Echo State Neural Network (ESN) has becoming much attractive in Artificial Neural Network (ANN) community since it can be easily constructed and adapted. Its superiority over traditional ANN has been demonstrated in many applications. The Wiener-Hopf solution is usually exploited to account for the adaptations. However, the solution can hardly ensure the Lyapunov stability of the trained Leaky-integrator ESNs (LiESNs), when they run in a closed-loop autonomously generative mode. LiESN is another type of ESN, which consists of leaky-integrator neurons. In this study, a sufficient condition of the Lyapunov stability for the autonomously running LiESNs is proposed and proved at first. And then, the output connection weight learning problem is translated into an optimization problem with a nonlinear restriction. Particle swarm optimization algorithm is explored to solve the optimization problem. The simulation experiment results show that the output weight adaptation algorithm, we proposed (we call it PSOESN) can effectively ensure the output precision as well as the Lyapunov stability of the trained LiESNs. It is concluded that the PSOESN is a more effective solution to the output connection weight adaptation problem of such autonomously running ESNs.