# A feasibility study of leveraging intermuscular coherence in EMG-driven neuromusculoskeletal modeling to improve muscle moment estimation

Emilie Mathieu, Sylvain Crémoux, David Gasq, Philippe Pudlo, David Amarantini

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## To cite this version:

Emilie Mathieu, Sylvain Crémoux, David Gasq, Philippe Pudlo, David Amarantini. A feasibility study of leveraging intermuscular coherence in EMG-driven neuromusculoskeletal modeling to improve muscle moment estimation. Biomedical Signal Processing and Control, 2025, 109, pp.108004. 10.1016/j.bspc.2025.108004. hal-05062997

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Submitted on 11 May 2025

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# A feasibility study of leveraging intermuscular coherence in EMG-driven neuromusculoskeletal modeling to improve muscle moment estimation

Emilie Mathieu<sup>a, *</sup>, Sylvain Crémoux<sup>b</sup>, David Gasq<sup>c,d</sup>, Philippe Pudlo<sup>a,1</sup>, David Amarantini<sup>c,1</sup>

## ARTICLE INFO

### Keywords:

EMG-driven models

Intermuscular coupling

EMG-EMG connectivity

Muscle moments

Co-contraction

Static optimization

## ABSTRACT

Current musculoskeletal models often oversimplify the neural strategies underlying muscle activation, potentially leading to unsatisfactory estimates of muscle forces. Numerous studies in motor control have established that the central nervous system synchronizes muscle activation by sending a common drive to synergistic muscles, measurable through intermuscular coherence – the frequency correlation between two EMG signals. As interest grows in understanding how muscles synchronize during movement coordination, leveraging inter-muscular coherence into musculoskeletal models represents an innovative approach. This could enhance the accuracy of muscle effort estimation and introduce a physiologically meaningful component of motor control. In this study, we introduce a new method that decomposes EMG signals into common and independent components, informed by intermuscular coherence, and integrates them into an EMG-driven model to estimate muscle moments. Using data from twenty-four healthy subjects performing horizontal upper limb extensions, we estimated moments of the four main muscles actuating the elbow and compared these estimations with those from a traditional EMG-driven model informed by full-wave rectified signal envelopes. Our results demonstrate that incorporating intermuscular coherence significantly enhanced kinetic data tracking and improved the robustness of muscle moment estimations against variations in model parameters, addressing a major limitation of traditional EMG-driven models. Furthermore, antagonist muscle moments were more accurately represented, resulting in more realistic co-contraction index values.

By integrating neural control strategies via intermuscular coherence into musculoskeletal models, the proposed approach offers a more accurate representation of muscle coordination. We recommend that future neuromusculoskeletal models incorporate intermuscular coherence to improve physiological realism of muscle effort estimations.

## 1. Introduction

Estimating muscle effort through musculoskeletal modelling provides essential information in various applications. For instance, it supports motor rehabilitation in clinical contexts, investigates the causes of musculoskeletal disorders in ergonomics, and enables control strategies in robotics for exoskeletons or prostheses [1,2]. A key problem in

musculoskeletal modelling is muscle redundancy, which refers to the possibility of generating the same net joint torque through numerous combinations of individual muscle efforts [3,4]. Consequently, the contributions of certain muscles, particularly antagonists, are often neglected or poorly estimated [5,6]. Musculoskeletal models are also highly sensitive to input parameters, limiting their practical applicability [3,7,8]. To address these limitations, recent research has focused

**Abbreviations:** EMG, electromyography; IMC, intermuscular coherence; $M_{inv}$, net elbow torque calculated with inverse dynamics; $M_{EMG}$, net elbow torque estimated with EMG-model or IMC-model; TB, long head of the triceps brachii; BB, biceps brachii; BA, brachialis; BR, brachioradialis; EMG_com, envelopes of the common EMG signals; EMG_indep, envelopes of the independent EMG signals; $M_{EMG_indep}$, muscle moment produced due to independent EMG signals; $M_{EMG_com}$, actuator moment related to common EMG signal; CCI, co-contraction index.

<footnote><sup>a</sup> Univ. Polytechnique Hauts-de-France, LAMIH, CNRS, UMR 8201, F-59313 Valenciennes, France</footnote>

<footnote><sup>b</sup> Univ Toulouse, CNRS, CerCo, Toulouse, France</footnote>

<footnote><sup>c</sup> ToNIC, Université de Toulouse, Inserm, Toulouse, France</footnote>

<footnote><sup>d</sup> Department of Clinical Physiological, Motion Analysis Center, Toulouse University Hospital, Toulouse, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: emilie.mathieu@uphf.fr (E. Mathieu).</footnote>

<footnote n="1">¹ These authors have contributed equally to this work and share last authorship.</footnote>
on integrating neural motor control strategies into musculoskeletal models.

