  Small continuous sensory and mechanical perturbations have often been used to
identify properties of the closed-loop neural control of posture and other
systems that are approximately linear time invariant. Here we extend this
approach to study the neural control of rhythmic behaviors such as walking. Our
method is based on the theory of linear time periodic systems, with
modifications to account for ability of perturbations to reset the phase of a
rhythmic behavior. We characterize responses to perturbations in the frequency
domain using harmonic transfer functions and then convert to the time domain to
obtain phase-dependent impulse response functions (IRFs) that describe the
response to a small brief perturbation at any phase of the rhythmic behavior.
IRFs describing responses of kinematic variables and muscle activations
(measured by EMG) to sensory and mechanical perturbations can be used to infer
properties of the plant, the mapping from muscle activation to movement, and of
neural feedback, the mapping from movement to muscle activation. We illustrate
our method by applying it to simulated data from a model and experimental data
of subjects walking on a treadmill perturbed by movement of the visual scene.
