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Essential tremor (ET) is defined as a neurological disorder that causes involuntary abnormal repetitive shaking. This shaking can appear in different parts of the body such as the hands, forearms, or head [

In this study, we use recordings from two patients (aged 64 and 53, both female) who underwent DBS surgery for medically refractory ET. Ethics approval for use of patients’ EEGs and LFPs for the development of new quantitative EEG (qEEG) methods was obtained both from the University of Sheffield and the NHS ethics committees (SMBRER207 and 11/YH/0414). The multichannel Natus Quantum Amplifier (Optima Medical Ltd.) at a sampling rate of 16,384 Hz was used for all EEG/LFP/EMG polygraphy recordings (analogue bandwidth 0.01–4000 Hz).

In our institution (Royall Hallamshire Hospital, Sheffield Teaching Hospitals, NHS Foundation Trust), every patient undergoing DBS surgery for tremor is offered, five days after implantation of the depth electrodes, a comprehensive electrophysiological analysis to include LFPs from the VIM thalamus, electroencephalography (EEG), and electromyographic (EMG) polygraphy recordings, measuring the electrical activity from the muscles, to look at the correlation between the tremor and the thalamic network oscillations. The cortical scalp EEG recordings are not used in this work. There are four electrode contacts on each macroelectrode placed in the thalamus in close proximity to each other (contacts 0, 1, 2, and 3) through which bipolar recordings of LFPs can be obtained. Many centres use an empirical approach in selecting the ideal electrode contact to stimulate after DBS surgery to obtain the ideal tremor suppression. This approach can be time-consuming, and it will become more of a problem as new multicontact directional leads make their way into clinical practice [

From the engineering perspective, the approaches to study connectivity between two signals can be classified as time- and frequency-domain-based methods. One of the classic linear measures to estimate similarity in time domain is cross-correlation that has been used to study ET [

In addition to the frequency- and time-based approaches, efforts have been made with the wavelet domain to study connectivity of brain networks. This approach aims to address limitations of the above methods on tackling dynamic systems by providing time-resolving value with accurate locality. Meanwhile, it is a model-free (nonparametric) measure, which reduces the requirement of a priori knowledge of the underlying model. Wavelet coherence has attracted increased interest on studying brain-related disorders. Jeong et al. [

Addressing these challenges, this paper develops a new wavelet-based correlation analysis framework combined by the estimation of connectivity strength, significance test, and phase-delay characterisation. It aims to better understand cerebromuscular interactions in a structured manner. It should be noted that this framework has the prospect to be applied on other applications of connectivity analysis, such as EEG-EEG and EEG-EMG.

Model-free (nonparametric) measures are chosen in this paper to study the correlation between LFPs and EMG because this biological system is so complex that it would require a substantial number of parameters and computation time to build a satisfactory parametric model. Additionally, there is no well-accepted analytical model to start with. A common deficiency when applying biomedical signal-processing tools is the assumption of stationarity. A typical practice for the estimation of spectrum distribution is segmenting long records of data and averaging calculations over segments. The main disadvantage of this practice is the inability of having time-resolved values. There is therefore no time resolution, and the dynamic behaviour of neuronal interactions cannot be revealed. One solution to overcome this issue is the use of wavelets.

Wavelet transformation makes a decomposition of a time series into a frequency-time domain. It uses convolution of a mother wavelet and its scaled and shifted versions. Among all the possible mother wavelets, the

Coherence is one of the most widely used methods for measuring linear interactions. It is based on the Pearson correlation coefficient used in statistics but in frequency and time domain. It measures the mean resultant vector length (or consistency) of the cross-spectral density between two signals. Its squared value varies from 0 to 1, meaning low and high linear frequential correlation. During this study, coherence is used as a reference standard for comparison to other methods. The wavelet formulation of coherence between two signals,

If multiple trials of both signals are available, square coherence can be estimated using (

The number of trails is denoted by

To assert significant values of wavelet coherence, the statistical methodology stablished by Gallego et al. [

In this paper, the parameter

Wavelet cross-spectrum (WCS) also provides information about linear synchronisation, but its values are not normalised as in wavelet coherence. Its calculation is written in (

It has been proven by Bigot et al

Apart from the quantification of the linear interaction strength and associated significance, it is also important to measure the time or phase lag between signals at a certain frequency. It is particularly important for studying corticomuscular interactions as they carry significant time delays.

