Synergy Coordination Index (SCI)

• Synergy Coordination Index (SCI) is an EMG-informed measure that quantifies lower-limb muscle coordination by averaging cosine similarities of NNMF-extracted, L2-normalized synergy vectors.
• A higher SCI indicates greater similarity among muscle synergies, reflecting a smaller synergy space and more efficient neuromuscular control during cycling.
• SCI is computed via non-negative matrix factorization on normalized EMG data, and it is integrated with SI and CI to provide both global and joint-specific insights.

The Synergy Coordination Index (SCI) is an electromyography-informed measure of lower-limb muscle coordination during cycling, defined in the study “Quantifying lower-limb muscle coordination during cycling using electromyography-informed muscle synergies” (Ahmadi et al., 25 Jul 2025). In that work, SCI is computed from non-negative matrix factorization (NNMF) synergy vectors and is interpreted as a measure of the size of the synergy space: higher SCI indicates greater cosine similarity among synergy vectors, a smaller synergy space, and, in the paper’s interpretation, more coordinated and economical neuromuscular control. The index was evaluated together with the Synergy Index (SI) and Coactivation Index (CI) across three cycling power levels in twenty recreational cyclists, with SCI increasing significantly as power increased (Ahmadi et al., 25 Jul 2025).

1. Formal definition and conceptual meaning

In the study, the preprocessed and normalized EMG matrix is denoted by $X \in \mathbb{R}^{m \times n}$, where $m$ is the number of muscles and $n$ is the number of samples. NNMF is used to approximate the EMG as

$X \approx WH,$

with $W \in \mathbb{R}^{m \times k}$ the synergy weight matrix and $H \in \mathbb{R}^{k \times n}$ the synergy activation matrix. The reconstruction is $\hat{X} = WH$, and the reconstruction error is $E = X - \hat{X}$ (Ahmadi et al., 25 Jul 2025).

For SCI, each synergy vector $W^{(i)}$ is first $L_2$-normalized,

$m$0

and the index is then defined as the average pairwise inner product over ordered pairs of distinct synergy vectors:

$m$1

where $m$2 is the number of $m$3-permutations of $m$4 elements (Ahmadi et al., 25 Jul 2025).

The paper gives the range and interpretation explicitly. SCI satisfies $m$5. An SCI of $m$6 means that all synergies are identical, so the synergy space collapses to a line and has minimal dimensionality. An SCI of $m$7 means that the synergies are mutually orthogonal, corresponding to maximal synergy space size and highest dimensionality. Larger SCI therefore indicates a smaller synergy space and, in the paper’s interpretation, more coordinated and economical neuromuscular control (Ahmadi et al., 25 Jul 2025).

A common misunderstanding is to read higher SCI as evidence of greater diversity among synergies. The definition implies the opposite: higher SCI means the column vectors of $m$8 are more similar, not more distinct. The study therefore treats SCI as a compactness measure of modular organization rather than a dispersion measure.

2. Relation to SI and CI

The study does not use SCI in isolation. It combines SCI with the Synergy Index (SI) and Coactivation Index (CI), which provide joint-specific information that complements the global geometry captured by SCI (Ahmadi et al., 25 Jul 2025).

Index	Exact basis	Interpretation
SCI	Average cosine similarity of the $m$9-normalized columns of $n$0 over ordered pairs	Higher SCI indicates a smaller synergy space
SI	Joint-level flexor-to-total ratio based on maximal synergy weights	$n$1 indicates flexor or dorsiflexor dominance
CI	Joint-level flexor-to-total ratio based on integrated EMG over predefined crank-angle intervals	$n$2 indicates flexor or dorsiflexor predominance

SI is defined from per-muscle maximal synergy contribution. For each muscle $n$3, the largest weight across synergies is taken:

$n$4

Joint-specific SI is then defined as a flexor-to-total ratio. For the hip, flexors are rectus femoris (RF) and extensors are biceps femoris long head (BF), giving

$n$5

For the knee, flexors are BF and gastrocnemius medialis (GM), and extensors are vastus medialis (VM), rectus femoris (RF), and vastus lateralis (VL):

$n$6

For the ankle, tibialis anterior (TA) is the dorsiflexor and GM and soleus (SOL) are plantarflexors:

$n$7

The interpretation is direct: $n$8 indicates flexor or dorsiflexor dominance, and $n$9 indicates extensor or plantarflexor dominance (Ahmadi et al., 25 Jul 2025).

