# Motor Unit Discharges from Multi-Kernel Deconvolution of Single Channel Surface Electromyogram

## Abstract

**:**

## 1. Introduction

- A single kernel is unlikely to be sufficient to represent a general EMG, including MUAPs corresponding to different conduction velocities (CV). Indeed, a widespread delay distribution is expected to be used to recover a MUAP with a larger support than the kernel (corresponding to a MU with a low muscle fibre CV), whereas, there will be problems in rebuilding MUAPs shorter than the kernel.
- Problems are expected if there are more innervation zones (IZs) and MUAPs are propagating in different directions under the detection point so that the single SD channel records waves with opposite phases.
- In ideal conditions, the deconvolution process would recover exactly the original data by convoluting the estimated cumulative firings with the selected kernel. As coherence is unaffected by filtering, it would be the same if applied to the original or the processed data. Thus, a generalization is needed to make the method applicable to important fields, such as intra- or inter-muscular coherence, overcoming the limitations of using the raw EMG.

## 2. Methods

#### 2.1. Signal Processing

- A large spread of IZs was assumed, so that MUAPs could propagate under the electrodes in two opposite directions. This happens in many different conditions, e.g., in sphincter muscles [25], in the case of fibre pinnation or, in general, when the distribution of IZs is not perpendicular to the fibre direction [26]. As a consequence, waveforms with opposite phases are recorded by the considered SD channel. In such a case, two kernels were considered, with the same PSD resembling that of the original data but with opposite phase. Specifically, the PSD of the first derivative of a Gaussian function is -4.6cm0cm$$\dot{G}\left(t\right)=\frac{d}{dt}\frac{{e}^{-\frac{{t}^{2}}{2{\sigma}^{2}}}}{\sqrt{2\pi {\sigma}^{2}}}\phantom{\rule{1.em}{0ex}}\to \phantom{\rule{1.em}{0ex}}F\left[\dot{G}\left(t\right)\right]=j2\pi f{e}^{-2{\pi}^{2}{f}^{2}{\sigma}^{2}}\phantom{\rule{1.em}{0ex}}\to \phantom{\rule{1.em}{0ex}}PSD={\left|F\left[\dot{G}\left(t\right)\right]\right|}^{2}=4{\pi}^{2}{f}^{2}{e}^{-4{\pi}^{2}{f}^{2}{\sigma}^{2}}$$$$\Gamma \left(t\right)=\left({f}^{2},log\frac{PSD}{4{\pi}^{2}{f}^{2}}\right)=({f}^{2},-4{\pi}^{2}{f}^{2}{\sigma}^{2})$$It is clear that ${\sigma}^{2}$ can be estimated by the slope of this curve divided by $-4{\pi}^{2}$. This procedure was applied to the PSD of the EMG, which is more complicated than the above expression, as different waveforms are summed, none of them are exactly obtained as a derivative of a Gaussian function, and noise is present. Thus, the PSD of the EMG was considered in a frequency range in which most of the power is found, i.e., in (${F}_{Med}-{F}_{std},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{F}_{Med}+2{F}_{std}$), where ${F}_{Med}$ is the median frequency and ${F}_{std}$ the standard deviation of the PSD (preliminary tests showed that this range provided stable results). Curve (5) was approximated by a straight line within this range and its slope was used to estimate ${\sigma}^{2}$. As detailed below, two different simulators were used to test this condition: a model with parallel fibres [27] and two different IZs and a simulator of pinnate muscle with fibres inclined with respect to the skin surface [28,29].
- A single direction of propagation was assumed, such as when electrodes are placed beyond the last IZ over a muscle with parallel fibre architecture. As MUAPs are generated by MUs with different CVs, the PSD of the EMG sometimes provides a curve (5) that is not well approximated by a straight line. The curve was then fit by a parabola, and its slopes in the 15th, 50th, and 85th percentile of the frequency range mentioned above were used to estimate the variances of three kernels. Those kernels ideally reflect MUAP prototypes of MUs with small, medium, and large values of CV. This way, the proposed method for the selection of the kernels adapts to the signal. Eventually, the method can come back to the single kernel case in the limit in which the curve (5) is linear, so that the three kernels are identical.

#### 2.2. Test Data

#### 2.3. Assessment of Performance

## 3. Results

## 4. Discussion

## 5. Conclusions and Further Work

## Funding

## Conflicts of Interest

## Abbreviations

CoV | Coefficient of Variation |

CV | Conduction Velocity |

CWF | Cumulative Weighted Firings |

EMG | ElectroMyoGram |

FR | Firing Rate |

ISI | Inter-Spike Interval |

MVC | Maximal Voluntary Contraction |

SD | Single Differential |

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**Figure 1.**Example of simulated data (muscle with 100 MUs) and estimation of cumulative firings. (

**A**) EMG from a muscle with two IZs (70% MVC, fat layer thickness of 3 mm, mean CV of 4 m/s, no synchronization between MU firings, CoV of ISI 10%, maximum FR 40 Hz), showing SD MUAPs with different phases. Small and large MUs are associated to different IZs (split into 75% of small MUs and 25% of large MUs, so that the amplitude of the two contributions is similar). The total signal and those corresponding to the activation of MUs innervated at different IZs are considered, showing (from up to down) a portion of EMG (0.5 s out of the simulated 10 s), the simulated CWF and the estimated CWF (considering two kernels, with opposite phase, with PSD fit to that of the data). Finally, the PSD of the simulated and estimated CWFs are shown. (

**B**) The same as in (

**A**) but considering a single IZ: in practice, the red signal is the same as in (

**A**), whereas the blue one has the opposite phase. Two kernels are considered, with the same phase but different time scales (0.9 and 1.1 rescaling with respect to the kernel fit to the data). Abbreviations: cumulative weighted firings—CWF; firing rate—FR; innervation zone—IZ; maximal voluntary contraction—MVC; power spectral density—PSD; single differential—SD; motor unit (MU) action potential—MUAP; and coefficient of variation of interspike interval—CoV of ISI.

