Pellionisz, A.J. (1991) Discovery of Neural Geometry by Neurobiology and its Utilization in Neurocomputer Theory and Development. Invited Lecture of the Chairman of Session "Neurobiological and Physiological Connection" at Internatl. Conf. on Artificial Neural Networks. Ed. by T. Kohonen, Helsinki, Finland p.485-493 Discovery of Neural Geometry by Neurobiology and its Utilization in Neurocomputer Theory and Development András J. Pellionisz New York University Medical Center, New York, NY, 10016 USA* With a geometrical approach to the brain becoming widespread, the exact nature of neural geometry is increasingly questioned. Certainly, neural geometry appears more sophisticated than being restricted to rigid Euclidean geometries spun by Cartesian frames of reference. Thus, new experimental methods are required to reveal neural geometries. The mathematical language of brain function can only be revealed based on information from neurobiology. In turn, implications of the discovery of non-Euclidean features of neural geometries are also becoming evident for the theory and development of artificial neural nets. This paper connects neurobiology to neurocomputer theory and development by showing that (1) multielectrode experimentation reveals how non-Euclidean distances in neural functional spaces can be calculated to measure external (physical) invariants by internal generalized coordinates (2) introduction of such generalized vectors leads to new avenues in the theory of pattern recognition and association (3) regarding non- metrical neural geometries, it is shown that morphology of single neurons reveals a fractal geometry (4) in turn, the existence of such "fractal templates", that comprise complexity, leads to the introduction of the concept that visual recognition of complex patterns and textures by the brain may be based on fractal geometrical primitives rather than customary Euclidean templates (such as points, lines, spheres and cylinders). A prototype of a new generation of electronic neurocomputers can thus be outlined, (e.g. a transputer based) parallel processor implementing neural geometries found in nature. 1. Introduction 1.1. Experimental Neurobiology and Neurocomputer Theory & Development Once the "boom cycle" of neurocomputing of the eighties will wane, two fields of activity will emerge significantly strengthened. On one hand, parallel computing by means of competing generations of "neurochips" will stay for good. On the other hand, a consensus will gather that "we still do not exactly know how the brain works". Thus a concentrated effort will emerge to establish the basic science foundation of neurocomputing industry by discerning from the living brain the mathematical-theoretical principles of operation of the CNS. This will require a truly interdisciplinary effort leading to development of "neurophysics" or "neuroinformatics". It ought to be remembered that there may be various successful applications of neurocomputer theory and development. One of the most important applications, however, remains the explanation how biological neural networks operate. Thus, a measure of maturity of neural net theory is the extent to which it yields an experimentally confirmed theory of brain function. For neurobiological experimentation and neurocomputer theory and development, to go beyond the initial stage of mutual token representation, two achievements have to be demonstrated. (1) Neurobiology must be shown to lead to mathematical conclusions that are usable for neurocomputer theory and development. (2) In turn, resulting neurocomputer theory must trigger experimentation to verify and/or improve mathematical theory of the biological brain. While these requirements appear ambitious, physics provides a workable precedent (leading perhaps to the axiomatic foundation of experimental-theoretical "neurophysics" as an academic discipline). There are notable examples on record for an active interaction of experimentation and theory. Rudimentary features of neural activity could be mathematically comprised (into Boolean algebra) a long time ago [1]. Likewise, the most basic synaptic learning algorithm could be promulgated several decades ago [2]. These mathematical-theoretical achievements based on primary experimental data both led to schools of research lasting for decades, as well as almost immediately resulted in secondary experimentation that in turn made the limitations of such "all or none" theoretical approaches obvious [3]. _____________________________________________________________ * This research was supported by the Alexander von Humboldt Prize for scientific research by Germany-W and was performed in Dept. Physics, Philipps Univ., Marburg D-3550. The author expresses gratitude for the generous hospitality of Prof. H.J.Reitboeck and for his invaluable contribution to the ideas outlined in this paper. The author is presently at NASA Ames Res. Ctr, Neurocomputing Lab, 261-3, Moffett Field, CA 94035 USA. (415) 604-4821, E-mail: Pellioni @ pioneer.arc.nasa.gov _____________________________________________________________ 1.2. A geometrical approach to brain function The geometrical approach to brain function (as opposed to algebra, e.g. the Boolean algebraic approach) provides with a more recent example for the advantages of theoretical-experimental interaction in neuroinformatics. Geometry is rapidly becoming the trend in neurocomputing not only at the daily agenda [4] but even at the "top" levels of neurophilosophy and epistemology [5],[6], see also [7]. When tracing the history of the emergence of a geometrical school, one finds that essential developments occurred when theory and experimentation cross-fertilized. While conceiving brain function geometrically is a classical idea by Descartes [8], brain theory only fairly recently reverted from Boolean algebra to a basically geometrical vector notation Ð when it became evident that the massively parallel organization of CNS necessitated the use of mathematical arrays. Thus theories of association, pattern recognition and learning were formulated in terms of (Cartesian) vectors and matrices [9]-[12]. Vector-matrix notation imported to brain theory features of the linear vectorspace with Euclidean geometry. The dogma was therefore tacitly established that the geometry of brain function is Euclidean (i.e. the