguments for the difference between an invariant Gaussian field with the measured spectrum (see for instance a discussion in [34]). In the successful longrange interaction model of Wolf, Geisel, Kaschube and coworkers, Gaussian fields turn out to be a better description of the initial stage of cortical map development than they are of the mature stage. We have two reasons for sticking to Gaussian fields in this paper: the first is that they are ideally suited to discussing and generalizing the concepts crucial to producing realistic maps, and the second is that a nonEuclidean version of the longrange interaction model can easily be written down in the upcoming Discussion.
2.3 Klein Geometries
What is a space M in which conditions (1) and (2) have a meaning? Condition (1) says we should look for Gaussian fields whose trajectories yield smooth maps, so M should be a smooth manifold. To generalize condition (2), we need a group of transformations acting on M, with respect to which the invariance condition is to be formulated. Felix Klein famously insisted [50] that the geometry of a smooth manifold M on which there is a transitive group action is completely determined by a pair \((G, K)\) in which G is a Lie group and K a closed subgroup of G. We shall recall here some aspects of Klein’s view, focusing for the twodimensional examples which we will use in the rest of this paper. This is famous and standard material; see the beautiful book by Sharpe [51], Chap. 4.
First, let us examine the previous construction and note that every geometrical entity we met can be defined in terms of the Euclidean group \(\mathit{SE}(2) \). write K for the subgroup of rotations around a given point, say o. If \(g = (R, \vec{x})\) is any element of G, the conjugate subgroup \(gKg^{1} = \{ (A, \vec{x} A \vec{x}), A \in K \}\) is the set of rotations around \(o+\vec{x}\). Now, the set \(gK = \{ (A, x), A \in K \}\) remembers x and only x, so we can recover the Euclidean plane by considering the family of all such cosets, that is, the set \(G/K = \{ gK, g \in G \}\).
Now when G is a general Lie group and K is a closed subgroup, the smooth manifold \(M = G/K\) comes with a natural transitive Gaction, and K is but the subset of transformations which do not move the point \(\{K\}\) of M. This is summarized by saying that M is a Ghomogeneous space.
With this in mind, we can rephrase our main objective in this paper: it is to show that some Klein pairs \((G, K)\) allow for a construction of V1like maps on the homogeneous space \(M = G/K\) , and a calculation of pinwheel densities in these V1like maps. We shall keep M twodimensional here, and stick to the three maximally symmetric spaces [52]—the Euclidean plane, the round sphere and the hyperbolic plane.
To recover the usual geometry of the round sphere \(\mathbb{S}^{2}\) from a Klein pair, we need the group of rotations around the origin in \(\mathbb {R}^{3}\), that is, \(G = \mathit{SO}(3) = \{ A \in\mathcal{M}_{3}(\mathbb {R})  ^{t}AA = I_{3} \text{ and } \det(A) = 1 \} \), and the closed subgroup \(K = \{ \bigl({\scriptsize\begin{matrix}R & 0\cr 0 & 1 \end{matrix}}\bigr) , R \in \mathit{SO}(2) \}\)—of course K is the group of rotations fixing \((0,0,1)\).
But now let us go forward to meet one of the many reasons why Klein’s description of spherical and hyperbolic geometries, far from being a matter of aesthetics, shows our concrete tools of map engineering the way to nonEuclidean places.
2.4 Group Representations and Noncommutative Harmonic Analysis
2.4.1 Unitary Representations
Assume we are given one of the two nonEuclidean Klein pairs \((G, K)\) above and we wish to build an orientation map with properties (1), (2), and (3) from Sect. 2.2. Conditions (1) and (2) say we should use a smooth complexvalued Gaussian random field that is invariant under the action of G. We shall come back to exploiting condition (2) in time. But condition (3) depends on classical Fourier analysis, which is based using plane waves and thus seems tied to \(\mathbb {R}^{n}\).
Fortunately there is a completely grouptheoretical description of classical Fourier analysis too: for details, we refer to the beautiful survey by Mackey [54]. One of its starting points is the fact that for functions defined on \(\mathbb {R}^{n}\), the Fourier transform turns a global translation of the variable (that is, passage from a function f to the function \(x \mapsto f(x  x_{0})\)) into multiplication by a universal (nonconstant) factor (the Fourier transform \(\widehat{f}\) is turned into \(k \mapsto e^{i k x_{0}} \widehat {f}(k)\)). From this behaviour of the Fourier transform under the action of the group of translations, some of those properties in Fourier analysis which are wonderful for engineering—like the formula for the Fourier transform of a derivative—follow immediately.
For many groups, including \(\mathit{SO}(3)\) and \(\mathit{SU}(1,1)\) which we will use in this paper, there is a “generalized Fourier transform” which gives rise to analogues of the property we just emphasized, although it is technically more sophisticated than classical Fourier analysis. It is best suited to analyzing functions defined on spaces with a Gaction, yielding concepts of “generalized frequencies” appropriate to the group G [55, 56].
It will then come as no surprise that the vocabulary of noncommutative harmonic analysis is well suited to describing the invariant Gaussian field model for orientation preference maps in V1, since the key features of this model rest on the action of \(\mathit{SE}(2) \) on the function space of orientation maps. As soon as we give details, it will also be apparent that an analogue of the monochromaticity condition (3) can be formulated in terms of these “generalized frequencies”.
Before we discuss its significance and its relevance to Euclidean (and nonEuclidean) orientation maps, we must set up the stage for harmonic analysis; so we beg our reader for a little mathematical patience until Sect. 2.4.3 brings us back to orientation maps.
Let G be a Lie group. Representation theory starts with two definitions: a unitary representation of G is a continuous homomorphism, say T, from G to the group \(\mathcal{U}(\mathcal{H})\) of linear isometries of a Hilbert space; we write \((\mathcal{H}, T)\) for it. This representation is irreducible when there is no \(T(G)\)invariant closed subspace of \(\mathcal{H}\) except \(\{0\}\) and \(\mathcal{H}\).
 1.
If p is a vector in \(\mathbb {R}^{n}\), define \(T_{p}(x) = e^{i p \cdot x}\) for each \(x \in G = \mathbb {R}^{n}\); this defines a continuous morphism from \(G = \mathbb {R}^{n}\) to the unit circle \(\mathbb {S}^{1}\) in \(\mathbb {C}\); by identifying this unit circle with the set of rotations of the complex line \(\mathbb {C}\), we may say that \((\mathbb {C}, T_{p})\) is an irreducible, unitary representation of \(\mathbb {R}^{n}\). In fact, every irreducible unitary representation of \(\mathbb {R}^{n}\) reads \((\mathbb {C}, T_{p})\) where p is a vector. Thus, the set of irreducible representations of the group of translations on the real line or of an ndimensional vector space is nothing else than the set of “time” or “space” frequencies in the usual sense of the word.^{4}
 2.
