Assessing neural connectivity and associated time delays of muscle responses to continuous position perturbations

Abstract

Both linear and nonlinear electromyographic (EMG) connectivity has been reported during the expression of stretch reflexes, though it is not clear whether they are generated by the same neural pathways. To answer this question, we aim to distinguish linear and nonlinear connectivity, as well as their delays in muscle responses, resulting from continuous elbow joint perturbations. We recorded EMG from Biceps Brachii muscle when eight able-bodied participants were performing a steady elbow flexion torque while simultaneously receiving a continuous position perturbation. Using a recently developed phase coupling metric, we estimated linear and nonlinear connectivity as well as their associated delays between Biceps EMG responses and perturbations. We found that the time delay for linear connectivity (24.5 ± 5.4 ms) is in the range of short-latency stretch reflex period (< 35 ms), while that for nonlinear connectivity (53.8 ± 3.2 ms) is in the range of long-latency stretch reflex period (40–70 ms). These results suggest that the estimated linear connectivity between EMG and perturbations is very likely generated by the mono-synaptic spinal stretch reflex loop, while the nonlinear connectivity may be associated with multi-synaptic supraspinal stretch reflex loops. As such, this study provides new evidence of the nature of neural connectivity related to the stretch reflex.

3. Introduction

Stretch reflexes are involuntary reactions and provide muscle activity with various latencies in response to unexpected mechanical perturbations. The short-latency component of the stretch reflex is governed by a mono-synaptic spinal loop with a latency of less than 35 ms^{17, 19}. Long latency (40–70 ms) responses are thought to be mediated by the supraspinal loops of the stretch reflex which involves multiple synaptic connections^{6, 19}. Although the refractory period of the motor neurons may cause a motor response in the long-latency response window based on a model simulation²⁶, numerous experimental studies reported evidence of supraspinal contributions to the long-latency response, as discussed in a recent review article¹².

A common way to investigate the stretch reflex is to apply a mechanical perturbation to the participant’s joint and measure the responses^{10, 11}. Many studies used transient perturbations with step or ramp perturbations^{5, 7, 9}. The use of transient perturbations requires multiple repetitions to improve the signal to noise ratio and extract the phase-locked responses. Alternatively, several studies used continuous but unpredictable perturbation signals, such as multi-sine perturbations, which is the sum of multiple random phase sinusoids, for the estimation of reflex gains from the mechanical admittance^23–25. Compared to transient perturbations, a multi-sine signal contains a finite number of sinusoidal components distributed in the frequency range of interest and therefore improves the signal-to-noise ratio for analyzing properties of a biological system in the frequency domain²³. Previous studies mainly assessed linear input-output relations of the short-latency mono-synaptic spinal loop of stretch reflex between mechanical joint perturbations and muscle responses (i.e. EMG signals)²⁶. Conversely, supraspinal loops could be highly nonlinear due to the multi-synaptic nature of the involved neural circuitry, as indicated by a recent review article³² and subsequently explored by a neural simulation model²⁷.

This study aims to assess linear versus nonlinear connectivity and their respective delays in the human stretch reflex in response to continuous elbow joint perturbations using a well-designed multi-sine perturbation signal and the proposed nonlinear connectivity analysis method³⁷. Previous neural simulation and in vivo studies indicate that the neural information can be transmitted in an approximately linear way (i.e. the output at the same frequency band as the input) to the output of the spinal motoneuron pool in a mono-synaptic motor pathway¹⁸, even though the behavior of each neuron itself is highly nonlinear⁸. However, in a multi-synaptic pathway, the nonlinear distortion from each synaptic connection can be progressively cumulated across multiple synaptic connection²⁷. Thus, we hypothesize that linear perturbation-EMG connectivity during the stretch reflex is originated from the mono-synaptic spinal loop, while the nonlinear connectivity is generated by multi-synaptic supraspinal loops. The assessment of time delays for linear vs. nonlinear connectivity in this study will provide us a way to test our hypothesis, since the muscle response from the mono-synaptic spinal loop (also known as the short-latency component of the stretch reflex) has a much shorter delay than that from multi-synaptic supraspinal loops (also known as the long-latency component of the stretch reflex).

4. Materials and Methods

4.1. Human Subject Experiment and Data Collection

We included eight able-bodied middle-to-old age (55–75 years old) participants (5 women) in this study. The Northwestern University Institutional Review Broad (IRB) approved the human subject protocol (study no.: STU00021840), and all participants provided informed consent prior to their participation in the study.

