Quantifying the nonlinear interaction in the nervous system based on phase-locked amplitude relationship

This paper introduces the Cross-frequency Amplitude Transfer Function (CATF), a model-free method for quantifying nonlinear stimulus-response interaction based on phase-locked amplitude relationship. The CATF estimates the amplitude transfer from input frequencies at stimulation signal to their harmonics/intermodulation at the response signal. We first verified the performance of CATF in simulation tests with systems containing a static nonlinear function and a linear dynamic, i.e., Hammerstein and Wiener systems. We then applied the CATF to investigate the second-order nonlinear amplitude transfer in the human proprioceptive system from the periphery to the cortex. The simulation demonstrated that the CATF is a general method which can well quantify nonlinear stimulus-response amplitude transfer for different orders of nonlinearity in Wiener or Hammerstein system configurations. Applied to the human proprioceptive system, we found a complicated nonlinear system behavior with substantial amplitude transfer from the periphery stimulation to cortical response signals in the alpha band. This complicated system behavior may be associated with the nonlinear behavior of the muscle spindle and the dynamic interaction in the thalamocortical radiation. This paper provides a new tool to identify nonlinear interaction in the nervous system. The results provide novel insight of nonlinear dynamics in the human proprioceptive system