Computer simulation of the motoneuron pool-muscle complex. I. Input system and motoneuron pool.

The aim of the present study was to simulate the input system and the motoneuron (MN) pool of the MN pool-muscle complex (MNPMC). Input fibers, which can originate from command centers in the central nervous system or from sensory organs, activate the MN pool. They generate sequences of action potentials, the frequency of which is proportional to a time-dependent activation factor (which is an input to the model). Different connection patterns between the input fibers and motor units (MUs) are allowed. For simplicity and since no precise experimental data are available, 70 input fibers and 4 boutons per fiber and MN are simulated (this corresponds approximately to the monosynaptic group-Ia input of the cat medial gastrocnemius muscle). Each bouton generates the same conductance change in the postsynaptic membrane. The MNs are modeled with a single compartment and a homogeneous membrane. According to experimental data, the membrane leakage conductance and capacitance are MU dependent. Since the precise relation is unknown: (a) the computed relation between MU contraction force and the MN leakage conductance was taken from a steady-state MNPMC model, and (b) the capacitance was assumed to be proportional to the leakage conductance. The MN membrane includes time- and voltage-dependent ionic channels (fast and slow K(+) and low- and high-threshold Ca(2+) channels). The density and time constant of the slow K(+) channels and the density of the Ca(2+) channels were fitted to approximate afterhyperpolarization characteristics and frequency-injected current relations of type-identified cat MNs. If the membrane reaches a voltage threshold the MNs generate action potentials, which were simulated by voltage pulses. The activation of the MN pool of the human first dorsal interosseus muscle was simulated with injected and synaptic currents in order to illustrate the size principle, synaptic noise, and other features of muscle activation. It is concluded that the present model reproduces the main properties of the input-output relations of different MN types within a muscle. Together with the simulation of the muscle force and the surface EMG, which will be published in subsequent papers, it will be a powerful tool for reproducing experiments on the motor system and investigating functional mechanisms of motor control.

A model for steady isometric muscle activation.

The present model of the motoneuronal (MN) pool-muscle complex (MNPMC) is deterministic and designed for steady isometric muscle activation. Time-dependent quantities are treated as time-averages. The character of the model is continuous in the sense that the motor unit (MU) population is described by a continuous density function. In contrast to most already published models, the wiring (synaptic weight) between the input fibers to the MNPMC and the MNs (about which no detailed data are known) is deduced, whereas the input-force relation is given. As suggested by experimental data, this relation is assumed to be linear during MU recruitment, but the model allows other, nonlinear relations. The input to the MN pool is defined as the number of action potentials per second in all input fibers, and the excitatory postsynaptic potential (EPSP) conductance in MNs evoked by the input is assumed to be proportional to the input. A single compartment model with a homogeneous membrane is used for a MN. The MNs start firing after passing a constant voltage threshold. The synaptic current-frequency relation is described by a linear function and the frequency-force transformation of a MU by an exponential function. The sum of the MU contraction forces is the muscle force, and the activation of the MUs obeys the size principle. The model parameters were determined a priori, i.e., the model was not used for their estimation. The analysis of the model reveals special features of the activation curve which we define as the relation between the input normalized by the threshold input of the MN pool and the force normalized by the maximal muscle force. This curve for any muscle turned out to be completely determined by the activation factor, the slope of the linear part of the activation curve (during MU recruitment). This factor determines quantitatively the relation between MU recruitment and rate modulation. This property of the model (the only known model with this property) allows a quantification of the recruitment gain (Kernell and Hultborn 1990). The interest of the activation factor is illustrated using two human muscles, namely the first dorsal interosseus muscle, a small muscle with a relatively small force at the end of recruitment, and the medial gastrocnemius muscle, a strong muscle with a relatively large force at the end of recruitment. It is concluded that the present model allows us to reproduce the main features of muscle activation in the steady state. Its analytical character facilitates a deeper understanding of these features.

Contribution of a Ia muscle afferent activation to the rise of H reflexes and somatosensory evoked potentials in man.

Electrical stimuli were applied to the tibial nerve in the popliteal fossa in man in order to investigate how information is transferred from group I muscle afferents to motoneurons and to the somatosensory cortex. For control purposes, identical stimuli were applied to the skin beside the electrode above the nerve. The somatosensory evoked potential (SEP) to skin stimulation alone had a peak latency which was 5 ms longer than the SEP to transcutaneous nerve stimulation. Influences by stimulation of the skin above the nerve could thus be excluded. The threshold intensity to evoke a liminal H reflex was at least two times higher than the threshold for a SEP. In most of the subjects, there was a correlation between the H reflex and the SEP size. If two identical stimuli were applied to the posterior tibial nerve with an interval of 1 s, the second H reflex was 30% smaller than the first one (postactivation depression). The corresponding SEPs were, however, only slightly reduced. Postactivation depression was probably caused by general intrinsic properties of synapses of group I muscle afferents. The results of this investigation indicate that: (1) a large volley in group I muscle afferents is necessary to evoke a liminal H reflex, whereas transmission from muscle afferents to the somatosensory cortex is very efficient; (2) these feedback signals to motoneurons and the somatosensory cortex are used independently.

Simulation of the surface EMG of an active muscle.

A model of the motoneuronal (MN) pool-muscle system was developed. The model consists of four modules: (1) the input to the MN pool, (2) the MN pool, (3) the muscle and (4) the surface electromyogram (EMG). A control parameter activates the input fibers and determines the activity level of the muscle. A single compartment model with a homogenous membrane was used to model the MNs. The trajectory between spikes is determined by two voltage-gated K(+)- and two voltage-gated Ca(2+)-channels. The size of the MNs is adjusted by the size of the leakage conductance. The model muscle is of circular cross section and with parallel fibers. The motor unit (MU) territories are of circular shape and their area is proportional to the MU contraction force. Action potentials propagated along the muscle fibers are approximated by a dipole with a current source and current sink. The potential evoked by the dipole at the recording site is computed. The surface EMG is obtained by summing up (1) the potentials of all fibers of the MU and (2) the MU action potentials of all active MUs. Numerical results show that the MUs are recruited with increasing contraction force and that the active MUs modulate their firing frequency similar as in real muscles. The model will be used for investigation of the motor system in man.