**Figure 2.**Estimation of three kernels, in the cases of (

**A**) no synchronization between MU discharges or (

**B**) high synchronization (signal with one IZ; 80% MVC, fat layer thickness of 3 mm, mean CV of 4 m/s, CoV of ISI 10%, and maximum FR 40 Hz). From top to bottom, the following panels are shown: the signal; its PSD (black) and the ones of the three estimated kernels (in red, blue, and green, keeping the same colours for indicating the same kernels in the following panels); the function of the PSD used to estimate the kernels (in black is the data in the range of interest, in gray is the part out of this range; the interpolation line is in yellow; the points used to estimate the kernels are coloured); and the kernels (with their colour) are superimposed to the MUAPs (in gray) whose CWF best correlates with its deconvolution signal.

**Figure 3.**Example of estimation of the CV from a signal with two IZs and MUAPs propagating in two directions. A contraction level of 30% MVC was simulated, considering a volume conductor with fat layer thickness of 3 mm, mean CV of 4 m/s, no synchronization between MU firings, CoV of ISI 10%, and maximum FR 40 Hz. The smallest MUs (which were the 75% of the active MUs) were innervated in one IZ and the others were under the second one. (

**A**) A portion of signal of half a second (10 s were simulated), showing three SD channels aligned to the muscle fibers. The contributions of MUAPs propagating in either of the two directions are also shown. Under the simulated signals, those obtained by deconvolution using two kernels with opposite phase are shown. (

**B**) CWFs, simulated (up) and estimated (down). The estimated CWFs were obtained processing the three SD channels using the same kernels (chosen based on the first of the three SD channels, i.e., the one shown above the others). (

**C**) Estimation of CV from adjacent, non overlapping epochs of 500 ms, using either the simulated or the estimated signals, corresponding to MUAPs propagating in single directions or to data deconvolved using each kernel, respectively. Both signed CV and its absolute value are considered.

**Figure 4.**Example of the estimation of coherence in different conditions. The same firings were generated and then applied to simulate different EMGs considering different sets of MUAPs obtained using three models. A contraction level of 40% MVC was simulated, considering a volume conductor with fat layer thickness of 3 mm, mean CV of 4 m/s, CoV of ISI 10%, maximum FR 40 Hz, and level of synchronization between MU firings of 10%. MUs were randomly split into two sets, used to generate two EMGs (as they were recorded by two SD channels, each one placed over one of the two different muscles, each constituted by 200 MUs). (

**A**) Simulated CWFs of the two muscles (portion of half a second on the left) and coherence (right, considering signals of 10 s duration). (

**B**) EMGs of two muscles with two IZs (left, same time range as for the CWFs in (

**A**) and estimated coherence (right), using either the raw signals or the estimated CWFs using two kernels with opposite phase, for each muscle. (

**C**) Same as (

**B**), but considering a signal with a single IZ and three kernels for the deconvolution. (

**D**) Same as (

**B**), but considering a simulated volume conductor with fibres inclined of 25° with respect to the skin surface.

**Figure 5.**Cross -correlation of the low frequency contributions (up to 50 Hz) of the simulated CWF and estimated by either the single-kernel (red) or multi-kernel method (black). Each estimation was obtained considering single channel SD EMG at 80% MVC with a duration of 10 s. Data are shown after pooling with respect to specific values of fat layer thickness, mean MU CV, maximal FR, level of synchronization of MU firings, and CoV of ISI. Different simulation models are considered: (

**A**) cylindrical volume conductor with parallel fibres and two IZs (50% of MUs innervated under each IZ; uniform distribution of MU size among different IZs); (

**B**) same volume conductor as in (

**A**), but considering a single IZ; (

**C**) pinnate muscle with fibre inclined with respect to the skin surface.

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**MDPI and ACS Style**

Mesin, L.
Motor Unit Discharges from Multi-Kernel Deconvolution of Single Channel Surface Electromyogram. *Electronics* **2021**, *10*, 2022.
https://doi.org/10.3390/electronics10162022

**AMA Style**

Mesin L.
Motor Unit Discharges from Multi-Kernel Deconvolution of Single Channel Surface Electromyogram. *Electronics*. 2021; 10(16):2022.
https://doi.org/10.3390/electronics10162022

**Chicago/Turabian Style**

Mesin, Luca.
2021. "Motor Unit Discharges from Multi-Kernel Deconvolution of Single Channel Surface Electromyogram" *Electronics* 10, no. 16: 2022.
https://doi.org/10.3390/electronics10162022