The paper deals with the problem of reconstructing the tree-like topological structure of a network of linear dynamical systems. A distance function is defined in order to evaluate the “closeness” of two processes and some useful mathematical properties are derived. Theoretical results to guarantee the correctness of the identification procedure for networked linear systems characterized by a tree topology are provided as well. The paper also suggests the approximation of a complex connected network with a tree in order to detect the most meaningful interconnections. The application of the techniques to the analysis of an actual complex network, i.e., to high frequency time series of the stock market, is extensively illustrated.

The dynamical phases of the Hindmarsh-Rose neuronal model are analyzed in detail by varying the external current I. For increasing current values, the model exhibits a peculiar cascade of nonchaotic and chaotic period-adding bifurcations leading the system from the silent regime to a chaotic state dominated by bursting events. At higher I-values, this phase is substituted by a regime of continuous chaotic spiking and finally via an inverse period doubling cascade the system returns to silence. The analysis is focused on the transition between the two chaotic phases displayed by the model: one dominated by spiking dynamics and the other by bursts. At the transition an abrupt shrinking of the attractor size associated with a sharp peak in the maximal Lyapunov exponent is observable. However, the transition appears to be continuous and smoothed out over a finite current interval, where bursts and spikes coexist. The beginning of the transition (from the bursting side) is signaled from a structural modification in the interspike interval return map. This change in the map shape is associated with the disappearance of the family of solutions responsible for the onset of the bursting chaos. The successive passage from bursting to spiking chaos is associated with a progressive pruning of unstable long-lasting bursts.

The paper considers the neuron model of Hindmarsh–Rose and studies in detail the system dynamics which controls the transition between the spiking and bursting regimes. In particular, such a passage occurs in a chaotic region and different explanations have been given in the literature to represent the process, generally based on a slow-fast decomposition of the neuron model. This paper proposes a novel view of the chaotic spiking-bursting transition exploiting the whole system dynamics and putting in evidence the essential role played in the phenomenon by the manifolds of the equilibrium point. An analytical approximation is developed for the related crucial elements and a subsequent numerical analysis signifies the properness of the suggested conjecture.