Dynamic behaviour of a model of the muscle stretch reflex

A General Approach for the Modelling of Negative Feedback Physiological Control Systems

Mathematical models can improve the understanding of physiological systems behaviour, which is a fundamental topic in the bioengineering field. Having a reliable model enables researchers to carry out in silico experiments, which require less time and resources compared to their in vivo and in vitro counterparts. This work's objective is to capture the characteristics that a nonlinear dynamical mathematical model should exhibit, in order to describe physiological control systems at different scales. The similarities among various negative feedback physiological systems have been investigated and a unique general framework to describe them has been proposed. Within such a framework, both the existence and stability of equilibrium points are investigated. The model here introduced is based on a closed-loop topology, on which the homeostatic process is based. Finally, to validate the model, three paradigmatic examples of physiological control systems are illustrated and discussed: the ultrasensitivity mechanism for achieving homeostasis in biomolecular circuits, the blood glucose regulation, and the neuromuscular reflex arc (also referred to as muscle stretch reflex). The results show that, by a suitable choice of the modelling functions, the dynamic evolution of the systems under study can be described through the proposed general nonlinear model. Furthermore, the analysis of the equilibrium points and dynamics of the above-mentioned systems are consistent with the literature.

A highly polarized excitable cell separates sodium channels from sodium-activated potassium channels by more than a millimeter

The bioelectrical properties and resulting metabolic demands of electrogenic cells are determined by their morphology and the subcellular localization of ion channels. The electric organ cells (electrocytes) of the electric fish Eigenmannia virescens generate action potentials (APs) with Na(+) currents >10 μA and repolarize the AP with Na(+)-activated K(+) (KNa) channels. To better understand the role of morphology and ion channel localization in determining the metabolic cost of electrocyte APs, we used two-photon three-dimensional imaging to determine the fine cellular morphology and immunohistochemistry to localize the electrocytes' ion channels, ionotropic receptors, and Na(+)-K(+)-ATPases. We found that electrocytes are highly polarized cells ∼ 1.5 mm in anterior-posterior length and ∼ 0.6 mm in diameter, containing ∼ 30,000 nuclei along the cell periphery. The cell's innervated posterior region is deeply invaginated and vascularized with complex ultrastructural features, whereas the anterior region is relatively smooth. Cholinergic receptors and Na(+) channels are restricted to the innervated posterior region, whereas inward rectifier K(+) channels and the KNa channels that terminate the electrocyte AP are localized to the anterior region, separated by >1 mm from the only sources of Na(+) influx. In other systems, submicrometer spatial coupling of Na(+) and KNa channels is necessary for KNa channel activation. However, our computational simulations showed that KNa channels at a great distance from Na(+) influx can still terminate the AP, suggesting that KNa channels can be activated by distant sources of Na(+) influx and overturning a long-standing assumption that AP-generating ion channels are restricted to the electrocyte's posterior face.

Alpha band cortico-muscular coherence occurs in healthy individuals during mechanically-induced tremor

The present work aimed at investigating the effects of mechanically amplified tremor on cortico-muscular coherence (CMC) in the alpha band. The study of CMC in this specific band is of particular interest because this coherence is usually absent in healthy individuals and it is an aberrant feature in patients affected by pathological tremors; understanding its mechanisms is therefore important. Thirteen healthy volunteers (23±4 years) performed elbow flexor sustained contractions both against a spring load and in isometric conditions at 20% of maximal voluntary isometric contraction (MVC). Spring stiffness was selected to induce instability in the stretch reflex servo loop. 64 EEG channels, surface EMG from the biceps brachii muscle and force were simultaneously recorded. Contractions against the spring resulted in greater fluctuations of the force signal and EMG amplitude compared to isometric conditions (p<.05). During isometric contractions CMC was systematically found in the beta band and sporadically observed in the alpha band. However, during the contractions against the spring load, CMC in the alpha band was observed in 12 out of 13 volunteers. Partial directed coherence (PDC) revealed an increased information flow in the EMG to EEG direction in the alpha band (p<.05). Therefore, coherence in the alpha band between the sensory-motor cortex and the biceps brachii muscle can be systematically induced in healthy individuals by mechanically amplifying tremor. The increased information flow in the EMG to EEG direction may reflect enhanced afferent activity from the muscle spindles. These results may contribute to the understanding of the presence of alpha band CMC in tremor related pathologies by suggesting that the origin of this phenomenon may not only be at cortical level but may also be affected by spinal circuit loops.