Suppose M is the real line, the Euclidean plane, the sphere or the hyperbolic plane, and G the corresponding isometry group. If f is a complexvalued function on M, define \(L(g) f:= x \mapsto f(g^{1} \cdot x)\). Then for every \(g \in G\), \(L(g)\) defines a unitary operator acting in the Hilbert space \(\mathbb{L}^{2}(M)\) (here integration is with respect to the measure determined by the metric we chose on M); so we get a canonical unitary representation \((\mathbb {L}^{2}(M), \mathcal {L})\) of G. It is very important to note that this representation is reducible in our four cases; we discuss its invariant subspaces in the next subsection.
A word of caution: our first example, although it is crucial to understanding how representation theory generalizes Fourier analysis, is much too simple to give an idea of what irreducible representations of nonabelian groups are like; for instance, the space \(\mathcal{H}\) of an irreducible representation very often happens to be infinitedimensional (the first and most famous examples are in [57]), and this will be crucial in our discussion of hyperbolic geometry.
2.4.2 Plancherel Decomposition
Suppose M is the Euclidean plane, the hyperbolic plane or the sphere. We shall now give an outline of the Plancherel decomposition of \(\mathbb{L}^{2}(M)\), which is crucial to our strategy for producing nonEuclidean orientation maps. This is standard material: for details, we refer to [53], Chap. 0.
When G is the rotation group \(\mathit{SO}(3)\) and M is the sphere, or more generally when G is compact, this is actually what happens, and it is a part of the Peter–Weyl theorem that the above direct sum decomposition holds. In the case of the sphere, all the \(m_{\gamma}\)s will be equal to 1 and we will describe the \(\mathcal{H}_{\gamma}\) in Sect. 3.2.1. But for the noncompact groups \(\mathit{SE}(2)\) and \(\mathit{SU}(1,1)\), the decomposition process turns out to degenerate.
A simpler example will help us understand the situation: consider the representation \(\mathcal {L}\) of \(\mathbb {R}\) on \(\mathbb{L}^{2}(\mathbb {R})\) (example 2 above). Since a change of origin induces but a (nonconstant) phase shift in the Fourier transform, the subspace \(\mathcal{F}_{I}\) of functions whose Fourier transform has support in interval I, is invariant by each of the \(\mathcal {R}(x)\), \(x \in \mathbb {R}\). But now it is true also that, say, \(\mathcal{F}_{[0,1]} = \mathcal{F}_{[2,2.5]} \oplus \mathcal{F}_{[2.5, 3]} = \mathcal{F}_{[2,2.25]} \oplus\mathcal{F}_{[2.25,2.5]} \oplus\mathcal{F}_{[2.5,2.75]} \oplus\mathcal{F}_{[2.75,3]}\), and so on. Since we can proceed to make the intervals smaller and smaller, we see that an irreducible subspace should be a onedimensional space of functions which have only one nonzero Fourier coefficient, in other words, each member of the irreducible subspaces should be a plane wave… which is not a squareintegrable function! So in this case, there is no invariant subspace of \(\mathbb{L}^{2}(\mathbb {R})\) that inherits an irreducible representation from \(\mathcal {R}\), and it is only by getting out of the original Hilbert space that we can identify irreducible “constituents” for \(\mathbb {L}^{2}(\mathbb {R})\).

are invariant under the canonical operators \(\mathcal {L}(g)\), \(g \in G\),

carry irreducible unitary representations of G,

and together give rise to the following version of the Plancherel formula: for each \(f \in\mathbb{L}^{2}(M)\) and for almost every x in M,where \(\mathcal{F}\) is some set of equivalence classes of representations of G (the “frequencies”), Π is a measure on \(\mathcal{F}\) (the “power spectrum”), and for each \(\omega\in\mathcal {F}\), \(f_{\omega}\) is a member of \(\mathcal{E}_{\omega}\) (a smooth function, then).$$f(x) = \int_{\omega\in\mathcal{F}} f_{\omega}(x)\, d\varPi(\omega), $$
Recall that our aim in introducing noncommutative harmonic analysis is to find an analogue of the monochromaticity condition (3), Sect. 2.2, in spherical and hyperbolic geometry. As we shall see presently, the situation in the Euclidean plane makes it reasonable to call an element of \(\mathcal{E}_{\omega}\) or \(\mathcal{H}_{\gamma}\) a monochromatic map. Belonging to one of the \(\mathcal{E}_{\omega}\), resp. one of the \(\mathcal{H}_{\gamma}\), will be our nonEuclidean analogue of the monochromaticity condition (3) in hyperbolic geometry, resp. spherical geometry. We shall see that a Gaussian random field providing orientationpreferencelike maps may be associated to each of these spaces of monochromatic maps, and that it yields quasiperiodic tilings of M with Euclideanlike pinwheel structures.
2.4.3 Relationship with Euclidean Symmetry in V1
 (3′):

\(\Delta\varGamma=  ( \frac{2\pi}{\varLambda} )^{2} \varGamma \)

if φ is in \(\mathcal {E}_{\varLambda}\), then \(g\cdot\varphi : x \mapsto\varphi(g^{1} x)\) is in \(\mathcal {E}_{\varLambda}\) for any \(g \in E(2)\); this means \(\mathcal {E}_{\varLambda}\) is an invariant subspace of the set of smooth maps;

\(\mathcal {E}_{\varLambda}\) has itself no closed invariant subspace if one uses the usual smooth topology for it: indeed if φ is any nonzero element in \(\mathcal {E}_{\varLambda}\), it may be shown that the family of maps \(g\cdot\varphi\), \(g \in G\), generates a dense subspace of \(\mathcal {E}_{\varLambda}\). Perhaps a word of caution is useful here: while Γ is rotationinvariant and determines a Ginvariant random field, it is certainly not itself invariant under the full group G of motions.
 (3″):

Γ belongs to one of the elementary invariant subspaces \(\mathcal {E}_{\varLambda}\).
This shows that the \(\mathcal{E}_{\varLambda}\) do provide the factors in the Plancherel decomposition of \(\mathbb{L}^{2}(\mathbb {R}^{2})\) described at the end of Sect. 2.4.2, and the equivalence between conditions (3) and (3″) shows how the spectral thinness condition found in models is related to the Plancherel decomposition of \(\mathbb{L}^{2}(\mathbb {R}^{2})\). In Sect. 2.2, we saw how each of this factors determines a unique Gaussian random field which provides realistic V1like maps.
