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

Statistical Mechanics of Neocortical ... - 25 - Lester Ingber5.2. Approaches to Source LocalizationAs described above, SMNI stresses that is natural to describe regional macroscopic dynamics in terms ofmesocolumnar interactions, i.e., a true “field” of firings is developed, which has been mathematicallyarticulated in the SMNI papers as defining the regional propagator in the path integral D ˜M over thespace-time volume dx − dt of the brain, in terms of the Lagrangian L each volume point. The spacevolume dx is where the head shape explicitly enters the calculation. For example, fitting L to actual datamight require discretization at electrode sites, as performed in this study, where actually the cost functionin terms of L to be fit contains the volume elements in dx. If a 3-D fit were being done, then dx would bethe head volume; if a 2-D surface fit were being done, dx would be the head/brain surface. In practice,we must rely on other methods, e.g., MRI, to determine the head shape to articulate the shape spanned bydx. A higher resolution algorithm, at the level of minicolumns, shows how short-ranged as well as longrangedinteractions among mesocolumns within and between regions can be naturally included in theSMNI theory [14].Such fits can be performed after other source localization techniques are applied to the data, e.g., headvolumeor Laplacian techniques. This still is very important, as identification of source(s), especiallythose that may be nonstationary, is not sufficient to describe their dynamical interaction. The fitted SMNIdynamics then can provide the dynamical description to predict future evolution of the interactions amongsources, e.g., to fill in gaps in data that might aid correlation with behavioral states.The SMNI dynamics can more directly be part of source localization algorithms. For example, if M/dt isidentified as the the mesocolumnar current (M is essentially the number of neuronal firings within thetime τ of about 5 msec; dt is the time resolution of the EEG, typically on this order or somewhat less),then we can identify M = B∇ 2 Φ, where ∇ 2 is the Laplacian and B is a constant included in the fit, similarto the relationship between M and Φ in this present project. Then, the Laplacian of Φ would be input intothe SMNI action A, and the parameters of the model would be fit as performed here. The ASA globaloptimization of the highly nonlinear SMNI finds the best fit among the combined local-global interactionsalgebraically described in the SMNI action A. The “curvatures” (second derivatives) of the parametersabout the global minimum, automatically returned by ASA, give a covariance matrix of the goodness offit about the global minimum.