^{2}, possibly tesselations analogous to lattices of 2-D plane. The standard representations for 2-D hyperbolic geometry are 2-D Poincare plane and Poincare disk. 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.

** Remark**: 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.

In TGD framework zero energy ontology (ZEO) suggests a generalization of replacing H^{2} with 3-D hyperbolic space H^{3}. The magnetic body (MB) of any system carrying dark matter as h_{eff}=nh_{0} provides 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 M^{4}? 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 H^{3} would define a tesselation of H^{3}. 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 E^{1}× H^{2} so that symmetry would be broken from SO(1,3) to SO(1,2).

M^{8}-H duality would allow to interpret the special value t=T in terms of special 6-D brane like solution of algebraic equations in M^{8} 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.

See the article Could brain be represented by a hyperbolic geometry?.

For a summary of earlier postings see Latest progress in TGD.