A custom robotic device³⁰ was used for the experiment. The device was connected to a Biodex pedestal attached to a Biodex track (System 3 Pro™; Shirley, NY, USA). The device contained a forearm linkage to support the weight of the participant’s arm and was instrumented with a 6-DOF load cell (JR3, Woodland, CA) to simultaneously measure the shoulder abduction (SABD) and elbow flexion (EF) torques generated by the participant. The robotic device was driven by an FHA-C rotary actuator (Harmonic Drive, Peabody, MA) to administer position perturbations to the participants’ elbow joint to induce stretch reflex responses.

Participants were seated in a Biodex chair with their trunk secured by belt restraints across the shoulders and lap. A fiberglass forearm-wrist-hand cast was made for each participant that allowed to rigidly attach the left forearm (the tested arm) to a customized beam connected to the JR3 6-DOF loadcell. The medial epicondyle of the humerus was aligned with the center of rotation of the actuator. The upper limb was positioned with 85° shoulder abduction, 45° shoulder flexion, 0° shoulder rotation, and 90° elbow flexion. The experimental setup is shown in Fig. 1a. Visual feedback was provided through a screen positioned in front of the participant as shown in Fig. 1b. Throughout the experiment, the controlling object was a red circle for elbow flexion (EF). The target EF level was indicated by a deep blue circle and tolerance thresholds (20% above and below the actual targets) were indicated by cyan circles. Appropriate audio cues using a pre-recorded voice were also provided to guide the participant with the tasks. The voice commands were pre-programmed to synchronize with the visual feedback appropriately. All study participants were asked and encouraged to meet the target. Based on our observation during the experiments and post experimental analysis, all the subjects were trying hard to stay at the target torque instead of only staying in the target zone.

Figure 1.

Experimental setup (a) and visual feedback (b).

Before the actual experiment, maximum voluntary torque (MVT) of elbow flexion was measured for each participant. The participant was asked to generate maximum elbow flexion (‘pull in’) for 5 seconds. Peak torque value was obtained from the data after filtering it with a moving median filter (250 ms window, 100 % overlap) online. Measurement of MVT was repeated until three consecutive maximum torques were within 5% of each other. The average of the last three trials is recorded as the final value of MVT. In the actual experiment, the participants were required to perform an elbow flexion torque of 22% MVT based on online visual feedback and hold there. Several training trials were conducted to help the participants understand the task and to maintain a constant elbow flexion torque based on the visual feedback. A previous study showed that the excitability of the corticospinal tract at this contraction level would ensure detectable supraspinal components of the stretch reflex when the elbow joint was perturbed¹⁶. At the beginning of each trial, once the participant achieved the desired level of EF, the voice command asked the participant to ‘HOLD’ at that position. Subsequently, the perturbations (i.e. the actual ‘trial’) began only if the participant was able to ‘HOLD’ at the desired EF and/or SABD level continuously for 2s. In case of failure, the participant was asked to relax and repeat the same task. After a 2 s stable holding period, participants were receiving small amplitude (peak-to-peak value 0.1 radians) continuous position perturbations at the elbow joint of their test arm while participants were instructed to maintain the same elbow flexion effort. The perturbations were given as a sum of 10 zero-centered sinusoids (with frequencies 0.8, 1.6, 2.4, 3.2, 4, 5.6, 7.2, 10.4, 12, 18.4 Hz) with a cycling period of 1.25 s. This well-designed multi-sine perturbation signal allows for the assessment of neural dynamics in the human sensorimotor system including stretch reflexes and its nonlinear connectivity without spectral leakage^{23, 29}. The perturbed holding period lasted for 15 s for each trial. Thus, there were 12 perturbation cycles within a single trial. The experiment consisted of 15 trials in total. Between trials, participants were required to relax their arm fully for 1 min to avoid muscle fatigue. Similar experimental designs have been previously applied to investigate stretch reflexes in other joints, such as the wrist joint, using linear and nonlinear approaches^{23–25, 36}.

The muscle activity at the Biceps (BIC) and Triceps (TRI) Brachii were recorded by active differential surface electromyography (EMG) electrodes with 1 cm inter-electronic distance using an 8 Channel Bagnoli™ EMG System (Delsys, Boston, MA) during the experiment. This device has a 60Hz line interference check for controlling the EMG signal quality. The collection of the EMG data was synchronized on-line with the perturbation and elbow flexion torque signal using a Transistor-Transistor Logic (TTL) pulse. All data were sampled and stored at 2048 Hz for the following off-line analysis.

