  We analyze the stability of recurrent networks, specifically, reservoir
computing models during training by evaluating the eigenvalue spectra of the
reservoir dynamics. To circumvent the instability arising in examining a closed
loop reservoir system with feedback, we propose to break the closed loop
system. Essentially, we unroll the reservoir dynamics over time while
incorporating the feedback effects that preserve the overall temporal integrity
of the system. We evaluate our methodology for fixed point and time varying
targets with least squares regression and FORCE training, respectively. Our
analysis establishes eigenvalue spectra (which is, shrinking of spectral circle
as training progresses) as a valid and effective metric to gauge the
convergence of training as well as the convergence of the chaotic activity of
the reservoir toward stable states.