CI is computed from integrated EMG over crank-angle intervals with high mechanical demand, defined as at least $X \approx WH,$0 peak net joint moment. The intervals used are hip $X \approx WH,$1, knee $X \approx WH,$2, and ankle $X \approx WH,$3. For each muscle $X \approx WH,$4,

$X \approx WH,$5

Joint-level CI is then defined as

$X \approx WH,$6

$X \approx WH,$7

$X \approx WH,$8

The paper emphasizes that CI here is not pairwise by muscle; it aggregates across flexor versus extensor sets per joint and integrates EMG over task-critical crank-angle ranges (Ahmadi et al., 25 Jul 2025).

Taken together, the three indices separate different aspects of coordination. SCI summarizes the size of the synergy space across the full muscle set, whereas SI and CI localize flexor-extensor balance and coactivation at specific joints. This suggests a multiscale description in which SCI captures global modular contraction while SI and CI identify the joint-level structure of that contraction.

3. Computational pipeline from EMG to SCI

The end-to-end workflow begins with EMG acquisition during a graded cycling test. Twenty recreational cyclists performed the protocol on an Echelon Connect EX-5 stationary trainer. The cadence and power protocol consisted of a warm-up at $X \approx WH,$9 W, $W \in \mathbb{R}^{m \times k}$0 rpm, for $W \in \mathbb{R}^{m \times k}$1 min, followed by stepwise increments of $W \in \mathbb{R}^{m \times k}$2 W each minute starting at $W \in \mathbb{R}^{m \times k}$3 W until task failure or voluntary stop. EMG was recorded bilaterally from seven lower-limb muscles—VM, RF, VL, BF, TA, GM, and SOL—using a Delsys Trigno wireless system sampled at $W \in \mathbb{R}^{m \times k}$4 kHz, with an RLS LM13 crank-angle encoder sampled at $W \in \mathbb{R}^{m \times k}$5 Hz for cycle normalization. The muscle set therefore comprised $W \in \mathbb{R}^{m \times k}$6 channels when both limbs were included (Ahmadi et al., 25 Jul 2025).

Preprocessing was standardized. At each power level, the first and last $W \in \mathbb{R}^{m \times k}$7 of cycles were discarded and the remaining $W \in \mathbb{R}^{m \times k}$8 analyzed. Dynamic MVC normalization was applied per muscle, defined as the mean of the highest $W \in \mathbb{R}^{m \times k}$9 EMG magnitudes across the test:

$H \in \mathbb{R}^{k \times n}$0

with normalized EMG

$H \in \mathbb{R}^{k \times n}$1

Filtering consisted of a $H \in \mathbb{R}^{k \times n}$2 Hz zero-lag fifth-order Butterworth high-pass filter, detrending, full-wave rectification, and a $H \in \mathbb{R}^{k \times n}$3 Hz zero-lag fourth-order Butterworth low-pass filter for the envelope. Data were downsampled from $H \in \mathbb{R}^{k \times n}$4 kHz to $H \in \mathbb{R}^{k \times n}$5 Hz. EMG envelopes were averaged over approximately $H \in \mathbb{R}^{k \times n}$6 cycles per power level aligned to crank angle with top dead center at $H \in \mathbb{R}^{k \times n}$7, and then unit-variance scaling was applied per muscle,

$H \in \mathbb{R}^{k \times n}$8

to ensure equal weighting across muscles in NNMF (Ahmadi et al., 25 Jul 2025).

Synergy extraction used NNMF with the model $H \in \mathbb{R}^{k \times n}$9, subject to $\hat{X} = WH$0 and $\hat{X} = WH$1. Two algorithms were used: multiplicative update with $\hat{X} = WH$2 iterations, $\hat{X} = WH$3, repeated $\hat{X} = WH$4 times for robust initialization, and alternating least squares (ALS) with $\hat{X} = WH$5 iterations and $\hat{X} = WH$6 repetitions to refine factors. Rank selection evaluated $\hat{X} = WH$7 using

$\hat{X} = WH$8

and

$\hat{X} = WH$9

with the smallest $E = X - \hat{X}$0 chosen such that $E = X - \hat{X}$1 and $E = X - \hat{X}$2 (Ahmadi et al., 25 Jul 2025).

