Could Brain Be Represented as a Hyperbolic Geometry?

Abstract

There are proposals that the lattice-like structures formed by neurons in some brain regions could be mapped to discrete sets of 2-D hyperbolic space H², possibly tesselations analogous to lattices of 2-D plane. The map is rather abstract: The points of tesselation would correlate with the  statistical properties of neurons rather than representing their geometric positions as such. In TGD framework, zero energy ontology (ZEO) suggests a generalization of replacing H² with 3-D hyperbolic space H³. The magnetic body (MB) of any system carrying dark matter as h_eff = nh₀provides a representation of any system (or perhaps vice versa). Could MB provide this kind of representation as a tesselation at 3-D hyperboloid of causal diamond (cd) defined as intersection of future and past directed light-cones of M⁴? The points of tesselation labelled by a subgroup of SL(2, Z) or it generalization replacing Z with algebraic integers for an extension of rationals  would be determined by its statistical properties. The positions of the magnetic images of neurons at H³ would define a tesselation of H³. The tesselation could be mapped to the analog of Poincare disk - Poincare ball - represented as t = T snapshot (t is the linear Minkowski time) of future light-cone. After t = T the neuronal system would not change in size. Tesselation could define cognitive representation as a discrete set of space-time points with coordinates in some extension of rationals assignable to the space-time surface representing MB. One can argue that MB has more naturally cylindrical instead of spherical symmetry so that one can consider also a cylindrical representation at E¹ x H² so that symmetry would be broken from SO(1,3) to SO(1,2). M⁸-H duality would allow to interpret the special value t = T in terms of special 6-D brane like solution of algebraic equations in M⁸ having interpretation as a "very special moment of consciousness" for self having CD as geometric correlate. Physically it could correspond to a (biological) quantum phase transition decreasing the value of length scale dependent cosmological constant Λ in which the size of the system increase by a factor, which is power of 2. This proposal is extremely general and would apply to cognitive representations at the MB of any system.