Fig 1.
A schematic presentation of the model idea and the necessary input data.
A. A single twitch force recording of a MU. The red lines indicate the parameters necessary to model the twitch (see Fig 2 for definitions of these parameters). B. The initial part of an unfused tetanic contraction recording (blue) and models of the successive contractions (black) obtained using a mathematical decomposition of the recording into force responses to individual stimuli. Green dots indicate force levels at the beginnings of the twitch-shape responses to successive stimuli. The red dots indicate the stimuli. C. A recording of a fused tetanic contraction. The red vertical line indicates the maximum tetanus force. D. The amplitudes of the decomposed twitches as a function of the force levels developed by a motor unit at the time moments when the contractions begin. The main aim of the modelling is to predict the course of an unfused tetanic contraction (blue recording) developed during stimulation at variable interpulse intervals using as input data the parameters indicated in red, i.e. the twitch time and force parameters, the maximum tetanus force and the applied pattern of stimuli. The previously observed correlation between the force of the decomposed twitches and the present MUs’ force level gave us a start point for the modelling.
Fig 2.
Description of the parameters used for the j-th MU recordings.
A. The model of the i-th decomposed contraction within the unfused tetanus is shown by a dashed line, the black solid line is a piece of the force obtained by subtraction of all previous (i-1) contraction models from the experimental tetanic force. The parameters of the i-th twitch-like contraction are: Fmax(j)(i)–the maximum twitch force; Tlead(j) (i)–the lead time, the time between the i-th stimulus (its time position is indicated by vertical arrow) and the start of the current i-th contraction; Thc(j)–the half-contraction time, the time from the start of the contraction to the moment when the twitch force reaches a half of its maximal value; Tc(j)–the contraction time, the time from the start of the contraction to the moment when the twitch amplitude reaches it maximal value Fmax(j)(i); Thr(j)(i)–the half-relaxation time, the time between the start of the contraction to the moment when during the relaxation, the twitch force decreases to Fmax(j)(i)/2; Ttw(j)(i)—the duration of the current contraction, from the time between the moment when the contraction starts and the moment when the force decreases to 0.01% of Fmax(j)(i). The equation describing this bell-shape 6-parameters curve are given in [9]; B. Parameters measured for tetanic contractions presented on a part of the unfused tetanic curve (left) and the maximum fused tetanus (right). Fmftf(j)—the maximal force that a MU develops during stimulation at 150 Hz stimulation frequency (in the fused tetanus). Ftetmin(j)(i)—the force level at which the i-th contraction starts; Fres(j)(i) = Fmftf(j)-Ftetmin(j)(i)—the residual force.
Fig 3.
Dependencies between two normalized parameters: Fmax(j)(i)/Fmax(j)(1) and Fres(j)(i)/Fmax(j)(1), for all 30 MUs.
The two parameters used to calculate data presented on the ordinate are illustrated in a frame left to the axis on an example of a train of decomposed twitches (red lines indicate amplitudes of the first and the i-th twitch). Additionally, the parameters used to calculate data presented on the abscissa are illustrated in a frame below the axis on a fragment of the unfused tetanus and the fused tetanus recordings (red lines indicate amplitudes of the first twitch and the residual force for the response to the i-th stimulus). The symbols on the main chart marked in blue present the data for slow MUs, in red—the data for FR MUs, and in green—the data for FF MUs. The data for each MU was approximated by straight lines in respective colors: blue for S MUs, red for FR MUs, and green for FF MUs. The angles α(j) (j = 1, 2,…, 30) for each MU were calculated between these lines and the ordinate.
Fig 4.
Approximation of the relationships between the angles α(j) and the parameter Fmftf(j) (reflecting the maximum force that the respective MU can develop in the fused tetanus) normalized to the amplitude of the first decomposed contraction.
The two parameters used to calculate data presented on the abscissa are illustrated in a frame below the axis on recordings of a fragment of an unfused tetanus and the fused tetanus (red lines indicate amplitudes of the first twitch and the maximum tetanus force). The data for the angles are given in the fourth column of Table 1. S MU—blue asterisks; FR MU—red asterisks; FF—green asterisks. The black dashed curves are different approximations: ‘o’—with a linear model: y = ax+b, a = -2.994, b = 91.96; ‘◊‘—with an exponential model from 1st type: y = aebx, a = 95.8, b = -0.04602; ‘✻‘—with an exponential model from 2nd type: y = aebx+cedx, a = 66.53, b = -0.2896, c = 54.6, d = 0.001297; ‘□’—with a power model (this model is chosen for further modeling and is marked with the bold dashed line): y = axb, a = 108.8, b = -0.2603. Here, y = α1(j) and x = Fmftf(j)/Fmax(j)(1).
Fig 5.
Linear approximation of the data for the contraction and the half-relaxation times.