4.2. Data Preprocessing and Pre-analysis

BIC EMG data were demeaned, high-pass filtered with a 20 Hz zero-phase shift filter, rectified, and normalized to the peak rectified BIC EMG obtained during EF MVT¹⁶. Despite the debate of justification of EMG rectification in the literature, our previous study showed no significant impact of EMG rectification on our linear/nonlinear connectivity analysis³⁶. However, high-pass filtering and rectification are recommended to eliminate motion artifacts from mechanical perturbation³⁸. All data were cut into epochs in a non-overlapping way based on the duration of the perturbation period (1.25 s). Thus, there were 15 epoch/trial × 15 trials = 225 perturbed epochs in total. EMG and torque signals were then evaluated to remove the bad epochs. The epochs with recorded EF torque clearly off the target (e.g. the participant gave up maintaining target torque) were removed, so as to ensure the tasks are performed properly and without fatigue in the epochs included for further analyses^{2, 31}.

We computed the amplitude spectra of the BIC EMG component that is phase-locked to the perturbation signal to examine the linear and nonlinear response to the perturbation. A linear interaction is known to generate the response in the same frequencies as the stimulus, i.e., the iso-frequency interaction between the stimulus and the response. Thus, in the frequency domain, the presence of nonlinearity can be detected by inspecting the harmonic (multiple of the stimulation frequency, e.g., 3f_i) and intermodulation (the sum or difference between stimulation frequencies) of stimulation frequencies in the amplitude spectral of the response. Shown in Fig. 2, we found that nonlinear muscle responses are mainly shown at the second-order harmonics (2f_i) and intermodulation frequencies (f₁ ± f₂, when f₁ > f₂); in comparison to higher-order nonlinearity. Thus, we focused on the second-order nonlinearity for the perturbation-EMG connectivity, in addition to its linear relation.

Figure 2.

Amplitude spectrum of phase-locked EMG signal at the Biceps Brachii. The symbols indicate the mean values across all participants and the error bars represent the standard deviation. The noise level was estimated from the recorded EMG when the participants kept their arm relax. The vast majority of nonlinear components higher than second-order are below noise level, and thus can be neglected in the analysis.

4.3. Linear/Nonlinear Connectivity and Time Delay Estimation

We used the multi-spectral phase coherency (MSPC)³⁷ to estimate linear and nonlinear connectivity and associated time delays in the muscle responses to continuous perturbations. MSPC is a general metric for analysis of various order nonlinear cross-frequency phase coupling between two signals, as well as the linear (1^st-order) iso-frequency phase coupling.

With the d-th order nonlinearity, the MSPC (denoted as Ψ, a complex number) for the two time series x(t) and y(t), X(f) and Y(f) are their Fourier transforms, with K epochs is defined by:

$${\psi }_{XY}{\left(f_1,f_2,\mathrm{\dots },f_R;a_1,a_2,\mathrm{\dots },a_R\right)}_d=\frac{1}{K}{\sum }_{k=1}^K\text{exp}\left(j\left({\sum }_{r=1}^R{a}_r{\varphi }_X{}_{{}_k}\left(f_r\right)-{\varphi }_Y{}_{{}_k}\left(f_{\sum }\right)\right)\right)$$ (1)

where f₁, f₂, …, f_R are frequencies of X(f); a₁, a₂,…, a_R are the integer weights of these frequencies, and ${\varphi }_X{}_{{}_k}\left(f_r\right)$ is the phase of X(f_r) at k-th epoch. The corresponding inspection output frequency of Y(f) is then $f_{\sum }={\sum }_{r=1}^R{a}_r{f}_r$ with R ≥ 1 and ${\sum }_{r=1}^R\left|a_r\right|=d$, and the order of nonlinearity is then $d={\sum }_{r=1}^R\left|a_r\right|>2$ while d = 1 indicates linear interaction. Typically, the d-th order MSPC can be used to quantify the d-th order harmonic (f_Σ= d · f_r) and intermodulation coupling ($f_{\sum }={\sum }_{R=1}^r{\text{a}}_{\text{r}}{\text{f}}_{\text{r}}$, R ≥ 2, ${\sum }_{\text{r}=1}^{\text{R}}\left|{\text{a}}_{\text{r}}\right|=\text{d}$) between x(t) and y(t).