The most straightforward approach would be calculating the time lag from the phase information in the complex data of the WCS results with a simple fraction

Illustration of the WPLI [

The proposed methodology can be illustrated by Figure

The flowchart of the proposed correlation analysis framework.

The simulation example aims to evaluate the performance of the proposed analysis framework on frequential and time resolution and robustness against noise. Considering a linear single input single output (SISO) system, the input signal is defined as

The input signal constitutes an ensemble of

One difference with the input is the change on the amplitude of each frequential component by setting

Example plots of the simulation signals (

If there is no noise, the measured frequency-time interaction after the significance test using WCS can be represented by Figure

Measured interaction after significance test for the simulation example without noise using WCS.

If there is noise involved, the number of trials is important and should be considered. Figure

Results of wavelet coherence associated with significance test for the simulation example where SNR = −5 dB and

Results of WCS associated with significance test for the simulation example where SNR = −5 dB and

To quantitatively study the influence of SNR and the number of trials on the results, the 2-D correlation coefficient between the significance tests of the ideal case (see Figure

The accuracy of connectivity detection using WCS, represented by the 2-D correlation coefficient, for various levels of noise and number of trials.

To complement the WCS results, Figure

The estimated weighted phase lag index for the simulation example.

All electrophysiological recordings were obtained with a multichannel Natus Quantum Amplifier (Optima Medical Ltd.). Four types of data were available for each recording: scalp electroencephalography (EEG), intracranial thalamic (VIM) local field potentials (LFPs), electromyography (EMG) with surface electrodes, and monoaxial accelerometer recordings from the hands and head. All data were sampled at 16.38 kHz and then downsampled to 2 kHz. This study focuses only on the interactions between thalamic LFPs and contralateral EMG (as each right and left half of the brain supplies the contralateral side of the body). LFPs refer to the summated electrical neuronal activity recorded with the DBS leads from the VIM thalamus. This activity was recorded by a quadripolar DBS lead with three possible input channels (0-1, 0–2, and 0–3) taking the pole 0 as a reference. Figure

Design of quadripolar DBS lead based on Medtronic DBS model 3387.

The simulation example has demonstrated that the use of multiple trials or realisations of an event was important to better reveal significant interactions. However, it is not possible to repeat the same ET event in the same subject with the same conditions such as timing within each tremorogenic oscillation. To overcome this issue, the tremorogenic EMG signal trials were substituted by time segments coming from a long single trial signal. The necessary condition for extracting statistical properties by analysing data over time instead of evaluating several data samples is called ergodicity [

For correcting this issue of the timing offset, a reference point, for each time segment extracted, is set as the closest peak of the EMG signal. Through this manner, phase cancellation artefacts can be avoided to some extent. To detect the peaks on EMG, the procedure follows three steps. The first step processes the data with a linear low-pass filter (passband edge frequency 15 Hz, stopband frequency 30 Hz, passband ripple 1 dB, and 60 dB of attenuation) since it is known that the tremor appears at low frequencies and the filtered signal is corrected with the corresponding delay of the filter. In the second step, the frequency component with highest magnitude is analysed. In the third step, a peak neighbourhood search is performed with a restriction based on the period of the fundamental frequency. Figure

Illustration of multiple-trial extraction.

Figure

Result of wavelet cross-spectrum of a 10 sec single trial between L0L3 LFPs and right triceps brachii EMG.

The colour scale establishes a range of values, in terms of WCS module, of fourth order of magnitude. However, it is not a reliable feature to compare with other combinations of signals, since the scale depends on the amplitude of the original signals that can differ depending on the impedance and position of the reference electrode. Instead, comparing different plots during a long period of time and checking which one is more consistent and regular will lead to better interpretation of the underlying interactions. Figure

Result of wavelet cross-spectrum of a 15 sec single trial between (a) left L0L3 LFPs and the right triceps brachii EMG, (b) L0L3-right biceps brachii, (c) L0L3-right extensor digitorum communis, (d) L0L3-right flexor carpi ulnaris, and (e) L0L3-right abductor pollicis brevis.

