A model of spindle afferent response to muscle stretch

Abstract

1. A unified model of the properties of stretch responses of mammalian spindle endings is proposed. This model encompasses the disparity between sensitivity of spindle endings to small and to large stretch of the muscle as well as the disparity in their dynamic responsiveness for different amplitudes of stretch. 2. In the model the mechanical properties of intrafusal fibers include a property akin to friction, which is hypothesized on the basis of reported observations on amphibian muscle. Transducer and encoder processes are modeled in the light of recent observations on isolated spindles. The model involves five unknown parameters whose values are selected by reference to certain reported observations on deefferented primary and secondary endings. The model can be used to predict responses to length changes of arbitrary time course. 3. Predicted responses to large ramp-and-hold stretch are quantitatively comparable to observations over a wide range of stretch velocities. The quantities compared include the increment in response during ramp stretch as well as the dynamic index, which is a measure of adaptation at stretch plateau. 4. At a fixed frequency of sinusoidal stretch, the relation between amplitudes of stretch and response is predicted in quantitative agreement with measurements. As the frequency of stretch is decreased, the predicted phase lead decreases and then increases, while the sensitivity decreases monotonically, in accord with observations. 5. In the model the high sensitivity for small stretch is not specific to any particular length of the muscle. When stretch is large, the region of high sensitivity is gradually reestablished at the new length, a phenomenon referred to as resetting. The dynamic response to a large stretch can be seen as arising, for the most part, from the dynamic process of resetting. 6. The influences of static or dynamic fusimotor activation on stretch responses of the primary ending are simulated by modifying the parameter values in the model. The modifications are such that static (dynamic) fusimotor activity speeds up (slows down) the resetting of the high-sensitivity region. The predictions mimic qualitatively the observed fusimotor effects not only on the response to large ramp stretch but also the contrasting effects seen with smaller, sinusoidal stretch.

Models of ensemble firing of muscle spindle afferents recorded during normal locomotion in cats

“…The pros and cons of different values of the exponent 'exp' in summed velocity and length models of the form f = constant1v exp + constantµd will be taken up again in the Discussion. Figure 3C and D shows predictions obtained with the use of two closely related models, those of Houk et al (1981) and Hasan (1983). It should be stressed that Houk et al (1981) did not define their model for negative velocities, nor did they claim its validity under these circumstances.…”

“…No clear recommendation was given for this parameter in the original study. The Hasan (1983) model is based on the Houk model, but incorporates position sensitivity (i.e. a length-dependent term independent of velocity), acceleration responses and fusimotor action.…”

“…Two additional software packages were used to analyse the averaged profiles on the basis of five mathematical models of spindle response from the literature (Matthews & Stein, 1969;Sch afer, 1974;Chen & Poppele, 1978;Houk et al 1981;Hasan, 1983) and an additional hybrid model of our own. The basic analysis was performed with the Matlab 4.2c and Simulink 1.3c graphical simulation program (Mathworks Inc., Natick, MA, USA) which allowed the models to be implemented in graphical form (Fig.…”

“…Starting in the late 1960s, several acute studies were aimed at deriving mathematical models of spindle response characteristics. The resulting models fall into three classes: linear transfer functions (Matthews & Stein, 1969;Chen & Poppele, 1978), non-linear functions based on curve-fitting (Houk, Rymer & Crago, 1981;Hasan, 1983) and non-linear models based on biological mechanisms of the muscle spindle (Otten, Scheepstra & Hulliger, 1995). The predictive power of the various models has never been directly compared, either in acute experiments or in relation to afferent recordings from awake animals or humans.…”

A closed loop musculoskeletal model of postural coordination dynamics

“…Many spindle models exist, at different level of physiology description, e.g. [23], but generally they are modelled with a pure gain and a small time delay [17] (35ms). Because SFR time delay is small and since no other time delays are introduced in our model, there will be neglected.…”