### Abstract

Feldman (1966) has proposed that a muscle endowed with its spinal reflex system behaves as a non-linear spring with an adjustable resting length. In contrast, because of the length-tension properties of muscles, many researchers have modeled them as non-linear springs with adjustable stiffness. Here we test the merits of each approach: Initially, it is proven that the adjustable stiffness model predicts that isometric muscle force and stiffness are linearly related. We show that this prediction is not supported by data on the static stiffness-force characteristics of reflexive muscles, where stiffness grows non-linearly with force. Therefore, an intact muscle-reflex system does not behave as a non-linear spring with an adjustable stiffness. However, when the same muscle is devoid of its reflexes, the data shows that stiffness grows linearly with force. We aim to understand the functional advantage of the non-linear stiffness-force relationship present in the reflexive muscle. Control of an inverted pendulum with a pair of antagonist muscles is considered. Using an active-state muscle model we describe force development in an areflexive muscle. From the data on the relationship of stiffness and force in the intact muscle we derive the length-tension properties of a reflexive muscle. It is shown that a muscle under the control of its spinal reflexes resembles a non-linear spring with an adjustable resting length. This provides independent evidence in support of the Feldman hypothesis of an adjustable resting length as the control parameter of a reflexive muscle, but it disagrees with his particular formulation. In order to maintain stability of the single joint system, we prove that a necessary condition is that muscle stiffness must grow at least linearly with force at isometric conditions. This shows that co-contraction of antagonist muscles may actually destabilize the limb if the slope of this stiffness-force relationship is less than an amount specified by the change in the moment arm of the muscle as a function of joint configuration. In a reflexive muscle where stiffness grows faster than linearly with force, co-contraction will always lead to an increase in stiffness. Furthermore, with the reflexive muscles, the same level of joint stiffness can be produced by much smaller muscle forces because of the non-linear stiffness-force relationship. This allows the joint to remain stable at a fraction of the metabolic energy cost associated with maintaining stability with areflexive muscles.

Original language | English (US) |
---|---|

Pages (from-to) | 463-477 |

Number of pages | 15 |

Journal | Biological Cybernetics |

Volume | 66 |

Issue number | 6 |

DOIs | |

State | Published - Apr 1992 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Biophysics

### Cite this

*Biological Cybernetics*,

*66*(6), 463-477. https://doi.org/10.1007/BF00204111

**A mathematical analysis of the force-stiffness characteristics of muscles in control of a single joint system.** / Shadmehr, Reza; Arbib, Michael A.

Research output: Contribution to journal › Article

*Biological Cybernetics*, vol. 66, no. 6, pp. 463-477. https://doi.org/10.1007/BF00204111

}

TY - JOUR

T1 - A mathematical analysis of the force-stiffness characteristics of muscles in control of a single joint system

AU - Shadmehr, Reza

AU - Arbib, Michael A.

PY - 1992/4

Y1 - 1992/4

N2 - Feldman (1966) has proposed that a muscle endowed with its spinal reflex system behaves as a non-linear spring with an adjustable resting length. In contrast, because of the length-tension properties of muscles, many researchers have modeled them as non-linear springs with adjustable stiffness. Here we test the merits of each approach: Initially, it is proven that the adjustable stiffness model predicts that isometric muscle force and stiffness are linearly related. We show that this prediction is not supported by data on the static stiffness-force characteristics of reflexive muscles, where stiffness grows non-linearly with force. Therefore, an intact muscle-reflex system does not behave as a non-linear spring with an adjustable stiffness. However, when the same muscle is devoid of its reflexes, the data shows that stiffness grows linearly with force. We aim to understand the functional advantage of the non-linear stiffness-force relationship present in the reflexive muscle. Control of an inverted pendulum with a pair of antagonist muscles is considered. Using an active-state muscle model we describe force development in an areflexive muscle. From the data on the relationship of stiffness and force in the intact muscle we derive the length-tension properties of a reflexive muscle. It is shown that a muscle under the control of its spinal reflexes resembles a non-linear spring with an adjustable resting length. This provides independent evidence in support of the Feldman hypothesis of an adjustable resting length as the control parameter of a reflexive muscle, but it disagrees with his particular formulation. In order to maintain stability of the single joint system, we prove that a necessary condition is that muscle stiffness must grow at least linearly with force at isometric conditions. This shows that co-contraction of antagonist muscles may actually destabilize the limb if the slope of this stiffness-force relationship is less than an amount specified by the change in the moment arm of the muscle as a function of joint configuration. In a reflexive muscle where stiffness grows faster than linearly with force, co-contraction will always lead to an increase in stiffness. Furthermore, with the reflexive muscles, the same level of joint stiffness can be produced by much smaller muscle forces because of the non-linear stiffness-force relationship. This allows the joint to remain stable at a fraction of the metabolic energy cost associated with maintaining stability with areflexive muscles.

AB - Feldman (1966) has proposed that a muscle endowed with its spinal reflex system behaves as a non-linear spring with an adjustable resting length. In contrast, because of the length-tension properties of muscles, many researchers have modeled them as non-linear springs with adjustable stiffness. Here we test the merits of each approach: Initially, it is proven that the adjustable stiffness model predicts that isometric muscle force and stiffness are linearly related. We show that this prediction is not supported by data on the static stiffness-force characteristics of reflexive muscles, where stiffness grows non-linearly with force. Therefore, an intact muscle-reflex system does not behave as a non-linear spring with an adjustable stiffness. However, when the same muscle is devoid of its reflexes, the data shows that stiffness grows linearly with force. We aim to understand the functional advantage of the non-linear stiffness-force relationship present in the reflexive muscle. Control of an inverted pendulum with a pair of antagonist muscles is considered. Using an active-state muscle model we describe force development in an areflexive muscle. From the data on the relationship of stiffness and force in the intact muscle we derive the length-tension properties of a reflexive muscle. It is shown that a muscle under the control of its spinal reflexes resembles a non-linear spring with an adjustable resting length. This provides independent evidence in support of the Feldman hypothesis of an adjustable resting length as the control parameter of a reflexive muscle, but it disagrees with his particular formulation. In order to maintain stability of the single joint system, we prove that a necessary condition is that muscle stiffness must grow at least linearly with force at isometric conditions. This shows that co-contraction of antagonist muscles may actually destabilize the limb if the slope of this stiffness-force relationship is less than an amount specified by the change in the moment arm of the muscle as a function of joint configuration. In a reflexive muscle where stiffness grows faster than linearly with force, co-contraction will always lead to an increase in stiffness. Furthermore, with the reflexive muscles, the same level of joint stiffness can be produced by much smaller muscle forces because of the non-linear stiffness-force relationship. This allows the joint to remain stable at a fraction of the metabolic energy cost associated with maintaining stability with areflexive muscles.

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U2 - 10.1007/BF00204111

DO - 10.1007/BF00204111

M3 - Article

VL - 66

SP - 463

EP - 477

JO - Biological Cybernetics

JF - Biological Cybernetics

SN - 0340-1200

IS - 6

ER -