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"2.1.2 The Neurogeometric Approach

We coined the term ‘neurogeometry’ to refer to this neural origin of perceived space. The aim in the present work is to take a first step in this direction, something made possible by the huge amount of new and fascinating experimental results now available thanks to new imaging techniques. As long as the brain remained, from an experimental point of view, a ‘black box’, there was no way of developing such an approach. What made this possible was thus that the brain became, at least to some extent, a ‘transparent box’. Brain imaging techniques are here the equivalent of the new observational methods that are always found to underlie any scientific revolution. We shall show that their results can be modelled using sophisticated mathematics that corresponds in the deepest possible ways to mathematics already invented by certain outstanding geometers like those already mentioned, and in particular Lie and Cartan, when they set out to understand mathematically how the geometry of the external world (Euclidean or otherwise) could come about.

...

2.2 Perceptual Geometry, Neurogeometry, and Gestalt Geometry

We stress that neurogeometry is about the internal geometry (already referred to here as ‘immanent’) of low-level vision, and not therefore the conventional ‘transcendent’ geometry of the perceived external 3D Euclidean space. It concerns a much more fundamental level, and to use the nice expression adopted by Misha Gromov to speak about sub-Riemannian geometry, it tries to understand perceived space from within.

In neurogeometry, anything that is not implemented neurally does not exist. This means that all the mathematical concepts used operationally in the models must have some material counterpart. There is a similar situation in computer science, where the software only works if it is compiled and realized materially in the physics of the hardware. It is not easy to implement this equivalence between geometric idealities and neural materialism. Indeed, on the one hand, trivial mathematical structures such as alignments, gluing of local charts, or direct products are implemented neurally in a very subtle way that is hard to study experimentally, and on the other hand, certain properties of the modelling structures will not be implemented and so will have no empirical meaning. The reader should bear this crucial point in mind: when a set of empirical phenomena is modelled by mathematical structures of a certain kind, only certain aspects of these structures will be open to empirical interpretation.7

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2.3 Geometry’s ‘Twofold Way’

Let us stress once more that, in neurogeometry, there is a twofold relationship between the geometry and neurophysiology of vision. As we shall explain in detail, it is the functional architecture of the visual areas, the precise organization of their neural connections, which generates the geometric properties of perceptual space, i.e. the perceived 3D space in which the objects of the external world are situated. We may thus envisage a ‘neural→spatial genesis’ of the kind ‘functional architecture→geometric properties of external space’. But as we shall see later, there exist geometric models of the functional architectures themselves; that is, the latter implement well-defined sui generis geometrical structures. It is important to distinguish carefully between the two levels at which geometry enters the discussion. The whole purpose of this book would become incomprehensible if they were confused. As we have seen, to formulate the distinction, we may return to the classical philosophical opposition between immanence and transcendence. The geometry of functional architectures is immanent in perception, internal and local, and its global structure is obtained by integration and coherent association of local data. In contrast, the geometry of perceived space is transcendent in the sense that it concerns the outside world and is given to us immediately as global.

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2.4 Idealities and Material Processes

To clarify this key point, let us make an analogy. Although it differs with regard to content, the new direction provided by neurogeometry is methodologically speaking of the same kind as the one taken during the last century with the advent of the Turing machine, λ-calculus, and computers. This computational revolution took the symbols that underlie logical idealities and turned them into material operations. It explained how the dominant logical idealism and analytic apriorism expounded from Bolzano to Frege could be naturalized and even physicalized. In other words, it explained how logical ‘software’ could be implemented in physical ‘hardware’. We are doing just the same here. The aim of the ‘neurogeometric’ approach is to obtain an explicit understanding of the material operations that underlie the geometric idealities of the synthetic a priori and to explain how some kind of geometric ‘software’ could be implemented in our neural ‘hardware’, hence the following analogy:

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2.5 Mathematical Prerequisites and the Nature of Model

By its very nature, the following will raise certain issues relating to didactic presentation, issues that might prove off-putting to some readers. Indeed, we shall use many mathematical concepts generally considered to be rather ‘advanced’: differential forms, connections, Lie groups, contact structures and symplectic structures, sub-Riemannian geometry, variational models, non-commutative harmonic analysis, and so on. We shall define these as we go along, assuming a basic understanding of differential and integral calculus, linear algebra, and elementary group theory. These are basic concepts that will be familiar to any science student and which are in any case easy to find in a good enyclopaedia. Having said that, the reader may wonder quite rightly why such mathematics is relevant here. Our long experience as teacher and researcher in cognitive science has shown us that biologists and psychologists are often intrigued, even shocked, by the idea that non-trivial mathematical models (going beyond simple methods of data analysis) should be needed in their field of study.

