Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 593et al. (1992); Ellis et al. (1999); Mavromatos and Nanopoulos (1995a);Mavromatos and Nanopoulos (1995b); Nanopoulos (1995)]:(1) Conservation of probability P∂ t P = ∂ t [Tr(ˆρ)] = 0.(2) Conservation of energy E, on the average(3) Monotonic increase in entropy∂ t 〈〈E〉〉 ≡ ∂ t [Tr(ˆρ E)] = 0.∂ t S = ∂ t [− Tr(ˆρ ln ˆρ)] ≥ 0,and thus automatically and naturally implies a microscopic arrow oftime, so essential in realistic biophysics of neural processes.Non–Critical Stringy MT–DynamicsIn EMN–language of non–critical (SUSY) bosonic strings, our MT–dynamics equation (4.108) reads∂ tˆρ = i{ˆρ, Ĥ} − iĝ ij{ˆρ, ˆq i }ˆ˙q j , (4.109)where the target–space density matrix ˆρ(ˆq i , ˆp i ) is viewed as a function ofcoordinates ˆq i that parameterize the couplings of the generalized σ−modelson the bosonic string world–sheet, and their conjugate momenta ˆp i , whileĝ ij = ĝ ij (ˆq i ) is the quantum operator of the positive definite metric in thespace of couplings. Therefore, the covariant quantum operator of externalfriction forces is in EMN–formulation given as ˆF i (ˆq i , ˆ˙q i ) = ĝ ij ˆ˙q j .Equation (4.109) establishes the conditions under which a large–scalecoherent state appearing in the MT–network, which can be considered responsiblefor loss–free energy transfer along the tubulins.Equivalence of Neurodynamic FormsIt is obvious that both the macroscopic NN–equation (4.103) and themicroscopic MT–equation (4.108) have the same open Liouville form, whichimplies the arrow of time. These proves the existence of the formal neuro–biological space–time self–similarity.In this way, we have described neurodynamics of both NN and MTensembles, belonging to completely different biophysical space–time scales,