In 1809, Carl Frederic Gauss  dealt in depth with problem of estimation of planetary and comet orbit parameters when the terrestrial observational data was sparse and imprecise. Gauss's problem bears strong resemblance to the problem we face in the estimation of neuronal state through electrical or optical measurements. Recent advances in Kalman filtering to estimate system state and parameters in nonlinear systems has offered the potential to apply such approaches to spatiotemporal neuronal systems. We here adapt the nonlinear method of unscented Kalman filtering (UKF) to observe the state and estimate parameters in a computational spatiotemporal excitable system that serves as a model of cerebral cortex.
We employ the modifications to the Wilson-Cowan equations as suggested by Pinto and Ermentrout , to model an experimental system in which we measure the wave dynamics of mammalian middle cortical layers using voltage sensitive dye. Employing a strategy of UKF suitable for neuronal models , we adapt covariance inflation from meteorological modeling to compensate for higher order errors known to occur in UKF .
We demonstrate the ability of an observer UKF (Figure 1A) to track spiral wave dynamics (Figure 1B), and to calculate control signals delivered through applied electrical fields in a manner that can be implemented experimentally. We demonstrate how this strategy can control the frequency of such a system, or quench the wave patterns, while minimizing the energy required for such results (Figure 1C).
Figure 1. A: UKF schematic, B: activity variable in 8 × 8 grid, and C: time trace of activity from one grid cell.
We have, to our knowledge, demonstrated the first computational framework to apply nonlinear Kalman filtering to track and control spatiotemporal cortical neuronal activity. The use of a Kalman observer permits state variable reconstruction and parameter estimation under conditions where direct proportional control calculated from measured variables would fail. In experimental applications, employing an observer UKF to generate the control signal can substantially reduce the control energy required to increase or decrease the frequency of oscillations, or to quench wave activity. We are adapting these findings to models of Parkinson's disease dynamics, and to experimental applications in control of cortical waves, control of hippocampal rhythms in navigating animals, and the extraction of features from EEG in brain machine interface paradigms. These results are in press in .
Supported by NIH Grants R01MH50006 and K02MH01493.
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