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For example, many differential equations can be applied to a whole range of different fields: Turing-type reaction–diffusion equations for morphogenetic processes, the Hodgkin–Huxley equation [20] for the propagation of action potentials, the spin glass equations of statistical physics for neural networks, the Lotka–Volterra equations for ecology, and so on. There is thus no deep reason why there should be any natural limit to the use of mathematical models. Another argument often put forward is that if we make the hypothesis that algorithms are implemented neurally, this would mean that neurons ‘calculate’, which is impossible. But this argument is also mistaken. In Mechanics the planets do not ‘calculate’ their trajectories. The only thing we can say is that theories based on laws involving global interactions (as is the case with Newton’s universal law of gravitation) are problematic and that the interactions must be localized (something achieved by general relativity). However, in neuroscience, we can be sure of the locality of the interactions, because these interactions occur through material connections between neurons. What passes for a neural ‘calculation’ is essentially the propagation of activity along connections, and this is a ‘calculation’ because the connections are organized into highly specific functional architectures. In other words, it is the structure of the functional architectures—in a sense, the ‘design’ of the neural ‘hardware’—which amounts to a calculation.

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2.7 Levels of Investigation: Micro, Meso, and Macro

Of course, we shall focus on modelling the functional architecture of the primary visual areas and in particular V1. But despite the apparently rather limited nature of the subject, we shall nevertheless only discuss a very small part of it. It is easy to understand why. To begin with, we shall only be dealing with the so-called functional, integrative, and computational neurosciences, and apart from the discussion in Sect. 5.12 of Chap. 5 which we shall explain when the time comes, we shall not be concerned with any aspect of molecular biology or genetics. This said there are still three levels of investigation for the purpose at hand: those of microneurophysiology, mesogeometry, and macrodynamics. These will receive differing amounts of attention. For instance, one of our basic experimental inputs (see Sect. 4.3 of Chap. 4) will be the fact that the single neurons in V1 detect a retinal position a = (x, y) and a preferred orientation p at a, although naturally at a certain scale. The data (a, p) is called a contact element in differential geometry, and we shall thus consider the single neurons of V1 as filters extracting contact elements from the optical signal. But just this simple claim is the subject of a huge experimental effort. For example, one needs to compare the situations for different species and take into account the fact that, in these results, neurons are treated as linear filters acting on stimuli reduced to single bars (simulating the edge of an object) or systems of parallel bars in motion (drifting gratings), while it is clear that there are significant nonlinearities and also that natural stimuli may have very different structures.9 One must also take into account the fact that the imaging techniques used do not have sufficient spatial resolution to distinguish individual neurons,10 whence one is in fact dealing with local averages over small groups of neurons, and a piece of geometric data like a contact element (a, p) reflects an average of the underlying activity. The geometric quantity we refer to as a ‘contact element’ thus represents a mesoscopic entity when compared with the microscopic level of individual neurons. One consequence of this choice of a mesoscopic level for neurogeometry is that what we shall call a ‘neuron’ will actually be a small patch of neurons, and we shall thus say little about true elementary neural circuits. There is an extensive literature on this subject and some sophisticated engineering, but we shall only refer to it from time to time. It should also be noted that even a very high resolution would not remove the problem of levels. Indeed, the neural code is a population coding, where each elementary operation activates a large number of neurons. A ‘high-resolution’ neurogeometry that was truly microscopic would therefore have to be based on the tools of stochastic differential geometry, something pointed out by specialists such as David Mumford, Jack Cowan, and Daniel Bennequin. So let us stress once again that the neurogeometry developed here will idealize things by sticking to a mesoscopic level. The global structures, processes, and dynamics that we shall study will thus be based on gluing together mesoscopic geometric elements. All the various aspects of the microlevel are currently the subject of ever more highly specialized studies. What this means is that, while our neurogeometric mesomodels are mathematically rather sophisticated, they concern only a very limited part of what contemporary neuroscience can teach us, and in a highly simplified way, so they are only a first step into this new field. What we would like to advocate in neuroscience is mainly the geometric framework, which seems relevant and natural for the mathematical modelling of functional architectures.

