Quantifying the Nonlinear Interaction in the Nervous System Based on Phase-Locked Amplitude Relationship

Abstract

This paper introduces the Crossfrequency Amplitude Transfer Function (CATF), a model-free method for quantifying nonlinear stimulus-response interaction based on phase-locked amplitude relationship. Method: 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 secondorder nonlinear amplitude transfer in the human proprioceptive system from the periphery to the cortex. Result: 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. Conclusion: This paper provides a new tool to identify nonlinear interaction in the nervous system. Significance: The results provide novel insight of nonlinear dynamics in the human proprioceptive system.

Fascicle dynamics of the tibialis anterior muscle reflect whole-body walking economy

“…This nonlinear, U-shaped relationship in which COT decreases then increases as a variable continually increases in value has been observed with walking speed 1 , 2 , stride frequency 3 , 5 , and TA muscle activity 42 . Prior studies have shown that muscle proprioceptive feedback is best modeled as a non-linear system 21 , 43 – 45 . So even as the amount of energy absorbed through muscle fascicle stretch may continually increase with walking speed, its relationship does not necessarily need to be linear to be useful to the nervous system.…”

Nonlinear System Identification of Neural Systems from Neurophysiological Signals

“…Several recent studies used sum-of-sinusoid stimulations to the wrist joint and found not only harmonics and subharmonics but also intermodulation patterns [6,29]. The majority of intermodulation responses presented the second-order nonlinearity, which is the sum or the difference between input frequencies [8]. These findings indicate that the nonlinearities in the somatosensory system allow the functional integration of input signals at different frequencies, and they can be transmitted in different somatosensory ascending pathways.…”

“…This effect typically occurs in a mono-synaptic neural system such as the corticospinal tract where the supraspinal motor command is linearly transferred to the motor output due to the pool effect of motor units [4]. However, multi-synaptic neural systems, such as the somatosensory system, have been reported highly nonlinear, showing harmonic responses to periodic stimuli [5][6][7][8]. Cross-frequency coupling in the corticothalamic interactions has also been reported when characterising essential tremor [9].…”

Nonlinear System Identification of Neural Systems from Neurophysiological Signals

“…Several recent studies used sum-of-sinusoid stimulations to the wrist joint and found not only harmonics and subharmonics but also intermodulation patterns [6, 29]. The majority of intermodulation responses presented the second-order nonlinearity, which is the sum or the difference between input frequencies [8]. These findings indicate that the nonlinearities in the somatosensory system allow the functional integration of input signals at different frequencies, and they can be transmitted in different somatosensory ascending pathways.…”

“…This effect typically occurs in a mono-synaptic neural system such as the corticospinal tract where the supraspinal motor command is linearly transferred to the motor output due to the pool effect of motor units [4]. However, multi-synaptic neural systems, such as the somatosensory system, have been reported highly nonlinear, showing harmonic responses to periodic stimuli [5][6][7][8]. Cross-frequency coupling in the corticothalamic interactions has also been reported when characterising essential tremor [9].…”