In the KMP model a positive energy $Z_i$ is associated with each site $i\in\{0,1,\dots,n\}$. When a Poisson clock rings at the bond $ij$ with energies $Z_i,Z_j$, those values are substituted by $U(Z_i+Z_j)$ and $(1-U)(Z_i+Z_j)$, respectively, where $U$ is a uniform random variable in $(0,1)$. The energy at the boundary vertices  $b\in\{0,1\}$ is always an exponential random variable with mean $T_b$, the fixed boundary condition. We show that the stationary measure for the resulting Markov process $Z(t)$ is the distribution of a vector $Z$ with coordinates $Z_i=T_iX_i$, where $X_{k}$ are iid exponential$(1)$ random variables, the law of $T$ is the invariant measure for an opinion model with the same boundary conditions, and $X,T$ are independent. The result confirms a conjecture based on the large deviations of the model. The discrete derivative of the opinion model behaves as a neural spiking process, which is also analysed. The hydrostatic limit shows that the empirical measure associated to the stationary distribution converges to the linear interpolation of the boundary values. Joint work with Anna de Masi and Davide Gabrielli, L'Aquila.