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2.8 The Context of Cognitive Science

We therefore stress once more that the cross-disciplinary nature of cognitive science is intrinsic and endogenous: it is imposed by the very nature of the entities, structures, and mental processes it investigates. An ability such as the perception of objects in three-dimensional space on the basis of ‘pixellated’ two-dimensional retinal data can be studied on a formal level (to identify the mathematical and formal features of the problem of constituting objects bounded by edges and filled with perceived qualities), on a behavioural level (studying the computational procedures, i.e. processes of integration, recognition, inference, and interpretation), and on the level of the biological substrate (investigation of neurophysiological mechanisms). This ability thus involves several levels of integration in both space and time. The cognitive sciences treat all these mental phenomena a priori as a broad class of natural phenomena. They do for the mental what biology has been doing for the living since the nineteenth century. Consequently, their status depends on the way we extend the concept of ‘nature’. If we understand ‘nature’ in the narrow (strictly physicalist) sense, this leads to a reductionist or ‘eliminativist’ understanding of the mental. But if ‘nature’ is taken in a broader sense, we arrive at an ‘emergentist’ understanding of the mental, e.g. emergence of macrostructures from microinteractions in complex systems, as in thermodynamics and sociology. But whichever option is chosen, the approach will be naturalistic and monistic, rejecting any Cartesian form of dualism between mind and body (two substances).

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2.9 Complex Systems and the Physics of the Mental

As ‘hard’ technosciences, the cognitive sciences are inextricably related to the study of complexity and derive from the intellectual environment that came into being in the 1940 to 50s, so admirably exemplified by exceptional scholars such as John von Neumann, Norbert Wiener, Warren McCulloch, and Walter Pitts. They belong to the movement that saw the joint emergence of the theories, techniques, and methods of computers, neural networks, cellular automata, information processing, and self-organizing, self-regulating complex systems.12 After several decades of progress in constant interaction with neuroscience, cognitive psychology, linguistics, and certain approaches to economics, these activities are now mature enough to justify referring to them as a ‘science’. This is part of a deep trend. There has been a gradual development of mathematical physics to treat the organizational complexity of material systems and the emergence of patterns and shapes, but also cognitive activities as ‘unphysical’ as conceptual categorization and learning. We began by understanding how shapes could ‘emerge’ and ‘self-organize’ in a stable manner on the macroscopic scale as causal consequences of complex interactions on the microscopic scale. Collective microphysical phenomena, both cooperative and competitive, provide the causal origin of joint behaviour on a macroscopic level which can break the homogeneity of a substrate. The classic physical example is provided by critical phenomena like phase transitions. It was then realized that neural networks are the same kind of system, but in which emergent shapes and structures can be interpreted as cognitive processes. If rather similar models crop up in rather disparate fields of empirical investigation, this is because complex systems possess certain relatively universal properties.13 By definition, these are large systems of interacting elementary units with emergent global macroscopic properties arising from cooperative or competitive collective interactions between these units. These systems contrast with classical deterministic mechanical systems in the following ways:

• They are singular and individuated, largely contingent, not concretely deterministic, even when they are ideally so: they are sensitive to tiny variations in their control parameters, a sensitivity that can induce divergence effects.

• They are historical products, resulting from processes of evolution and adaptation

• They are out of equilibrium and have an internal regulation that keeps them within their range of viability.

They have little to do with classical mechanistic determinism. They are analyzed using new physical and mathematical theories and a computational approach making heavy use of computer simulation. The role of nonlinear dynamical systems (attractors, structural stability properties, and bifurcations), chaos theory, fractals, statistical physics (renormalization group), self-organized criticality, algorithmic complexity, genetic algorithms, and cellular automata has become key to understanding their statistical and computational properties. In short, through the engineering of self-organized, non-hierarchical, distributed, and acentered artificial systems, we are beginning to be able to model and simulate reasonably well biological systems (immunological systems, neural networks, evolutionary processes), ecological systems, cognitive systems, social systems, and economic systems.

