A general mathematical algorithm for predicting the course of unfused tetanic contractions of motor units in rat muscle

Abstract

An unfused tetanus of a motor unit (MU) evoked by a train of pulses at variable interpulse intervals is the sum of non-equal twitch-like responses to these stimuli. A tool for a precise prediction of these successive contractions for MUs of different physiological types with different contractile properties is crucial for modeling the wholemuscle behavior during various types of activity. The aim of this paper is to develop such a general mathematical algorithm for the MUs of the medial gastrocnemiusmuscle of rats. For this purpose, tetanic curves recorded for 30 MUs (10 slow, 10 fast fatigue-resistant and 10 fast fatigable) were mathematically decomposed into twitch-like contractions. Each contractionwas modeled by the previously proposed 6-parameter analytical function, and the analysis of these six parameters allowed us to develop a prediction algorithm based on the following input data: parameters of the initial twitch, the maximum force of a MUand the series of pulses. Linear relationshipwas found between the normalized amplitudes of the successive contractions and the remainder between the actual force levels at which the contraction startedand the maximum tetanic force. The normalizationwas made according to the amplitude of the first decomposed twitch. However, the respective approximation lines had different specific angles with respect to the ordinate. These angles had different and non-overlapping ranges for slow and fast MUs. A sensitivity analysis concerning this slope was performedand the dependence between the angles and the maximal fused tetanic force normalized to the amplitude of the first contractionwas approximated by a power function. The normalizedMU contraction and half-relaxation times were approximated by linear functions depending on the normalized actual force levels at which each contraction starts. The normalization was made according to the contraction time of the first contraction. The actual force levels were calculated initially from the recorded tetanic curves and subsequently from the modeled curves obtained from the summation of all models of the preceding contractions (the so called "full prediction"). The preciseness of the predictionwas verified by two coefficients estimating the error between the modeled and the experimentally recorded curves. The proposed approach was tested for 30 MUs from the database and for three additional MUs, not included in the initial set. It was concluded that this general algorithm can be successfully used for modeling of a unfused tetanus course of a singleMUof fast and slow type.