On the dynamical stability of skeletal muscle

TL;DR

The paper tackles whether activation-independent tissue properties can stabilize contractions in the dip region of the skeletal muscle force-length curve, a question tied to dynamic stability. It first shows that a 1D Hill-type model with in-series elements exhibits dynamic instability when the slope of the active force-length relation is negative, and confirms this via eigenvalue analysis and a continuum limit. The authors then stabilize the system by introducing a small base material (Neo-Hookean) component, yielding a convex total force-length relationship and robust stability for active stretch across tested lengths, with stability characterized by the sign of $∂F/∂λ$. This work suggests that intrinsic 3D tissue mechanics can provide activation-independent stabilization, informing robust, multi-scale muscle simulations and reducing reliance on activation-dependent elements like actin-titin interactions.

Abstract

There has been debate for over 70-years about whether active skeletal muscle is dynamically stable at lengths greater than its optimal length. The stability of computational muscle models is a critical issue, as it directly affects our ability to simulate muscle deformation across different operating lengths, especially at lengths where muscles are known to remain functional despite model-predicted instabilities. In this study, we revisit the question of dynamical stability of ODE-based models of skeletal muscle. In particular, we investigate whether activation-independent tissue properties can provide stability to contractions along the dip region of the total force-length curve. First, using a combination of analytical tools (eigenvalue analysis and non-dimensionalization) and numerical simulations, we confirm that traditional Hill-type muscle models can display divergent dynamics in this region. Then, we propose a stabilized version of a 1D Hill-type muscle model that incorporates the 3D nature of skeletal muscle deformation. This results in a completely convex force-length relationship that can bring robustness to numerical simulations, while preserving the computational efficiency of 1D models. Our findings suggest that activation-independent intrinsic mechanical properties of muscle are sufficient to stabilize contractions even in the dip region, offering new insight into how muscles maintain functional integrity during active stretch.

Figures (7)

• Figure 1: (a) Active and passive force-length (FL) relationships. The region $\lambda_M < 1$ corresponds to the ascending limb (in blue) since the curve has a positive slope. In turn, the region $\lambda_M > 1$ corresponds to the descending limb (in ochre) which has a distinctive negative slope. (b) When active and passive forces are added, they form the total FL relationship, which typically has a dip region (i.e. a region with negative slope) around $1 < \lambda_M < 1.35$.
• Figure 1: Mass enhanced multi-body muscle model. The muscle is divided into N in-series segment, each containing a contractile element (CE), a parallel elastic element (PEE), and a point mass m.
• Figure 1: (a) Forces in the stabilized model ($\chi_{BM} = 0.0015$). (b) Normalized total force for different values of $\chi_{BM}$.
• Figure 2: Stability of an isometric contraction according to the system of ODEs \ref{['eq:odes_hill_only']}. At the beginning of the simulation, the contractile units (segments) are nonuniformly stretched at around 85% (left), 115% (middle), and 145% (right) of the muscle's optimal length. At an initial stretch of 115% ($\lambda_M = 1.15$), the descending limb instability causes the segments to bifurcate to two different lengths.
• Figure 2: Stability of an isometric contraction according to the system of ODEs \ref{['eq:odes_hill_stabilized']}. As before, the elements (as well as the muscle) are stretched initially at around 85% (left), 115% (middle), and 145% (right) of the muscle's optimal length. In all cases, the contractions are stable since the derivative of the total force (made of active, passive, and base material components) is always positive. The stability conferred by the base material properties in the dip-region of the force-length relationship (stretch of 1.15) should be contrasted with the instability shown in Fig.3

Theorems & Definitions (1)

• Proof 1: Proof of Claim 1