functional space is erected by Cartesian orthogonal and thus separable coordinate axes). This rudimentary axiom was useful for some time for both theory and experimentation. Theory could employ arrays, representing the whole. Experimentation could concentrate, on a one-by-one manner, on (supposedly) separable vectorial components such as horizontal and vertical sensory and motor systems in vestibulo-ocular and gaze research. By the late seventies, the dogma implicit in the Euclidean- Cartesian geometrical approach became a limit to progress. Thus, as early as in 1980 great effort was spent to pioneer the concept that the coordinates intrinsic to brain function are expressed in generalized frames (with non-orthogonal, typically overcomplete axes [4]). In theory of neural net function, the concept of distance is essential for pattern recognition, association, sensory-motor coordination etc. Thus, it is noteworthy that when working with nature's coordinates intrinsic to CNS, geometrical features such as distances, angles, geodesics, etc. can only be calculated via the metric tensor, instead of the use of classical algorithm to calculate the Euclidean distance of vectors. Basic algorithms for calculating distances of Cartesian and non-Cartesian (generalized) vectors are shown in Fig.1. (after [4],[13],[14]). The required axiomatic change from extrinsic Cartesian vectors to non-Cartesian intrinsic coordinates (for which a non-Euclidean measure of distance is valid) mathematically meant a change from orthogonal to generalized vectors (tensors, expressed in non- orthogonal coordinate systems; [4]). Most importantly, however, by this mathematically exact generalization (which is trivial in retrospect ) a new avenue of experimentation was opened. Investigation first focused on quantitatively establishing nature's coordinate systems [15]-[19], and on working out neural network models for transformation of coordinates [13],[20]-[23]. This trend established the "coordinate system approach" in experimental neuroscience [24]-[31]. The concept of generalized coordinate system transformations intrinsic to biological neural nets is therefore an example for a specific mathematical lesson directly learnt from experimental neuroscience. Based on this concept, a concise mathematical theory of the biological brain could be developed (Tensor Network Theory; [32]). Theory, in turn, exerted its influence on experimental research by triggering development of the coordinate system approach which resulted in experimentally testable and confirmed quantitative hypotheses [33],[34],[23]. For neurocomputer development, mathematical theory yielded a generalized neural net learning paradigm for the internalization of external geometries into intrinsic functional geometries of neural networks ([35], reprinted in [36]). This led to a neural network approach featuring the vestibular apparatus as one of the most prominent biological neurocomputer [34],[37],[38]. The generalized coordinate system approach, by targeting a very specific part of the CNS (sensorimotor systems such as vestibulo-cerebellar coordination of gaze) had the advantage that it could be quantitatively elaborated for particular CNS subsystems. For the same reason, it was greatly hampered by the initial resistance from the specific established approaches in gaze control. It is therefore significant that in the course of a single decade from the initiation of an approach of modeling e.g. the vestibulo-ocular reflex as a parallel distributed system [4] the field of research of gaze control first gradually turned around [39]-[41] and before the decade was out parallel distributed approaches were heralded to be superior to
Figure 1.
Fig.1. (after [4],[13]). Fundamental differences of vectors expressed in Cartesian orthogonal system of coordinates (panel on left) and generalized (co- and contravariant) vectors expressed in non- Cartesian frames (panel on right). Most importantly, for Cartesian vectors distance can be calculated by the Euclidean measure, while for generalized vectors, found in natural neural networks, the establishment of the metric is needed to measure distance. _______ early models which "have generally taken the form of black-box diagrams (for example [42]) representing the flow of hypothetical signals between idealized signal-processing blocks...but indicated little about how real eye-movement signals would be carried and processed by real neural networks " [43]. The task in the nineties is admittedly to "construct parallel, distributed models of the vestibulo- oculomotor system" [43]. With the coordinate system approach on record for having turned a whole field of experimental neurobiological research around in a single decade, and early "critiques" having failed to register comparable competition during the same period, the geometrical approach is now internationally recognized and the trend is focused on geometry as a mainstay of neural net research [5],[6]. In the nineties, it is evident in retrospect that orthogonality or overcompleteness of nature's coordinate systems were not the central issues. It is a somewhat more fundamental question if theorists and modelists of the biological brain accept and adopt the use of coordinates intrinsic to neural nets with any implications that it may entail. Most importantly, however, the question is if we "bite the finger" that singles out metrical geometries (erected by generalized coordinate systems) as nature's perhaps simplest neural geometries or, rather, we "look where the geometrical approach is pointing", i.e. that the brain is a geometrical object for which our Euclidean intuitions do not necessarily apply. There is today an "explosion" in discovering nature's geometries in general [44]. Also, evidence rapidly gathers that grossly non-Euclidean fractal and chaotic geometries are also manifest on neurons and neural systems [45]-[47]. These developments mandate a massive effort to ensure that neurocomputer theory and development goes beyond the stage of vector-matrix notation assuming Euclidean geometry of the vectorspace. As outlined earlier [48], similar to earlier advances in brain theory, progress is hinged on an interaction of experimental neuroscience which discovers the nature of such geometries and theory of neural nets which appropriately uses such understanding.
