The echo state network (ESN) is a recurrent neural network with a sparsely connected hidden layer (with typically 1% connectivity). The connectivity and weights of hidden neurons are randomly assigned and are fixed. The weights of output neurons can be learned so that the network can (re)produce specific temporal patterns.
The main interest of this network is that although its behaviour is non-linear, the only parameters are the weights of the output layer. The error function is thus quadratic with respect to the parameter vector and can be differentiated easily to a linear system.