To improve muscle effort estimation, electromyographic signals (EMG) are commonly used to drive [3,9], calibrate models [6] or track physiological data [10]. However, EMG is incorporated into these models as linear envelopes reflecting the level of muscle activation but not the underlying neural control mechanisms [11].

Overcoming this limitation requires consideration of motor control theories. The central nervous system generates rhythmic neural activity in the motor cortex that propagates to peripheral structures, facilitating efficient information transmission within a complex neural network [12]. To simplify the control of multiple effectors and to synchronize their activation, the nervous system generates common drives that are sent to synergistic muscles [13–16]. Some musculoskeletal model have turned to the concept of muscle synergy to estimate the activity of un-measured muscles [17,18]. In this context, Pizzolato et al. [19] devel-oped an OpenSim toolbox where muscle activations, derived by decomposing EMG signals into synergies, serve as inputs to musculo-skeletal model. Other researchers have integrated muscle synergies into simplified models establishing linear relationships between isometric force and EMG signals [20]. Such integration addresses multicollinearity issues caused by noise, crosstalk, and physiological factors inherent in EMG recordings. Synergy-based approaches enhance the consistency of EMG-to-force mapping across sessions conducted under identical con-ditions. However, they also show limitations when applied to varying experimental contexts.

While the inclusion of muscle synergies has improved EMG-to-force consistency and compensated for missing data, it remains limited. Muscle synergies are thought to reflect motor commands and are ob-tained by decomposing EMG signals to capture some invariant charac-teristics of muscle activities shared in the control of movements [15]. Therefore, different muscle synergies can recruit the same two muscles, leading into a division of the information. Interestingly, neural communication is thought to rely on the modulation of coherence, that is the frequency correlation between two signals [21]. Thus, the coherence between two EMG signals (IMC) allows to quantify in a single measure all the motor commands shared between the two muscles, to control of muscle force, movement amplitude or direction [16,22–24]. IMC is a measure of functional connectivity that reflects neurophysio-logical mechanisms underlying muscle synergy formation [25].

Significant IMC has been observed between agonist [24,26] and antagonist muscles [27,28] for both the upper [15] and lower limbs [13], across frequency bands thought to.

reflect distinct neural control mechanisms [23,29]. Low-frequency oscillations (<15 Hz) are generally attributed to subcortical structures, such as brainstem and spinal cord, and contribute to rhythmic and reflex motor control. In contrast, the beta (15–30 Hz) and gamma (30–80 Hz) band oscillations are primarily associated with cortical regions, partic-ularly the sensorimotor areas involved in voluntary and fine motor control [30–32]. Additionally, oscillations around 80 Hz have been recorded in the motor cortex during dynamic tasks [33]. Although the role of IMC above 60 Hz remains poorly understood, some evidence suggests it may support maximal force production with reduced energy cost, though its precise neural origin remains uncertain [34].

Numerous studies have shown that the IMC is sensitive to changes in force output [23,35,36], fatigue [37] and pathological conditions [38]. For example, Delcamp et al. [38] demonstrated increased IMC between antagonist elbow muscles in stroke patients exhibiting excessive co-contraction. They suggested that this increase in IMC between antago-nist elbow muscles could result from a reduced neural drive selectivity. Collectively, these findings support the idea that different neural structures communicate common motor commands via coherence to control muscle force [13,24]. By considering such time-frequency domain interactions, the interpretability of antagonist muscle activity could be improved, while simultaneously reducing the dimensionality of the modeling problem.

EMG-driven models and synergy-based models typically rely on forward solving strategies that require extensive anthropometric parameterization, including estimates of maximum force, force-length and force-velocity relationships, and moment arms [3]. These param-eters can vary significantly between individuals and are adjusted during a calibration process [7,39] that may involve spline regression [40,41]. A major limitation of these models is their sensitivity to initial parameter values, which can reduce the reliability of estimations [3,7,8]. For instance, the model proposed by [42] requires estimating 15 parameters per muscle. In contrast, model with inverse solving strategies, such as the EMG-driven model proposed by Amarantini & Martin [9], seek to reduce this dependency. Their model distributes the net joint torque among muscles based on EMG activity and includes parameters related to biarticularity, joint stiffness, and viscosity, estimated through nu-merical optimization. While reducing reliance on anthropometric pa-rameters and eliminating the need for prior calibration, this model still overlooks the common neural drive between muscles. However, its inverse-structure framework, driven directly by EMG signals, makes it well suited for the integration of IMC, potentially enhancing the robustness of estimations.

Our study aims to test the feasibility of incorporating IMC as a biomarker of neural strategy into a musculoskeletal model to refine muscle effort estimations.

Specifically, we aim to address two limitations of current EMG-driven models: (1) the underestimation of antagonist muscle contributions, and (2) the lack of robustness in estimations due to sensitivity to parameter initialization. Integrating data linking the efforts of synergistic muscles into a model requiring few anthropometric parameters could address these limitations.