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2.10 The Philosophical Problem of Cognitive Science

But in the physical sciences, there is also modelling in a stronger sense which is of a quite different kind. For this modelling in the strong sense, methods are specific to the theoretical conceptualization of a particular kind of object and can be used to reconstruct the phenomena in some real field from its constitutive theoretical concepts. Mathematical physics is able to reconstruct the whole diversity of physical phenomena from its theoretical concepts. This completely changes the status and function of concepts. We no longer subsume empirical diversity by abstraction under the unity of theoretical categories and concepts. Rather, concepts are transformed into algorithms for reconstructing the diversity of phenomena. Put another way, conceptual analysis is converted into a computational synthesis. At the present time, the ideal of a computational synthesis of phenomena has only really been achieved in physics, which is restricted to a very narrow and highly constrained region of empirical reality. Huge regions of phenomena have been left outside the reconstruction zone, even though a fair number of these regions have been studied in detail by many empirical and descriptive disciplines. Here, we may cite:

• The whole macroscopic organizational and morphological complexity of material systems.

• All cognitive operations, including categorization, inference, induction, learning.

• The whole semiotic and linguistic dimension of meaning.

• And in fact anything having to do with phenomenality itself as a process of phenomenalization of an underlying physical objectivity.

In other words, it is only by restricting phenomenal reality to its most elementary form (essentially, the trajectories of material bodies, fluids, particles, and fields) that we have been able to carry through the programme of reconstruction and computational synthesis. For the other classes of phenomena, this project has long come up against unsurmountable epistemological obstacles. At this point, it was taken as self-evident that there was an unavoidable scission between phenomenology (being as it appears to us in the perceived world and the cognitive faculties that process it) and physics (the objective being of the material world). However, we may say that it is not so much self-evident as a straightforward prejudice. In any case, this disjunction transformed the perceived world into a world of subjective-relative appearances—mental projections—with no objective content and belonging to psychology. Beyond psychology, the most that could be attributed to these appearances in the way of objectivity was a logical form of objectivity to be found in the theories of meaning and mental contents, from Bolzano and Frege, Husserl and Russell, to contemporary analytical philosophy. We may say that the current work aims to go beyond this scission by developing a mathematical neurophysics of the phenomenology of the perceived world and common sense. The neurogeometry of vision presented here will be one aspect of this.

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2.11 Some Examples

2.11.4 The Cut Locus

Our last example concerns the cut locus of a figure, also called the generalized symmetry axis or ‘skeleton’. Following the psychologist Blum [26], Thom [27] always stressed its fundamental role in perception (see Fig. 2.5). Once again, imaging can show us the neural reality of the construction of this inner skeleton, for which there is no trace whatever in the sensory input, the latter consisting merely of an outer contour. Figures 2.6 and 2.7, produced by David Mumford’s disciple Tai Sing Lee, illustrate the response of a population of simple V1 neurons, whose preferred orientation is vertical, to textures with edges specified by opposing orientations. Up to around 80–100 ms, the early response involves only the local orientation of the stimulus. Between 100 and 300 ms, the response concerns the overall perceptual structure and the cut locus appears. These experiments are rather delicate to carry out, and they are much debated, but the detection of cut loci seems to be well demonstrated experimentally

All these examples share the fact that the geometry of the percept is constructed— Husserl would say ‘constituted’—from sensory data which do not contain it, whence it must originate somewhere else. Put another way, they all involve subjective Gestalts. This is indeed why we chose them, because, as claimed by Jancke et al. [30], these subjective global structures ‘reveal fundamental principles of cortical processing’, the kind of principles that interest us here. The origins of visual perceptual geometry can be found in the functional architecture which implements an immanent geometry, and it is the latter that provides the focus of neurogeometry."

-Jean Petitot, Elements of Neurogeometry: Functional Architectures of Vision

https://www.springer.com/gp/book/9783319655895

Eric Schmid

Eric Schmid

Eric Schmid

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Eric Schmid

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G. GUERINI) - https://youtu.be/hqRU0UUHrhE (
P. BUCK) - https://youtu.be/cuLxx5D2eDI (
K. EDWARDS) - https://youtu.be/-Rw1paGHp_E (
BLUBBER) - https://youtu.be/HNw3mGyx0fc (
CHURCH ORGAN) - https://youtu.be/Fn0aUSBqKXw (
M. TSCHIEMER) - https://youtu.be/B-IqRjbVcI0 (
P. HOROBIN) - https://youtu.be/rdhqfBrManA (
GREIGE TRAVAIL) - https://youtu.be/3vu6jsHadnc (
T. FUGATE, D. LEE) - https://youtu.be/cV7ypBDbnoA (
UNKNOWN) - https://youtu.be/DzXYaePjVrY (
CANWLL CORFE) - https://youtu.be/eHY3ajA1zZU (
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