A. The dependence between the contraction time Tc(j)(i) and the parameter Ftetmin(j) (reflecting the MU force level at which the next contraction starts), normalized according to the parameters of the first decomposed twitch-like contraction Tc(j)(1) and Fmax(j)(1), respectively. The equation of the straight line is: y = p1+p2x, and p1 = 1.104, p2 = 0.274; B. The dependence between the half-relaxation time Thr(j)(i) and the parameter Ftetmin(j)(i), normalized to the parameters of the first decomposed twitch-like contraction Thr(j)(1) and Fmax(j)(1), respectively. The equation of the straight line is: y = p1+p2x, and p1 = 2.397, p2 = 0.3509; blue circles—S MUs; red circles—FR MUs; green circles—FF MUs. The two twitch time parameters used to calculate the data presented on the ordinate are illustrated in frames left to the axis on the example of a series of decomposed twitches (red lines indicate the contraction time for the first and the i-th twitch in A, and the half-relaxation time for the first and for the i-th twitch in B). Additionally, the parameters used to calculate data presented on the abscissa are illustrated in a frame below the axis on a fragment of the unfused tetanus recording (red lines indicate amplitudes of the first twitch and the force level at which the i-th contraction starts).
Table 1.
Calculated angles and coefficients reflecting the similarity between the experimental curves and the predicted curves for all 33 MUs.
FitCo1 and AreaCo1 are these coefficients when the modeled curve is obtained as a sum of equal to the model of the first contraction twitches, according to the respective stimulation pattern. α1(j) is the angle calculated by using the Eq (1). FitCo2 and AreaCo2 are the coefficients for the experimental and modeled curve (the calculated experimental values of Ftetmin (j)(i) are used as input parameters for the prediction). FitCo3 and AreaCo3 are the coefficients for the experimental and modeled curve using the same angles α1(j) but for full prediction algorithm (i.e. values of Ftetmin (j)(i) are also predicted consecutively). α2(j) is the angle obtained by a sensitivity analysis so that the modeled curve is the most similar to the experimental one, and FitCo4 and AreaCo4 are the respective coefficients. α3(j) is the improved angle using the Eq (11) (see Fig 8), and FitCo5 and AreaCo5 are the respective coefficients.
Fig 6.
Illustration of the calculation of the parameter Fprtetmin(j)(i) for the 23th MU after adding the models of all preceding contractions.
The first five models (plotted by using black dashed lines) are summed and the accumulated force is plotted as a solid black line; the value of the force Fprtetmin(23) (6) is computed at the moment when the 6th pulse comes and the next, 6th contraction, is calculated using this value. The model of this 6th contraction is red dashed line. The solid black line after the appearance of the 6-th pulse shows the addition of the force evoked by the 6th pulse to the previous five contractions.
Fig 7.
The effect of the application of the new approach for prediction of the successive contraction for three MUs.
Comparison between the recorded tetanic curves (red), the force curves obtained by summation of equal twitches according to the same stimulation pattern (green), and the curves predicted by the approximation approach (blue) using the angles α1(i) from Table 1. A. A slow MU (S1 in Table 1), the stimulation pattern with interpulse intervals IPI1 is used; B. A FR MU (FR4 in Table 1), the stimulation pattern with interpulse intervals IPI2 is used; C. A FF MU (FF2 in Table 1), the stimulation pattern with interpulse intervals IPI3 is used. IPIs are given in S1 File Table B. Note that the time and the force scales are different for the three MUs.
Fig 8.
Comparison between experimental tetani and the force curves, obtained through the new approximation approach.
Full prediction of three additional tetanic curves, obtained by stimulation with three new patterns, applied to three MUs not included in the input database. Red color—the recorded curve; blue color—the predicted force curve. A. A slow MU stimulated with the mean frequency of 20 Hz (S11 in Table 1), the stimulation pattern with interpulse intervals IPI4 is used; B. A FR MU stimulated with the mean frequency of 50 Hz (FR11 in Table 1), the stimulation pattern with interpulse intervals IPI5 is used; C. A FF MU stimulated with the mean frequency of 33.3 Hz (FF11 in Table 1), the stimulation pattern with interpulse interval IPI6 is used. The values of the angles α1(i) are shown in Table 1. IPIs are given in S1 File Table B. Note that the time and the force scales are different for the three MUs.
Fig 9.
Approximation of the relationships between the improved angles, i.e. α2(j) in Table 1, and the parameter Fmftf(j) normalized to the amplitude of the first decomposed contraction of a MU.
S MUs—blue asterisks; FR MUs—red asterisks; FF MUs—green asterisks. The squares present the three additional MUs (S11, FR11 and FF11). The black dashed curve is the new approximation with a power model y = axb, where a = 117.2, b = -0.3144, y = α2 (j) and x = Fmftf(j)/Fmax(j)(1).
Fig 10.
Relationship between the two parameters, Ftetmin(j)(i) and Fmax(j)(i), both normalized according to the first twitch amplitude, presented for three border values of the angle α.
The Eq (11) is used for calculations. This dependence is linear and is shown by blue asterisks. If Ftetmin(j)(i) = 0, i.e. the contraction starts from a fully relaxed MU, the amplitude of the evoked contraction will be equal to the amplitude of the individual twitch. A. α = 450 –the amplitudes of the successive contractions increase when the levels of the force at which a contraction starts increase, and Fmax(j)(i) are always bigger than Fmax(j)(1). B. α = 900 –the successive contractions always have amplitudes equal to the maximal force of the single twitch. C. α = 117.20 –the amplitudes of the successive contractions decrease when Ftetmin(j)(i) increase. Since Fmax(j)(i)≥0 the normalized value of Ftetmin(j)(i) cannot exceed the value of 1.94 (the crossed point of the dotted vertical and horizontal lines). When Ftetmin(j)(i)/ Fmax(j)(1) = 1.94, a MU is unable to respond with a new contraction.