The net joint torque, the estimation of muscle efforts and co-contraction index derived from estimations were obtained using data from a healthy population performing a repetitive elbow extension task and were compared to those from an EMG-driven model previously published [9]. We hypothesized that incorporating IMC would improve physiological realism by enhancing net joint torque tracking, increasing robustness to parameter variations, and yielding more accurate antagonist moment estimations, thus affecting the co-contraction index.

## 2. Methods

This study utilized data collected as part of previous investigations into neuromuscular adaptations in stroke patients [38,43]. This study included data from nine healthy subjects enrolled in a preliminary study (approved by the Research Ethical Committee of Toulouse University Hospitals No. 07-0716) and fifteen healthy subjects from an interventional protocol (No. ID-RCB: 2017-A01616-47).

### 2.1. Participants

The data of the twenty-four healthy subjects (51 ± 14 years) of the control groups were used in this study. All participants provided written informed consent prior to participation.

### 2.2. Protocol

The experimental procedure can be found in more detail in [38,43]. Briefly, it included 10 active self-paced full elbow extensions with the right upper limb at a stable distance from a table to achieve in-plane movement.

### 2.3. Material

Kinematic data were recorded at 125 Hz (system OptiTrack; NaturalPoint Inc., Corvallis, OR, USA). Reflective markers were placed on the acromia, the lateral epicondyle, ulnar styloid and metacarpus 2 (Fig. 1) of the observed arm.
<figure><img src="image_5.png" /><figcaption>Fig. 1. Experimental setup representation, with body segments and angles. The elbow extension was performed in the horizontal plane above a table and was modelled in two dimensions. The arrows indicate the position of the reflective markers: LACR for left acromion, RACR for right acromion, REL for right lateral epicondyle, RUS for right ulnar styloid and RMET2 for right metacarpus 2. Shoulder (θs), elbow (θe) and wrist (θw) angles were calculated in the horizontal plane.</figcaption></figure>

EMG data from the long head of the triceps brachii (TB), as well as the biceps brachii (BB), brachialis (BA) and brachioradialis (BR) were collected at 1000 Hz (Biopac Systems Inc., Goleta, CA, USA).

## 2.4. Data processing

All analyses were conducted with MATLAB (R2019b, 9.7.0, The MathWorks Inc, Natick, MA). Because the modelling approaches implemented do not require calibration, all recorded data were used for estimation purposes.

The following sections outline the data processing steps for 1) the EMG-driven model (EMG-model) based on [9] (section 2.4.2), and 2) the “IMC-model” integrating functional coupling in muscle pairs as assessed by IMC (section 2.4.3). As a first step, both models use inverse dynamics to determine the net joint torque.

### 2.4.1. Net joint torque calculation with inverse dynamics

Raw kinematic data were 6 Hz low-pass filtered (Butterworth, zero-lag, 4th order) [44]. For each trial, the elbow joint angle in the horizontal plane was computed from the Cartesian coordinates of the anatomical landmarks using a sign convention in which flexion is positive. Spline functions were fitted to the joint angle data using MATLAB's `csaps` function, and the first and second derivatives were computed to obtain angular velocity and acceleration. Elbow joint angle, velocity and acceleration data were normalized to 100 time points and averaged over the 10 trials. Net elbow joint torque ($M_{inv}$) was calculated using inverse dynamics with the Lagrange formalism [9]. As recommended by Rao et al. [45], moments of inertia, center of gravity positions and segment masses were calculated from Zatsiorsky's anthropometric table [46].

### 2.4.2. EMG-model

The proposed modelling approach is based on Amarantini and Martin's model [9], further developed by [47,48].

The input data for this model are the net joint torque and the envelopes of the EMG signals from the muscles actuating this joint. Each envelope is calculated from the entire EMG signal, which is filtered and rectified. This process is performed independently for each muscle. Finally, the aim of this model is to distribute the net joint torque into different muscle moments estimated by applying coefficients $α, β, δ$ and $ω$ to EMG envelopes.

Each muscle moment can be estimated by solving the following min/max optimization problem:

Find $α_i, ω_i(t), β = \{β_{wrist}, β_{elbow}, β_{shoulder}\}, δ = \{δ_{wrist}, δ_{elbow}, δ_{shoulder}\}$
Where $i, j = \{BA, BB, BR, TB\}$

That minimize $F_{obj}$,

$$F_{obj} = \max(M_{EMG\_i}(t)) \quad (1)$$

$$M_{EMG\_i}(t) = (α_i \bullet ω_i(t) \bullet EMG_i(t))^T (1 + E \bullet (β \bullet Δθ) - E \bullet (δ \bullet ˙θ)) \quad (2)$$

$$\text{And } M_{EMG}(t) = \sum M_{EMG\_i}(t) \quad (3)$$

with constraints:

$$\begin{cases}
\begin{array}{l}
α_{BA}, α_{BB}, α_{BR} \ge 0 \\
α_{TB} \le 0 \\
β, δ \ge 0
\end{array} \\
M_{EMG\_BA}(t), M_{EMG\_BB}(t), M_{EMG\_BR}(t) \ge 0; \text{ et } M_{EMG\_TB}(t) \le 0 \\
0 \le ω_{BA}(t), ω_{BB}(t), ω_{BR}(t), ω_{TB}(t) \le 1 \\
M_{inv}(t) = M_{EMG}(t) \pm ε
\end{cases}$$

