Nonlinear Filters for Tracking Chaos in Neurobiological Time Series

Summary

A nonlinear digital filtering approach to tracking the nonstationary dynamical behavior of neurobiological time series is described and evaluated. Nonlinear digital filters were designed with the objective of achieving high resolution tracking of the changing properties of complex nonstationary time series. Three simple nonlinear autoregressive models; the logistic, quadratic, and cubic models, were chosen because they each exhibit a rich variety of behavior including simple periodic, complex periodic, and chaotic behavior depending upon the value of a critical model parameter (the bifurcation parameter). The sensitivity of the time series to the value of the critical bifurcation parameter in each model is shown in the bifurcation diagrams Figs, la, lb, and lc.

Digital filters based on the three models were derived based on the least-squares estimate of the critical parameter. The tracking capability of these filters was evaluated using nonlinear, nonstationary test data. The nonstationarity was introduced by continuous time modulation of the bifurcation parameter in each model. The parameter was modulated over a range which results in the production of both periodic and chaotic behavior.

The test results obtained with these filter models demonstrate that accurate high resolution tracking of the critical bifurcation parameters can be achieved in essentially real-time using a standard laboratory computer. The high resolution capability of the filter is particularly useful for identifying chaotically stationary data epochs, an essential condition for estimating other chaotic features such as the Hausdorff dimension, the Lyapunov exponent or the attractor topography. Since the critical bifurcation parameter values increase with data complexity (i.e., increasing when data in the filter window changes from periodic to

chaotic), such filters can provide an effective economical means of monitoring the nonlinear and nonstationary chaotic properties of a long time series.