Estimation of nonlinear psychophysical kernels

Reverse correlation techniques have been extensively used in physiology (Marmarelis & Marmarelis 1978; Sakai, Naka, & Korenberg, 1988), allowing characterization of both linear and nonlinear aspects of neuronal processing (e.g., Emerson, Bergen, & Adelson, 1992; Emerson & Citron 1992). Over the past decades, Ahumada (1996) developed a psychophysical reverse correlation technique, termed noise image classification (NIC), for deriving the linear properties of sensory filters in the context of audition first (Ahumada, 1967; Ahumada, Marken, & Sandusky, 1975), and then vision (Ahumada, 1996, 2002; Beard & Ahumada, 1998). This work explores ways of characterizing nonlinear aspects of psychophysical filters. One approach consists of an extension of the NIC technique (ExtNIC), whereby second-order (rather than just first-order) statistics in the classified noise are used to derive sensory kernels. It is shown that under some conditions, this procedure yields a good estimate of second-order kernels. A second, different approach is also considered. This method uses functional minimization (fMin) to generate kernels that best simulate psychophysical responses for a given set of stimuli. Advantages and disadvantages of the two approaches are discussed. A mathematical appendix shows some interesting facts: (1) that nonlinearities affect the linear estimate (particularly target-present averages) obtained from the NIC method, providing a rationale for some related observations made by Ahumada (1967); (2) that for a linear filter followed by a static nonlinearity (LN system), the ExtNIC estimate of the second-order nonlinear kernel is correctly null, provided the criterion is unbiased; (3) that for a biased criterion, such an estimate may contain predictable modulations related to the linear filter; and (4) that under certain assumptions and conditions, ExtNIC does return a correct estimate for the second-order nonlinear kernel.