$M_{inv}$ is the elbow net joint torque determined with inverse dynamics and $M_{EMG}$ is the estimated net joint torque using EMG. $α$ is the matrix of the EMG-moment relationship in a static condition and $ω(t)$ represents the matrix of individual muscle gains. The model also considers biarticularity (E), stiffness ($β$), and viscosity ($δ$). $Δθ(t)$ is the joint angle vector computed relative to half of the movement excursion at each joint and $˙θ(t)$ is the joint angular velocity vector. The nonlinear constraints impose a positive sign for the flexor muscle moments and a negative sign for the extensor muscle moment. Furthermore, $M_{inv}$ was tracked with a tolerance $ε$ set to 0.015. Matlab's `fminimax` function was used to solve the optimization problem. $α, β, δ$ and $ω$ were initially set to 15, 0.1, 0.1 and 0.1 respectively. To improve computation time and smoothness of $ω$ parameters, the optimization problem was solved on a reduced number of points (29 ± 12 points) corresponding to the times where first and second derivatives of $M_{inv}$ are null. The estimated moments were then interpolated over 100 points.

### 2.4.3. IMC-model

The new proposed model uses the same structure as the EMG-model, but the input data consider the functional coupling between the different muscles through IMC. The following sections and Fig. 2 describe the different steps of data processing to estimate muscle moments with the IMC-model. The initial step is to determine the input data for the model through the decomposition of each EMG signal into signals shared with other muscles, designated as “common signals,” and signals specific to each muscle, called “independent signals”.

### 2.4.4. “Common” and “independent” EMG signals

A muscle pair can be modeled as three actuators: two independent actuators reflected by EMG activation derived from independent drives and one actuator controlled commonly reflected by EMG activations derived from common drive (Fig. 3). From a kinetic point of view, this common actuator develops a mechanical action specifically associated with the common command. The production of a moment by two synergistic muscles can therefore be divided into 4 actuator moments: two independent moments produced by the two muscle actuators controlled
<figure><img src="image_6.png" /><figcaption>**Fig. 2.** Schematic representation of successive data processing steps for IMC-model. Kinematic data are required to calculate elbow net joint torques. EMG data undergo different processing and are split into common and independent commands to be integrated into the model with intermuscular coherence (IMC-model).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 3.** A. Schematic representation of the common drive theory [16]. B. Schematic representation of our proposal to group part of muscle activation controlled by common drive into a common actuator (in blue) as proposed in common drive theory. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

individually, one common moment produced by common actuator motor units belonging to muscle 1 and one common moment produced by common actuator motor units belonging to muscle 2.

To integrate these common actuators into an EMG-driven model, it was necessary to determine the corresponding EMG signals (Fig. 4). Given the significant relationship between IMC and force production

across a broad frequency range, the EMG signals were filtered to preserve the relevant frequency band used for IMC calculation (Butterworth low-pass filter, 80 Hz, zero-lag, 4th order). The autospectra of each EMG signal and IMC (see next section for IMC calculation) between each muscle pair were then computed.

The next step involved calculating the common signal autospectra by
<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Representation of the different steps to separate EMG signals into common and independent signals. First, the frequency content of the two EMG signals over time, i.e., the autospectra were determined. Secondly, the common drive proportion was estimated by calculating the magnitude of intermuscular coherence. Third, the spectrum of a common drive between two muscles was determined by multiplying the EMG autospectrum by intermuscular coherence. Similarly, it is possible to determine the spectrum of the independent drive by multiplying the EMG autospectrum by one minus intermuscular coherence. Finally, to obtain the EMG signals corresponding to these common and independent autospectra, an inverse transformation was carried out.</figcaption></figure>

multiplying the EMG autospectra by the IMC. Similarly, the independent signal autospectra were obtained by multiplying the EMG autospectra by 1 minus the IMC. An inverse transformation of the common and independent autospectra was then performed to retrieve the corresponding EMG signals. Finally, the envelopes of the common (EMG_com) and

independent (EMG_indep) EMG signals were calculated.

### 2.4.5. Intermuscular coherence

The calculation of IMC is an important step, as it enables the determination of the proportion of common drive received by each muscle

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Successive data processing steps to calculate intermuscular coherence with WavCrossSpec toolbox. The first row shows the mean EMG signal of the right BB and right TB muscles across 10 trials, depicted in the left and right figures respectively. The second row displays the autospectra of the EMG signals. The third row shows the cross-spectrum, which quantifies the common energy between the two EMG signals. The red-circled areas indicate where the cross-spectrum values are significant. The fourth row presents the time-frequency map of coherences. The final row shows the coherence within the regions where the cross-spectrum values are significant. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
pair. The steps involved in calculating IMC are illustrated in Fig. 5.

