tozziarturo at libero.it tozziarturo at libero.it
Wed Nov 23 18:03:13 CET 2016

Inviato da Libero Mail per Android -------- Messaggio inoltrato --------
Da:  tozziarturo at libero.it A:  fis at listas.unizar.es Data: mercoledì, 23 novembre 2016, 07:55AM +01:00

>>>Dear FIS, 
>>>thanks for this precious oppurtinity.
>>>Here you find the text in order to start the discussion.
>>>Arturo Tozzi 
            for Nonlinear Science, University of North Texas
            Union Circle, #311427
            TX 76203-5017, USA, and
>>>tozziarturo at libero.it
>>>James F. Peters
            of Electrical and Computer Engineering, University of Manitoba
            Chancellor ’ s Circle, Winnipeg,
              MB R3T 5V6
>>>james.peters3 at umanitoba.ca
>>>This discussion aims to throw a bridge between
            neuroscience and the far-flung
            branch of topology.  Indeed, topology,
            assessing the properties that are preserved through
            deformations, stretchings
            and twistings of objects, is a underrated methodological
            approach with
            countless possible applications.  In
            particular, the  Borsuk-Ulam
            theorem ( BUT),  cast in a quantitative fashion which has the
            potential of being operationalized,
            stands for a
            universal principle underlying a number of natural 
            Here w e
            want to investigate whether  BUT and its recently
              developed variants might allow the assessment of brain
            function in terms
            of affinities and projections from real spaces to abstract
>>>Borsuk-Ulam theorem .  (See note at the end)  BUT
              states that, if a single point on a circumference
              projects to a higher spatial dimension, it gives rise to
              two antipodal points
              with matching description on a sphere, and vice versa.   Points on
            a sphere are “antipodal”,
            provided they are diametrically opposite , such as, for
            example,  the
              poles of a sphere.  This  means, e.g., that there
            exist on the earth
            surface at least two antipodal points with the same
            temperature and pressure.  BUT looks like a translucent
            glass sphere
            between a light source and our eyes: we watch two lights on
            the sphere surface
            instead of one.  But the two lights are
            not just images, they are also real with observable
            properties, such as
            intensity and diameter.  This means that  two antipodal points can be
            described at one level of
            observation, while just a single point at a lower level. 
>>>Variants of the
              Borsuk-Ulam theorem.   The concept of
            antipodal points can be generalized to
            countless types of system signals, by introducing novel BUT
            variants.  The two opposite points can be used not just
            for the description of simple topological points, but also
            of   lines, or perimeters, areas, regions, spatial
            patterns, images, temporal
            patterns, movements, paths, particle trajectories, vectors,
            tensors, functions,
            algorithms, parameters, groups, range of data, symbols,
            signs, thermodynamic
            parameters, or, in general, signals.   If we
            simply evaluate nervous activity
            instead of “signals”,  BUT
            leads naturally to the possibility of a region-based,
              not simply point-based, brain geometry.   For
            example, a
            brain region assessed through fMRI can have features such as
            area, diameter,
            average signal value, entropy and so on.   We
            are t hus allowed to describe
            brain functional
            and/or anatomical features in terms of 
            antipodal points on a sphere .  It is
            noteworthy that the  BUT
            allows also the evaluation of the energetic nervous
            requirements.  Indeed,
            there exists a physical link
            between the  two
            spheres of different
            dimensions  and  their energetic features.  When  two
            antipodal functions on  a higher –dimensional structure, 
            project to a lower-dimensional structure, a single function
            is achieved.
            This means that the single mapping function on the
            lower-dimensional structure
            displays values of energy parameters lower than the sum of
            two corresponding
            antipodal functions on the higher-dimensional structure. 
            Therefore, in a metastable brain formed by
            structures with different dimensions, a decrease in
            dimensions gives rise to a
            decrease in energy. We achieve a metastable brain/mind in
            which the energetic
            changes do not depend anymore on thermodynamic parameters,
            but rather on affine
            connections, homotopies and continuous functions. An example
            is provided by a
            recent paper, where BUT allows the  detection
            of Bayesian Kullback-Leibler divergence  during unsure
              perception  (Tozzi
            and Peters, 2016b) .
            similar  points and regions do
            not need necessarily
            to be opposite (antipodal), or embedded on the same
            structure.   Therefore, t he applications of BUT can be generalized also for
            neighbouring points (and/or regions) on an sphere.  In
            effect, it is possible t o evaluate
              matching signals, even if they are not exaclty
              “opposite” each other.  As a result, t he
            points restriction from the “standard” BUT is no longer
            needed, and  w e can also consider regions
            that are either
            adjacent or far apart.   This BUT variant applies, provided
            there is a
            pair of regions on the sphere with the same feature value.  We are thus allowed to say that the two points (or
            regions, or whatsoever)
            do not need necessarily to be antipodal, in order to be
            labeled together.  In brain terms, this means, for example,
            two regions on the cortical surface with the same entropy
            values can be
            described together.  
