Mathematical Models of Proprioceptors. II. Structure and Function of the Golgi Tendon Organ

Abstract

We developed a physiologically realistic mathematical model of the Golgi tendon organ (GTO) whose elements correspond to anatomical features of the biological receptor. The mechanical interactions of these elements enable it to capture all salient aspects of GTO afferent behavior reported in the literature. The model accurately describes the GTO's static and dynamic responses to activation of single motor units whose muscle fibers insert into the GTO, including the different static and dynamic sensitivities that exist for different types of muscle fibers (S, FR, and FF). Furthermore, it captures the phenomena of self- and cross-adaptation wherein the GTO dynamic response during motor unit activation is reduced by prior activation of the same or a different motor unit, respectively. The model demonstrates various degrees of nonlinear summation of GTO responses resulting from simultaneous activation of multiple motor units. Similarly to the biological GTO, the model suggests that the activation of every additional motor unit to already active motor units that influence the receptor will have a progressively weaker incremental effect on the GTO afferent activity. Finally, the proportional relationship between the cross-adaptation and summation recorded for various pairs of motor units was captured by the model, but only by incorporating a particular type of occlusion between multiple transduction regions that were previously suggested. This occlusion mechanism is consistent with the anatomy of the afferent innervation and its arrangement with respect to the collagen strands inserting into the GTO.

The optimal neural strategy for a stable motor task requires a compromise between level of muscle cocontraction and synaptic gain of afferent feedback

“…Ib afferent activity was simulated based on active muscle forces, according to the model developed by Mileusnic and Loeb (2006). The set of nonlinear equations in this model were solved in MATLAB 2013a with the function "fsolve."…”

Physiologically Based Control Laws Featuring Antagonistic Muscle Co-Activation for Stable Compliant Joint Drives

“…In our model, all GTOs are assumed to provide the total force at joint level as a feedback signal τ L to the CNS [36].…”

Model-Based Sensorimotor Integration for Multi-Joint Control: Development of a Virtual Arm Model

“…The higher slope at the lower levels of muscle forces is due to the disproportionate influence of the early-recruited slow-twitch muscle fibers to GTO output. 13,38 Model Integration in Matlab/Simulink Integration of individual model components, such as the VM, the spindle model, the GTO model and the skeletal SIMM model, was accomplished using a software tool-musculoskeletal modeling in simulink (MMS). 14 MMS is a C program that automatically converts a SIMM model of a sensorimotor system into a Simulink block that embodies its mechanical dynamics.…”

“…There have been many recent advances in the sophistication and completeness of individual model components, 10,35,37,38,52 and in the tools necessary to embody specific model structures in the computational environment in which the models run. 3,15,24 All of these now make it feasible to develop a realistic multi-joint, multi-muscle virtual arm (VA) model for computational studies of human motor control and learning.…”