Dario Madeo

Game Interactions and Dynamics on Networked Populations

A new mathematical formulation of evolutionary game dynamics on networked populations is proposed. The model extends the standard replicator equation to a finite set of players organized on an arbitrary network of connections (graph). Classical results of multipopulation evolutionary game theory are used in combination with graph theory to obtain the mathematical model. Specifically, the players, located at the vertices of the graph, are interpreted as subpopulations of a multipopulation dynamical game. The members of each subpopulation are replicators, engaged at each time instant into 2-player games with the members of other connected subpopulations. This idea allows us to write an extended equation describing the game dynamics of a finite set of players connected by a graph. The obtained equation does not require any assumption on the game payoff matrices nor graph topology. Stability of steady states, Nash equilibria and the relationship of the proposed model to the standard replicator equation are discussed. The dynamical behavior of the model over different graphs is also investigated by means of extended simulations.

Hypnotic assessment based on the Recurrence Quantification Analysis of EEG recorded in the ordinary state of consciousness.

The cerebral cortical correlates of the susceptibility to hypnosis in the ordinary states of consciousness have not been clarified. Aim of the study was to characterize the EEG dynamics of subjects with high (highs) and low hypnotisability (lows) through the non-linear method of Recurrence Quantification Analysis (RQA). The EEG of 16 males--8 highs and 8 lows--was monitored for 1min without instructions other than keeping the eyes closed, being silent and avoiding movements (short resting), and during 15 min of simple relaxation, that is with the instruction to relax at their best. Highs and lows were compared on the RQA measures of Determinism (DET) and Entropy (ENT), which are related to the signal determinism and complexity. In the short resting condition discriminant analysis could classify highs and lows on the basis of DET and ENT values at temporo-parietal sites. Many differences in DET and all differences in ENT disappeared during simple relaxation, although DET still separated the two groups in the earliest 6min of relaxation at temporo-parietal sites. Our RQA based approach allows to develop computer-based methods of hypnotic assessment using short-lasting, single channel EEG recordings analyzed through standard mathematical methods.

Pain perception and EEG dynamics: Does hypnotizability account for the efficacy of the suggestions of analgesia?

We report novel findings concerning the role of hypnotizability, suggestions of analgesia and the activity of the Behavioral Inhibition/Activation System (BIS/BAS) in the modulation of the subjective experience of pain and of the associated EEG dynamics. The EEG of high (highs) and low hypnotizable participants (lows) who completed the BIS/BAS questionnaire was recorded during basal conditions, tonic nociceptive stimulation without (PAIN) and with suggestions for analgesia (AN). Participants scored the perceived pain intensity at the end of PAIN and AN. The EEG midline dynamics was characterized by indices indicating the signal predictability (Determinism) and complexity (Entropy) obtained through the Recurrence Quantification Analysis. The reduced pain intensity reported by highs during AN was partially accounted for by the activity of the Behavioral Activation System. The decreased midline cortical Determinism observed during nociceptive stimulation in both groups independently of suggestions remained significantly reduced only in lows after controlling for the activity of the Behavioral Activation System. Finally, controlling for the activity of the Behavioral Inhibition System abolished stimulation, suggestions and hypnotizability-related differences. Results indicate that the BIS/BAS activity may be more important than hypnotizability itself in pain modulation and in the associated EEG dynamics.

Original article: Linear least squares parameter estimation of nonlinear reaction diffusion equations

Abstract: This paper concerns with the development of a direct parameter identification procedure for a class of nonlinear reaction-diffusion equations. We assume to know the model equations with the exception of a set of constant parameters, such as diffusivity and reaction term parameters. Using the finite element method the original partial differential equation is transformed into a set of ordinary differential equations. A linear least squares method is then applied to estimate the unknown parameters by using normal equations. The measurements errors obtained following this approach are significantly lower than the error obtained by a nonlinear least squares identification procedure. In order to better understand the differences between the two approaches, a sensitivity analysis with respect to initial conditions and mesh dimension is performed. The robustness of the method is tested on noise corrupted data, showing that the linear least square method may be sensitive to perturbations in the data. The procedure is applied to two ecological models describing the dynamics of population growth.

Lumping evolutionary game dynamics on networks

We study evolutionary game dynamics on networks (EGN), where players reside in the vertices of a graph, and games are played between neighboring vertices. The model is described by a system of ordinary differential equations which depends on players payoff functions, as well as on the adjacency matrix of the underlying graph. Since the number of differential equations increases with the number of vertices in the graph, the analysis of EGN becomes hard for large graphs. Building on the notion of lumpability for Markov chains, we identify conditions on the network structure allowing to reduce the original graph. In particular, we identify a partition of the vertex set of the graph and show that players in the same block of a lumpable partition have equivalent dynamical behaviors, whenever their payoff functions and initial conditions are equivalent. Therefore, vertices belonging to the same partition block can be merged into a single vertex, giving rise to a reduced graph and consequently to a simplified system of equations. We also introduce a tighter condition, called strong lumpability, which can be used to identify dynamical symmetries in EGN which are related to the interchangeability of players in the system.

Cross-evidence for hypnotic susceptibility through nonlinear measures on EEGs of non-hypnotized subjects

Assessment of hypnotic susceptibility is usually obtained through the application of psychological instruments. A satisfying classification obtained through quantitative measures is still missing, although it would be very useful for both diagnostic and clinical purposes. Aiming at investigating the relationship between the cortical brain activity and the hypnotic susceptibility level, we propose the combined use of two methodologies - Recurrence Quantification Analysis and Detrended Fluctuation Analysis - both inherited from nonlinear dynamics. Indicators obtained through the application of these techniques to EEG signals of individuals in their ordinary state of consciousness allowed us to obtain a clear discrimination between subjects with high and low susceptibility to hypnosis. Finally a neural network approach was used to perform classification analysis.