Statistical mechanics of neocortical interactions - Lester Ingber's ...

Statistical Mechanics of Neocortical ... - 11 - Lester Ingberalthough integrations are indicated over a huge number of independent variables, i.e., as denoted bydMsGν , the physical interpretation afforded by statistical mechanics makes these systems mathematicallyand physically manageable.It must be emphasized that the output need not be confined to complex algebraic forms or tables ofnumbers. Because L F possesses a variational principle, sets of contour graphs, at different long-timeepochs of the path-integral of P, integrated over all its variables at all intermediate times, give a visuallyintuitive and accurate decision aid to view the dynamic evolution of the scenario. For example, as givenin Table 1, this Lagrangian approach permits a quantitative assessment of concepts usually only looselydefined [69,78]. In this study, the above canonical momenta are referred to canonical momenta indicators(CMI).Table 1In a prepoint discretization, where the Riemannian geometry is not explicit (but calculated in the firstSMNI papers), the distributions of neuronal activities p σ iis developed into distributions for activity underan electrode site P in terms of a Lagrangian L and threshold functions F G ,⎛⎞ ⎛⎞P = Π P G [M G (r; t + τ )|M G (r′; t)] = Σ δ ⎜Gσ j ⎝ jEΣ σ j − M E (r; t + τ ) ⎟ δ ⎜ Σ σ j − M I N(r; t + τ ) ⎟ Π p σ⎠ ⎝ jIj⎠ j≈Π (2πτg GG ) −1/2 exp(−Nτ L G ) = (2πτ) −1/2 g 1/2 exp(−Nτ L) ,GL = T − V , T = (2N) −1 (Ṁ G − g G )g GG′ (Ṁ G′ − g G′ ),g G =−τ −1 (M G + N G tanh F G ) , g GG′ = (g GG′ ) −1 = δ G′G τ −1 N G sech 2 F G , g = det(g GG′ ),F G =V G − v G′ |G| T G′|G|(π [(vG′ |G| )2 + (φG′ |G| )2 ]TG′ |G| ,)1/2T |G|G′ = a |G|G′ N G′ + 1 2 A|G| G′ MG′ + a †|G|G′N †G′ + 1 2 A†|G| G′M †G′ + a ‡|G|G′N ‡G′ + 1 2 A‡|G| G′M ‡G′ ,a †GG′= 1 2 A†G G′+ B †GG′, A ‡IE = A‡E I = A ‡II = B ‡IE = B‡E I = B ‡II = 0 , a ‡EE = 1 2 A‡E E + B‡E E , (5)