**Shadmehr R,
Arbib MA (1992), A
mathematical analysis of the force-stiffness characteristics of muscles and the
role of reflexes in control of a single joint system. Biological Cybernetics
66:463-477.**

**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.