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-de pendent quantities are treated as time-averages. The character of the model is continuous in the sense that the motor unit (MU) population is describe d by a continuous density function. In contrast to most already published m odels, the wiring (synaptic weight) between the input fibers to the MNPMC a nd the MNs (about which no detailed data are known) is deduced, whereas the input-force relation is given. As suggested by experimental data, this rel ation is assumed to be linear during MU recruitment, but the model allows o ther, nonlinear relations. The input to the MN pool is defined as the numbe r of action potentials per second in all input fibers, and the excitatory p ostsynaptic potential (EPSP) conductance in MNs evoked by the input is assu med to be proportional to the input. A single compartment model with a homo geneous membrane is used for a MN. The MNs start firing after passing a con stant voltage threshold. The synaptic current-frequency relation is describ ed by a linear function and the frequency-force transformation of a MU by a n 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 pa rameters were determined a priori, i.e., the model was not used for their e stimation. The analysis of the model reveals special features of the activa tion 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 determi ned by the activation factor, the slope of the linear part of the activatio n curve (during MU recruitment). This factor determines quantitatively the relation between MU recruitment and rate modulation. This property of the m odel (the only known model with this property) allows a quantification of t he recruitment gain (Kernell and Hultborn 1990). The interest of the activa tion factor is illustrated using two human muscles, namely the first dorsal interosseus muscle, a small muscle with a relatively small force at the en d of recruitment, and the medial gastrocnemius muscle, a strong muscle with a relatively large force at the end of recruitment. It is concluded that t he present model allows us to reproduce the main features of muscle activat ion in the steady state. Its analytical character facilitates a deeper unde rstanding of these features.