Ivancevic_Applied-Diff-Geom

588 Applied Differential Geometry: A Modern Introductionergy transfer across the cells, without dissipation, had been first conjecturedto occur in biological matter by [Frölich and Kremer (1983)]. Thephenomenon conjectured by them was based on their 1D superconductivitymodel: in 1D electron systems with holes, the formation of solitonic structuresdue to electron–hole pairing results in the transfer of electric currentwithout dissipation. In a similar manner, Frölich and Kremer conjecturedthat energy in biological matter could be transferred without dissipation,if appropriate solitonic structures are formed inside the cells. This idea haslead theorists to construct various models for the energy transfer across thecell, based on the formation of kink classical solutions (see [Satarić et al.(1993); Satarić et al. (1998)].The interior of living cells is structurally and dynamically organizedby cytoskeletons, i.e., networks of protein polymers. Of these structures,microtubules (MTs, for short) appear to be the most fundamental(see [Dustin (1984)]). Their dynamics has been studied by a numberof authors in connection with the mechanism responsible for dissipation–free energy transfer. Hameroff and Penrose [Hameroff (1987)] have conjecturedanother fundamental role for the MTs, namely being responsiblefor quantum computations in the human neurons. [Penrose (1989);Penrose (1994); Penrose (1997)] further argued that the latter is associatedwith certain aspects of quantum theory that are believed to occurin the cytoskeleton MTs, in particular quantum superposition and subsequentcollapse of the wave function of coherent MT networks. Theseideas have been elaborated by [Mavromatos and Nanopoulos (1995a);Mavromatos and Nanopoulos (1995b)] and [Nanopoulos (1995)], based onthe quantum–gravity EMN–language of [Ellis et al. (1992); Ellis et al.(1999)] where MTs have been physically modelled as non-critical (SUSY)bosonic strings. It has been suggested that the neural MTs are the micrositesfor the emergence of stable, macroscopic quantum coherent states,identifiable with the preconscious states; stringy–quantum space–time effectstrigger an organized collapse of the coherent states down to a specificor conscious state. More recently, [Tabony et al. (1999)] have presentedthe evidence for biological self–organization and pattern formation duringembryogenesis.Now, we have two space–time biophysical scales of neurodynamics. Naturallythe question arises: are these two scales somehow inter–related, isthere a space–time self–similarity between them?The purpose of this subsection is to prove the formal positive answerto the self–similarity question. We try to describe neurodynamics on both