Comparison of the energetic properties and the dynamical stability in a mathematical model of the stretch reflex

The previous studies on finite-time thermodynamic engines have shown that some of the parameters affecting their thermodynamic performance also affect their stability. Moreover, such parameters have to be tuned to reach an optimal trade-off between these two generic properties. In the present work we carry out a similar analysis on a mathematical model of the stretch reflex regulatory pathway, which is a simplified version of a previously published model. We show that the model has a unique stable fixed point in the absence of time delays. However, when the system inherent time delays are considered, they can destabilize the fixed point and engender a stable limit cycle. We further explore the parameter space to analyse the sensitivity of the system stability to variations in the parameter values. Particular attention is paid to the parameter here denoted as α, which has been shown to determine the muscle thermodynamic properties during steady-state contractions: larger values of α mean more powerful and less efficient muscles. Our results indicate that the stretch reflex pathway is less stable in the more powerful and less efficient muscles. We finally compare these observations with those obtained on thermal engines.