The preprocessing of EEG signals was performed with EEGLAB toolbox 14.0.0b (http://www.sccn.ucsd.edu/eeglab). The EEG filter was set between 1 Hz and 45 Hz. The common average reference was chosen and the independent component analysis (ICA) method was used to remove the artifacts. The EEG data were divided into segments of 2 s and the power spectral density (PSD) of each segment was then estimated using Welch method (pWelch algorithm, 0.5 Hz frequency resolution, an overlapping 1-second hanning window, no phase shift). According to our previous studies [26, 27], FV led to a desynchronized power at C3 or C4 for stroke patients in these sub-beta bands, including beta1 (13, 18 Hz), beta2 (18, 21 Hz), and beta3 (21, 30 Hz). Therefore, these sub-beta bands above were selected to calculate the relative power in this study. The relative power was estimated as follows:

(1)$$\mathrm{RP}(\mathrm{f1},\mathrm{f2})=\frac{{\int }_{\mathrm{f1}}^{\mathrm{f2}}\mathrm{PSD}(\mathrm{f1},\mathrm{f2})\mathrm{df}}{{\int }_1^{45}\mathrm{PSD}(1,45)\mathrm{df}}$$

where RP(f1, f2) standed for the relative power of the specified frequency band ([f1, f2]), in which f1 and f2 indicated the low and high frequency band respectively. The changes of beta power at C3 and C4 in different FV phases compared to baseline were described as the motor-related power desynchronization (MRPD), which was calculated as follows:

(2)$${\mathrm{MRPD}}_{\{\mathrm{S1},\mathrm{S2},\mathrm{S3}\}}=\frac{{\mathrm{RP}}_{\{\mathrm{S1},\mathrm{S2},\mathrm{S3}\}}-{\mathrm{RP}}_{\text{Baseline}}}{{\mathrm{RP}}_{\text{Baseline}}}$$

The negative MRPD value, which was similar to the event-related desynchronization (ERD), reflected the activation of the sensorimotor cortex. In our study, the averaged RP and averaged MRPD in all the FV phases were calculated. To describe the diversity of the activation between the left and the right primary sensorimotor cortex (S1–M1), the difference index of the activation of bilateral S1–M1 (${\delta }_{\{\mathrm{beta1},\mathrm{beta2},\mathrm{beta3}\}}$) was estimated to be:

(3)${\delta }_{\{\text{beta1},\text{beta}2,\text{beta}3\}}={\mathrm{MRPD}}_{\{\text{beta}1,\text{beta}2,\text{beta}3\}}^{\mathrm{C4}}-{\mathrm{MRPD}}_{\{\text{beta}1,\text{beta}2,\text{beta}3\}}^{\mathrm{C3}}$ $={\mathrm{MRPD}}_{\{\text{beta}1,\text{beta}2,\text{beta}3\}}^{\text{Contra}}-{\mathrm{MRPD}}_{\{\text{beta}1,\text{beta}2,\text{beta}3\}}^{\text{Ipsi}}$

where ${\mathrm{MRPD}}_{\{beta1,beta2,beta3\}}^{Ipsi}$ indicated MRPD of the ipsilesional S1-M1 (also referred to S1-M1 located at the same side as the vibrated limb) and ${\mathrm{MRPD}}_{\{beta1,beta2,beta3\}}^{\mathrm{Contra}}$ indicated MRPD of the contralesional S1-M1 (also referred to S1-M1 located at the same side as the vibrated limb) in the beta1, beta2, and beta3 band. The activation of ipsilesional S1-M1 was higher than that of contralesional S1-M1 according to $\delta$ > 0, whilst the activation of ipsilesional S1-M1 was lower than that of contralesional S1-M1 according to $\delta$ < 0.

As is well known for us, the coordination of the agonist and antagonist muscles can ensure the smoothness of joint movement. As for patients with PSS, the impairment of motor function was induced by the coexistence of the weakness of the antagonist muscle and the over-activity of spastic muscle. The persistence of this phenomenon would exacerbate spasticity. Therefore, in our previous study, the index, which was used to assess the relief of spasticity, was put forward as follows [29]:

(4)$${\lambda }_{\{\text{ext,fle}\}}^{{\text{AUC}}_{-}\text{muscle}}=\frac{{\text{AUC}}_{-}{\text{muscle}}_{\{\text{ext},\text{fle}\}}^{\text{bicps}}}{{\text{AUC}}_{-}{\text{muscle}}_{\{\text{ext},\text{fle}\}}^{\text{bicps}}+{\text{AUC}}_{-}{\text{muscle}}_{\{\text{fle,ext}\}}^{\text{triceps}}}$$

(5)$${\lambda }_{\{\text{ext,fle}\}}^{\text{Compliance}}=\frac{{\text{Compliance}}_{\{\text{ext,fle}\}}^{\text{biceps}}}{{\text{Compliance}}_{\{\text{ext},\text{fle}\}}^{\text{biceps}}+{\text{Compliance}}_{\{\text{fle,ext}\}}^{\text{triceps}}}$$

where ${\lambda }_{\{\mathrm{ext},\mathrm{fle}\}}^{\mathrm{Compliance}}$, ${\lambda }_{\{\mathrm{ext},\mathrm{fle}\}}^{\mathrm{AUC}\mathrm{\_}\mathrm{muscle}}$ represented the normalized compliance, AUC_muscle of biceps muscle following the elbow joint’s flexion and extension, respectively. The AUC_muscle represented the area under the curve for muscle. The muscle compliance, as well as AUC_muscle, was estimated at the force of 1–2 kg. The changes of ${\lambda }_{\{\mathrm{ext},\mathrm{fle}\}}^{\mathrm{Compliance}}$ and ${\lambda }_{\{\mathrm{ext},\mathrm{fle}\}}^{\mathrm{AUC}\mathrm{\_}\mathrm{muscle}}$ induced by FV compared to before-FV ($\mathrm{\Delta }{\lambda }_{\{\mathrm{ext},\mathrm{fle}\}}^{\mathrm{AUC}\mathrm{\_}\mathrm{muscle}}$, $\mathrm{\Delta }{\lambda }_{\{\mathrm{ext},\mathrm{fle}\}}^{\mathrm{Compliance}}$) were estimated as follows:

(6)$$\mathrm{\Delta }{\lambda }_{\{\text{ext,fle}\}}^{{\text{AUC}}_{-}\text{muscle}}={{\lambda }_{\{\text{ext,fle}\}}^{{\text{AUC}}_{-}\text{muscle}}|}_{\text{after-FV}}-{{\lambda }_{\{\text{ext,fle}\}}^{{\text{AUC}}_{-}\text{muscle}}|}_{\text{before-FV}}$$

(7)$$\mathrm{\Delta }{\lambda }_{\{\text{ext},\text{fle}\}}^{\text{Compliance}}={{\lambda }_{\{\text{ext},\text{fle}\}}^{\text{Complince}}|}_{\text{after-FV}}-{{\lambda }_{\{\text{ext},\text{fle}\}}^{\text{Compliance}}|}_{\text{before-FV}}$$