Since the use of a high-pass filter can limit EMG cancellation effect [49], raw EMG data were filtered using a 3 Hz high-pass Butterworth filter (zero-lag, 4th order) [27,50,51]. The CleanLineNoise function [52] was used to remove 50 Hz electrical noise. Filtered EMG signals were not rectified [27,38,49].

To account for the non-stationary aspect of EMG signals, IMC was computed in the time-frequency domain from a wavelet transformation using WaveCrossSpec toolbox [53]. The choice of wavelet parameters was made to offer the best compromise between temporal and frequency resolution. The WaveCrossSpec toolbox enables 3 parameters to be adjusted: the nvoice parameter, which concerns frequency resolution, the J1 parameter, which determines the frequency range over which the analysis is performed, and the wavenumber parameter, which defines the number of cycles of the mother wavelet. Most studies on IMC report significant IMC in frequency bands ranging from 5 to 60 Hz [29] and some up to 100 Hz [34]. IMC at frequencies above 60 Hz has been little explored, although oscillatory activity around 80 Hz is strongly present in the motor cortex during dynamic tasks. To avoid losing potentially relevant information, IMC values were retained up to 80 Hz [33,54]. As in [27], we set nvoice at 7, J1 at 50 and wavenumber at 10 to effectively detect the different frequency components of EMG signals. This parameterization enables analysis over a frequency band ranging from 0 to 80 Hz in steps of 0.22 Hz. A time normalization was conducted to quantify the IMC from trials of different lengths [55]. IMC magnitude was only quantified where significant interactions were identified on the cross-spectrum [53], reflecting the common drive received by each pair of muscles.

### 2.4.6. Muscle moment estimation

The input data for the IMC-model differs from that of the EMG-model. Rather than using the EMG envelopes calculated independently for each muscle, the IMC-model utilizes the common and independent signals, which are calculated from IMC.

As EMG-model, optimized parameters $\alpha$, $\beta$, $\delta$ and $\omega$ are associated with each common and independent signal to determine the common and independent moments while ensuring a good fit between the sum of these moments and the net joint torque determined with inverse dynamics.

$$ \text{And } M_{EMG\_com\_ij}(t) = (\alpha_{i-j} \bullet EMG\_com_{ij}(t))^T (1 + E \bullet (\beta \bullet \Delta\theta) - E \bullet (\delta \bullet \dot{\theta})) \quad (6) $$

$$ M_{EMG\_i}(t) = M_{EMG\_indep\_i}(t) + \sum M_{EMG\_com\_ij}(t) \quad (7) $$

$$ \text{And } M_{EMG}(t) = \sum M_{EMG\_i}(t) \quad (8) $$

with constraints:

$$ \left\{ \begin{array}{l} \alpha_{BA}, \alpha_{BB}, \alpha_{BR}, \alpha_{BA-j}, \alpha_{BB-j}, \alpha_{BR-j} \ge 0 \\ \alpha_{TB}, \alpha_{TB-j} \le 0 \\ \beta, \delta \ge 0 \\ M_{EMG\_indep\_BA}(t), M_{EMG\_indep\_BB}(t), M_{EMG\_indep\_BR}(t) \ge 0; et M_{EMG\_indep\_TB}(t) \le 0 \\ M_{EMG\_com\_BAJ}(t), M_{EMG\_com\_BBJ}(t), M_{EMG\_com\_BRJ}(t) \ge 0; et M_{EMG\_com\_TBJ}(t) \le 0 \\ 0 \le \omega_{BA}(t), \omega_{BB}(t), \omega_{BR}(t), \omega_{TB}(t) \le 1 \\ M_{inv}(t) = M_{EMG}(t) \pm \epsilon \end{array} \right. $$

$M_{EMG\_indep}$ represents the muscle moment produced due to independent signal and $M_{EMG\_com}$, the common actuator moment related to common signal. Each muscle moment is made up of one moment linked to the independent neural drive specific to that muscle and three moments linked to the common neural drive shared with the other muscles. The optimization procedure minimizes the maximum of all these force moments, while ensuring equality between the moment determined by inverse dynamics and the sum of the moments of the four muscles.

All the initial parameter values, constraints and number of optimized points have been kept identical to the EMG-model.

### 2.5. Statistical analysis

Parametric or non-parametric statistical mapping (SPM or SnPM) [56,57] was used to compare the results of IMC-model with those of EMG-model.

