Use of Meixner functions in estimation of Volterra kernels of nonlinear systems with delay

Abstract

Volterra series representation of nonlinear systems is a mathematical analysis tool that has been successfully applied in many areas of biological sciences, especially in the area of modeling of hemodynamic response. In this study, we explored the possibility of using discrete time Meixner basis functions (MBFs) in estimating Volterra kernels of nonlinear systems. The problem of estimation of Volterra kernels can be formulated as a multiple regression problem and solved using least squares estimation. By expanding system kernels with some suitable basis functions, it is possible to reduce the number of parameters to be estimated and obtain better kernel estimates. Thus far, Laguerre basis functions have been widely used in this framework. However, research in signal processing indicates that when the kernels have a slow initial onset or delay, Meixner functions, which can be made to have a slow start, are more suitable in terms of providing a more accurate approximation to the kernels. We, therefore, compared the performance of Meixner functions, in kernel estimation, to that of Laguerre functions in some test cases that we constructed and in a real experimental case where we studied photoreceptor responses of photoreceptor cells of adult fruitflies (Drosophila melanogaster). Our results indicate that when there is a slow initial onset or delay, MBF expansion provides better kernel estimates.