[Fis] Fwd: NEW DISCUSSION SESSION--TOPOLOGICAL BRAIN
tozziarturo at libero.it
tozziarturo at libero.it
Wed Nov 23 18:03:13 CET 2016
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Da: tozziarturo at libero.it A: fis at listas.unizar.es Data: mercoledì, 23 novembre 2016, 07:55AM +01:00
Oggetto: NEW DISCUSSION SESSION--TOPOLOGICAL BRAIN
>>
>>>
>>>
>>>Dear FIS,
>>>thanks for this precious oppurtinity.
>>>Here you find the text in order to start the discussion.
>>>
>>>
>>>TOPOLOGY AND BRAIN FUNCTION
>>>
>>>Arturo Tozzi
>>>Center
for Nonlinear Science, University of North Texas
>>>1155
Union Circle, #311427
>>>Denton,
TX 76203-5017, USA, and
>>>tozziarturo at libero.it
>>>James F. Peters
>>>Department
of Electrical and Computer Engineering, University of Manitoba
>>>75A
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
phenomena.
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
ones.
>>>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) .
>>>Descriptively
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
non-antipodal
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
antipodal
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,
that
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
the
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
structures
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’
features.
>>>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
just
does exist by - and on – itself.
>>>Symmetries
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
matching
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.
>>>Questions.
>>>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
“neuro-philosophers”?
>>>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
procedures?
>>>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
Youtube:
>>>https://www.youtube.com/watch?v=oxfqraR1bIg
>>>
>>>
>>>Best wishes
>>>
>>>
>>>Arturo Tozzi
>>>AA Professor Physics, University North Texas
>>>Pediatrician ASL
Na2Nord, Italy
>>>Comput Intell
Lab, University Manitoba
>>>http://arturotozzi.webnode.it/
>>>
>
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