When d = 1, MSPC quantifies linear connectivity (f_Σ = fr), i.e., iso-frequency phase coupling. Then, the eq. (1) can be simplified as

$${\psi }_{XY}\left(f\right)=\frac{1}{K}\sum {}_{k=1}^K\text{exp}(j\left({\varphi }_X{}_{{}_k}\left(f\right)-{\varphi }_Y{}_{{}_k}\left(f\right)\right)$$ (2)

When d = 2, MSPC quantifies the second-order nonlinear connectivity:

$${\psi }_{XY}{\left(f_1,f_2,;a_1,a_2\right)}_2=\frac{1}{k}{\sum }_{k=1}^K\text{exp}\left(j\left({\sum }_{r=1}^2{a}_r{\varphi }_X{}_{{}_k}\left(f_r\right)-{\varphi }_Y{}_{{}_k}\left(f_{\sum }\right)\right)\right)$$ (3)

We computed MSPC based on the Fourier Transform and the expressions: exp(ϕ₁ + ϕ₂) = exp ϕ₁ · exp ϕ₂, $\text{exp}j{\varphi }_X{}_{{}_k}=X_k\left(f\right)/\left|X_k\left(f\right)\right|$ and X(−f)= X*(f) (see Ref.³⁷ Appendix A. for the details of Fourier Transform based MSPC computation), where * represents a complex conjugate. The magnitude of MSPC is then defined as the multi-spectral phase coherence and denoted as ψ = |Ψ|. The multi-spectral phase coherence reflects the strength of nonlinear phase coupling (which is a real number). The value of ψ varies between 0 and 1, where 1 indicates that the nonlinear phase relationship is perfectly consistent across epochs, and 0 indicates that the nonlinear phase relationship is random. The first-order multi-spectral phase coherence is also known as phase locking value (PLV), and the second order is known as bi-phase locking value (bPLV)³. We used a 95% confidence threshold (α = 0.05) to determine the significance of MSPC³⁷, which is square root of 3/K (K is the total number of epochs). The details of MPSC statistics is available in Ref.³⁷ Appendix A and B.

Given an input-output relationship X → Y with a time delay τ, the phase lag between the input X and the output Y, Δϕ_XY = 2πf_Στ can be extracted from the MSPC: exp(jΔϕ_XY) = Ψ_XY/ψ_XY. Thus, the time delay for a given order MPSC estimated connectivity can be estimated by the cost function:

$${\tau }_{est}=\underset{\tau }{\min }(\sum {}_{f_{\sum }={\sum }_{r=1}^R{a}_r{f}_r}\left|\text{exp}(j\left(2\pi f_{\sum }\tau \right)-\frac{{\psi }_{\text{XY}}\left(f_1,f_2,\dots ,f_R;a_1,a_2,\dots ,a_R\right)}{{\psi }_{XY}\left(f_1,f_2,\dots ,f_R;a_1,a_2,\dots ,a_R\right)}\right|)$$ (4)

In this study, we estimated the time delays for the linear connectivity and the second-order nonlinear connectivity in separate using significant MSPC values. A grid search algorithm (with a resolution of 0.1 ms) is applied to find the local minima of the cost function for estimating the delay.

To verify the effectiveness of the used connectivity and time delay estimation methods, we also tested the methods in a simulated system containing both linear and nonlinear pathways with different time delays (18.1 ms for the linear pathway, 32.7 ms for the nonlinear pathway), using the same ten-frequency multi-sine signal as the input to the simulated system. We considered the 2nd order nonlinearity in this simulation for the nonlinear part, since a majority of signal power in the real data is shown in the stimulated frequency (linear part) and 2nd order nonlinearity (see Fig. 2). The used power function x² can generate both 2nd order harmonics and intermodulation, without inducing other irrelevant nonlinear frequency components in the output. This simplification allows us to verify the effectiveness of the used connectivity and time delay estimation methods with a low computational cost.

The error of time delay estimation is defined as

$$\Delta \tau =\frac{{\tau }_{set}-{\tau }_{est}}{{\tau }_{set}}\times 100\%$$

where τ_set is the real delay set in the simulation, τ_est is the estimated delay.