Figure 2.
Fig.2. (after [37]). Proposed theoretical-experimental paradigm to measure by multi-unit recording technique the functional vectors intrinsic to natural neuronal network transformations. Neural expression of head-acceleration is in the form of covariant vector at the vestibular nuclei (E) and at the cerebellar Purkinje cells (PC). From measurements of covariant expression of physical invariants the metric tensor of the geometry of the n-space can be established as shown in Fig. 3.A,D-H. _______ 2. Discovery of Neural Geometry by Neurobiology With the question no longer if the mathematical language of brain function is geometry but "what is the exact nature of such neural geometry?" , experimental investigation is likely to proceed on two tracks. First, it is important to qualitatively explore the drastically different manifestations of neural geometries (metrical and non-metrical fractal and chaotic geometries), as well as their connections, in order to gain a perspective and to assess the scope of the task. Second, starting with perhaps the most rigid and closest to conventional geometries of smooth (derivable) metrical functional manifolds it is essential to elaborate experimental paradigms by which actual measurements of neural geometry could take place (see Fig.2.). 2.1. Measurement of Metrical Neural Geometry by Multi-Unit Recording As originally introduced [14], the so-called multi-unit recording technique provides both an opportunity and a need for a geometrical analysis. Multielectrode techniques have been pioneered through the past decades [49]-[63]. While processing multielectrode data most commonly relies on correlation analysis [64]-[66], lately a geometrical approach was taken [67] proposing an interpretation of neuronal activities of n neurons as a point in the n-dimensional vectorspace in which a Euclidean geometry governs. It was, however, pointed out [14], that instead of the automatic assumption of a Euclidean metric tensor of neural n-spaces (on which calculation of geometrical features, such as distances, would be based) the approach needs to be reversed. Geometrical analysis of multi-unit data cannot start with a well-defined geometry since it is basically unknown. Instead, the multielectrode analysis method must be the means of resulting in a geometry as discerned from experimental data. This appeal was well taken by the community concerned with multielectrode recording techniques, since it is readily accepted that the assumption of Euclidean geometry is arbitrary Ð however, for some time, no specific experimental paradigm was forwarded to offer a concrete procedure to define the geometry (measure the metric tensor) of the neural n -space.
Figure 3.
Fig. 3 (after [14] and [46]). Two major avenues for the exploration and measurement of non-metrical fractal (BC) and metrical (D-H) neural geometries. Panels A,D-H from [14] show that r, the cross- correlogram of firing frequencies, approximates the g covariant metric tensor, yielding the geometry of the curved functional manifold. Panels BC from [46] show that individual neural arbors can be well approximated with a deterministic fractal model, comprising complexity to a template (C). _______ The cooperative project of A.J.P and Dr. H. Reitboeck (supported by Alexander von Humboldt Foundation, manuscript in preparation) has led to an experimental paradigm by which the geometry (metric tensor) of the neural n-space underlying vestibular and cerebellar functional spaces can be quantitatively established. Fig.2. outlines a skeletomuscular model of the eye-head-neck system coupled with a neural network model of the vestibulocerebellar intrinsic coordinate system transformation (after [37]). As shown in inset 2A, at two stages of the transformation-chain of intrinsic vectors, the experimentally presented physical invariants (such as distances in the angular acceleration-space) are represented as covariant, sensory-type vectors. This is physically proven at the first stage, at primary vestibular neurons, as the vestibular semicircular canals take orthogonal projection components of the head-acceleration (each in its own plain). Tensor network theory predicts that the cerebellar input (intercepted at the level of cerebellar Purkinje cells) is also a covariant (sensory-type) intention-vector (although it is expressed in the motor frame of reference). Fig.3E shows (for elaboration, see [14]) that in case of such covariant vectors the cross-correlogram table of firing frequencies converges to the covariant metric tensor, from which the contravariant metric can be calculated by the Moore- Penrose generalized inverse [68]. The proposed procedure of multi- unit data analysis, therefore, yields a specific measure of both metric tensors of the neural n-space, by which the internal geometrical representation of external physical invariants can be appropriately calculated (Fig.4.) Fig.4D also demonstrates that the direct use of Euclidean distance-algorithm on such covariant vectorial expressions would yield a completely erroneous and non-unique "measurement' of such distances. Since most sensorimotor coordination events are based on "navigation" of animals in the extrinsic physical spacetime manifold, it can be expected that the above type of analysis will reveal how intrinsic metrical geometries (embodied in sensorimotor neural nets) "embed" the external physical space. 