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@en

Background: Prior work indicates that 50–75% of individuals post-hemiparetic stroke have upper-extremity weakness and, in turn, inaccurately judge the relative torques that their arms generate during a bimanual task. Recent findings also reveal that these individuals judge the relative torques their arms generate differently depending on whether they reference their paretic vs. non-paretic arm. Objective: Our goal was to determine whether individuals with hemiparetic stroke inaccurately matched torques between arms, regardless of the arm that they referenced. Methods: Fifteen participants with hemiparetic stroke and 10 right-hand dominant controls matched torques between arms. Participants performed this task with their right arm referencing their left arm, and vice versa. Participants generated (1) 5 Nm and (2) 25% of their reference elbow’s maximum voluntary torque (MVT) in flexion and extension using their reference arm while receiving audiovisual feedback. Then, participants matched the reference torque using their opposite arm without receiving feedback on their matching performance. Results: Participants with stroke had greater magnitudes of error in matching torques than controls when referencing their paretic arm (p < 0.050), yet not when referencing their non-paretic arm (p > 0.050). The mean magnitude of error when participants with stroke referenced their paretic and non-paretic arm and controls referenced their dominant and non-dominant arm to generate 5 Nm in flexion was 9.4, 2.6, 4.2, and 2.5 Nm, respectively, and in extension was 5.3, 2.8, 2.5, and 2.3 Nm, respectively. However, when the torques generated at each arm were normalized by the corresponding MVT, no differences were found in matching errors regardless of the arm participants referenced (p > 0.050). Conclusions: Results demonstrate the importance of the arm referenced, i.e., paretic vs. non-paretic, on how accurately individuals post-hemiparetic stroke judge their torques during a bimanual task. Results also indicate that individuals with hemiparetic stroke judge torques primarily based on their perceived effort. Finally, findings support the notion that training individuals post-hemiparetic stroke to accurately perceive their self-generated torques, with a focus of their non-paretic arm in relation to their paretic arm, may lead to an improved ability to perform bimanual activities of daily living.

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