Elsevier

Journal of Biomechanics

Volume 48, Issue 4, 26 February 2015, Pages 712-715
Journal of Biomechanics

Short communication
Modelling suppressed muscle activation by means of an exponential sigmoid function: Validation and bounds

https://doi.org/10.1016/j.jbiomech.2015.01.009Get rights and content

Abstract

The aim of this study was to establish how well a three-parameter sigmoid exponential function, DIFACT, follows experimentally obtained voluntary neural activation-angular velocity profiles and how robust it is to perturbed levels of maximal activation. Six male volunteers (age 26.3±2.73 years) were tested before and after an 8-session, 3-week training protocol. Torque–angular velocity (Tω) and experimental voluntary neural drive–angular velocity (%VA–ω) datasets, obtained via the interpolated twitch technique, were determined from pre- and post-training testing sessions. Non-linear regression fits of the product of DIFACT and a Hill type tetanic torque function and of the DIFACT function only were performed on the pre- and post-training Tω and %VA–ω datasets for three different values of the DIFACT upper bound, αmax, 100%, 95% & 90%. The determination coefficients, R2, and the RMS of the fits were compared using a two way mixed ANOVA and results showed that there was no significant difference (p<0.05) due to changing αmax values indicating the DIFACT remains robust to changes in maximal activation. Mean R2 values of 0.95 and 0.96 for pre- and post-training sessions show that the maximal voluntary torque function successfully reproduces the Tω raw dataset.

Introduction

In vivo measurements of the maximum voluntary force–velocity relationship show differences to the in vitro tetanic profile, with eccentric forces not increasing much above isometric and tending to decline with increasing lengthening velocity (Westing, 1988, Dudley et al., 1990, Weber and Kriellaars, 1997). This difference could be due to a neural, tension-limiting mechanism that reduces maximal neural drive at high levels of muscular tension (Westing et al., 1990; Westing et al., 1991, Pain and Forrester, 2009, Pain et al., 2013). Yeadon et al. (2006) represented the in vivo maximum voluntary torque–velocity relationship as a product of a theoretical four parameter Hill-type tetanic torque function, and a three parameter differential activation function (DIFACT). The latter representing the net reduction in neural drive to the muscle with low neural activation at high eccentric velocities to full activation at high concentric velocities. However, the DIFACT function was not explicitly based on measured neural changes and its validity was implicitly assumed through the ability of the combined seven parameter function to reproduce the in vivo torque-velocity profiles. Furthermore, due to its quadratic form, the DIFACT function had multiple equivalent solutions and is difficult to manipulate algebraically. Pain and Forrester (2009) used a sigmoid exponential function to represent the DIFACT function in order to simplify mathematical manipulation when finding solutions for the seven parameter MVC torque function (MVC). Again the function was only implicitly shown to be successful through scaling of voluntary EMG signals (Pain and Forrester, 2009).

Therefore, although now used repeatedly (Lewis et al., 2012, Forrester et al., 2011, Tillin et al., 2012, Pain et al., 2013) in the literature the DIFACT function has yet to be verified in an explicit way. The aims of this study were (i) to establish experimentally how well the DIFACT function follows the in vivo voluntary neural activation–angular velocity profiles in a group of subjects; and (ii) to test the robustness of the exponential DIFACT function to perturbed upper levels of maximal activation.

Section snippets

Method

Measurements from six male volunteers (age 26.3±2.7 years, body mass 72.9±11.7 kg, height, 172.2±8.4 cm; mean±SD) tested before and after eight sessions (over three weeks) of high velocity strength training on an isovelocity dynamometer were used as the raw data to address the aims of this study. They all gave written, informed consent and the study was conducted in accordance with the approval given by Loughborough University Ethical Advisory Committee. In brief, testing followed similar methods

Results

Applying the extra-sum-of-squares F-Test on the seven parameter MVC function fit to the torque–angular velocity dataset, for αmax=100%, showed that 3 out of 6 subjects had a significant (p<0.05) higher torque output post-testing. The same outcome was obtained when the αmax values were set equal to 95% and 90%.

There was no significant difference between the R2 values of the three fits with different αmax values for both pre- and post-training datasets (p=0.95 & p=0.99 respectively) for any of

Discussion

The aim of this work was to determine how well the three-parameter exponential differential activation function DIFACT (Pain and Forrester, 2009) reproduces the in-vivo T–ω and %VA–ω profiles and whether changing the value of the maximum activation level, αmax, in DIFACT (Pain and Forrester, 2009) would affect its robustness. Results show that the MVC torque function reproduces the Tω raw data set very well irrespective of the αmax value. The DIFACT function is also successful in reproducing

Conflict of interest statement

Neither author has any conflict of interests.

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