5. Results

5.1. Simulation Results

The linear connectivity (i.e. iso-frequency phase coupling) is shown in Fig. 4 (a) as a spectrum of input frequency, and nonlinear connectivity is shown in Fig. 4 (b) as a map of input frequency pairs. The statistically significant linear connectivity is only shown in the ten input frequencies as expected. The significant nonlinear connectivity is shown in 10 second-order harmonics and 90 $\left(C_{10}^2\right)$ intermodulation combinations of the ten input frequencies. The time delays were indicated by the local minima (or nadir) of the cost functions as shown in Fig. 5. The estimated delay for linear connectivity is 18.1 ms (zero error), and for nonlinear connectivity is 32.6 ms (0.306% error).

Figure 4.

Simulation results: linear (a) and nonlinear connectivity (b). (a) Linear connectivity is given as a spectrum of the input frequency. (b) Nonlinear connectivity is shown as a map of input frequency pairs. Harmonics are indicated by the dashed line, where f₁ = f_2, while intermodulation is where f₁ ≠ f₂ and (f_1, f₂) for f₁ + f_2, (f_1, − f₂) for f₁ – f_2.

Figure 5.

Simulation results: time delay cost function. The estimated delay is indicated by the local minima of each curve.

5.2. Stimulus-response Connectivity and Delays in Stretch Reflexes

Both linear and nonlinear connectivity were detected in the BIC EMG responses to the perturbations (see Fig. 6). The high linear connectivity (MSPC > 0.5) is detected in the range of 3.2–10.4 Hz with the peak value at 5.6 Hz. The nonlinear connectivity is shown at both harmonics and intermodulation frequency pairs. The time delay for linear connectivity is 24.5 ± 5.4 ms (mean ± std.), while the time delay for nonlinear connectivity is around 54.8 ± 3.2 ms. The delay for nonlinear connectivity is around 30 ms longer than that for linear connectivity (paired t-test: p < 0.0001).

Figure 6.

Linear and nonlinear connectivity in the muscle’s responses to the perturbations (mean results across all subjects). (a) Linear connectivity is given as a spectrum of the input frequency. (b) Nonlinear connectivity is shown as a map of input frequency pairs. Harmonics are indicated by the dashed line, where intermodulation is where f₁ = f_2, while intermodulation is where f₁ ≠ f₂ and (f_1, f₂) for f₁ + f_2, (f_1, − f₂) for f₁ – f_2.

6. Discussion

This study, for the first time, quantitatively distinguished linear and nonlinear connectivity as well as their time delays in the human stretch reflexes at the elbow joint, in response to continuous perturbations. The subjects performed elbow flexion torque generation tasks while receiving continuous position perturbations. The position perturbations are encoded by muscle spindles and transmitted via Ia afferents to the motoneuron pool via stretch reflex loops. The nonlinear component of the stretch reflex has been previously reported³⁸. In the present study, the used connectivity method, i.e. MSPC, allows for the quantification of the amplitude-independent linear (iso-frequency) and nonlinear (cross-frequency) phase coupling including harmonic and intermodulation coupling. A model simulation in a previous study demonstrated the effectiveness of MPSC in assessing input-output phase relations and the time delay in a monosynaptic system with a single time delay³⁷. The results from the current study provided new evidence showing that MPSC can quantitatively distinguish linear and nonlinear connectivity as well as their time delays in a precise way even when the tested system contains different pathways with different time delays.

Applying MSPC to investigate the human stretch reflex at the elbow joint, we detected both linear and second-order nonlinear connectivity. The connectivity result is consistent with a previous study on stretch reflexes at the wrist joint, showing the dominance of linear connectivity as well as the second-order nonlinear connectivity³⁸. However, the previous study did not separately estimate the time delays for linear connectivity and nonlinear connectivity. The overall time delay estimated from all significant MSPC (including both linear and nonlinear parts) is around 33 ms. Thus, they assumed that both linear connectivity and nonlinear connectivity are generated by the mono-synaptic spinal stretch reflex loop, since the estimated time delay is too short to be from the supraspinal reflex loops^{15, 38}. Different from the previous study, we separately estimated the time delays for linear connectivity and nonlinear connectivity. The time delay for linear connectivity (24.5 ± 5.4 ms) is in the range of short-latency stretch reflex period for the elbow (< 35 ms), while the delay for nonlinear connectivity (53.8 ± 3.2 ms) is in the range of long-latency stretch reflex period (40–70 ms)^{13, 17}.

