Could brain have a hyperbolic geometry in some sense?

There are proposals that neurons in brain could have hyperbolic geometry (see this) in the sense that neurons would form 2-D lattice like structure imbeddable to 2-D hyperbolic geometry H². The standard representations for 2-D hyperbolic geometry are 2-D Poincare plane and Poincare disk. Poincare disk is claimed to be natural imbedding space for the lattice like structure of neutrons. These lattice structures of H² are known as tesselations. There is a painting of Escher visualizing Poincare disk. From this painting one learns that the density of points of the tesselation increases without limit as one approaches the boundary of the Poincare disk.

Already 2-D hyperbolic plane allows infinite number of tesselations as left coset spaces G\H² of H²= SO(1,2)/SO(1,1). G is here some infinite discrete subgroup G⊂ SO(1,2)= SL(2,R) of Lorentz group SO(1,2) of 3-D Minkowski space M³ satisfying some additional conditions. SO(1,2) acts as isometries of hyperbolic geometry preserving thus distances. For ordinary sphere S² the analogs of tesselations are finite lattices correspond to Platonic solids – tetrahedron, octahedron and cube, and icosahedron and dodecahedron. Tesselations would define hyperbolic analogs of Platonic solids.

In fact, Lorentz group SO(1,3) of 4-D Minkowski space acts as Möbius transformations z→ (az+b)/(cz+d), ad-bc=1, of complex plane. For SL(2,R) the parameters (a,b,c,d) are real.

The obvious question is whether one could generalize the hypothesis.

1. Could 3-D lattice-like structure formed by neurons correspond to 3-D hyperbolic space H³ representable as hyperboloid t²-x²-y²-z²== t²-r_M²=a². a has interpretation as light-cone proper time and in TGD inspired cosmology it corresponds to cosmic time. 2-D hyperbolic space could be seen as subspace of H³. Now infinite discrete subgroups of SO(1,3) would define tesselations as lattice-like structures. They would serve as 3-D analogs of Platonic solids.

2. The first thing to notice is that if the brain region would correspond to a region of H³, it would participate inc cosmic expansion. This cannot true. Quite generally all known astrophysical objects participate in cosmological expansion by receding from each other as the cosmic redshifts show but do not experience cosmological expansion themselves. TGD solves this paradox by the assumption that cosmic expansion takes place as quantum phase transitions in which expansion occurs in rapid jerks, which correspond to reductions of length scale dependent cosmological constant Λ by a power of 2 if p-adic length scale hypothesis is accepted.

There is evidence that even Earth has experienced this kind of expansion during Cambrian Explosion, which would have increased the radius of Earth by factor 2 (see this). An interesting question is whether one could see also the biological growth and development of organs and organelles as a sequence of this kind of phase transitions.

For light-cone M⁴₊ one has a natural slicing is by using the hyperboloids a= constant. This slicing would define a natural time coordinate as analog of cosmic time. The usual linear Minkowski coordinates define a second natural natural slicing by t=constant sections, where t is the linear Minkowski time.

One can define the standard hyperbolic coordinates of M⁴₊ by the line element

ds²= da²- a²(dη² +sinh²(η)dΩ²) .

dΩ² = dθ²+sin²(theta) dφ² is the line element of unit sphere S². η is the hyperbolic angle identifiaable as analog of ordinary angle and having expression

tanh(η)= r_M/t== β

having an interpretation as velocity β =v/c n radial direction satisfying β≤ 1: one has t= acosh(η) and r_M=asinh(η).

What could the precise correspondence between 3-D surfaces and H³ be?

1. The space-time surface itself is not hyperbolic space but could in some sense have discrete subgroup of G⊂ H³ as its symmetries: a possible interpretation would be as cognitive representations consisting of points of H with coordinates in extension of rationals defining the adele. The lattice-like structure associated with 3-surfaces could be mappable to this kind of hyperboloid for some value of a.

Could the subsystem of brain in question could be seen as an intersection of the with t=T section of M⁴₊ with the slicing of M⁴₊ by a= constant hyperboloids such that neurons as points of the tesselation of H³ defining cognitive representation would belong to the intersection? For t>T the 3-D structure would be preserved in good approximation.

The discrete geometries of neural sub-system as tesselations would naturally correspond to discrete subgroups of G⊂ SO(1,3) as analogs G\ H³ of Platonic solids. There is infinite number of them. One obtains also their 2-D variants as 2-D planar slices consistent with the symmetries just like one can have 2-D lattices as sub-lattices of 3-D lattices in E³.

In the case of neuronal system one can estimate the T from the size R of the system, which could that of brain hemisphere at most (at the level of magnetic body (MB) the size could be much larger). This would give R≤ 9 cm and T ≤ R/c≤ 3 × 10^-10 s. The part of neuronal system considered could be the above described intersection corresponding to time t=T. After this no expansion would take place and the 3-D analog of Poincare ball would be preserved.

This picture suggests an interesting connection to TGD based view about quantum measuremrent theory, which actually extends physics to a theory of consciousness.

1. In zero energy ontology (ZEO) replacing ordinary ontology of quantum theory the notion of causal diamond (CD) plays a central role. CDs for a length scale hierarchy and CDs have sub-CDs. Space-time surfaces for given CD have ends at the upper and lower boundary of CD. In this picture the appearance of hyperbolic geometry would be very natural. I have indeed speculated with the possibility that the tesselations of H³ could explain the quantization of redshift in cosmology for which there is evidence and might play a role also in quantum biology.

2. M⁸-H correspondence states that space-time surfaces could be regarded either as algebraic surfaces in M⁸ or as preferred extremals of action in H=M⁴× CP₂ reducing to minimal surface satisfying infinite number of additional conditions. Otherwise the consistency of dynamics in H dictated by partial differential equations with algebraic dynamics in M⁸ dictated by algebraic equations would not be possible.

One can say that space-time surfaces are roots of an octonionic polynomial obtained as an algebraic continuation of a real polynomial with rational coefficients to octonionic polynomial. This in the sense that either imaginary or real part of P in quaternionic sense vanishes and gives rise to 4-D surface in the generic case.

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