In the study, $E = X - \hat{X}$3 was the minimum satisfying the criteria across power levels. The reported reconstruction exceeded $E = X - \hat{X}$4 and achieved $E = X - \hat{X}$5, $E = X - \hat{X}$6, satisfying the “or” condition. Synergies were labeled across participants and power levels by K-means clustering on columns of $E = X - \hat{X}$7, repeated $E = X - \hat{X}$8 times with random initial centroids and retaining the most frequent solution. Participant data for a cluster were excluded if two synergies mapped to the same cluster or if a synergy’s correlation with the cluster mean was below $E = X - \hat{X}$9, per $W^{(i)}$0 for $W^{(i)}$1. Bootstrapped $W^{(i)}$2 confidence intervals on the muscle weights in $W^{(i)}$3 were used to determine whether a muscle was a significant contributor to a synergy (Ahmadi et al., 25 Jul 2025).

After NNMF, SCI is computed directly from the normalized columns of $W^{(i)}$4. The paper’s worked example takes six pairwise cosine similarities among four normalized synergies, sums them over unordered pairs to $W^{(i)}$5, doubles to $W^{(i)}$6 for ordered pairs, and divides by $W^{(i)}$7, yielding $W^{(i)}$8. The example is presented as consistent with the higher-power range reported in the data (Ahmadi et al., 25 Jul 2025).

4. Empirical behavior across cycling power levels

Four muscle synergies were identified consistently across power levels, with changes in synergy composition and activation timing correlated with increased muscular demands. SCI increased with power level: low power level (LPL) $W^{(i)}$9, moderate power level (MPL) $L_2$0, and high power level (HPL) $L_2$1. One-way repeated-measures ANOVA revealed significant differences among all three power levels, and paired $L_2$2-tests showed significant differences across all pairs, with $L_2$3 (Ahmadi et al., 25 Jul 2025).

The paper interprets this increase according to the SCI definition. Higher SCI means greater cosine similarity among synergy vectors and a smaller synergy space. The authors therefore interpret the observed increase as improved neuromuscular coordination and a more focused, efficient control strategy at higher mechanical demand (Ahmadi et al., 25 Jul 2025).

The SCI findings were accompanied by systematic changes in SI and CI. At the knee, $L_2$4 decreased from flexor-dominant at LPL to extensor-dominant at higher power, with dominant-knee values reported as $L_2$5. The study links this shift to greater reliance on quadriceps during downstroke at high workloads, consistent with a more focused synergy space. At the ankle, $L_2$6 increased with power, with non-dominant values $L_2$7, indicating increasing dorsiflexor contribution to stabilize the ankle while plantarflexors generate power. At the hip, $L_2$8 remained near $L_2$9, indicating balanced flexor-extensor synergy contributions (Ahmadi et al., 25 Jul 2025).

CI showed a different but complementary pattern. Knee CI decreased with power, with non-dominant values $m$00, indicating reduced antagonist activity and more efficient torque production. Ankle CI increased with power, with non-dominant values $m$01, reflecting joint stabilization via co-contraction at high load. Hip CI remained relatively constant, although asymmetry emerged at higher loads (Ahmadi et al., 25 Jul 2025).

Representative synergy descriptions further contextualize the SCI increase. Four synergies were consistently matched across power levels: Synergy 1 in the ED phase, Synergy 2 in the LD phase, Synergy 3 in the EU phase, and Synergy 4 in the LU phase. The paper reports modest timing advance for Synergy 2 from LPL to HPL, with phase shift $m$02, $m$03, and marked timing advance for Synergy 4, with phase shift $m$04, $m$05. Activation profile correlations across power levels remained high but decreased with load; for Synergy 4, the correlation was $m$06 for LPL to MPL and $m$07 for LPL to HPL. These changes were presented as consistent with activation adjustments coinciding with increased SCI (Ahmadi et al., 25 Jul 2025).