Muscle proprioceptive feedback and spinal networks

This review revolves primarily around segmental feedback systems established by muscle spindle and Golgi tendon organ afferents, as well as spinal recurrent inhibition via Renshaw cells. These networks are considered as to their potential contributions to the following functions: (i) generation of anti-gravity thrust during quiet upright stance and the stance phase of locomotion; (ii) timing of locomotor phases; (iii) linearization and correction for muscle nonlinearities; (iv) compensation for muscle lever-arm variations; (v) stabilization of inherently unstable systems; (vi) compensation for muscle fatigue; (vii) synergy formation; (viii) selection of appropriate responses to perturbations; (ix) correction for intersegmental interaction forces; (x) sensory-motor transformations; (xi) plasticity and motor learning. The scope will at times extend beyond the narrow confines of spinal circuits in order to integrate them into wider contexts and concepts.

Effects of spinal recurrent inhibition on motoneuron short-term synchronization

Spinal recurrent inhibition linking skeleto- motoneurons (alpha-MNs) via Renshaw cells (RCs) has been variously proposed to increase or decrease tendencies toward synchronous discharges between alpha-MNs. This controversy is not easy to settle experimentally in animal or human paradigms because RCs receive, in addition to excitatory input from alpha-MNs, many other modulating influences which may change their mode of operation. Computer simulations help to artificially isolate the recurrent inhibitory circuit and thus to study its effects on alpha-MN synchronization under conditions not achievable in natural experiments. We present here such a study which was designed to specifically test the following hypothesis. Since many alpha-MNs excite any particular Renshaw cell, which in turn inhibits many alpha-MNs, this convergence-divergence pattern establishes a random network whose random discharge patterns inject uncorrelated noise into alpha-MNs, and this noise counteracts any synchronization potentially arising from other sources, e.g., common inputs (Adam et al. in Biol Cybern 29:229-235, 1978). We investigated the short-term synchronization of alpha-MNs with two types of excitatory input signals to alpha-MNs (random and sinusoidally modulated random patterns). The main results showed that, while recurrent inhibitory inputs to different alpha-MNs were indeed different, recurrent inhibition (1) exerted rather small effects on the modulation of alpha-MN discharge, (2) tended to increase the short-term synchronization of alpha-MN discharge, and (3) did not generate secondary peaks in alpha-MN-alpha-MN cross-correlograms associated with alpha-MN rhythmicity.

Entrainment to extinction of physiological tremor by spindle afferent input

In this study the systematic modulation of wrist flexor muscle activity by imposed joint movement was examined. Ten subjects maintained a constant contraction level (25% of the maximum; trial duration: 20 s) in flexor carpi radialis while their wrists were perturbed with 50 different quasi-sinusoidal signals (frequency range: 0.5-9.5 Hz; amplitude: 0.3-4.2 degrees ). The frequency spectra of wrist position and the rectified and filtered electromyogram (EMG) were determined. The muscle activity was only weakly entrained to imposed movements of small amplitude and low frequency, as shown by a small peak in the EMG spectrum at the frequency of movement, while the most prominent peak in the spectrum was between 9 and 15 Hz, corresponding to the frequency range of physiological tremor. The entrainment of muscle activity increased markedly as the amplitude and frequency of the imposed movement increased, to the point of saturation of modulation and harmonic peaks in the spectrum. In parallel with this increase in entrainment, the 9-15 Hz tremor peak was progressively extinguished. The results are consistent with a coupled oscillator model in which the central oscillatory source(s) of tremor became fully entrained to the imposed movement at the highest amplitudes and frequencies. Such coupling depends on communication between the external forcing oscillator and the central oscillator(s), the I (a) afferent signal from the imposed movement being the most likely candidate to provide the entraining signal for the central oscillator(s).

