**Appendix B**

In the case of a system without dissipation, *R*2=0, the general equations become

$$I\_{E-} = \kappa \Pi\_1^- I\_2 + \frac{\Pi\_1^+ - \Pi\_1^-}{R\_1} \,\,\,\,\,\tag{A6}$$

$$
\Pi\_1^- = \mathcal{R}\_- I\_{E-} + \Pi\_{1'}^S \tag{A7}
$$

$$I\_{E+} = \alpha \Pi\_1^+ I\_2 + \frac{\Pi\_1^+ - \Pi\_1^-}{R\_1},\tag{A8}$$

$$
\Pi\_1^+ = \Pi\_1^R - \mathcal{R}\_+ I\_{E+\prime} \tag{A9}
$$

which leads to

$$I\_{E+} = \frac{aI\_2 \frac{R\_1}{\mathcal{R}\_+} \Pi\_1^S + \frac{\Delta \Pi\_1}{\mathcal{R}\_+} + aA I\_2 \frac{R\_1}{\mathcal{R}\_-} \Pi\_1^R + A \frac{\Delta \Pi\_1}{\mathcal{R}\_-}}{1 + AB},\tag{A10}$$

$$I\_{E-} = \frac{aI\_2 \frac{R\_1}{R\_-} \Pi\_1^R + \frac{\Delta \Pi\_1}{R\_-} - aBI\_2 \frac{R\_1}{R\_+} \Pi\_1^S - B\frac{\Delta \Pi\_1}{R\_+}}{1 + AB},\tag{A11}$$

with

$$\begin{aligned} A &= \left( aI\_2 R\_1 \frac{R\_-}{R\_+} - \frac{R\_1}{R\_+} - \frac{R\_-}{R\_+} \right), \\ B &= \left( aI\_2 R\_1 \frac{R\_+}{R\_-} + \frac{R\_+}{R\_-} + \frac{R\_1}{R\_-} \right). \end{aligned}$$
