S.R. Lopes

Delayed feedback control of bursting synchronization in a scale-free neuronal network

Several neurological diseases (e.g. essential tremor and Parkinsons disease) are related to pathologically enhanced synchronization of bursting neurons. Suppression of these synchronized rhythms has potential implications in electrical deep-brain stimulation research. We consider a simplified model of a neuronal network where the local dynamics presents a bursting timescale, and the connection architecture displays the scale-free property (power-law distribution of connectivity). The networks exhibit collective oscillations in the form of synchronized bursting rhythms, without affecting the fast timescale dynamics. We investigate the suppression of these synchronized oscillations using a feedback control in the form of a time-delayed signal. We located domains of bursting synchronization suppression in terms of perturbation strength and time delay, and present computational evidence that synchronization suppression is easier in scale-free networks than in the more commonly studied global (mean-field) networks.

Collective behavior in a chain of van der Pol oscillators with power-law coupling

We consider the synchronization properties of a one-dimensional chain of coupled van der Pol oscillators. The uncoupled oscillators have an attracting limit-cycle with a given normal mode frequency. We introduce a given level of randomness in the normal mode distribution of the oscillators and study the conditions under which the chain synchronizes. The coupling depends on the distance along the lattice in a power-law fashion. There is a frequency synchronization transition as we pass continuously from a global to a local coupling. We observe phase and lag synchronization transition for other coupling regimes.

Transport properties in nontwist area-preserving maps.

Nontwist systems, common in the dynamical descriptions of fluids and plasmas, possess a shearless curve with a concomitant transport barrier that eliminates or reduces chaotic transport, even after its breakdown. In order to investigate the transport properties of nontwist systems, we analyze the barrier escape time and barrier transmissivity for the standard nontwist map, a paradigm of such systems. We interpret the sensitive dependence of these quantities upon map parameters by investigating chaotic orbit stickiness and the associated role played by the dominant crossing of stable and unstable manifolds.

Nonlinear three-mode interaction and drift-wave turbulence in a tokamak edge plasma

A three-wave interaction model with quadratic nonlinearities and linear growth/decay rates is used to investigate the occurrence of drift-wave turbulence driven by pressure gradients in the edge plasma of a tokamak. Model parameters are taken from a typical set of measurements of the floating electrostatic potential in the tokamak edge region. Some aspects of the temporal dynamics exhibited by the three-wave interaction model are investigated, with special emphasis on a chaotic regime found for a wide range of the wave decay rate. An intermittent transition from periodic to chaotic behavior is observed and some statistical properties, such as the interburst and laminar length interval durations, are explored.

Phase synchronization of coupled bursting neurons and the generalized Kuramoto model.

Bursting neurons fire rapid sequences of action potential spikes followed by a quiescent period. The basic dynamical mechanism of bursting is the slow currents that modulate a fast spiking activity caused by rapid ionic currents. Minimal models of bursting neurons must include both effects. We considered one of these models and its relation with a generalized Kuramoto model, thanks to the definition of a geometrical phase for bursting and a corresponding frequency. We considered neuronal networks with different connection topologies and investigated the transition from a non-synchronized to a partially phase-synchronized state as the coupling strength is varied. The numerically determined critical coupling strength value for this transition to occur is compared with theoretical results valid for the generalized Kuramoto model.

Model for tumour growth with treatment by continuous and pulsed chemotherapy

In this work we investigate a mathematical model describing tumour growth under a treatment by chemotherapy that incorporates time-delay related to the conversion from resting to hunting cells. We study the model using values for the parameters according to experimental results and vary some parameters relevant to the treatment of cancer. We find that our model exhibits a dynamical behaviour associated with the suppression of cancer cells, when either continuous or pulsed chemotherapy is applied according to clinical protocols, for a large range of relevant parameters. When the chemotherapy is successful, the predation coefficient of the chemotherapic agent acting on cancer cells varies with the infusion rate of chemotherapy according to an inverse relation. Finally, our model was able to reproduce the experimental results obtained by Michor and collaborators [Nature 435 (2005) 1267] about the exponential decline of cancer cells when patients are treated with the drug glivec.

Effective transport barriers in nontwist systems

In fluids and plasmas with zonal flow reversed shear, a peculiar kind of transport barrier appears in the shearless region, one that is associated with a proper route of transition to chaos. These barriers have been identified in symplectic nontwist maps that model such zonal flows. We use the so-called standard nontwist map, a paradigmatic example of nontwist systems, to analyze the parameter dependence of the transport through a broken shearless barrier. On varying a proper control parameter, we identify the onset of structures with high stickiness that give rise to an effective barrier near the broken shearless curve. Moreover, we show how these stickiness structures, and the concomitant transport reduction in the shearless region, are determined by a homoclinic tangle of the remaining dominant twin island chains. We use the finite-time rotation number, a recently proposed diagnostic, to identify transport barriers that separate different regions of stickiness. The identified barriers are comparable to those obtained by using finite-time Lyapunov exponents.

Mode locking in small-world networks of coupled circle maps

We investigate the emergence of mode locking, or frequency synchronization, in a chain of coupled sine-circle maps with randomly distributed parameters, and exhibiting the small-world property. The coupling prescription we adopt considers the nearest and next-to-the-nearest neighbors of a given site, as well as randomly chosen non-local shortcuts, according to a given probability. A transition between synchronized and non-synchronized patterns is observed as this probability is varied. We also study the statistics of the synchronization plateaus, evidencing a Poisson-type distribution.

On a cellular automaton with time delay for modelling cancer tumors

In this work we considered cellular automaton model with time delay. Time delay included in this model reflects the delay between the time in which the site is affected and the time in which its variable is updated. We analyzed the effect of the rules on the dynamics through the cluster counting. According to this cluster counting, the dynamics behavior is investigated. We verified periodic oscillations same as delay differential equation. We also studied the relation between the time delay in the cell cycle and the time to start the metastasis, using suitable numerical diagnostics.

Transition to chaos in the conservative four-wave parametric interactions

In this work, we analyze the transition from regular to chaotic states in the parametric four-wave interactions. The temporal evolution describing the coupling of two sets of three-waves with quadratic nonlinearity is considered. This system is shown to undergo a chaotic transition via the separatrix chaos scenario, where a soliton-like solution (separatrix) that is found for the integrable (perfect matched) case becomes irregular as a small mismatch is turned on. As the mismatch is increased the separatrix chaotic layer spreads along the phase space, eventually engrossing most part of it. This scenario is typical of low-dimensional Hamiltonian systems.