We now have gathered all the ingredients for building twodimensional V1like maps with nonEuclidean symmetries. But before we leave the Euclidean setting, let us remark that the irreducible representation of \(\mathit{SE}(2)\) carried by the \(\mathcal{E}_{\varLambda}\) has been used in [39], although the presentation there is rather different.^{7} While the approach of [39], which brings the horizontal connectivity to the fore and uses Heisenberg’s uncertainty principle to exploit the noncommutativity of \(\mathit{SE}(2)\), has notable differences with using Gaussian random fields, it is very interesting and defines realvalued random fields which are good candidates for the maps \(a_{\vartheta}\) of Sect. 2.1. To the author’s knowledge this is the first time irreducible representations of \(\mathit{SE}(2)\) were explicitly used to study V1, and reading this paper was the starting point for the present study.
3 Results
3.1 Hyperbolic Geometry
Let us now turn to plane hyperbolic geometry. The relevant groups for capturing the global properties of the hyperbolic plane assemble in the Klein pair \((G,K) = (\mathit{SU}(1,1), \mathit{SO}(2))\) as described in Sect. 2.
If we are to look for pinwheellike arrangements lurking in the representation theory of \(\mathit{SU}(1,1)\), we need a familiarity with some irreducible representations. We shall use the next paragraph to give the necessary details on the geometry of the unit disk; the description of all unitary representations of \(\mathit{SU}(1,1)\), however, we shall skip over^{8} in order to focus on the Plancherel decomposition of \(\mathbb{L}^{2}(G/K)\).
We should note at this point that hyperbolic geometry and \(SL_{2}\)invariance^{9} have been used by Chossat and Faugeras for a different purpose [59]; the same basic ingredients will appear here.
3.1.1 Geometrical Preliminaries
Just as a family of parallel lines in \(\mathbb {R}^{2}\) has an associated family of parallel hyperplanes that are orthogonal to each line in the family, the set of Aorbits has an associated family of parallel horocycles: writing \(b_{0}\) for the point of the boundary \(\partial \mathbb{D}\) that is in the closure of every Aorbit (i.e. the point \(1+0i\) in \(B = \partial\mathbb{D}\)), a circle that is tangent to \(\partial\mathbb{D}\) at \(b_{0}\) meets every Aorbit orthogonally. What is more, given two such circles, there is on any Aorbit a unique segment that meets them both orthogonally; the length of this hyperbolic geodesic segment does not depend on the Aorbit chosen, so it is very reasonable indeed to call our two circles parallel. Circles tangent to \(\partial\mathbb{D}\) were named horocycles by Poincaré, so we have been looking at the (parallel) family of those horocycles that are tangent to \(\partial\mathbb{D}\) at \(b_{0}\).
There is one more definition that we shall need: it is closely linked to an important theorem in the structure of semisimple Lie groups [52, 60].
Theorem
Every element \(g \in G = \mathit{SU}(1,1)\) may be written uniquely as a product ^{10} \(k a n\), where \(k \in K\), \(a \in A\), and \(n \in N\). This is known as an Iwasawa decomposition for G.
Note that K, A, and N are onedimensional subgroups of G, but that the existence of a unique factorization \(G = KAN\) does not mean at all that G is isomorphic with the direct product of K, A, and N.
Even if this is a very famous result, an idea of the proof will be useful for us. Note first that any point \(x \in\mathbb{D}\) may be reached from O by following the horizontal geodesic for a while (forwards or backwards) until one reaches the point of the horizontal axis which is on the same horocycle in the family of Norbits, then going for x along this circle; this means that we can write \(x = n\cdot (a \cdot O)\), where \(n \in N\) and \(a \in A\); now if g is any element of G, we may consider \(x = g \cdot O\) and write it \(x = n \cdot(a \cdot O) = (na) \cdot O\); then \((na)^{1}g\) sends O to O, so it is an element of K. This proves the existence statement; uniqueness is easy but more technical.
Now if b is a point of the boundary \(\mathbb{D}\) that has principal argument θ, we may view it as an element of K by assigning to it the element \(\bigl({\scriptsize\begin{matrix} e^{i \theta} & 0 \cr 0 & e^{i\theta} \end{matrix}}\bigr) \); note that this element acts on \(\mathbb{D}\) as a rotation of angle 2θ! So beware, diametrally opposite elements b and −b of the boundary define the same rotation.
3.1.2 Helgason Waves and Harmonic Analysis
At first sight, there is no reason why harmonic analysis on the hyperbolic plane should “look like” Euclidean harmonic analysis: their invariance groups are apparently quite different and there is nothing like an abelian “hyperbolic translation group” whose characters may obviously be taken as a basis for building representation theory. So it may come as a surprise that there are analogues of plane waves in hyperbolic space, and (more importantly) that these enjoy much the same relationship to hyperbolic harmonic analysis as Fourier components do to Euclidean analysis. The discovery of these plane waves can be traced back to the seminal work of HarishChandra [61] on spherical functions of semi simple Lie groups (we shall come back to this in a moment), and their systematic use in nonEuclidean harmonic analysis is due to Helgason [53, 62]. Since they will be a key ingredient in the rest of this section, let us now describe these waves.
This is indeed the Laplace operator for \(\mathbb{D}\): it can be defined from grouptheoretical analysis alone, in much the same way we obtained the Poincaré metric in Sect. 2.3 and the Appendix. Theoretical questions aside, the reader may check easily that this new Laplacian is Ginvariant, that is, \(\Delta_{\mathbb{D}} [ f(g^{1} \cdot) ] = [ \Delta_{\mathbb{D}} f ](g^{1} \cdot)\).
For each continuous function \(\mu: B \rightarrow \mathbb {R}\), we then know that \(\mathcal{P}_{\omega}(\mu)\) belongs to \(\mathcal{E}_{\omega }(\mathbb{D})\), and in fact the image of \(\mathcal{P}_{\omega}\) is dense in \(\mathcal{E}_{\omega}(\mathbb{D})\) for several natural topologies (see [53], Chap. 0, Theorem 4.3, Lemma 4.20). Since \(\Delta_{\mathbb{D}}\) is Ginvariant, \(\mathcal{E}_{\omega}(\mathbb {D})\) is a stable subspace of \(\mathcal{C}^{\infty}(\mathbb{D})\); by studying \(\mathcal{P}_{\omega}\), Helgason was able to prove that the \(\mathcal{E}_{\omega}(\mathbb{D})\) is irreducible ([53], Chap. 0, Theorem 4.4). The following theorem then achieves the Plancherel decomposition of \(\mathbb{L}^{2}(\mathbb{D})\) in the sense of Sect. 2.4.2, and is a cornerstone of harmonic analysis on the unit disk (see [53], Chap. 0, Theorem 4.2; the extension to \(\mathbb{L}^{2}\) is proved there also):
Theorem
(HarishChandra, Helgason)
When we build our hyperbolic maps in the next section, the \(\mathcal {E}_{\omega}(\mathbb{D})\) are the only representations we shall need. We will come back to this shortly.
