Table 1.
The patient details for the seven patients with ET who underwent DBS surgery and whose local field potential data was used in the study. We include age at time of DBS surgery, gender and tremor grading scores for right and left arms in four positions: rest, held at nose, held outstretched and whilst making a reaching movement (intention). All signals were down sampled to a sampling frequency of 64 Hz. The power spectra of the filtered LFP signals were obtained using a Fourier transform (fft function in Matlab). Cross-coherence was performed between the LFP and EMG signals (mscohere function in Matlab, Mathworks), using a periodic Hamming window, with the number of steps set to 32 and 50% overlap. Spectra were averaged across electrodes and sides of the brain and then across patients for each condition. The results obtained from this data analysis were used to constrain the behaviour of the computational model, by comparing the peak frequency of the cross-coherence to the frequency of oscillations produced by the model.
Fig 1.
Schematic representation of the network modelled in the study.
(A) The model contains four populations, the motor cortex, the Vim nucleus of the thalamus, the reticular nucleus of the thalamus (nRT) and the deep cerebellar nuclei (DCN). In addition, there are two additional inputs, a constant external input in through the cerebellar population and the DBS input into the Vim. (B) The DBS input, unlike the external input, is not tonic, but oscillatory over time. We used a biphasic square pulse to mirror the pulses used in clinical therapy. The waveform is defined by the following parameters: frequency (f), amplitude (A), pulse width (PW) and the multiple for the biphasic phase (m).
Fig 2.
A representative simultaneous recording of EMG and LFP from one ET patient.
In (A), the power spectra are shown for the EMG and the LFP across the entire recording. The EMG spectrum clearly shows a tremor band peak at 4Hz and subsequent peaks at the harmonic frequencies. The LFP also shows a 4Hz peak, although with smaller power. (B) The Cross-coherence between the EMG and LFP reveals a peak at 4Hz and the harmonic frequencies.
Fig 3.
The EMG and LFP data over epochs.
(A) The data for each of the patients was first split into different behavioural epochs: rest, when the patient was asked to refrain from moving (seven patients, 14 spectra); moving, when the patient was asked to make self-paced repetitive movements such as opening and closing the hand (five patients, nine spectra); and posture, when the patient was asked to hold their arm out (three patients, four spectra). We can see that in all epochs there is an increase in tremor-band coherence, with a more tuned increase in the moving and posture conditions. (B) In addition to the mean cross coherence, in order to further demonstrate the variability across patients, we show the cross-coherence for the posture epoch for all nine spectra and a histogram summarising the peaks of the cross-coherence spectra in the 3–11 Hz range.
Fig 4.
The oscillatory activity observed in the model.
The activity of the Vim population alone (A) is shown over 5 seconds of simulated time, and for the three populations (B) displaying oscillations for approximately one cycle (0.25 seconds). The oscillation amplitude is stable over time for any one population, but varies across populations. The oscillation is lead by the Vim population (blue), then followed by the nRT (red) and finally the cortex (green).
Table 2.
The free parameters used in the Wilson-Cowan model of the thalamocortical-cerebellar network for essential tremor. The weight parameters for the six connections and the time constants in the model network are given here, and all other parameters are listed in the text. These parameters were selected prior to the bifurcation analysis, by sweeping the parameter space and were chosen to be close to the bifurcation point where the oscillation exists with the correct frequency. The w symbols are those appearing in the four equations describing the network in the methods section.
Fig 5.
Bifurcation analysis of the connection weights.
The relationship between the six connection weights can be examined by using bifurcation analysis, which allows us to co-vary any two parameters at a time and trace out the region in parameter space where the bifurcation leading to oscillatory activity occurs. In this way, we can split the parameter space into oscillatory (shaded) and non-oscillatory regions and therefore make predictions about the network structure in the pathological state. Furthermore, we examined the frequency of the oscillations throughout these shaded regions and found that the frequency of oscillations does not remain constant. In fact, as parameters varied from the default values (black circle), the frequency increased (solid line) or decreased (dashed line) as shown in the plots. The frequency of oscillations we observed within these regions however, was between 2 Hz and 8 Hz.
Fig 6.
DBS effects on oscillatory activity.
When the DBS input is applied to the Vim, the effect on the oscillatory activity is both amplitude and frequency dependent. (A) At low amplitudes, DBS changes the frequency of the oscillation, and the relationship between the applied and the resulting frequency is linear. (B) At higher DBS amplitudes, this relationship changes, and the higher the DBS frequency the lower the frequency of the high amplitude oscillation, which is eventually replaced by the low-amplitude, high frequency activity. When the DBS amplitude increases further (C), this switch from low-frequency, high-amplitude activity to low-amplitude, high-frequency activity occurs at a lower stimulation frequency.
Fig 7.
The change in oscillations with increasing DBS amplitude.
For a single DBS frequency, the change in Vim activity is shown as the amplitude of the stimulation increases. The baseline large amplitude tremor band activity is shown at the top, and the low-amplitude high frequency activity at the bottom. In between, the activity gradually switches from one to the other, with an initial increase in frequency at low amplitudes, followed by a decrease in frequency as the amplitude increases further.
Fig 8.
The baseline oscillation in the network is described as a limit cycle, which can be seen when the activities of two populations (Vim and cortex) are plotted against one another over time (A). When the DBS input is applied to the network, the limit cycle is no longer constant and deviates from a perfect cycle over time (B). This is caused by a saddle-node on an invariant circle (SNIC) bifurcation, which can be seen here when the ext parameter value is gradually increased to its value in Table 2 (C).