5. Reliability, limitations, and implementation considerations

The study includes several reliability and consistency checks. Rank selection used $m$08 and local criteria $m$09 to ensure reconstruction fidelity. K-means clustering was repeated $m$10 times, exclusions were applied for low cluster correlation or duplicated cluster assignments, and bootstrapped $m$11 confidence intervals on $m$12 weights were used to confirm significant muscle contributors within each synergy (Ahmadi et al., 25 Jul 2025).

Sensitivity considerations were also reported. The number of synergies was chosen via VAF-based criteria, with the authors reporting robust reconstruction using $m$13 across all power levels. Preprocessing choices—filters, MVC normalization, and unit-variance scaling—were standardized and consistent across subjects and conditions to ensure comparability. The paper states explicitly that unit-variance scaling prevents dominance by high-SNR muscles during NNMF. Limb asymmetries were present in SI and CI, particularly at the knee and ankle, whereas SCI was computed on the full $m$14-muscle set across both limbs at each power level (Ahmadi et al., 25 Jul 2025).

Several limitations and potential confounders are noted. Cadence was controlled at a $m$15 rpm target, but real-world variability was absent and the indoor ergometer context may not generalize to outdoor cycling. Surface EMG was limited to superficial muscles, key muscles such as gluteus maximus were not recorded, and crosstalk together with EMG-to-force nonlinearity limits direct mechanical inference. No concurrent kinematics or kinetics were collected, so the indices were derived purely from EMG; the authors state that future work should integrate motion, force, and musculoskeletal modeling. The cohort consisted of young recreational cyclists, and fatigue together with bilateral asymmetries may influence synergy structure and CI or SI, especially at high power (Ahmadi et al., 25 Jul 2025).

For implementation, the paper recommends standardizing preprocessing and time-normalization to crank angle, using unit-variance scaling per channel before NNMF, applying VAF-based rank selection, and verifying stability via repeated NNMF initializations and clustering. It also recommends reporting SCI together with confidence intervals across sessions or participants if possible, for example by bootstrapping $m$16 columns, and matching muscle sets and preprocessing choices when comparing cohorts or conditions (Ahmadi et al., 25 Jul 2025).

A plausible implication is that SCI is most informative when treated as one component of a coordinated analysis protocol rather than as a standalone scalar. The study’s own design supports this reading, because SCI is interpreted alongside SI, CI, clustering consistency, and reconstruction-fidelity diagnostics.

6. Terminological scope and distinctions from other uses of “SCI”

In the cycling study, SCI unequivocally denotes the Synergy Coordination Index and is defined as the average cosine similarity of the $m$17-normalized synergy vectors extracted from NNMF (Ahmadi et al., 25 Jul 2025). The paper further states that this SCI follows Alnajjar et al. (2013) and Wojtara et al. (2014) in form and that no subspace-angle or eigenvalue-based variants are used in the study. This is important because “SCI” is not a unique acronym across the literature.

One source of possible confusion is “A Measure of Synergy in Coalitions” (Rahwan et al., 2014). That paper develops a formal measure of synergy in characteristic-function cooperative games, denoted by the synergy value $m$18, defined as deviation from an average-impact baseline built from average Shapley values. The paper explicitly does not use or define the term “Synergy Coordination Index (SCI)” (Rahwan et al., 2014).

A second source of ambiguity is “The Signal Credibility Index for Prediction Markets” (Nechepurenko, 29 Apr 2026). In that paper, SCI stands for Signal Credibility Index and is defined through persistence, two-sidedness, and flow-breadth components in a microstructure setting. The paper states directly that this usage is unrelated to any “Synergy Coordination Index” appearing in other contexts and should not be conflated with the EMG-based SCI used in cycling (Nechepurenko, 29 Apr 2026).

These distinctions matter because the acronym SCI spans at least three different conceptual domains: EMG-informed muscle-synergy geometry in cycling, coalition performance deviation in cooperative game theory, and coordination credibility in prediction-market microstructure. Only the first of these corresponds to the Synergy Coordination Index as formally defined in the cycling study (Ahmadi et al., 25 Jul 2025).