A second order mechanical model of muscle spindle primary endings

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.List of Symbols Used 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 - (s) Laplace transform of the length of the elastic component of the proposed Maxwell branch of the sensory region of the model - (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 - (s) Laplace transform of the length of the purely elastic branch of the model - The transducer gain of the output from the -branch relative to that of the -branch - v(s) Laplace transform of the total output signal (s) + (s) - Time constant defined by or =RC for the mechanical and the electrical system respectively - Angular frequency equal to 2f - Rate constant describing the relation between the lead and the lag time constant of a first order lead-lag filter network     

A mechanical model of muscle and its application to the intrafusal fibres of the mammalian muscle spindle

A three-component model of the muscle is used in which the components change their values when the muscle is stimulated. In particular the elastic components change not only their moduli of elasticity but the unstretched lengths decrease when the muscle is in the active state. When the model is extended to the intrafusal muscle fibres of the mammalian muscle spindle it can reproduce some of the observed responses to mechanical stretch and fusimotor stimulation.     

Viscoelastic and dynamic nonlinear properties of airway smooth muscle tissue: roles of mechanical force and the cytoskeleton

The viscoelastic and dynamic nonlinear properties of guinea pig tracheal smooth muscle tissues were investigated by measuring the storage (G') and loss (G") moduli using pseudorandom small-amplitude length oscillations between 0.12 and 3.5 Hz superimposed on static strains of either 10 or 20% of initial length. The G" and G' spectra were interpreted using a linear viscoelastic model incorporating damping (G) and stiffness (H), respectively. Both G and H were elevated following an increase in strain from 10 to 20%. There was no change in harmonic distortion (K(d)), an index of dynamic nonlinearity, between 10 and 20% strains. Application of methacholine at 10% strain significantly increased G and H while it decreased K(d). Cytochalasin D, isoproterenol, and HA-1077, a Rho-kinase inhibitor, significantly decreased both G and H but increased K(d). Following cytochalasin D, G, H, and K(d) were all elevated when mean strain increased from 10 to 20%. There were no changes in hysteresivity, G/H, under any condition. We conclude that not all aspects of the viscoelastic properties of tracheal smooth muscle strips are similar to those previously observed in cultured cells. We attribute these differences to the contribution of the extracellular matrix. Additionally, using a network model, we show that the dynamic nonlinear behavior, which has not been observed in cell culture, is associated with the state of the contractile stress and may derive from active polymerization within the cytoskeleton.     

A nonlinear model of passive muscle viscosity

The material properties of passive skeletal muscle are critical to proper function and are frequently a target for therapeutic and interventional strategies. Investigations into the passive viscoelasticity of muscle have primarily focused on characterizing the elastic behavior, largely neglecting the viscous component. However, viscosity is a sizeable contributor to muscle stress and extensibility during passive stretch and thus there is a need for characterization of the viscous as well as the elastic components of muscle viscoelasticity. Single mouse muscle fibers were subjected to incremental stress relaxation tests to characterize the dependence of passive muscle stress on time, strain and strain rate. A model was then developed to describe fiber viscoelasticity incorporating the observed nonlinearities. The results of this model were compared with two commonly used linear viscoelastic models in their ability to represent fiber stress relaxation and strain rate sensitivity. The viscous component of mouse muscle fiber stress was not linear as is typically assumed, but rather a more complex function of time, strain and strain rate. The model developed here, which incorporates these nonlinearities, was better able to represent the stress relaxation behavior of fibers under the conditions tested than commonly used models with linear viscosity. It presents a new tool to investigate the changes in muscle viscous stresses with age, injury and disuse.     

Directly probing the mechanical properties of the spindle and its matrix

Several recent models for spindle length regulation propose an elastic pole to pole spindle matrix that is sufficiently strong to bear or antagonize forces generated by microtubules and microtubule motors. We tested this hypothesis using microneedles to skewer metaphase spindles in Xenopus laevis egg extracts. Microneedle tips inserted into a spindle just outside the metaphase plate resulted in spindle movement along the interpolar axis at a velocity slightly slower than microtubule poleward flux, bringing the nearest pole toward the needle. Spindle velocity decreased near the pole, which often split apart slowly, eventually letting the spindle move completely off the needle. When two needles were inserted on either side of the metaphase plate and rapidly moved apart, there was minimal spindle deformation until they reached the poles. In contrast, needle separation in the equatorial direction rapidly increased spindle width as constant length spindle fibers pulled the poles together. These observations indicate that an isotropic spindle matrix does not make a significant mechanical contribution to metaphase spindle length determination.     

