Nonlinear Analysis of Neuronal Systems

TransNav, the International Journal on Marine Navigation and Safety of Sea Transportation

This is a short tutorial on Volterra and Wiener series applications to modelling of nonlinear systems and phenomena, and also a survey of the recent achievements in this area. In particular, we show here how the philosophies standing behind each of the above theories differ from each other. On the other hand, we discuss also mathematical relationships between Volterra and Wiener kernels and operators. Also, the problem of a best approximation of large-scale nonlinear systems using Volterra operators in weighted Fock spaces is described. Examples of applications considered are the following: Volterra series use in description of nonlinear distortions in satellite systems and their equalization or compensation, exploiting Wiener kernels to modelling of biological systems, the use of both Volterra and Wiener theories in description of ocean waves and in magnetic resonance spectroscopy. Moreover, connections between Volterra series and neural network models, and also input-output descriptions of quantum systems by Volterra series are discussed. Finally, we consider application of Volterra series to solving some nonlinear problems occurring in hydrology, navigation, and transportation. http://www.transnav.eu the International Journal on Marine Navigation and Safety of Sea Transportation Volume 9 Number 1

Mathematical and Computer Modelling, 1998

Abstra&-Methods for nonlinear system identification are often classified, based on the employed model form, into parametric (nonlinear differential or difference equations) and nonparametric (functional expansions). These methods exhibit distinct sets of advantages and disadvantages that have motivated comparative studies and point to potential benefits from combined use. Fundamental to these studies are the mathematical relations between nonlinear differential (or difference, in discrete time) equations (NDE) and Volterra functional expansions (VFE) of the class of nonlinear systems for which both model forms exist, in continuous or discrete time. Considerable work has bean done in obtaining the VFE's of a broad class of NDE's, which can be used to make the transition from nonparametric models (obtained from experimental input-output data) to more compact parametric models. This paper presents a methodology by which this transition can be made in discrete time. Specifically, a method is proposed for obtaining a parametric NARMAX (Nonlinear Auto-Regressive Moving-Average with exogenous input) model from Volterra kernels estimated by use of input-output data.

Mathematical and Computer Modelling, 1989

This paper presents an identification procedure based on Volterra-Wiener functional expansion representations of a class of nonlinear systems often encountered in practice. Approximate analytical expressions are derived for the Volterra and Wiener kernels of these systems in the case of weak nonlinearities, i.e. for small magnitude coefficients of the polynomial nonlinearity. This identification method is based on Wierner kernel measurements obtained from input-output data when the input is Gaussian white noise. This method can be used for modelling and validation studies of this class of systems from experimental input-output data.

Biological Cybernetics, 1977

A Volterra-like polynomial representation is derived and its convergence discussed for two neuronal models in which subthreshold inputs are integrated either without loss (integrate and fire) or with a decay which follows an exponential time course (leaky integrator). This polynomial representation provides a kind of "nonlinear transfer function" for the nonlinear encoding process. Standard formulae are used to derive explicitely the output for various inputs as in linear system theory. Moreover, the nonlinear transfer function associated with cascades or networks of neurons can be also obtained. Finally, extensions and implications of these results are discussed.

IEEE Transactions on Biomedical Engineering, 2005

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.

Nonlinear Dynamics, 2005

Aeroelastic dynamics must be accurately known to ensure safe and efficient flight testing. Unfortunately, most models of aircraft systems typically describe only the linear dynamics. These models are inadequate for predicting behaviors, such as limit cycle oscillations, resulting from nonlinearities. This paper presents an approach to augment a linear model by identifying associated nonlinear operators. Essentially, the difference between a

IEEE Transactions on Biomedical Engineering, 1992

A nonlinear system may be characterized by an orthogonal functional power series (FPS) computed from cross correlations between input and output variables. "Is the response changing over the course of the experiment?" is a fundamental question encountered in the analysis of both FPS and evoked potentials (EP's). Regression on closed-form functions of time produces a time-varying FPS or EP. Evaluation of these functions at a specified time point produces a system characterization for that instant.

Neural Networks, 2000

This paper address the issue of nonlinear model estimation for neural systems with arbitrary point-process inputs using a novel network that is composed of a pre-processing stage of a Laguerre filter bank followed by a single hidden layer with polynomial activation functions. The nonlinear modeling problem for neural systems has been attempted thus far only with Poisson point-process inputs and using crosscorrelation methods to estimate low-order nonlinearities. The specific contribution of this paper is the use of the described novel network to achieve practical estimation of the requisite nonlinear model in the case of arbitrary (i.e. non-Poisson) point-process inputs and high-order nonlinearities. The success of this approach has critical implications for the study of neuronal ensembles, for which nonlinear modeling has been hindered by the requirement of Poisson process inputs and by the presence of high-order nonlinearities. The proposed methodology yields accurate models even for short input-output data records and in the presence of considerable noise. The efficacy of this approach is demonstrated with computer-simulated examples having continuous output and point-process output, and with real data from the dentate gyrus of the hippocampus. ᭧

ArXiv, 2014

The Volterra series can be used to model a large subset of nonlinear, dynamic systems. A major drawback is the number of coefficients required model such systems. In order to reduce the number of required coefficients, Laguerre polynomials are used to estimate the Volterra kernels. Existing literature proposes algorithms for a fixed number of Volterra kernels, and Laguerre series. This paper presents a novel algorithm for generalized calculation of the finite order Volterra-Laguerre (VL) series for a MIMO system. An example addresses the utility of the algorithm in practical application.

Algorithms, 2020

In the paper we deal with the problem of non-linear dynamic system identification in the presence of random noise. The class of considered systems is relatively general, in the sense that it is not limited to block-oriented structures such as Hammerstein or Wiener models. It is shown that the proposed algorithm can be generalized for two-stage strategy. In step 1 (non-parametric) the system is approximated by multi-dimensional regression functions for a given set of excitations, treated as representative set of points in multi-dimensional space. ‘Curse of dimensionality problem’ is solved by using specific (quantized or periodic) input sequences. Next, in step 2, non-parametric estimates can be plugged into least squares criterion and support model selection and estimation of system parameters. The proposed strategy allows decomposition of the identification problem, which can be of crucial meaning from the numerical point of view. The “estimation points” in step 1 are selected to e...