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

Statistical Mechanics of Neocortical ... -9- Lester Ingbermathematical physics used to develop SMNI has been utilized to describe the model in terms of rigorousCMI that provide an immediate intuitive portrait of the EEG data, faithfully describing the neocorticalsystem being measured. The CMI give an enhanced signal over the raw data, and give some insights intothe underlying columnar interactions.3.1. CMI, Information, EnergyIn the first SMNI papers, it was noted that this approach permitted the calculation of a true nonlinearnonequilibrium “information” entity at columnar scales. With reference to a steady state P( ˜M) for ashort-time Gaussian-Markovian conditional probability distribution P of variables ˜M, when it exists, ananalytic definition of the information gain ˆϒ in state ˜P( ˜M) over the entire neocortical volume is definedby [73,74]ˆϒ[ ˜P] =∫ ... ∫ D ˜M ˜P ln( ˜P/P) , DM = (2π ĝ 2 0∆t) −1/2uΠ (2π ĝ 2 s∆t) −1/2 dM s , (1)where a path integral is defined such that all intermediate-time values of ˜M appearing in the folded shorttimedistributions ˜P are integrated over. This is quite general for any system that can be described asGaussian-Markovian [75], even if only in the short-time limit, e.g., the SMNI theory.As time evolves, the distribution likely no longer behaves in a Gaussian manner, and the apparentsimplicity of the short-time distribution must be supplanted by numerical calculations. The FeynmanLagrangian is written in the midpoint discretization, for a specific macrocolumn corresponding tos=1M(t s ) = 1 2 [M(t s+1) + M(t s )] . (2)This discretization defines a covariant Lagrangian L F that possesses a variational principle for arbitrarynoise, and that explicitly portrays the underlying Riemannian geometry induced by the metric tensor g GG′ ,calculated to be the inverse of the covariance matrix g GG′ . Using the Einstein summation convention,P =∫ ... ∫ DM exp ⎛ ⎝ − u⎞Σ ∆tL Fs⎠ ,s=0DM = g 1/20 +(2π ∆t) −Θ/2 Π ug 1/2 Θs + Π (2π ∆t) −1/2 dMs G ,s=1G=1