Two parameters to be determined for multitrial analysis for real data include the number of trials and the overlap rate. With sampling starting from the 6th sec of the data of L0L3 and right triceps brachii with a window length of 1 sec epoch, the correlation was estimated where the trial number was set as 10 and 20, and the overlap rate was set as 0%, 50%, and 75%. For the trial number of 10 and the overlap rate of 0%, the sampling windows are [6 s, 7 s], [7 s, 8 s], … , [15 s, 16 s]. It should be noted that the overlap rates of 50% and 75% are approximate values. The true overlap rates are determined by the references points based on the closest peak of the EMG signal. For example, the second window of the overlap rate of 50% is not necessary to start exactly from 6.5 s. It starts from the closest EMG peak around 6.5 s. Figures

Result of wavelet cross-spectrum for left L0L3 versus right triceps brachii based on multitrials with different parameter settings (number of trials and degree of overlap).

Significance test result of wavelet cross-spectrum for left L0L3 versus right triceps brachii based on multitrial analysis with different parameter settings (number of trials and degree of overlap).

Result of wavelet cross-spectrum for L0L3-right triceps brachii based on a single trial over a 1-second epoch.

To evaluate the time lag of the significant interactions, the WPLI approach was applied and the result is shown in Figure

Result of WPLI for L0L3 versus right-triceps brachii based on 30 trials and 80% overlap.

This paper proposes a novel data analysis framework to study thalamomuscular associations in essential tremor involving three steps: correlation strength estimation, significance test, and phase lag characterisation. This framework aims to improve the robustness and reliability of correlation analysis between the local field potential recordings from the brain and the tremulous electrical activity recorded on EMG. It has been shown in the simulation example that the proposed approach can effectively evaluate the linear interaction between two signals. The sensitivity analysis studies show how the number of trials and noise level of measurement affect the results. For data with noise level < −5 dB or >10 dB, a significant number of trials produce much better results. However, for data with noise level > −5 dB and <5 dB, the number of trials has less influence on the findings. The application of the method, on real data from two patients with ET undergoing DBS surgery for tremor suppression, demonstrates the validity of the proposed approach through segmenting a long single epoch into a number of overlapped windows to produce the averaged strength of associations. One limitation of this approach is that the result is difficult to be quantified due to the complexity of WCS patterns if the ground truth is unknown. Another potential limitation is that the number of trials plays an important role in improving the performance of this approach. With a single trial, the significance test cannot be constructed. In the real data application, it is not possible to repeat the same ET event (i.e., tremorogenic oscillation) in the same subject with the same conditions. Future work therefore will focus on quantification of the results and reduce the dependency from the number of trials.

It should be noted that wavelet cross-spectrum and phase lag characterisation used in this framework are not novel. However, combining them together along with a significance test is new. Furthermore, this is the first attempt to apply wavelet-based correlation analysis on patients with medically refractory essential tremor undergoing surgery. This paper shows a clear association between the thalamic local field potential recordings and the contralateral tremorogenic EMG oscillations, at the frequency of the tremor and its first harmonic (Figure

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported in part by the Through-Life Engineering Services Centre at Cranfield University (UK), in part by Royal Hallamshire Hospital at Sheffield (UK), in part by Zhejiang University of Technology (China), and in part by the Computational and Software Techniques in Engineering MSc Course at Cranfield University (UK). The authors thank Neurocare for the purchase of the medical equipment used for the polygraphy recordings on their patients. This is research was carried out in part at the National Institute for Health Research (NIHR) Sheffield Biomedical Research Centre (Translational Neuroscience)/NIHR Sheffield Clinical Research Facility. This work was also supported by the National Science Foundation Program of China (61601029), a grant of the Ningbo 3315 Innovation Team, and the China Association for Science and Technology (2016QNRC001).