Volterra models and three-layer perceptrons

This paper proposes the use of a class of feedforward artificial neural networks with polynomial activation functions (distinct for each hidden unit) for practical modeling of high-order Volterra systems. Discrete-time Volterra models (DVM's) are often used in the study of nonlinear physical and physiological systems using stimulus-response data. However, their practical use has been hindered by computational limitations that confine them to low-order nonlinearities (i.e., only estimation of low-order kernels is practically feasible). Since three-layer perceptrons (TLP's) can be used to represent input - output nonlinear mappings of arbitrary order, this paper explores the basic relations between DVM and TLP with tapped-delay inputs in the context of nonlinear system modeling. A variant of TLP with polynomial activation functions - termed 'separable Volterra networks' (SVN's) - is found particularly useful in deriving explicit relations with DVM and in obtaining practicable models of highly nonlinear systems from stimulus-response data. The conditions under which the two approaches yield equivalent representations of the input-output relation are explored, and the feasibility of DVM estimation via equivalent SVN training using backpropagation is demonstrated by computer-simulated examples and compared with results from the Laguerre expansion technique (LET). The use of SVN models allows practicable modeling of high-order nonlinear systems, thus removing the main practical limitation of the DVM approach. Index Terms - Laguerre kernel expansion, nonlinear system modeling, polynomial activation functions, separable Volterra network, three-layer perceptrons, Volterra kernels, Volterra models.