Recursive Genome Function of the Cerebellum: Geometric Unification of Neuroscience and Genomics; Pellionisz AJ, Graham R, Pellionisz PA, Perez JC. Chapter 14. In: Handbook of the Cerebellum and Cerebellar Disorders, Eds. By Manto E, Gruol D, Schmahmann J, Koibuchi N, Rossi F, 2013, Springer: Chapter 61, p 1381-1422, DOI 10.1007/978-94-007-1333-8_61


Andras J. Pellionisz, Roy Graham, Peter A. Pellionisz, Jean-Claude Perez


Recursive Fractal Genome Function in the geometric mind frame of Tensor Network Theory (TNT) leads through FractoGene to a mathematical unification of physiological and pathological development of neural structure and function as governed by the genome. The cerebellum serves as the best platform for unification of neuroscience and genomics. The matrix of massively parallel neural nets of fractal Purkinje brain cells explains the sensorimotor, multidimensional non-Euclidean coordination by the cerebellum acting as a space-time metric tensor. In TNT, the recursion of covariant sensory vectors into contravariant motor executions converges into Eigenstates composing the cerebellar metric as a Moore-Penrose Pseudo-Inverse.

The Principle of Recursion is generalized to genomic systems with the realization that the assembly of proteins from nucleic acids as governed by regulation of coding RNA (cRNA) is a contravariant multi-component functor, where in turn the quantum states of resulting protein structures both in intergenic and intronic sequences are measured in a covariant manner by non-coding RNA (ncRNA) arising as a result of proteins binding with ncDNA modulated by transcription factors. Thus, cRNA and ncRNA vectors by their interference constitute a genomic metric. Recursion through massively parallel neural network and genomic systems raises the question if it obeys the Weyl law of Fractal Quantum Eigenstates, or when derailed, pathologically results in aberrant methylation or chromatin modulation; the root cause of cancerous growth. The growth of fractal Purkinje neurons of the cerebellum is governed by the aperiodical discrete quantum system of sequences of DNA bases, codons and motifs. The full genome is fractal; the discrete quantum system of pyknon-like elements follows the Zipf-Mandelbrot Parabolic Fractal Distribution curve.

The Fractal Approach to Recursive Iteration has been used to identify fractal defects causing a cerebellar disease, the Friedreich Spinocerebellar Ataxia - in this case as runs disrupting a fractal regulatory sequence. Massive deployment starts by an open domain collaborative definition of a standard for fractal genome dimension in the embedding spaces of the genome-epigenome-methylome to optimally diagnose cancerous hologenome in the nucleotide, codon or motif-hyperspaces. Recursion is parallelized both by open domain algorithms, and also by proprietary FractoGene algorithms on high performance computing platforms, for genome analytics on accelerated private hybrid clouds with PDA personal interfaces, becoming the mainstay of clinical genomic measures prior and post cancer intervention in hospitals and serve consumers at large as Personal Genome Assistants.



Abstract and Clickable References in .pdf format

Citation before publication by co-Author (Perez, Nov. 2011, .pdf)

Independent citation before publication (Petoukhov, Dec. 2011. pdf) See also illustration of quarternion fractals and excerpts here

Dr. Anderson's lucid explanation of Pellionisz' use of covariance and contravariance, in "Neurocomputing 2", Anderson, Pellionisz and Rosenfeld, MIT1990 (free .pdf)

Cerebellum Acting via a Metric Tensor (Pellionisz and Llinas, 1980 full free text .pdf)

Dual (covariant and contravariant) valence explained in Wikipedia

Covariant and Contravariant Functors explained in Wikipedia

FractoGene Timeline (Andras J. Pellionisz)

Fractals (with contribution by Peter A. Pellionisz)

Authors' acknowledgement: "to Academician Prof. Sergey Petoukhov, Moscow, for reference to Gazalé and appreciative comments of the Chapter on dual valence, the RNA system serving as a 'Genomic Cerebellum'"

Classic Textbooks on Covariant and Contravariant representation:

Hay G.E. (1953) Vector and Tensor Analysis. New York: Dover
Coburn N. (1955) Vector and Tensor Analysis, Dover, New York
Bickley W.G. and Gibson R.E. (1962) Via Vector to Tensor. New York: Wiley
Hoffmann B. (1966) About Vectors. Prentice Hall, New Jersey
Kay D.C. (1988) Tensor Calculus. New York: Schaum's Outlines, McGraw-Hill

Modern Academic and Lucid Book (eminently available in print-form AND on the web) on Covariant and Contravariant representation:

Kevin Brown
Section 5.2. is on Tensors, Covariant, Contravariant
[Excerpts from Kevin Brown's Print/Web-book, Section 5.2]:

"Ten masts at each make not the altitude
which thou hast perpendicularly fell.
Thy life’s a miracle. Speak yet again."

...In general, any given vector or tensor can be expressed in both contravariant and covariant form with respect to any given coordinate system. For example, consider the vector P shown below.

Figure 1 shows an arbitrary coordinate system with the axes X1 and X2, and the contravariant and covariant components of the position vector P with respect to these coordinates. As can be seen, the jth contravariant component consists of the projection of P onto the jth axis parallel to the other axis, whereas the jth covariant component consists of the projection of P into the jth axis perpendicular to that axis. This is the essential distinction (up to scale factors) between the contravariant and covariant ways of expressing a vector or, more generally, a tensor. ...If the coordinate system is "orthogonal" (meaning that the coordinate axes are mutually perpendicular) then the contravariant and covariant interpretations are identical...

[The "existence proof" that the two proundly different representations of MAKING an invariant from components and MEASURING components of the emerging invariant can be understood by biologists is exemplified the great number of co-authors and followers of Tensor Network Theory over a 30-year lead-time. Likewise, some will instantly grasp that MAKING proteins from cRNA and MEASURING attributes of proteins by ncRNA are fundamentally different procedures and representations - AJP]