A SnPM paired t-test was used to compare the absolute difference between the net joint torque determined with inverse dynamics ($M_{inv}$) and the sum of estimated muscle moments ($M_{EMG}$) for IMC-model and EMG-model. We also compared the normalized root mean squared error for both models, calculated as follows:

$$ nRMS = \frac{\sqrt{\frac{1}{T} \int_{0}^{T} (M_{inv}(t) - M_{EMG}(t))^2 dt}}{\frac{1}{2} ((Max_{0<t<T} M_{inv}(t) - Min_{0<t<T} M_{inv}(t)) + (Max_{0<t<T} M_{EMG}(t) - Min_{0<t<T} M_{EMG}(t))) \times 100 \quad (9) $$

Thus, each muscle moment can be estimated by solving the following min/max optimization problem:

Find $\alpha_i$, $\alpha_{i-j}$, $\omega_i(t)$, $\beta = \{\beta_{wrist}, \beta_{elbow}, \beta_{shoulder}\}$, $\delta = \{\delta_{wrist}, \delta_{elbow}, \delta_{shoulder}\}$

Where i, j = {BA, BB, BR, TB}, i ≠ j.

That minimize $F_{obj}$,

$$ F_{obj} = \max(M_{EMG\_i}(t)) \quad (4) $$

This objective function is identical to that of the EMG-model, but the estimation of muscle moments is separated into moments related to common signals and moments related to independent signals.

$$ \text{With } M_{EMG\_indep\_i}(t) = (\alpha_i \bullet \omega_i(t) \bullet EMG\_indep\_i(t))^T (1 + E \bullet (\beta \bullet \Delta\theta) - E \bullet (\delta \bullet \dot{\theta})) \quad (5) $$

We also compared the cocontraction index (CCI), which represents the ratio between antagonist moments and the sum of agonist and antagonist moments. This index is commonly used in the biomechanical analysis of movement [58] and in rehabilitation [59] to estimate the contribution of antagonistic muscle to movement. A SPM paired t-test was conducted to compare the CCI computed with agonist and antagonist moment estimations, with EMG-model and IMC-Model. The CCI was computed at each individual time node using the method proposed by [60]:

$$ CCI = 2 \times \left( \frac{\text{meanantagonistmoment}}{\text{meanantagonistmoment} + \text{meanagonistmoment}} \right) \times 100 \quad (10) $$

The size of each cluster where the scalar output statistic, SnPM(t) or SPM(t), exceeds the critical threshold was used to calculate cluster p-values [61].

A SnPM repeated measures ANOVA (2 Models X 4 Muscles) was performed to evaluate net joint torque repartition into muscle moments.
SnPM post-hoc t-tests with Bonferroni correction for multiple comparisons were used when significant interactions were detected.

Hicks et al. [8] emphasize the importance of ensuring that model parameterization does not influence estimation. So, the estimation robustness to the variation of initial values of α and ω was assessed. Data from only 15 subjects were analyzed due to difficulties in achieving optimization convergence for the EMG-model. Initial values for α (10, 15, 30) and ω (0.1, 0.5, 0.9) were selected to ensure optimization convergence for the maximum number of subjects. Four repeated measures ANOVAs were conducted to compare muscle moments across the three initial conditions for α and ω separately, for each model. Since achieving optimization convergence with alternative values for β and δ was impossible for most subjects (more than 15), the estimation robustness depending on these parameters was not tested, assuming the initially chosen values were optimal.

All SPM and SnPM analyses were implemented using the open source spm1d code (v.0.4.8, https://www.spm1d.org) in MATLAB.

## 3. Results

As a first qualitative report, it is interesting to note that the optimization process converged for all subjects with the IMC-model but did not converge for 4 subjects for the EMG-model. These subjects were removed from all statistical analyses, except for IMC-model estimation robustness testing.

### 3.1. Net joint torque tracking

No difference was found between $M_{inv}$ and $M_{EMG}$ of both models (Fig. 6). The mean nRMS was $2.71 \pm 1.11\%$ for both models, without a significant difference between EMG-model and IMC-model ($p > 0.05$).

### 3.2. Estimation robustness

For the EMG-model, the SnPM ANOVA revealed significant interactions between α's initial values and muscle moment estimations at ~54–60 % and ~94–100 % ($p < 0.001$ for all clusters) of the extension

movement. Similarly, significant interactions between ω during initial conditions and muscle moment estimations were revealed at ~0–92 % and ~95–100 % ($p < 0.001$ for all clusters) of the extension movement.

For the IMC-model, no significant interaction was found between α and ω initial values and muscle moments estimations.

### 3.3. Estimation of muscle moments

The SnPM ANOVA revealed a significant interaction between models and muscle moment estimations at ~2–41 %, ~47–84 % and ~96–100 % of the extension movement ($p = 0.005$ for all clusters). Post-hoc analysis revealed that the BB moment estimation was significantly increased with the IMC-model at ~11–37 % ($p < 0.001$) and ~55–74 % ($p < 0.001$) of the extension movement (Fig. 7B). Likewise, the BR moment estimation was significantly increased with the IMC-model at ~7–25 % ($p < 0.001$), ~69–75 % ($p = 0.006$) and ~97–100 % ($p = 0.006$) of the extension movement (Fig. 7C). No significant difference was revealed for BA muscle moment (Fig. 7A).

