Equation of motion and general solution for the one-dimensional complex cell response in the signal-tuned approach

A signal-tuned approach has been recently introduced for modeling stimulus-dependent cortical receptive fields. The approach is based on signal-tuned Gabor functions, which are Gaussian-modulated sinusoids whose parameters are obtained from a "tuning" signal. Given a stimulus to a cell, it is taken as the tuning signal for the Gabor function modeling the cell's receptive field, and the inner product of the stimulus and the stimulus-dependent field produces the cell's response. Here, we derive and solve the equation of motion for the signal-tuned complex cell response r(x,τ), where x and τ are receptive-field parameters: its center, and the delay with which it adapts to a change in input. The motion equation can be mapped onto the Schrödinger equation for a system with time-dependent imaginary mass and time-dependent complex potential, and yields a plane-wave solution and an Airy-packet solution. The plane-wave solution replicates responses previously obtained for temporally modulated and translating signals, and yields responses which seem compatible with apparent-motion effects, when the stimulus is a pair of alternating pulses. The Airy-packet solution can lead to long-range propagating responses.