3.1.3 Hyperbolic Orientation Maps
It is time to build our hyperbolic analogue of Orientation Preference Maps. Suppose we wish to arrange sensors on \(\mathbb{D}\) so that each point of \(\mathbb{D}\) is equipped with a receptive profile which has an orientation preference and a selectivity. This may be local model for an arrangement of V1like receptive profiles on a negatively curved region of the cortical surface, and though its primary interest is probably in clarifying the role of symmetries in discussions, the construction to come can be thought of in this way.
We shall require that this arrangement have the same randomness structure (condition (1)) as the Euclidean model of Sect. 2, that is, be a “typical” realization of a standard complexvalued Gaussian random field on the space \(\mathbb{D}\), say z. If it is to have an analogous invariance structure (conditions (2) and (3)), it should, first, be assumed to be Ginvariant; what is more, we should look for a field that probes an irreducible factor of the representation of G on \(\mathbb{L}^{2}(\mathbb {D})\) (see Sect. 2.4.2); as a consequence, any realization of z should be an eigenfunction of \(\Delta_{\mathbb{D}}\), with the eigenvalue determined by z. Remembering the Euclidean terminology we used in Sect. 2, let us introduce the following notion.
Definition
A monochromatic Gaussian field on \(\mathbb {D}\) is a complexvalued Gaussian random field on \(\mathbb{D}\) whose probability distribution is \(\mathit{SU}(1,1)\)invariant and which takes values in one of the \(\mathcal{E}_{\omega}\), \(\omega>0\). If z is such a field, the positive number ω will be called the spectral parameter of z.
To see how to build such a monochromatic field, we should translate our requirements into a statement about its covariance function; luckily there is a theorem here ([63], see the discussion surrounding Theorem 6′ and Theorem 7, in particular Eq. (3.20) there) that says our conditions on z are fulfilled if, and only if, the covariance function of z, when turned thanks to the Ginvariance of z into a function from \(\mathbb{D}\) to \(\mathbb {C}\), is an elementary spherical function for \(\mathbb{D}\) (a radial function on \(\mathbb{D}\) which is an eigenfunction \(\Delta_{\mathbb{D}}\)). What does this mean?
First, note that the covariance function \(C: \mathbb{D}^{2} \rightarrow \mathbb {C}\) of our field may be seen as a function \(\tilde{C} = G^{2} \rightarrow \mathbb {C}\): we need only set \(\tilde{C}(g_{1}, g_{2}) = C(g_{1} \cdot O, g_{2} \cdot O)\). Now, that z should be Ginvariant means that for every \(g_{0} \in G\), \(\tilde{C}(g g_{1}, g g_{2})\) should be equal to \(C(g_{1}, g_{2})\); in particular, writing \(\varGamma(g)\) for \(\tilde{C}(g, 1_{G})\), we get \(\tilde{C}(g_{1}, g_{2}) = \varGamma(g_{2}^{1} \cdot g_{1})\). The whole of the correlation structure of the field is summed up in this Γ, which is a function from G to \(\mathbb {C}\).
Now, not every function from G to \(\mathbb {C}\) can be obtained in this way: since it should come from a function C which is defined on \(\mathbb {D}^{2}\) and thus satisfies \(C(g_{1}k_{1},g_{2}k_{2}) = C(g_{1}, g_{2})\) when \(k_{1}\), \(k_{2}\) is in K, it should certainly satisfy \(\varGamma(k_{1} g k_{2}) = \varGamma(g)\) for \(k_{1}, k_{2} \in K\); so Γ does in fact define a function on \(\mathbb{D}\) and this function is leftKinvariant, that is, radial in the usual sense of the word (property (A)). What is more, since the field is assumed to have variance 1 everywhere, it should also satisfy (B) \(\varGamma(\mathrm{Id}_{G}) =1\) (property (B)).
Let us now add that monochromaticity for z is equivalent to Γ being an eigenfunction of \(\Delta_{\mathbb{D}}\) (property (C)).
Functions on \(\mathbb{D}\) with properties (A), (B), and (C) are called elementary spherical functions for \(\mathbb{D}\).
Now, we stumbled upon these (following Yaglom) while looking for pinwheellike structures, but spherical functions (and their generalizations to semisimple symmetric spaces) have been intensely studied in the last half of a century. In fact, they were defined by Elie Cartan as early as 1929 with the explicit objective of determining the irreducible components of \(\mathbb{L}^{2}(G/K)\) for a large class of Klein pairs \((G,K)\). The following theorem will look like an easy consequence of everything we discussed earlier, but history went the other way and it is in looking for spherical functions that HarishChandra discovered what we called Helgason waves.
Theorem
(HarishChandra 1958, [61])
If we plot \(\varphi_{\omega}\) it will resemble the Euclidean Besselkind covariance functions; only there is a marked difficulty in dealing with the growth at infinity of these functions, which accounts for some (not all, of course) of the many difficulties HarishChandra and Helgason had to overcome in developing harmonic analysis on \(\mathbb{D}\).
The properties of elementary spherical functions include the conditions which guarantee, thanks to the existence theorem by Kolmogorov mentioned in Sect. 2, that each of the \(\varphi_{\omega}\) really is the covariance function of a Gaussian field on \(\mathbb{D}\). So we can summarize the preceding discussion with the following statement.
Proposition A
For each \(\omega>0\), there is exactly one monochromatic Gaussian field on \(\mathbb{D}\) with spectral parameter ω.
These are our candidates for providing V1like maps on \(\mathbb{D}\). We now need to see, by plotting one, whether a “typical” sample of a monochromatic field looks like a hyperbolic V1like map, hence we need to go from the covariance function to a plot of the field itself. All technical details aside, Euclidean and hyperbolic spherical functions are close enough for the transition from a spherical function to the associated Gaussian field to be exactly the same in both cases. We said nothing of this step in Sect. 2, so let us come back to the Euclidean setting for a second.
This construction depends only on properties which are common to the Euclidean and hyperbolic plane; thus, it transfers unimpaired to the hyperbolic plane.
So there does appear a quasiperiodic tiling of the unit disk; it should not of course be forgotten that this quasiperiodicity holds only when the area of an “elementary cell” is measured in the appropriate hyperbolic units (see the previous section and the next).
3.1.4 Hyperbolic Pinwheel Density
What is this area σ of an elementary cell, by the way, and can we estimate the density of pinwheels per area σ?