Despite the ongoing debate of the origin of long-latency stretch reflex component^{20, 26, 28}, most studies suggested that the long-latency stretch reflex component is generated by the supraspinal reflex loop^{12, 19, 20}. In vivo experiments recording from corticomotoneural cells in Macaque monkeys found a cortical contribution to the long-latency stretch reflex component¹, similar results have also been demonstrated in human beings by measuring event-related potential (ERP) from the sensorimotor area during the long-latency stretch reflex period^{15, 33}. Furthermore, a recent study showed that a subthreshold transcranial magnetic stimulation (TMS) over the contralateral primary motor cortex only modulates the long-latency stretch reflex component rather than the short-latency component²⁰. Recent studies suggested that reticular formation may also contribute to the long-latency stretch reflex component via the reticulospinal tracts^{12, 22, 32}. This subcortical, supraspinal contribution may play a critical role in hyperactive stretch reflex in unilateral brain injury (such as hemiparetic stroke) where the corticospinal tract is damaged^{14, 16}. Other supraspinal contributors, such as the cerebellum, basal ganglia, red nucleus, etc., have also been previously reported, indicating that the long-latency stretch reflex component is a mixture of responses from multiple supraspinal loops¹². It is a part of a sophisticated feedback control loop system that enables additional longer-latency reflex modulation after the short-latency mono-synaptic stretch response²¹.

Compared to the monosynaptic spinal stretch reflex loop, supraspinal loops involve multiple synapses in the ascending dorsal columns passing through medulla and thalamus and followed by additional synapses in the descending motor pathways onto alpha motor neurons in the spinal cord³². Our previous model simulation indicates that the nonlinear connectivity increases with the number of synapses in a motor pathway²⁷. Multi-synaptic connections can generate nonlinear connectivity between the input stimulation and the output neural responses^{27, 29, 34}, while a mono-synaptic connection mainly results in linear connectivity^{18, 27}. Thus, we suggested that the detected linear connectivity with a short time delay is very likely generated by the mono-synaptic stretch reflex loop, while the nonlinear connectivity with a long time delay may be associated with the multi-synaptic stretch reflex loop. A previous model simulation study indicates that the refractory period of motor neurons can also cause a motor response in the long-latency response window when a ramp perturbation is used²⁶. However, with our method and experiment design, the influence of this factor has been minimized in this study. Different from ramp perturbation, multi-sine perturbation generates continuous input to the muscle, thus it eliminating the effects of refractory period on the response latency.

By examining the frequency contents of the connectivity, we found that the high linear connectivity is detected in the range of 3.2–10.4 Hz with the peak value at 5.6 Hz, which likely captures the dynamics of the arm around its eigenfrequency⁴. The large nonlinear connectivity is shown around a lower input frequency around 2.4 Hz. This lower frequency nonlinear response may be related to the neural dynamics of multi-synaptic pathways, since our recent study found that increased usage of multi-synaptic pathways in the stretch reflexes can lead to a shift of EMG responses towards a lower frequency³⁵.

In conclusion, this study provides new insights on the neural connectivity in the stretch reflexes, especially the origin of linear vs. nonlinear connectivity. The used approach has the potential to become a useful quantitative tool to determine functional connectivity and the associated neural transmission delays in various neural pathways including multi-synaptic and mono-synaptic pathways.

Limitation and Future work:

In this study, we focused on the linear and the second-order nonlinear connectivity. Neglecting higher order (or other types of) nonlinearity may cause a bias of estimation of latency for nonlinear components. However, our recent study showed that the majority of nonlinear transfer occurs at the second-order nonlinear term in the human sensorimotor system³⁹. The bias is then negligible since high order nonlinear components are around the noise level as shown in Figure 2. In the future, electrical brain signal (electroencephalogram, EEG) recording and TMS will be added to the current protocol to further investigate the potential cortical influence on the nonlinear long-latency component in the stretch reflex. We also plan to apply the proposed approach to individuals with a unilateral brain injury to better understand the contributions from cortical and subcortical areas to the nonlinear long-latency stretch reflex component, in order to develop biomarkers based on our nonlinear connectivity analysis.

Figure 3.

Simulated system with linear and nonlinear pathways. The weights for linear (w_l) and nonlinear pathways (w_n) were set to ensure their outputs are in the same scale.

Table 1.

Estimated time delays for linear and nonlinear connectivity in BIC EMG responses to the perturbations (unit: ms)

Participant	P1	P2	P3	P4	P5	P6	P7	P8	Mean	Std.
Linear	18.9	26.6	26.2	24.4	16.4	33.2	21.7	28.4	24.5	5.4
Nonlinear	57.7	51.2	54.6	52.1	48.6	56.0	53.1	57.4	53.8	3.2