Dynamic neural network models of the premotoneuronal circuitry controlling wrist movements in primates

Dynamic recurrent neural networks were derived to simulate neuronal populations generating bidirectional wrist movements in the monkey. The models incorporate anatomical connections of cortical and rubral neurons, muscle afferents, segmental interneurons and motoneurons; they also incorporate the response profiles of four populations of neurons observed in behaving monkeys. The networks were derived by gradient descent algorithms to generate the eight characteristic patterns of motor unit activations observed during alternating flexion-extension wrist movements. The resulting model generated the appropriate input-output transforms and developed connection strengths resembling those in physiological pathways. We found that this network could be further trained to simulate additional tasks, such as experimentally observed reflex responses to limb perturbations that stretched or shortened the active muscles, and scaling of response amplitudes in proportion to inputs. In the final comprehensive network, motor units are driven by the combined activity of cortical, rubral, spinal and afferent units during step tracking and perturbations. The model displayed many emergent properties corresponding to physiological characteristics. The resulting neural network provides a working model of premotoneuronal circuitry and elucidates the neural mechanisms controlling motoneuron activity. It also predicts several features to be experimentally tested, for example the consequences of eliminating inhibitory connections in cortex and red nucleus. It also reveals that co-contraction can be achieved by simultaneous activation of the flexor and extensor circuits without invoking features specific to co-contraction.

Oscillations in a Simple Neuromechanical System: Underlying Mechanisms

A half-center neural oscillator was coupled to a simple mechanical system to study the closed-loop interactions between a central pattern generator and its effector muscles. After a review of the open-loop mechanisms that were previously introduced by Skinner et al. (1994), we extend their geometric approach and introduce four additional closed-loop mechanisms by the inclusion of an antagonistic muscle pair acting on a mass and connected to the half-center neural oscillator ipsilaterally. Two of the closed-loop mechanisms, mechanical release mechanisms, have close resemblance to open-loop release mechanisms whereas the latter two, afferent mechanisms, have a strong dependence on the mechanical properties of the system. The results also show that stable oscillations can emerge in the presence of sensory feedback even if the neural system is not oscillatory. Finally, the feasibility of the closed-loop mechanisms was shown by weakening the idealized assumptions of the synaptic and the feedback connections as well as the rapidity of the oscillations.

On the role of recurrent inhibitory feedback in motor control

This article reviews presumed roles of recurrent inhibition in motor control, that have been proposed over the past five decades. The discussion is structured in an order of increasing complexity. It starts out with the simplest and earliest circuit, that is recurrent self-inhibition of skeleto-motoneurons, and related functions. It soon becomes clear that in order to understand recurrent inhibition, we must look beyond the simple self-inhibitory CNS circuit. First, recurrent inhibition must be seen in the context of other neural circuits. Second, some quantitative features appear to be correlated with features of the neuromusculo-skeletal periphery. Third, the aspect of lateral inhibition between different members of a motoneuron pool as well as between different motoneuron pools points to the essential multiple input-multiple output structure of recurrent inhibition that again can be understood only by correlating it with features of the neuromusculo-skeletal periphery. Another extension results from the discovery that recurrent inhibition affects not only skeleto-motoneurons, but also gamma-motoneurons, Ia inhibitory interneurons mediating reciprocal inhibition between antagonist motoneurons, other Renshaw cells and cells of origin of the ventral spinocerebellar tract (VSCT). Then the view broadens again, investigating the potential role that recurrent inhibition plays in two far-ranging theories of motor control, the inverse-dynamics approach and the equilibrium-point hypothesis. Finally, the present author tries to formulate, in broad strokes, a personal functional interpretation of recurrent inhibition. All the functional considerations, right or wrong, should yield ideas for new experiments, and this then is the last objective of this review.

Neuronal network models of motor generation and control

Computer simulation of neuronal networks is rapidly becoming accepted as a powerful tool in neuroscience. We illustrate the trends in this field by looking at motor generation and control, with examples from recent modeling studies of different systems, including the spinal swimming rhythm generator of the lamprey.

How different afferent inputs control motoneuron discharge and the output of the motoneuron pool

In theory, there are at least two distinct mechanisms by which afferent inputs could alter motoneuron discharge and shape the output of a motoneuron pool: either by delivering synaptic current to the motoneurons' somata ('classic' synaptic transduction); or by altering the motoneurons' voltage-sensitive conductances (neuromodulation). Recent work has confirmed the operation of both of these mechanisms. It has been shown that the effect of a 'classic' synaptic input on motoneuron firing rate is predicted by the product of the effective synaptic current and the slope of the motoneuron's frequency-current relation. It has also been shown that neuromodulators can alter both the slope of a motoneuron's frequency-current relation and its threshold for repetitive firing. It is argued here, however, that when two or more sources of synaptic input are activated concurrently, the distinction between these two mechanisms is blurred. Computer simulations of motoneuron and motor pool behavior have proved extremely useful in understanding these processes.