Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 591open Liouville equation:∂ t ρ = −[ρ, H] + Φ . (4.103)In particular, the scalar force term can be cast as a linear Poisson–bracket formΦ = F i [A, q i ] , with [A, q i ] = − ∂A∂p i. (4.104)Now, in a similar way as the conservative Liouville equation (4.98) resemblesthe continuity equation (4.99) from continuum dynamics, also theopen Liouville equation (4.103) resembles the probabilistic Fokker–Planckequation from statistical mechanics. If we have a ND stochastic processx(t) = (q i (t), p i (t)) defined by the vector Itô SDEdx(t) = f(x, t) dt + G(x, t) dW,where f is a ND vector function, W is a KD Wiener process, and G is aN ×KD matrix valued function, then the corresponding probability densityfunction ρ = ρ(x, t|ẋ, t ′ ) is defined by the ND Fokker–Planck equation (see,e.g., [Gardiner (1985)])∂ 2∂ t ρ = − div[ρ f(x, t)] + 1 (Q ij ρ) , (4.105)2 ∂x i ∂x jwhere Q ij = ( G(x, t) G T (x, t) ) . It is obvious that the Fokker–Planckijequation (4.105) represents the particular, stochastic form of our generalopen Liouville equation (4.103), in which the scalar force term is given bythe (second–derivative) noise term∂ 2Φ = 1 (Q ij ρ) .2 ∂x i ∂x jEquation (4.103) will represent the open classical model of our macroscopicNN–dynamics.Continuous Neural Network DynamicsThe generalized NN–dynamics, including two special cases of gradedresponse neurons (GRN) and coupled neural oscillators (CNO), can bepresented in the form of a stochastic Langevin rate equation˙σ i = f i + η i (t), (4.106)