>>>T he
            original formulation of the BUT describes the presence of
            antipodal points on
            spatial manifolds in every dimension, provided the manifold
            is a convex,
            positive-curvature structure ( i.e , a
            ball).  However, many brain functions are
            believed to occur on functional hyperbolic manifolds in
            guise of a saddle,
            i.e., equipped with negative-curvature and concave shape. 
            Therefore, we are allowed to look for
            antipodal points also on structures equipped with curvatures
            other than the
            convex one.  Whether a system structure
            displays a concave, convex or flat appearance, does not
            matter: we may always
            find the points with matching description predicted by BUT .  A
            single description on a plane can be projected to higher
            dimensional donut-like
            structures, in order that a torus stands for the most
            general structure which
            permits the description of matching points .
>>>Although BUT has been originally described just in
            case of  n being a natural number which expresses
            a structure embedded in a spatial dimension, nevertheless  n can also stand for other types of numbers, when assessing
            brain  sphere. The BUT  can be
            used not just  for  the description of “spatial”
            dimensions equipped with natural numbers, but also of antipodal regions
            on brain spheres
            equipped with other  kinds of
            dimensions, such as a temporal or a fractal one .  This means, e.g., that spherical
              can be made not just of space, but also of time.  The
              dimension  n might stand not just for a natural
              number, but also for an
              integer, a rational, an irrational or an imaginary one. 
              For example,  n   may stand for a fractal
              dimension, which is generally expressed by a rational
              number.    This makes it possible for us t o use the n parameter as a versatile tool for the description
            of systems’
>>>Furthermore, matching
            points (or regions)
            might project to lower dimensions on the same structure.   A sphere may map on
            itself: the projection of two
            antipodal points to a single point into a dimension lower
            can be internal to
            the same sphere .   In this case,
              matching descriptions are assessed at one dimension of
              observation, while
              single descriptions at a lower one, and vice versa.   Such
              correlations are based on mappings,
              e.g., projections from one dimension to another.   I n
            many applications (for example, in fractal systems), we do
            not need the
            Euclidean space (the ball) at all:  a system may
              display an intrinsic,  internal point
              of view,  and does not need to lie in any
            dimensional space.  Therefore, we may think that the system
            does exist by  - and on – itself. 
              and BUT.   Symmetries are the most
            general features
            of mathematical, physical and biological entities and
            provide a very broad
            approach, explaining also how network communities integrate
            or segregate
            information.  S ymmetries  may
            be regarded as the most general feature of systems, perhaps
            more general than
            free-energy and entropy constraints too. 
            Indeed, recent data suggest that thermodynamic requirements
            have close
            relationships with symmetries.   A
              symmetry break occurs when the symmetry is present at one
              level of observation, but “hidden” at another level.  A
              symmetry break is detectable at a lower
              dimension of observation.  Thus, we can
              state that single descriptions are broken (or hidden)
              symmetries, while
              matching descriptions are restored symmetries.  
              In other words, a symmetry can be hidden at the lower
              dimension and
              restored when going one dimension higher. 
              If we assess just single descriptions, we cannot see their
              descriptions: when we evaluate instead systems one
              dimension higher, we are
              able to see their hidden symmetries.  
              This also means, that, going from a lower to an higher
              level of
              assessment, we find more information: indeed, to make an
              example, a
              three-dimensional image encompassess more information than
              a two-dimensional
              one.  In sum, symmetries, single and
              matching descriptions stand for  the
              common language able to describe the metastable brain.
>>>1)        Could we use projections and
            mappings, in order to
            describe brain activity?
>>>2)        Is such a topological approach
            linked with previous
            claims of old “epistemologists” of recent
>>>3)        Is such a topological approach
            linked with current
            neuroscientific models?
>>>4)        The BUT and its variants display
            four ingredients,
            e.g., a continuous function, antipodal points, changes of
            dimensions and the
            possibility of types of dimensions other than the spatial
            ones. Is it feasible
            to assess brain function in terms of BUT and its variants?
>>>5)        How to operationalize the
>>>6)        Is it possible to build a general
            topological theory
            of the brain?
>>>7)        Our “from afar” 
            approach takes into account the dictates of far-flung
            branches, from
            mathematics to physics, from algebraic topology, to
            neuroscience.  Do you think that such broad
            multidisciplinary tactics could be the key able to unlock
            the mysteries of the
            brain, or do you think that more specific and “on focus”
            approaches could give
            us more chances? 
>>>NOTE: A simple
            explanation of BUT and its
            novel variants
>>> (with
            the proper bibliography) can be found in this short movie on
>>>Best wishes
>>>Arturo Tozzi
>>>AA Professor Physics, University North Texas
>>>Pediatrician ASL
              Na2Nord, Italy
>>>Comput Intell
              Lab, University Manitoba
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