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

Statistical Mechanics of Neocortical ... - 15 - Lester Ingberwhere the M G -space drifts g G , and diffusions g GG′ , are given above. Note that the macroscopic drifts anddiffusions of the Φ’s are simply linearly related to the mesoscopic drifts and diffusions of the M G ’s. Forthe prepoint M G (t) firings, the same linear relationship in terms of { φ, a, b } is assumed.The mesoscopic probability distributions, P, are scaled and aggregated over this columnar firing space toobtain the macroscopic probability distribution over the scalp-potential space:P Φ [Φ] =∫ dM E dM I P[M E , M I ]δ [Φ−Φ′(M E , M I )] . (16)The parabolic trough described above justifies a formP Φ = (2πσ 2 ) −1/2 exp(− ∆t2σ 2 ∫ dx L Φ),L Φ = α 2 |∂Φ/∂t|2 + β 2 |∂Φ/∂x|2 + γ 2 |Φ|2 + F(Φ) , (17)where F(Φ) contains nonlinearities away from the trough, σ 2 is on the order of 1/N given the derivationof L above, and the integral over x is taken over the spatial region of interest. In general, there also willbe terms linear in ∂Φ/∂t and in ∂Φ/∂x.Previous calculations of EEG phenomena [5], show that the short-fiber contribution to the α frequencyand the movement of attention across the visual field are consistent with the assumption that the EEGphysics is derived from an average over the fluctuations of the system, e.g., represented by σ in the aboveequation. I.e., this is described by the Euler-Lagrange equations derived from the variational principlepossessed by L Φ (essentially the counterpart to force equals mass times acceleration), more properly bythe ‘‘midpoint-discretized’’ Feynman L Φ , with its Riemannian terms [2,3,11], Hence, the variationalprinciple applies,0 = ∂ ∂tThe result isα ∂2 Φ∂t 2∂L Φ∂(∂Φ/∂t) + ∂∂x∂L Φ∂(∂Φ/∂x) − ∂L Φ∂Φ . (18)+ β∂2 Φ ∂F+ γ Φ−∂x2 ∂Φ = 0 . (19)If there exist regions in neocortical parameter space such that β /α =−c 2 , γ /α = ω 2 0,1 ∂F=−Φf (Φ) , (20)α ∂Φ