In many realistic networks, the edges representing the interactions between nodes are time varying. Evidence is growing that the complex network that models the dynamics of the human brain has time-varying interconnections, that is, the network is evolving. Based on this evidence, we construct a patient- and data-specific evolving network model (comprising discrete-time dynamical systems) in which epileptic seizures or their terminations in the brain are also determined by the nature of the time-varying interconnections between the nodes. A novel and unique feature of our methodology is that the evolving network model remembers the data from which it was conceived from, in the sense that it evolves to almost regenerate the patient data even on presenting an arbitrary initial condition to it. We illustrate a potential utility of our methodology by constructing an evolving network from clinical data that aids in identifying an approximate seizure focus; nodes in such a theoretically determined seizure focus are outgoing hubs that apparently act as spreaders of seizures. We also point out the efficacy of removal of such spreaders in limiting seizures.

The echo state property is a key for the design and training of recurrent neural networks within the paradigm of reservoir computing. In intuitive terms, this is a passivity condition: a network having this property, when driven by an input signal, will become entrained by the input and develop an internal response signal. This excited internal dynamics can be seen as a high-dimensional, nonlinear, unique transform of the input with a rich memory content. This view has implications for understanding neural dynamics beyond the field of reservoir computing. Available definitions and theorems concerning the echo state property, however, are of little practical use because they do not relate the network response to temporal or statistical properties of the driving input. Here we present a new definition of the echo state property that directly connects it to such properties. We derive a fundamental 0-1 law: if the input comes from an ergodic source, the network response has the echo state property with probability one or zero, independent of the given network. Furthermore, we give a sufficient condition for the echo state property that connects statistical characteristics of the input to algebraic properties of the network connection matrix. The mathematical methods that we employ are freshly imported from the young field of nonautonomous dynamical systems theory. Since these methods are not yet well known in neural computation research, we introduce them in some detail. As a side story, we hope to demonstrate the eminent usefulness of these methods.