In the Euclidean case, we used results from physics that originally dealt with superpositions of Euclidean waves. Of course singularities in superpositions of random waves do occur in many interesting physical problems: interest first came from the study of waves traveling through the (irregular) arctic surface [65]; quantum physics has naturally been providing many interesting random superpositions: they occur in laser optics [66], superfluids [67]… . This has prompted recent mathematical developments. In this section, we would like to point out that these are now sharp enough to allow for calculations outside Euclidean geometry.
Consider an invariant monochromatic random field z, and write ω for the corresponding “wavenumber” (so that z belongs to \(\mathcal{E}_{\omega}\)). We would like to evaluate the expectation for the number of pinwheels (zeroes of z) in a given domain \(\mathcal {A}\) of the unit disk. Let us write \(\mathcal {N}_{\mathcal {A}}\) for the random variable recording the number of pinwheels in \(\mathcal {A}\). We will now evaluate the expectation of this random variable, and the result will be summarized as Theorem A below.
Theorem
(see [48], Theorem 6.2)
To use this theorem, we should note (see [38]) that in a field with constant variance, at each point p the value any derivative of any component of the field is independent (as a random variable) from the value of the field at p; so the two variables \(\operatorname{Det}[d\mathbf {z}](p)\) and \(\mathbf{z}(p)\) are independent too; thus for invariant fields on \(\mathbb{D}\) we know that the hypotheses in the theorem are satisfied, and that in addition we may remove the conditioning in the expectation formula. So we are left with evaluating the mean determinant of a matrix whose columns are independent Gaussian vectors, with zero mean and the same variance \(V_{p}\) as \(\partial_{x} \mathop{\mathfrak{Re}}(\mathbf{z}) (p)\). We are left with evaluating the Euclidean area of the random parallelogram generated by these random vectors, and using the “base times height” formula it is easy to prove this mean area is \(2V_{p}\).
So we need to see that \(\int_{\mathcal {A}} \frac{V_{p}}{\pi}\, dp\) is equal to \(\mathcal {A}_{h} \frac{V_{0}}{\pi}\). But this is easy: when the realvalued Gaussian field \(\zeta= \mathop{\mathfrak{Re}} \mathbf{z} \) is Ginvariant, we can define a Ginvariant Riemannian metric on \(\mathbb{D}\) by setting \(g^{\zeta}_{ij}(p) = \mathbb{E} \{ \partial_{i} \zeta(p) \partial _{j} \zeta(p) \}\); as we said in Sect. 2, this must be a constant multiple of the Poincaré metric. It follows that \(V_{p}\) is equal to \(\eta(p) V_{0}\), while \(\eta(p)\) is the hyperbolic surface element. This proves the announced formula \(\mathbb{E} \{ \mathcal {N}_{\mathcal {A}} \} ={\mathcal {A}_{h}} \frac{V_{0}}{\pi} \).
Theorem A
It is worth pointing out that the proof above no longer features any reference to wave propagation; we just needed the invariance properties of our covariance function and a nice property of our new Laplacian. This means that calculations should travel unimpaired to geometries where nothing like wave propagation is available for building spherical functions and representation theory. We shall see this at work on the sphere in the next section.
But let us linger a moment in the hyperbolic plane, for our new monochromatic maps do exhibit a rather unexpected feature: while samephase wavefronts in \(e_{\omega,b}\) line up at hyperbolic distance \(\frac{2\pi}{\omega}\), it seems like the right hyperbolic area for a “hyperbolic hypercolumn”, that area which we called σ at the beginning of this subsection, should be \(\frac{4\pi^{2}}{\omega^{2} + 1}\). In fact, we claim that the typical hyperbolic distance between two points in the map that have the same orientation preference is not \(\frac{2\pi}{\omega}\) as we would guess by thinking in Euclidean terms, but \(\frac{2\pi}{\sqrt{\omega^{2} + 1}}\). There is something of course to support of this claim: we can evaluate the typical spacing by selecting a portion of a geodesic and evaluate the mean number of points with a given orientation preference. To motivate the statement of our result, see the discussion preceding Theorem C below, and also [34].
Theorem B
Note that the zeroes considered here are points on Σ where the preferred orientation is the vertical, and have nothing to do with pinwheel centers (which were the zeroes considered in Theorem A).
3.2 Spherical Geometry
Let us now examine the positivelycurved case, viz. the sphere \(\mathbb {S}^{2}\). Recall from Sect. 2 that the geometry of the sphere is captured by the Klein pair \((\mathit{SO}(3), \mathit{SO}(2))\).
We will start by looking for an orientation preferencelike map on the sphere. Let us therefore look for an arrangement z with our usual randomness structure, that is, for a complexvalued standard Gaussian random field on the space \(\mathbb {S}^{2}\); let us further assume that the field z is Ginvariant and probes an irreducible factor of the natural representation of \(\mathit{SO}(3)\) on \(\mathbb{L}^{2}(\mathbb {S}^{2})\) (see Sect. 2.4.2). The arguments we used for the hyperbolic plane go through, so we are now looking for a Gaussian random field whose covariance function is an elementary spherical function for \(\mathbb {S}^{2}\).
In the last section, we built these out of hyperbolic analogues of Euclidean plane waves; here there is no obvious “plane wave” candidate for carrying the torch. However, it is quite easy to find alternative buildingblocks for the irreducible factors of the representation of G on \(\mathbb{L}^{2}(\mathbb{S}^{2})\): these are the familiar spherical harmonics; since there will be a significant difference between the maps we shall describe and those we encountered on nonpositively curved spaces of the preceding sections, we shall use the next paragraph to examine their rôle in representation theory even if this is famous textbook material; see [68], Chap. 7.
3.2.1 Preliminaries on the Spherical Harmonics and the Plancherel Decomposition of \(\mathbb{L}^{2}(\mathbb {S}^{2})\)
We should add at this point that the natural representation of \(\mathit{SO}(3)\) on \(\mathcal{H}_{\ell}\) is indeed irreducible; in the next section we shall associate an orientation preference map to each of the \(\mathcal {H}_{\ell}\). But before we close this section, let us see how this relates to the decomposition of \(\mathbb{L}^{2}(\mathbb {S}^{2})\).
Notice that all analytic difficulties in the decomposition have vanished (the irreducible factors \(\mathcal{H}_{\ell}\) are really spaces of squareintegrable functions), and that Fourier series are enough to reconstruct a function, which means here that a countable set of irreducible representations is enough to decompose \(\mathbb {L}^{2}(\mathbb {S}^{2})\). Recall from Sect. 2.4.2. that there is a simple reason for the marked differences between what happens on the sphere and what happens in our previous examples: Hermann Weyl proved that when the group G is compact, there is but a countable set of equivalence classes of irreducible representations.