Finally, agonist moment estimation was significantly increased with the IMC-model at ~14–16 % ($p = 0.005$), ~59–74 % ($p < 0.001$) and ~98–100 % ($p = 0.005$) of the extension movement (Fig. 7D).

### 3.4. Estimation of cocontraction index

The cocontraction index was significantly higher at ~7–37 % ($p < 0.001$) and lower at ~77–83 % ($p < 0.001$) and ~97–100 % ($p = 0.02$) of the extension movement (Fig. 8) for the IMC-model compared with the EMG-model.

This lower cocontraction with the EMG-model at the beginning of the movement is particularly visible in muscle moment estimations for 14 subjects. As an example (Fig. 9), BB and BR moments estimated with the EMG-model are null for more than half of the movement. With the IMC-model, none of the three antagonist muscle moments are null from the beginning of the movement and increase with a peak during deceleration phase (Fig. 9A).

<figure><img src="image_10.png" /><figcaption>**Fig. 6. Net joint torque tracking.** Mean and standard deviation of $M_{inv}$ (green solid line) and $M_{EMG}$ estimated with the IMC-model (grey dotted line; left panel) and the EMG model (red dotted line; right panel). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 7. Muscle moment estimations.** Muscle moments (mean ± sd) estimated with the IMC-model (grey solid line) and EMG-model (orange dotted line). A. BA muscle moment, B. BB muscle moment, C. BR muscle moment, D. TB muscle moment. Statistically significant differences between models are represented with grey bars on the top of each figure. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 8. Cocontraction index.** Mean and standard deviation of the cocontraction index computed with the IMC-model (grey solid line) and EMG-model (orange dotted line). Statistically significant differences between models are represented with grey bars on the top of the figure. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

## 4. Discussion

This study aimed to develop and evaluate the effectiveness of a neuro-musculoskeletal model integrating neural motor control strategies through IMC. EMG signals were decomposed into common and independent signals and used as input into an EMG-driven model to

estimate muscle moments acting around the elbow. In comparison to the previous EMG-driven model, the estimation robustness was improved, and the antagonist moments were more realistically modeled with the IMC-assisted model, which consequently adjusted the values of the co-contraction index.

A major difficulty encountered in musculoskeletal modeling to
<figure><img src="image_13.png" /><figcaption>Fig. 9. Agonist moment (blue), antagonist moment (green), and net joint torque (black) estimations for one subject with the IMC-model (A) and EMG-model (B). Individual BA (Dash-dotted lines), BB (dotted lines), and BR (dashed line) moments are also depicted. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

estimate muscle effort is the influence of the initial parameter values, such as maximal muscle forces, maximal muscle stress or initial values of optimized parameters [8]. This difficulty was particularly pronounced for the muscle moments estimated with the EMG-model. In both models compared, the optimization procedure is constrained only by the sign convention (corresponding to the direction of muscle action) and the net joint torque tracking. Despite this high degree of freedom, the muscle moments estimated with the IMC-model showed no significant influence from the initial parameter values. Previous studies have shown that integrating neural strategies into a musculoskeletal model improves the estimation of muscle moments [3,19,62,63]. Our results reveal that accounting for neural strategies enabling the functional coupling of muscles, through the decomposition of EMG into common and independent signals, improves estimation robustness with respect to model parameterization. In both models, the parameters α and ω likely exert a strong influence on the estimation. The parameter α is a fixed multiplicative coefficient that proportionally scales the estimated moment throughout the entire movement – higher α values result in higher estimated moments for the same EMG amplitude. In contrast, ω is a time-varying coefficient, bounded between 0 and 1, that modulates the temporal variations of the estimated moment. The higher estimation robustness to initial parameters in the IMC-model can be explained by the introduction of links between all muscles through common signals which reduces problem dimensionality. While the optimization of the EMG-model is free to distribute the antagonist moment to any of the 3 elbow flexors, the integration of common and independent signals limits the number of optimal solutions. This approach enables the model to converge towards a single solution regardless of initialization while ensuring an optimal solution for all subjects.

Due to the optimization constraints implemented on net joint torque tracking, both models give an equivalent fitting of the torque determined with inverse dynamics. However, the EMG-model failed to converge for four subjects. This lack of convergence likely stems from the inability to satisfy the strict constraints related to net joint torque tracking. Conversely, the IMC-model converged for all participants further supporting its use as a more reliable approach for accurate net joint torque tracking.

