A model for steady isometric muscle activation

Abstract

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.

A Mathematical Model for the Steady Activation of a Skeletal Muscle

“…Equation (12) contains the parameters α, c, Δ, and h [42]. Experimental observations suggest that α [24,26,2] and c [27] are MU-independent, and, in the present approach, we assume muscle independence.…”

“…It evokes an EPSP equal to the firing threshold voltage V T and therefore satisfies V EP SP (In T ) = V T . Assuming that all MNs of the pool have the same firing threshold [42], (6) at threshold provides…”

“…Fitting the frequency-injected current relations with a straight line with slope κ [42,24], the MN firing frequency ν(G EP SP , i inj ) as a function of G EP SP and i inj is given by…”

“…Since no additional force due to newly recruited MUs is available, it is an integral of strictly concave functions given in (10). Consequently, the affine ("linear" in [42]) relation can hold exclusively during recruitment. In a forthcoming paper, we will show that the possible activation curves are not limited to affine functions.…”

“…Experimental observations suggest that α [24,26,2] and c [27] are MU-independent, and, in the present approach, we assume muscle independence. On the basis of experimental data, α was set to 1.14 and c to 0.9 [42], but the forthcoming general discussion is valid for all α > 0, 0 < c < 1, and Δ > 0. A muscle is thus specified by Δ and h. In 0 cannot be measured experimentally, and in [42] it was assumed to be the same for all muscles.…”