3.2.2 Spherical Orientation Maps
We have now at our disposal everything that is needed for orientation preferencelike maps on the sphere, and on top of it, one important observation: our set of spherical maps, unlike the set of its Euclidean or Hyperbolic analogues, is discrete in nature. Out of the spherical harmonics \(Y_{\ell m}\) arises one irreducible factor of \(\mathbb{L}^{2}(\mathbb {S}^{2})\) per ℓ; we feel it is appropriate to name the corresponding invariant Gaussian random field a spin ℓ monochromatic field.
3.2.3 Spherical Pinwheel Density
Expectation values for pinwheel densities in spherical maps may be evaluated with the same methods we used in the previous sections. Here, however, there appears a significant difference with the Euclidean and Hyperbolic cases: while monochromatic fields in those cases were indexed by a continuous parameter that is easily interpreted as a wavelength, there is apparently no natural scale for writing pinwheel densities.
 (a)
What is the mean (spherical) distance Λ between isoorientation domains in a field that probes \(\mathcal{H}_{\ell}\)?
 (b)
What is the mean number of pinwheels within a given subset of the sphere, relative to the (spherical) area of this subset? Is it \(\frac {\pi}{\varLambda^{2}}\)?
To answer the first question, let us select a geodesic segment on \(\mathbb {S}^{2}\), that is, a portion of a great circle. What is, on this segment, the mean number of points where \(\varPhi_{\ell}\) exhibits a given orientation? Since standard Gaussian fields are shiftinvariant, we can consider a fixed value of the orientation, say the vertical. Points where \(\varPhi_{\ell}\) exhibits this orientation are points where \(\mathop{\mathfrak{Re}}(\varPhi_{\ell})\) vanishes; so let us define \(\varPsi_{\ell} = \mathop{\mathfrak{Re}}(\varPhi_{\ell})\) and look for its zeroes on the given great circle.
Theorem C
Thus, if the mean number of points on Σ to which \(\varPhi_{\ell}\) attributes a given orientation preference is to be no less or no more than one, the length of Σ must be Λ. This answers question (a).
Now, if \(\mathcal{A}\) is a subset of the sphere, denote by \(A_{s}\) its spherical area and by \(\mathcal{N}_{\mathcal{A}}\) the random variable recording the number of pinwheels of \(\varPhi_{\ell}\) in A. Then, as in the previous cases, we observe a scaled density of π:
Theorem D
Let us give a sketch of proof of Theorem D: since the only difference with the hyperbolic case is the lack of global coordinates which simplified the presentation there, we think it is better to keep this proof short and refer to our upcoming Ph.D. thesis for full details. A first step is to adapt the formula by Azais and Wschebor (the version in Sect. 3.1.4 holds when the field is defined on an open subset of \(\mathbb {R}^{n}\)) to prove that \(\mathbb{E} \{ \mathcal {N}_{\mathcal {A}} \} ={\mathcal {A}_{s}} \frac{V_{0}}{\pi}\), where \(V_{0}\) is the variance of any derivative of \(\mathop{\mathfrak{Re}}(\mathbf{z})\) at a point \(p_{0}\) on \(\mathbb{S}^{2}\). Now to evaluate \(V_{0}\), we use the fact that it is equal to the expectation for the second partial derivative (in any direction) at \(p_{0}\) of the covariance function Γ of \(\mathop{\mathfrak{Re}}(\mathbf{z})\). This expectation does not depend on the chosen direction, and to adapt the arguments in the proof of Theorem A we can use the grouptheoretical interpretation of \(\Delta_{\mathbb {S}^{2}}\) as the Casimir operator associated to the action of \(\mathit{SO}(3)\) on \(\mathbb{S}^{2}\) (see [43], Sect. 5.7.7). As in the proof of Theorem A, we can then evaluate \(V_{0}\) as half the value of \(\Delta_{\mathbb{S}^{2}}(\varGamma)\) at \(p_{0}\), but because \(\Delta_{\mathbb {S}^{2}} \mathbf{z} = (\frac{2\pi}{\varLambda} )^{2} \mathbf{z}\) this halfvalue turns out to be \(\frac{\pi}{\varLambda}^{2}\), proving Theorem D.
3.2.4 An Alternative Orientation Map, with ShiftTwist Symmetry
We have so far been looking for arrangements of V1like receptive profiles on curved (homogeneous) surfaces; for this we used complexvalued random fields. We shall now look for a pinwheellike structure on the sphere which is of a slightly different kind, perhaps more likely to be of use in discussions which include horizontal connectivity, or which relate to the vestibular system and its interaction with vision. We will also provide a simple criterion on pinwheel densities to distinguish between our two types of spherical maps.
In this subsection \(\mathbb {S}^{2}\) sits as the unit sphere in \(\mathbb {R}^{3}\), and we try to arrange threedimensional abelian Fourier coefficients on the sphere: in other words, we assume each point \(\vec{u}\) on \(\mathbb {S}^{2}\) is equipped with a sensor whose receptive profile depends on a plane wave \(x \in \mathbb {R}^{3} \mapsto \operatorname{exp} \langle\omega(\vec{u}) \cdot x\rangle\) (this profile could be a threedimensional Gabor wavelet). Here \(\omega(\vec{u}) \in \mathbb {R}^{3}\) is a linear form on \(\mathbb {R}^{3}\) (so it may be thought of as a vector). Let us assume further that at each point \(\vec{u}\), the corresponding sensor neglects everything that happens in directions collinear to \(\vec{u}\), so that \(\omega(\vec{u})\cdot v = 0\) as soon as \(\vec{v} \perp\vec{u}\).
This kind of arrangement does not seem very interesting if (a part of) the sphere is thought of as a piece of cortical surface, and we do not set it forth as a model for a visual area; yet it would not be completely unreasonable to think of an arrangement like this if \(\vec {u}\) were to stand for gaze direction, and it makes sense (not to say that it is useful) to consider a remapping of this structure across the cortical surface (this would displace the interpretation of the pinwheellike layout, which would only exist at a functional level).
This formula is familiar from differential geometry; in fact, our set of maps is precisely the set \(\varOmega^{1}(\mathbb {S}^{2})\) of (vector fields or, more accurately) differential 1forms on the sphere. Now, let us come back to \(\varOmega^{1}(\mathbb {S}^{2})\): we can add two such maps, so \(\varOmega ^{1}(\mathbb {S}^{2})\) is a vector space. After a suitable completion, we may consider the Hilbert space \(\varOmega^{1}_{\mathbb{L}^{2}}(\mathbb {S}^{2})\) of forms which are squareintegrable, and since rotations are unitary maps, and writing \(P(g)\) for the map \(\omega\mapsto g_{\star} \omega\) whenever g is a rotation, we get a unitary representation \((\varOmega^{1}_{\mathbb {L}^{2}}(\mathbb {S}^{2}), P)\) of the rotation group.
