Summing responses of cat soleus muscle spindles to combined static and dynamic fusimotor stimulation(1)

A physiologically enhanced muscle spindle model: using a Hill-type model for extrafusal fibers as template for intrafusal fibers

The muscle spindle is an essential proprioceptor, significantly involved in sensing limb position and movement. Although biological spindle models exist for years, the gold-standard for motor control in biomechanics are still sensors built of homogenized spindle output models due to their simpler combination with neuro-musculoskeletal models. Aiming to improve biomechanical simulations, this work establishes a more physiological model of the muscle spindle, aligned to the advantage of easy integration into large-scale musculoskeletal models. We implemented four variations of a spindle model in Matlab/Simulink®: the Mileusnic et al. (2006) model, Mileusnic model without mass, our enhanced Hill-type model, and our enhanced Hill-type model with parallel damping element (PDE). Different stretches in the intrafusal fibers were simulated in all model variations following the spindle afferent recorded in previous experiments in feline soleus muscle. Additionally, the enhanced Hill-type models had their parameters extensively optimized to match the experimental conditions, and the resulting model was validated against data from rats' triceps surae muscle. As result, the Mileusnic models present a better overall performance generating the afferent firings compared to the common data evaluated. However, the enhanced Hill-type model with PDE exhibits a more stable performance than the original Mileusnic model, at the same time that presents a well-tuned Hill-type model as muscle spindle fibers, and also accounts for real sarcomere force-length and force-velocity aspects. Finally, our activation dynamics is similar to the one applied to Hill-type model for extrafusal fibers, making our proposed model more easily integrated in multi-body simulations.

Information processing in tree networks of excitable elements

We study the collective response of small random tree networks of diffusively coupled excitable elements to stimuli applied to leaf nodes. Such networks model the morphology of certain sensory neurons that possess branched myelinated dendrites with excitable nodes of Ranvier at every branch point and at leaf nodes. Leaf nodes receive random inputs along with a stimulus and initiate action potentials that propagate through the tree. We quantify the collective response registered at the central node using mutual information. We show that in the strong-coupling limit, the statistics of the number of nodes and leaves determines the mutual information. At the same time, the collective response is insensitive to particular node connectivity and distribution of stimulus over leaf nodes. However, for intermediate coupling, the mutual information may strongly depend on the stimulus distribution among leaf nodes. We identify a mechanism behind the competition of leaf nodes that leads to nonmonotonous dependence of mutual information on coupling strength. We show that a localized stimulus given to a tree branch can be occluded by the background firing of unstimulated branches, thus suppressing mutual information. Nonetheless, the mutual information can be enhanced by a proper stimulus localization and tuning of coupling strength.

Proprioceptive Recognition with Artificial Neural Networks Based on Organizations of Spinocerebellar Tract and Cerebellum

Muscle kinematics and kinetics are nonlinearly encoded by proprioceptors, and the changes in muscle length and velocity are integrated into Ia afferent. Besides, proprioceptive signals from multiple muscles are probably mixed in afferent pathways, which all lead to difficulties in proprioceptive recognition for the cerebellum. In this study, the artificial neural networks, whose organizations are biologically based on the spinocerebellar tract and cerebellum, are utilized to decode the proprioceptive signals. Consistent with the controversy of the proprioceptive division in the dorsal spinocerebellar tract, the spinocerebellar tract networks incorporated two distinct inferences, (1) the centralized networks, which mixed Ia, II, and Ib and processed them together; (2) the decentralized networks, which processed Ia, II, and Ib afferents separately. The cerebellar networks were based on the Marr-Albus model to recognize the kinematic states. The networks were trained by a specific movement, and the trained networks were subsequently required to predict kinematic states of six movements. The results demonstrated that the centralized networks, which were more consistent with the physiological findings in recent years, had better recognition accuracy than the decentralized networks, and the networks were still effective even when proprioceptive afferents from multiple muscles were integrated.

Linear and quadratic models of point process systems: contributions of patterned input to output

In the 1880's Volterra characterised a nonlinear system using a functional series connecting continuous input and continuous output. Norbert Wiener, in the 1940's, circumvented problems associated with the application of Volterra series to physical problems by deriving from it a new series of terms that are mutually uncorrelated with respect to Gaussian processes. Subsequently, Brillinger, in the 1970's, introduced a point-process analogue of Volterra's series connecting point-process inputs to the instantaneous rate of point-process output. We derive here a new series from this analogue in which its terms are mutually uncorrelated with respect to Poisson processes. This new series expresses how patterned input in a spike train, represented by third-order cross-cumulants, is converted into the instantaneous rate of an output point-process. Given experimental records of suitable duration, the contribution of arbitrary patterned input to an output process can, in principle, be determined. Solutions for linear and quadratic point-process models with one and two inputs and a single output are investigated. Our theoretical results are applied to isolated muscle spindle data in which the spike trains from the primary and secondary endings from the same muscle spindle are recorded in response to stimulation of one and then two static fusimotor axons in the absence and presence of a random length change imposed on the parent muscle. For a fixed mean rate of input spikes, the analysis of the experimental data makes explicit which patterns of two input spikes contribute to an output spike.

