Ivancevic_Applied-Diff-Geom

592 Applied Differential Geometry: A Modern Introductionwhere σ i = σ i (t) are the continual neuronal variables of ith neurons(representing either membrane action potentials in case of GRN, or oscillatorphases in case of CNO); J ij are individual synaptic weights;f i = f i (σ i , J ij ) are the deterministic forces (given, in GRN–case, byf i = ∑ j J ij tanh[γσ j ] − σ i + θ i , with γ > 0 and with the θ i representinginjected currents, and in CNO–case, by f i = ∑ j J ij sin(σ j − σ i ) + ω i ,with ω i representing the natural frequencies of the individual oscillators);the noise variables are given as η i (t) = lim ∆→0 ζ i (t) √ 2T/∆ where ζ i (t)denote uncorrelated Gaussian distributed random forces and the parameterT controls the amount of noise in the system, ranging from T = 0(deterministic dynamics) to T = ∞ (completely random dynamics).More convenient description of the neural random process (4.106) isprovided by the Fokker–Planck equation describing the time evolution ofthe probability density P (σ i )∂ t P (σ i ) = − ∂ (f i P (σ i )) + T ∂2∂σ i ∂σ 2 P (σ i ). (4.107)iNow, in the case of deterministic dynamics T = 0, equation (4.107)can be put into the form of the conservative Liouville equation (4.98), bymaking the substitutions: P (σ i ) → ρ, f i = ˙σ i , and [ρ, H] = div(ρ ˙σ i ) ≡∑i∂∂σ i(ρ ˙σ i ), where H = H(σ i , J ij ). Further, we can formally identifythe stochastic forces, i.e., the second–order noise–term T ∑ ∂ 2iρ with∂σ 2 iF i [ρ, σ i ] , to get the open Liouville equation (4.103).Therefore, on the NN–level deterministic dynamics corresponds to theconservative system (4.98). Inclusion of stochastic forces corresponds tothe system opening (4.103), implying the macroscopic arrow of time.Open Quantum SystemBy formal quantization of equation (4.103) with the scalar force termdefined by (4.104), in the same way as in the case of the conservativedynamics, we get the quantum open Liouville equation∂ tˆρ = i{ˆρ, Ĥ} + ˆΦ, with ˆΦ = −i ˆFi {ˆρ, ˆq i }, (4.108)where ˆF i = ˆF i (ˆq i , ˆp i , t) represents the covariant quantum operator of externalfriction forces in the Hilbert state–space H = Hˆq i ⊗ Hˆpi .Equation (4.108) will represent the open quantum–friction model of ourmicroscopic MT–dynamics. Its system–independent properties are [Ellis