Using this representation may look rather unnatural in biology; but corresponding transformations have been discussed in the flat case, though with a very different language: in [27, 31] they are called Shifttwist transformations. Indeed, differential forms on \(\mathbb {R}^{2}\) can be identified with functions from \(\mathbb {R}^{2}\) to \(\mathbb {C}\), and the natural action on differential forms of a rotation around the origin^{15} \((A, 0) \in \mathit{SE}(2)\) is turned in this way into the operation \(f \mapsto A f( A^{1} \cdot)\) on complexvalued functions, which is exactly the shifttwist transformation considered in [27, 31] (compare Sect. 2.3 in [27]).
Bringing the horizontal connectivity and notions like the association field into the picture ([13, 30], Chap. 4), it seems natural to introduce the (co)tangent bundle of the surface on which orientation maps are to be developed.
Now, the unitary representation \((\varOmega^{1}_{\mathbb{L}^{2}}(\mathbb {S}^{2}), P)\) is of course not irreducible; so in order to get “elementary arrangements”, we may look for its irreducible constituents as we did for \(\mathbb{L}^{2}(\mathbb {S}^{2})\) and hope that pinwheellike structures are to be found there.
There is a useful remark here: if f is a realvalued smooth function on the sphere, its derivative df provides us with an element of \(\varOmega^{1}_{\mathbb{L}^{2}}(\mathbb {S}^{2})\). What is more, if g is a rotation, then \(P(g) \,df = d [ \vec{u} \mapsto f(g^{1} \vec{u}) ]\). So any Ginvariant irreducible subspace \(\mathcal{H}_{\ell}\) of \(\mathbb {L}^{2}(\mathbb {S}^{2})\) yields a Ginvariant irreducible subspace \(\mathcal{H}_{\ell}^{\mathrm{exact}}\) of \(\varOmega^{1}_{\mathbb{L}^{2}}(\mathbb {S}^{2})\): we need only consider the derivatives of real parts of elements of \(\mathcal{H}_{\ell}\).
Is there more to fields probing other irreducible factors of \(\varOmega^{1}_{\mathbb{L}^{2}}(\mathbb {S}^{2})\) than what we see on the \(\mathcal{H}_{\ell}^{\mathrm{exact}}\)? There is not, for there is a duality operation on \(\varOmega^{1}_{\mathbb{L}^{2}}(\mathbb {S}^{2})\) which will allow us to describe all the other irreducible factors. This is the Hodge star: to define it in our very particular case, notice first that if \(\vec{u}\) is a unit vector, we get an notion of oriented bases on the plane \(\vec{u}^{\perp}\) from “the” usual notion of oriented basis in the ambient space. Then, start with a differential form ω, and shift each of the (co)vectors \(\omega(\vec{u})\) with a rotation of angle \(+\frac{\pi}{2}\) in each (co) tangent space; this gives a new form ✩ω. Obviously it is orthogonal to ω, and what is more, it commutes with rotations: ${g}_{\star}(\text{\u2729}\omega )=\text{\u2729}({g}_{\star}\omega )$ for any rotation g.
Random differential forms probing the \(\mathcal{H}_{\ell}^{\mathrm{coexact}}\) have exactly the same orientation preference layout as those we have already met, except for a difference of “chirality” that corresponds to a global shift of the orientations. We should note here that (the probability distributions for) our nontwisted fields \(\varPhi_{\ell}\) were unchanged under a global shift of the orientations.
While our new maps do resemble the nontwisted orientation maps of the previous paragraph, looking at pinwheel densities will reveal a notable difference. Indeed, although there is a formula of Kac–Rice type for the mean number of critical points of an invariant monochromatic field like \(\mathop{\mathfrak{Re}}(\varPhi_{\ell})\), it involves a Hessian determinant at the place where we earlier met the Jacobian determinant of \(\varPhi_{\ell}\)—this was the determinant of a random matrix with independent coefficients, which is not the case for any Hessian (symmetric!) matrix.
4 Discussion
In this paper, we started from a reformulation of existing work by Wolf, Geisel and colleagues, with the aim to understand the crucial symmetry arguments used in models with the help of noncommutative harmonic analysis, which is often a very wellsuited tool for using symmetry arguments in analysis and probability. Understanding these Euclidean symmetry arguments from a conceptual standpoint showed us that Euclidean geometry at the cortical level is a way to enforce conditions that are not specific to Euclidean geometry but have a meaning on every “symmetric enough” space, and we thus saw how a unique Gaussian random field providing V1like maps can be associated to each irreducible “factor” in the Plancherel decomposition of the Hilbert space of squareintegrable functions on the Euclidean plane, the hyperbolic plane and the sphere. We proved that in these three settings, when scaled with the typical value of column spacing, monochromatic invariant fields exhibit a pinwheel density of π. Theorems A′ and D′ in the Appendix prove that the same result holds when the monochromaticity condition is dropped: in other words, a pinwheel density of π appears as a signature of (shift) symmetry. Since pinwheel densities can be measured in individual sample maps thanks to the ergodicity properties of invariant Gaussian fields (see [38], Sect. 6.5), this yields a criterion to see whether an individual map (which cannot be itself invariant!) is likely to be a sample from a field with an invariance property, whether the map be drawn on a flat region or on a curved, homogeneous enough region. In the spherical case, also we saw that the number of pinwheels in the map can in principle distinguish between rotationinvariance and shifttwist symmetry; to see whether this observation can be turned into a precise criterion distinguishing the various kinds of invariance from actual measurements on individual sample maps, it would probably be interesting to see whether there is anything to be said of pinwheel densities in Euclidean or hyperbolic maps with shifttwist symmetry, and of the mean column spacing in shifttwist symmetric maps.