3. Use of Neural Geometry in Neurocomputer Theory and Development This section of the paper is largely a prediction for the future, although in the specific field of sensorimotor operations the use of non-Cartesian frames of reference (erecting spaces in which the metric tensor is not a Kronecker-delta as in Euclidean geometries) has already led to neural net learning algorithms [35] as well as to a patent for a cerebellar coordinator [69]. It is also very likely that a premature generalization of the ubiquitous vector-notation in neural net theory towards "points in vectorpaces with non-Euclidean measures of distance" would trigger undue resistance by the already established approaches, and thus they should run their course until limitations become apparent. 3.1. Theory: Association and Recognition of Patterns as Generalized Vectors Some impact of the use of generalized vectors on the fields of research of association and pattern recognition is already clear [48] as these categories are rather squarely based on the concept of "distance". Fig.1. provides a glimpse how vector formalisms of theories of association and pattern recognition is likely to be rejuvenated by the use of generalized vectors once the underlying metric tensor becomes experimentally accessible. The cooperation of A.J.P. and H.J.Reitboeck has led to a re-thinking of pattern recognition which customarily starts with considering n-tuplets as points in regular vectorspace separated by Euclidean distances [10]. 3.2. Theory: Fractal primitives for Recognition of Textured Patterns A geometrical re-thinking by A.J.P. and H.J.R. of pattern recognition is also under way in an even more radical sense than allowing "pattern vectors" to be generalized. Fig.3. also symbolizes that neural networks and neurons themselves open up an avenue for neural geometry that is drastically different from that of metrical spaces. Multi-unit recording technique is illustrated in Fig.A (from [70]) by two neural dendritic trees, and it is shown in Fig.3.BC (from [46]) that the complexity of the dendritic arbor of Purkinje nerve cells can be rather closely approximated by deterministic fractal geometry. It is noteworthy that the complex tree (Fig.3B) is fully determined by the fractal template (Fig.3C). Thus it appears that neural systems do use a fractal geometry for information compression. Remarkably, in technological research and development fractal geometry already found its way to utilization in image compression [71]. The above facts taken together trigger the germinal idea exposed here that neural systems might themselves utilize a fractal geometrical information compression in (visual) recognition of patterns, especially of textured images. This idea is obvious in the sense that there is no philosophical reason to limit the brain to the use of Euclidean geometrical primitives in vision [72]. A specific suggestion following from the above idea is to experimentally employ in probing the visual system fractal geometrical primitives. Testing the visual system by fractal geometrical primitives (beyond Euclidean primitives such as points, line segments and directions as it was done in classical studies [73]) is prone to become a new approach in experimental-theoretical investigation. Already, in an aesthetic sense, it is evident that some fractal images are naturally used in pattern recognition (see "letter Y" in Fig.3C). If the role of the CNS is to reflect, by an internal geometrical model, the external geometry, it does not escape one's attention that fractal patterns have a natural and pleasing appeal to our brain [74].
Figure 4.
Fig.4. How does the brain geometrically represent the external world? Schematic diagram of the theoretical-experimental reconstruction of external invariants such as distances and directions (panel A) by neural networks of the brain (panel E). Panel B illustrates a sensory coordinate system intrinsic to neural measure- ments of external invariants. In the specific case of the vestibular semicircular canal apparatus, such physical invariants are points (distances) in the angular acceleration space, which are measured by the non-orthogonal coordinate system of the vestibulum. Panel C illustrates a firing frequency response of an array of neurons (detectable from the vestibular nuclei). Given that such an activity vector is covariant, Euclidean measurement and reconstruction of distances is grossly erroneous and non-unique (see "re-construction" of input A in panel D). Using the proposed paradigm for the geometrical analysis of multi-unit recordings, the cross-correlation analysis of firing frequencies yields the metric tensor of the functional space by which the external invariants can be faithfully reconstructed (panel E). _______ 3.3. Development: Neurocomputers as Geometrical Machines Given the explosive rate of progress of neurocomputer theory and development it is reasonable to expect that electronic implementation of "geometrical machines" will proceed before the book on Neural Geometry is completed. Indeed, novel generations of neurocomputers are likely to fully take advantages both of the functionality that is not attainable by limiting massively parallel array processing to Cartesian vectors (such as the distinction of sensory and motor type expressions) as well as the flexibility of the architecture designed to accommodate generalized vectorial operations (which fully include the rather specific Cartesian vectorial operations). 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