  Volterra analysis and its variants have long been prominent among methods for
modeling multi-input non-linear systems. The product of Volterra analysis, the
Volterra kernels, are particularly suited to quantifying intra- and inter-input
interactions. They are also readily interpretable, which means that they can be
related directly to physical behaviors, and more distantly, to the underlying
processing mechanisms of the system being tested. However, accurate estimation
of a sufficient set of classical kernels is often not possible for complex
systems because the number of kernels that need to be determined, and hence
experiment time, increases radically with system memory, response frequency
bandwidth, and non-linear interaction order. Practical approaches to kernel
estimation often involve forced reductions of the generality of the analysis
that in turn compromise interpretability.
  Here we illustrate the effects on kernel interpretability of two common
reductions, slow-stimulation and the use of binary inputs, using both numerical
simulations and data from a Visual Evoked Potential experiment. We show how a
non-standard version of binary analysis, involving a different coding of the
inputs and the use of particular groupings of kernel slices, improves kernel
interpretability. We bring together, in a comprehensive fashion, all of the
mathematical considerations needed to apply and interpret the results of
Volterra kernel analyses. The input-coding method we describe allows one to
correctly quantify multi-input interactive effects that occur both within the
separate input channels and across them.
