Cross-correlation methods for quantification of nonlinear input-output transformations of neural systems using a Poisson random test input

We show that input-output cross-correlation can identify the nonlinear input-output transformation of a system by estimating the kernels of the Discrete Volterra Series (DVS). The DVS kernels have a simple interpretation and provide a useful characterization of the system nonlinearities. We present two complementary methods for computing the DVS kernels. The first method uses the formula for the Discrete Poisson Orthogonal Series (DPOS) kernels. We derive the DPOS by orthogonalizing the DVS for the Poisson Random Event Sequence (PRES) test input. Although we show that the DPOS kernels contain components involving all the DVS kernels, we also show that we can isolate a single DVS kernel if at least one of three conditions is satisfied. The second method is a general method that involves solving a set of simultaneous linear equations to isolate each DVS kernel from other system nonlinearities. We use simulations to demonstrate our theoretical results.