Previous models often neglected antagonist muscles [64]. By incorporating EMG signals into these models, researchers have significantly improved antagonist muscle moment estimation [9,48]. However, the optimization procedure remained free to set the parameters α and/or ω to zero, potentially neglecting antagonist muscle actions entirely. For some subjects, the optimization procedure of the EMG-model resulted in non-significant BB and BR moments for more than half of the extension movement (Fig. 9). This outcome is not physiologically realistic because none of the 3 elbow flexors is completely inactive during elbow extension [6]. Using independent and common EMG signals determined with IMC appropriately modifies the distribution of net joint torque into muscle moments. The estimations obtained with the IMC-model indicate a significant contribution of each antagonist muscle over the entire movement with more pronounced involvement at the end of the extension. This latter finding agrees with work from [6] and supports the IMC-model’s approach of considering neural strategies through common and independent drives to obtain physiologically realistic muscle moment estimations.

The lower cocontraction index observed with the EMG-model in the first half of the movement is linked to the underestimation of certain antagonist moments. Antagonist muscles play significant roles in joint stabilization and protection, and in movement deceleration [65,66]. Therefore, accurately estimating their contribution is important for comprehensive biomechanical analysis. Although previous research has improved cocontraction estimation by adding a cocontraction parameter representing muscle cooperation [6], this parameter was not derived directly from physiological data. The authors acknowledged that such an approach might be insufficient to account for inter-individual differences. In contrast, IMC provides direct information on common and individual neural drives from neural structures involved in muscle contractions. By integrating common EMG signals to link the force production of each muscle involved in the motor action, our approach enables more comprehensive consideration of all activated muscles.

This feasibility study demonstrates that incorporating IMC into modeling enhances both the realism and robustness of estimates; however, it also presents certain limitations that should be addressed in
future studies. In this study, both models were compared using identical experimental data, thereby minimizing the influence of uncertainties related to EMG acquisition and inverse dynamics computation on our overall conclusions. Additionally, we adopted a simplified approach to integrate IMC by assuming that common neural drives were shared only between pairs of muscles, while disregarding more globally distributed inputs. A more comprehensive integration of neural strategies would require accounting for shared commands across all muscles involved. Furthermore, we chose to integrate IMC over a broad frequency range (0–80 Hz), considering the oscillatory activity observed in cortical areas involved in motor control. Future work could focus on narrowing the frequency band by specifically targeting frequencies associated with motor unit synchronization. While oscillations in the beta frequency band (15–30 Hz) are generally associated with the control of isometric contractions [32] and show a strong correspondence between cortical activity and motor unit synchronization [67], an IMC-model focusing on this frequency range could provide further insights into the role of these oscillations in force production control during dynamic movements. Interestingly, [68] suggested that gamma-band oscillations (30–60 Hz) are more frequently observed during dynamic motor tasks and proposed that they may serve as a mechanism for rapidly generating motor commands in such contexts. An IMC-model incorporating only gamma-band IMC could enable a more detailed exploration of the role of these oscillations in various dynamic tasks, for example in the context of motor recovery after stroke [69]. Analyzing these different frequency bands could also facilitate a more nuanced investigation of how specific frequency ranges influence muscle joint moment estimations.

To advance the methodological development of the IMC- model, machine learning approaches could serve as alternatives to numerical optimization methods, potentially improving the efficiency and perfor- mance of the search for optimal solutions [70,71]. Additionally, we chose to use the method proposed by [53] to detect significant IMC values, although alternative approaches, such as surrogate analysis, could also be employed [72,73].

Since direct measurement of muscle moments at the elbow joint is not feasible in vivo, further studies are needed to confirm the accuracy of cocontraction estimation using the IMC-model and to evaluate the effectiveness of integrating common and independent EMG signals into other modeling approaches that already incorporate neural strategies such as those involving muscle synergies [19].

## 5. Conclusion

Finally, we integrated EMG signals separated into common and in- dependent signals with the information provided by IMC into a neuro- musculoskeletal model to better estimate individual muscle moments. Our results revealed that the net joint torque distribution into different muscle moments is realistic while ensuring higher estimation robustness to model parameterization compared to a previously published EMG- driven model. This procedure can be used in any musculoskeletal model using EMG signals, regardless of the solving strategy and joint of interest.

## CRediT authorship contribution statement

**Emilie Mathieu:** Writing – review & editing, Writing – original draft, Methodology, Data curation, Conceptualization. **Sylvain Crémoux:** Writing – review & editing, Methodology, Conceptualization. **David Gasq:** Writing – review & editing, Supervision, Project administration, Investigation. **Philippe Pudlo:** Writing – review & editing, Supervision, Methodology, Conceptualization. **David Amarantini:** Writing – review & editing, Supervision, Methodology, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial

interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

The authors would like to acknowledge native English-speaking and biomechanics specialist Julia Greenfield for her assistance in English proofreading during the writing process of the manuscript. The authors would also like to thank Célia Delcamp and Camille Cormier for their valuable contribution in collecting and managing the data.

## Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.bspc.2025.108004.

## Data availability

Data will be made available on request.

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