## 2 Peripheral Ultrasonic Neuromodulation--Technical Specifications
### Intensity  
The strength of an ultrasonic wave is characterized by its intensity, usually in the unit of watts per centimeter square (W/cm\({}^{2}\)), i.e., the average power per unit cross-sectional area evaluated over a surface perpendicular to the propagation direction. For acoustic plane waves, the intensity is related to the pressure amplitude by:  
\[I\left(W/\text{cm}^{2}\right)=\frac{p^{2}}{2}\sqrt{\frac{\kappa}{\rho}}=\frac {p^{2}}{2\rho c}\]  
As shown in Figure 1A, the spatial distribution of US intensity peaks at the focal location and attenuates quickly outside the focus. To evaluate US intensity in neuromodulation, spatial peak (SP) intensity is more widely used than spatial average (SA) intensity. In the time domain, the instantaneous intensity can be calculated from the pressure plot, as shown in Figure 1B. The temporal peak (TP) intensity is the maximum US intensity, the pulse average (PA) intensity is the average intensity within the pulse width, and the temporal average (TA) is the average intensity for several pulse repetition cycles. In experimental and clinical studies, US intensity is generally quantified as spatial peak-temporal peak (SPTP), spatial peak-pulse average (SPPA), and spatial peak-temporal average (SPTA) intensities. In order to compare studies, it is worth emphasizing that, for the same US wave, the magnitude of \(I_{SPTP}\), \(I_{SPPA}\), and \(I_{SPTA}\) is in descending order, and \(I_{SPTP}\) can be orders of magnitude higher than \(I_{SPTA}\) for US pulses with low duty cycle. Physically, \(I_{SPTP}\) reflects the highest spatial intensity in the US beam and is closely related to potential mechanical effects and cavitation in the tissue. \(I_{SPTA}\) measures the highest spatial intensity averaged over the pulse repetition period and is related to the magnitude of the thermal effect. US neuromodulation generally operates at low-intensity levels to avoid cavitation and is more concerned with the local thermal effect. Thus, \(I_{SPTA}\) appears to  
Figure 1: Schematics of spatial and temporal ultrasound (US) intensity. (**A**) The spatial intensity distribution peaks at the focal location and attenuates quickly outside the focus. (**B**) Three different temporal averages of US intensity. \(p\): pressure; \(I\): intensity; SP: spatial peak; SA: spatial average; TP: temporal peak; PA: pulse average; TA: temporal average.  
be a more suitable intensity indicator for peripheral neuromodulation studies to avoid any prominent thermal effectors.  
US stimulation of \(I_{SPTA}\) below 1 W/cm\({}^{2}\) is generally considered as low-intensity, and the FDA has approved the application of US in patients with a maximum \(I_{SPTA}\) of 0.72 W/cm\({}^{2}\) for diagnostic purposes, which is, presumably, safe enough also for therapeutics [42]. Several studies have reported neuromodulatory effects on the CNS, with US intensity \(<\)1 W/cm\({}^{2}\)[15; 20; 21; 22; 28]. So far, there seem to be no studies reporting appreciable neuromodulatory effects on the PNS with US intensity \(<\)1 W/cm\({}^{2}\)[43]. On the other hand, \(I_{SPTA}\) over 200 W/cm\({}^{2}\) is generally considered as high-intensity US and has been tested in a number of clinical trials [44; 45], after being approved by the FDA, for the ablation of cancer cells in patients via local elevation of temperature up to 85 \({}^{\circ}\)C [44; 46]. High-intensity focused US is also an approved tool by the FDA for coagulative necrosis in the brain to create stereotactic lesions, also an irreversible ablation process [26].  
We, along with some other research groups, have demonstrated that US stimulation on the PNS, with \(I_{SPTA}\) between 1 and 200 W/cm\({}^{2}\), is unlikely to induce a sufficient temperature change in the target region to elicit temperature-driven neuromodulation [43; 47]. This intermediate intensity range has been explored on the PNS by several neuromodulation studies, which have been systematically reviewed in the subsequent sections.