A cross-bridge model that is able to explain mechanical and energetic properties of shortening muscle.

The responses of muscle to steady and stepwise shortening are simulated with a model in which actin-myosin cross-bridges cycle through two pathways distinct for the attachment-detachment kinetics and for the proportion of energy converted into work. Small step releases and steady shortening at low velocity (high load) favor the cycle implying approximately 5 nm sliding per cross-bridge interaction and approximately 100/s detachment-reattachment process; large step releases and steady shortening at high velocity (low load) favor the cycle implying approximately 10 nm sliding per cross-bridge interaction and approximately 20/s detachment-reattachment process. The model satisfactorily predicts specific mechanical properties of frog skeletal muscle, such as the rate of regeneration of the working stroke as measured by double-step release experiments and the transition to steady state during multiple step releases (staircase shortening). The rate of energy liberation under different mechanical conditions is correctly reproduced by the model. During steady shortening, the relation of energy liberation rate versus shortening speed attains a maximum (approximately 6 times the isometric rate) for shortening velocities lower than half the maximum velocity of shortening and declines for higher velocities. In addition, the model provides a clue for explaining how, in different muscle types, the higher the isometric maintenance heat, the higher the power output during steady shortening.     

Modelling the mechanical properties of cardiac muscle

A model of passive and active cardiac muscle mechanics is presented, suitable for use in continuum mechanics models of the whole heart. The model is based on an extensive review of experimental data from a variety of preparations (intact trabeculae, skinned fibres and myofibrils) and species (mainly rat and ferret) at temperatures from 20 to 27°C. Experimental tests include isometric tension development, isotonic loading, quick-release/restretch, length step and sinusoidal perturbations. We show that all of these experiments can be interpreted with a four state variable model which includes (i) the passive elasticity of myocardial tissue, (ii) the rapid binding of Ca²⁺ to troponin C and its slower tension-dependent release, (iii) the kinetics of tropomyosin movement and availability of crossbridge binding sites and the length dependence of this process and (iv) the kinetics of crossbridge tension development under perturbations of myofilament length.     

Comparison of motor endings of the cat's muscle spindle stained for NADH-tetrazolium reductase and cholinesterase

Summary Cat muscle spindles were examined histochemically in serial transverse sections of tenuissimus muscles stained for ATPase, NADH-TR and ChE alternating sequentially. Motor nerve terminals on nuclear bag₁, bag₂ and nuclear chain intrafusal muscle fibers were identified in periodic sections stained for ChE. Intrafusal fiber regions that carried ChE-active areas were then examined in staining for NADH-TR. The motor endings on the three types of intrafusal fiber differed in their apparent histochemical content of both ChE and NADH-TR. The observations suggest that functional differences may exist among motor nerve terminals on the various intrafusal fiber types.     

The motor units of cat medial gastrocnemius: electrical and mechanical properties as a function of muscle length.

The effects of changing muscle length on the mechanical properties of 89 motor units from adult cat medial gastrocnemius have been studied in eight experiments. Few differences were found between the effects of length on tetanic tension, twitch tension, twitch-tetanus ratio, twitch contraction time, twitch half relaxation time, rate of force development and electrical activity for fast contracting (twitch contraction time less than or equal to 45 msec) and slowly contracting (greater than 45 msec) units. Those differences that did appear did not persist when these two groups were matched by tetanic tension. It is concluded that the biophysical mechanisms responsible for the changes in mechanical and electrical properties with length must be similar for fast and slow twitch units and not related to potential differences in their muscle fiber type. The effects of changing muscle length on the mechanical properties of the eight whole muscles suggest that changes in force output with length are of minor importance during normal movements as the muscle is found to be electrically active over a relatively narrow range of lengths close to the optimum length for tetanus of the whole muscle. The very shortest muscle lengths at which there is only minimal force development are not used in natural movements, while the declining limb of the length tension curve is at muscle lengths beyond the maximum in situ length.     