Since our aim was to understand the role of symmetry arguments, one aspect restricting the scope of our constructions in a fundamental way is our focusing on homogeneous spaces rather than spaces with variable curvature. Of course, we have good technical reasons for this: the way symmetry arguments are used in existing discussions made it natural to focus on those twodimensional spaces which have a large enough symmetry group, and our constructions are entirely based on exploiting the presence of this symmetry group. One might wish to make the setting less restrictive, especially since the places where the surface of real brains is closest to a homogeneous space are likely to be the flat parts. But using analogues of symmetry arguments on nonsymmetric spaces is a major challenge in (quantum) field theory, and if one wished to start from the reformulation we gave of Wolf and Geisel’s work in Sect. 2.4, generalizing the arguments of this paper to find V1like maps on Riemannian manifolds on nonconstant curvature would be formally analogous to adapting Wigner’s description of elementary particles on Minkowski spacetime to a general curved spacetime—a challenge indeed! Answering this challenge would bring us close to the twodimensional models from quantum field theory or statistical mechanics, and make us jump to infinitedimensional “phase spaces” (and wouldbe groups). This is a step the author is not ready to take, and it is likely that simpler ways to study the nonhomogeneous case would come with shifting the focus from mature maps back to development models.
Indeed, readers familiar with development models have perhaps been puzzled by another aspect of our paper, which is the fact that we used Gaussian random fields as the setting for our constructions: Gaussian fields provide sample maps which look very much like orientation maps, and as we emphasized the statistical properties of their zero set are very strikingly reminiscent of what is to be found in real maps, but there are appreciable and measurable differences between the output of invariant Gaussian fields and real orientation maps (see for instance a discussion in [34]). As we recalled in the Introduction, it is likely that Gaussian fields provide a better description for the early stage of cortical map development, but that the Gaussian description later acquires drawbacks because it is not compatible with the nonlinearities essential to realistic development scenarii.
Allowing z to evolve from an initial fluctuation, when \(\gamma< 1\) and when \(\sigma/\varLambda\) is large enough Eq. (1) leads first to an invariant, approximately Gaussian field (thanks to an application of the central limit theorem to a linearized version of Eq. (1); see [26]), then to nonGaussian quasiperiodic V1like random fields.
As Wolf and coworkers point out (see for instance the supplementary material in [11], Sect. 2), this partial differential equation is the Euler–Lagrange equation of a variational problem, so solutions are guaranteed to converge to stable stationary states as time wears on. Wolf and colleagues showed (using numerical studies) that when \(\sigma/\varLambda\) is large enough, V1like maps are among the stable solutions in the Euclidean case. On arbitrary Riemannian manifolds, however, there is no way to guarantee that structurerich stable solutions of the above PDE exist; it would certainly be worth examining, at least with numerical simulations, but this is beyond the author’s strengths at present. It is perhaps natural to imagine that given the analogy between maps obtained by truncation from invariant GRFs and the output of the longrange interaction model, the constructions in this paper are a strong indication that on symmetric spaces, the stable solutions of (1) include maps which look like those of Figs. 5–7, and that the difference between those and the monochromatic invariant fields studied in this paper is analogous to the difference between experimental maps, or at least the output of (1) in the Euclidean case, and maps sampled from invariant Gaussian fields on \(\mathbb {R}^{2}\).
In adult animals measurements seem to indicate that the structure of mature maps departs from that of maps sampled from invariant Gaussian fields; remarkably, there is experimental evidence for the fact that a pinwheel density of π, which in a Gaussian initial stage appears as a signature of Euclidean symmetry as we saw, is maintained during development in spite of the important refinements in cortical circuitry and the departure from Gaussianity that they induce [11]. Independently of modeling details, we see that geometrical invariance can be measured in principle, even on individual maps: upon evaluating local column spacings (with respect to geodesic length in the curved case) and performing space averaging, one gets a length scale Λ; when scaling pinwheel density with respect to Λ, observing a value of π is a strong indication that geometrical invariance on the cortical surface is an important ingredient in development.
In addition to this, one might think that arranging neurons and their receptive profiles on a homogeneous enough space has benefits from the point of view of information processing: by allowing the conditions of homogeneity and isotropy to make sense, a constant curvature could help distribute the information about the stimulus in a more uniform way (note that as the eyes move constantly, a given image is processed by many different areas in V1 in a relatively short time). Neurons receiving inputs from several adjacent regions of V1 could then handle spike statistics which vary little as the sensors move and have a more stable worldview.
There are some conditions for this, called positivedefiniteness, but they are automatically satisfied by correlation functions obtained from experimental data.
Likewise, if n is an integer, define \(T(u) = u^{n} = e^{i n \vartheta }\) for every element \(u = e^{i \vartheta} \in \mathbb {S}^{1}\); the circle \(\mathbb {S}^{1}\) is a group under complex multiplication and \((\mathbb {C}, T)\) provides an irreducible unitary representation of \(\mathbb {S}^{1}\).
Two given irreducible unitary representations \((\mathcal{H}_{1}, T_{1})\) and \((\mathcal{H}_{2}, T_{2})\) are equivalent if there is a unitary map U from \(\mathcal{H}_{1}\) to \(\mathcal{H}_{2}\) such that \(UT_{1}(g)\) and \(T_{2}(g) U\) coincide for every \(g \in G\). In example 1, it is very easy to check that there is no such unitary map intertwining \(T_{p}\) and \(T_{q}\) if \(p \neq q\).
To be precise, we can multiply \(\widehat{f}\) with the Dirac distribution on the circle of radius K, obtaining a tempered distribution on \(\mathbb {R}^{2}\), and define \(f_{K}\) as the inverse Fourier transform of this multiplication: it is automatically a smooth function.
For completeness we recall that there are unitary irreducible representations of \(\mathit{SU}(1,1)\) which do not enter the Plancherel decomposition of \(\mathbb{L}^{2}(G/K)\); the deep and beautiful work by Bargmann and HarishChandra on these representations will not appear in this paper.
The groups \(\mathit{SU}(1,1)\) and \(SL_{2}(\mathbb {R})\) are famously isomorphic; see for instance [59].
That the eigenvalue should be real is the technical reason why the growth factor in the modulus is needed.
This means each \(\mathbf{z}(x)\) is a complexvalued Gaussian random variable with zero mean and variance one.
For legibility we rewrote the derivative in the horizontal direction as \(\partial_{1}\) in the next formula: so if φ is a function of two variables \(x_{1}, x_{2} \in\mathbb{D}\), \(\partial_{1,x_{1}}\) denotes the derivative in the horizontal direction of \(x_{1} \mapsto \varphi(x_{1}, x_{2})\).
Because translations have zero derivative, a general element \((A, v)\) of \(\mathit{SE}(2)\) then acts on complexvalued functions on \(\mathbb {R}^{2}\) as \(f \mapsto ( x \mapsto A f( A^{1} v + A^{1} x ) )\).
Notes
Declarations
Acknowledgements
I owe many warm thanks to my doctoral advisor Daniel Bennequin, who suggested this work and has been of much help in designing and writing this paper. I am grateful to the referees whose help was precious for making this paper more readable. I also thank Alessandro Sarti for discussions of Euclidean symmetry in V1, and Michel Duflo for his kind interest in the results of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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