Mathematical models of proprioceptors. II. Structure and function of the Golgi tendon organ

We developed a physiologically realistic mathematical model of the Golgi tendon organ (GTO) whose elements correspond to anatomical features of the biological receptor. The mechanical interactions of these elements enable it to capture all salient aspects of GTO afferent behavior reported in the literature. The model accurately describes the GTO's static and dynamic responses to activation of single motor units whose muscle fibers insert into the GTO, including the different static and dynamic sensitivities that exist for different types of muscle fibers (S, FR, and FF). Furthermore, it captures the phenomena of self- and cross-adaptation wherein the GTO dynamic response during motor unit activation is reduced by prior activation of the same or a different motor unit, respectively. The model demonstrates various degrees of nonlinear summation of GTO responses resulting from simultaneous activation of multiple motor units. Similarly to the biological GTO, the model suggests that the activation of every additional motor unit to already active motor units that influence the receptor will have a progressively weaker incremental effect on the GTO afferent activity. Finally, the proportional relationship between the cross-adaptation and summation recorded for various pairs of motor units was captured by the model, but only by incorporating a particular type of occlusion between multiple transduction regions that were previously suggested. This occlusion mechanism is consistent with the anatomy of the afferent innervation and its arrangement with respect to the collagen strands inserting into the GTO.

Mathematical models of proprioceptors. I. Control and transduction in the muscle spindle

We constructed a physiologically realistic model of a lower-limb, mammalian muscle spindle composed of mathematical elements closely related to the anatomical components found in the biological spindle. The spindle model incorporates three nonlinear intrafusal fiber models (bag(1), bag(2), and chain) that contribute variously to action potential generation of primary and secondary afferents. A single set of model parameters was optimized on a number of data sets collected from feline soleus muscle, accounting accurately for afferent activity during a variety of ramp, triangular, and sinusoidal stretches. We also incorporated the different temporal properties of fusimotor activation as observed in the twitchlike chain fibers versus the toniclike bag fibers. The model captures the spindle's behavior both in the absence of fusimotor stimulation and during activation of static or dynamic fusimotor efferents. In the case of simultaneous static and dynamic fusimotor efferent stimulation, we demonstrated the importance of including the experimentally observed effect of partial occlusion. The model was validated against data that originated from the cat's medial gastrocnemius muscle and were different from the data used for the parameter determination purposes. The validation record included recently published experiments in which fusimotor efferent and spindle afferent activities were recorded simultaneously during decerebrate locomotion in the cat. This model will be useful in understanding the role of the muscle spindle and its fusimotor control during both natural and pathological motor behavior.

Estimation of muscle spindle information rate by pattern matching and the effect of gamma system activity on parallel spindles

The information transmission properties of ensembles of MSs and the effect of the gamma system on these properties were studied. Three converging lines of research were taken: (1) the development of information theoretic estimation tools, and the formulation of an "operational" interpretation for the information rate; (2) animal experiments in which the mutual information rate was estimated and the effect of the gamma system was quantified; (3) simulation of a muscle spindle model with gamma activation in order to corroborate the results of the animal experiments. The main hypothesis was that the gamma system will enhance information theoretic measures that quantify the quality of the sensory neural channel comprised from an ensemble of primary muscle spindle afferents. A random stimulus was applied to a muscle in the hind limb of a cat, while spike trains from several primary MS afferents were recorded simultaneously. The stimulus was administered twice, with an operative and a disconnected gamma system. The mutual information rate between the stimulus and spike trains, as well as other information theoretic measures, was estimated. The information rate of ensembles of MSs increased with increasing ensemble size. However, with an operative gamma system the "ensemble effect" was much higher. In addition, the ensemble effect was influenced by the stimulus spectrum. A muscle spindle population model with gamma activation was simulated with stimuli that were identical to that of the animal experiments. The simulation results supported the experimental results and corroborated the main hypothesis. The results indicate that the gamma system has an important role in enhancing information transmission from ensembles of MSs to the spinal cord.

Spindle model responsive to mixed fusimotor inputs and testable predictions of beta feedback effects

Skeletofusimotor (beta) motoneurons innervate both extrafusal muscle units and muscle fibers within muscle spindle stretch receptors. By receiving excitation from group Ia muscle spindle afferents and driving the muscle spindle afferents that excite them, they form a positive feedback loop of unknown function. To study it, we developed a computationally efficient model of group Ia afferent behavior, capable of responding to multiple fusimotor inputs, that matched experimental data. This spindle model was then incorporated into a simulation of group Ia feedback during ramp/hold and triangular stretches with and without closure of the beta loop, assuming that gamma and beta fusimotor drives of the same type (static or dynamic) have identical effects on spindle afferent firing. The effects of beta feedback were implemented by driving a fusimotor input with a delayed and filtered fraction of the spindle afferent output. During triangular stretches, feedback through static beta motoneurons enhanced Ia afferent firing during shortening of the spindle. In contrast, closure of a dynamic beta loop increased Ia firing during lengthening. The strength of beta feedback, estimated as a "loop gain" was comparable to experimental estimates. The loop gain increased with velocity and amplitude of stretch but decreased with increased superimposed gamma fusimotor rates. The strongest loop gains were seen when the beta loop and the gamma bias were of different types (static vs. dynamic).

Signalling properties of muscle spindles and tendon organs

Some important issues for muscle receptors remain unresolved. For muscle spindles it is uncertain how responses to combined static and dynamic fusimotor stimulation may summate. Such summation may occur during certain phases of locomotion. Two mechanisms considered here include electrotonic spread of generator current between sources of impulse activity and mechanical interactions between contracting intrafusal fibres. For tendon organs, it remains uncertain what aspects of muscle tension they signal. Here they were tested for their ability to respond to rises in whole-muscle passive tension after eccentric contractions. It was found that only when motor units were contracted which had a direct action on a tendon organ did it signal the rise in tension. The finding raises questions about the role of tendon organs as monitors of muscle tension.