Summary
With reference to experimental data and the failure of earlier proposed first order linear models of mammalian muscle spindles, a second order mechanical model of de-efferented primary endings is studied. The model takes into account the presence of two different types of intrafusal muscle fibres in a complete spindle organ. It further allows the incorporation of different gain of the mechano-electric conversion into a depolarization of the sensory terminals innervating the two types of fibres.
It is shown that a closer approximation to the behaviour of the biological prototype is obtained if the transducer gain of the branch corresponding to the nuclear bag intrafusal fibres is chosen significantly higher than that corresponding to the nuclear chain branch.
The marked nonlinear behaviour of muscle spindle primary endings as recently reported by Matthews and Stein (1968, 1969) is interpreted as a saturation effect of the high gain mechano-electric transducer of the nuclear bag branch. The saturation is considered to reflect a condition of complete depolarization of these sensory terminals. If a higher transducer gain actually is present, a complete depolarization of these terminals would occur at a lower degree of mechanical deformation than for the nuclear chain terminals. The mechano-electric transducer system of the nuclear chain fibres might thus behave approximately linearly within a larger range of input amplitudes. The greatly reduced gain of the primary endings at large emplitudes of imposed muscle vibrations as observed experimentally (Matthews and Stein, 1968, 1969) may thus be accounted for by the transducer gain of the nuclear chain fibres alone.
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Abbreviations
- C :
-
Viscous stiffness, or electrical capacitance
- f :
-
Frequency of a signal
- K :
-
Static gain of a system
- k :
-
Elastic stiffness
- R :
-
Electrical resistance
- s :
-
The Laplace operator
- H(s) :
-
General transfer function of a system
- X (s) :
-
Laplace transform of the difference between instantanous and resting length of the complete intrafusal muscle fibre, according to the suggested model shown in Fig. 2
- α(s) :
-
Laplace transform of the length of the elastic component of the proposed Maxwell branch of the sensory region of the model
- ψ(s) :
-
Laplace transform of the difference between instantanous and resting length of the lumped model of the polar (non-sensory) region of the intrafusal fibres
- λ(s) :
-
Laplace transform of the length of the purely elastic branch of the model
- μ α :
-
The transducer gain of the output from the α-branch relative to that of the λ-branch
- v(s) :
-
Laplace transform of the total output signal λ(s) +μ α α(s)
- τ :
-
Time constant defined by \(\tau = \frac{C}{k}\) or τ=RC for the mechanical and the electrical system respectively
- ω :
-
Angular frequency equal to 2πf
- η :
-
Rate constant describing the relation between the lead and the lag time constant of a first order lead-lag filter network
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Rudjord, T. A second order mechanical model of muscle spindle primary endings. Kybernetik 6, 205–213 (1970). https://doi.org/10.1007/BF00276721
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DOI: https://doi.org/10.1007/BF00276721