Methods for Parameter Identification in Oscillatory Networks

Directed information transfer in the brain is presumably arranged by coupled oscillators. A suitable model is provided by oscillatory networks. Generally, an oscillatory network is described by a set of coupled oscillators, where each component is given by an ordinary differential equation. Usually, these equations contain unknown parameters like frequency, phase or damping quantities. Furthermore, the network contains parameters describing couplings between the oscillators. A general parameter identification task is the estimation of an optimal parameter set by fitting the network solution to measured data. This results mostly in least squares problems. Least squares problems can be solved by general optimization methods, but it is shown that special techniques were superior to these general approaches. All methods for non-linear optimization are iterative. From a given starting parameter vector, the methods produce a series of vectors which converge to a local minimum of a certain objective function. In principle, there are two classes of optimization procedures, the local and the global ones. The local procedures are mostly gradient descent algorithms, and find a local optimum in dependence on starting parameters. In comparison to global ones, they are characterized by a faster convergence. Global search algorithms tend to a reduced dependence of the optimal solution on the starting conditions. In many cases, the resulting optimization problems could be solved easily by a modified Levenberg-Marquardt method. When this method failed, an improvement of the results could be reached by Bock's multiple shooting method. It attempts to reduce the problems of local convergence in the presence of a vast number of local minima of the objective function. Thereby, the time domain of the network evolution is divided into many intervals, and the parameter identification is performed on each of these intervals with adapted initial values. The disadvantage is that the resulting solution is not necessary continuous, and some effort is required to find an adequate continuous solution. Furthermore, stochastic algorithms like simulated annealing can be useful at least for the determination of suitable starting parameters. The result of a parameter identification is demonstrated on a network model for the description of the spatio-temporal behavior of cortical 600Hz oscillations within the Brodmann areas 3b and 1 after electrical stimulation of nervus medianus.