Incorporating the length-dependent passive-force generating muscle properties of the extrinsic finger muscles into a wrist and finger biomechanical musculoskeletal model

Dynamic movement trajectories of low mass systems have been shown to be predominantly influenced by passive viscoelastic joint forces and torques compared to momentum and inertia. The hand is comprised of 27 small mass segments. Because of the influence of the extrinsic finger muscles, the passive torques about each finger joint become a complex function dependent on the posture of multiple joints of the distal upper limb. However, biomechanical models implemented for the dynamic simulation of hand movements generally don’t extend proximally to include the wrist and distal upper limb. Thus, they cannot accurately represent these complex passive torques. The purpose of this short communication is to both describe a method to incorporate the length-dependent passive properties of the extrinsic index finger muscles into a biomechanical model of the upper limb and to demonstrate their influence on combined movement of the wrist and fingers. Leveraging a unique set of experimental data, that describes the net passive torque contributed by the extrinsic finger muscles about the metacarpophalangeal joint of the index finger as a function of both metacarpophalangeal and wrist postures, we simulated the length-dependent passive properties of the extrinsic finger muscles. Dynamic forward simulations demonstrate that a model including these properties passively exhibits coordinated movement between the wrist and finger joints, mimicking tenodesis, a behavior that is absent when the length-dependent properties are removed. This work emphasizes the importance of incorporating the length-dependent properties of the extrinsic finger muscles into biomechanical models to study healthy and impaired hand movements.     

Separation and estimation of muscle spindle and tension receptor populations by vibration of the biceps muscle in the frog

Frog spinal cord reflex behaviors have been used to test the idea of spinal primitives. We have suggested a significant role for proprioception in regulation of primitives. However the in vivo behavior of spindle and golgi tendon receptors in frogs in response to vibration are not well described and the proportions of these proprioceptors are not established. In this study, we examine the selectivity of muscle vibration in the spinal frog. The aim of the study was (1) to examine how hindlimb muscle spindles and GTO receptors are activated by muscle vibration and (2) to estimate the relative numbers of GTO receptors and spindle afferents in a selected muscle, for comparison with the mammal. Single muscle afferents from the biceps muscle were identified in the dorsal roots. These were tested in response to biceps vibration, intramuscular stimulation and biceps nerve stimulation. Biceps units were categorized into two types: First, spindle afferents which had a high conduction velocity (approximately 20-30 m/s), responded reliably (were entrained 1:1) to muscle vibration, and exhibited distinct pauses to shortening muscle contractions. Second, golgi tendon organ afferents, which had a lower conduction velocity (approximately 10-20 m/s), responded less reliably to muscle vibration at physiologic muscle lengths, but responded more reliably at extended lengths or with background muscle contraction, and exhibited distinct bursts to shortening muscle contractions. Vibration responses of these units were tested with and without muscle curarization. Ensemble (suction electrode) recordings from the dorsal roots were used to provide rough estimates of the proportions of the two muscle afferent types.     

A mathematical model for the mechanical properties of scallop shells

The mechanical behaviour of one or both shell valves of five species of pectinacean and one anomiacean was determined using strain gauge rosettes attached to the inner surfaces of the valves. A multiple linear regression analysis accounted for over 64% of the total vanance in tensile shell compliance. The important shell architectural variables were thickness, corrugation and, much less important, convexity and shell length. Calculation showed that in general corrugation of the shell was easily the cheapest way of producing stiffness, in terms of the amount of shell material required. Suggestions are offered as to why not all scallops are scalloped.     

Effect of a polarizing current on receptor potential of the isolated frog muscle spindle

The effect of a polarizing current on electrical activity of the isolated frog muscle spindle was studied. A depolarizing current increased the frequency and reduced the amplitude of afferent spike activity, both spontaneous and evoked by mechanical stimulation. A hyperpolarizing current produced the opposite effect. The amplitude of the receptor potential in response to a mechanical stimulus varied as a linear function of the intensity of the polarizing currentA. A. Ukhtomskii Physiological Institute, Leningrad State University. Translated from Neirofiziologiya, Vol. 5, No. 1, pp. 95–101, January–February, 1973.