A. M. Batista

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.

Analytical results for coupled-map lattices with long-range interactions

We obtain exact analytical results for lattices of maps with couplings that decay with distance as r(-alpha). We analyze the effect of the coupling range on the system dynamics through the Lyapunov spectrum. For lattices whose elements are piecewise linear maps, we get an algebraic expression for the Lyapunov spectrum. When the local dynamics is given by a nonlinear map, the Lyapunov spectrum for a completely synchronized state is analytically obtained. The critical line characterizing the synchronization transition is determined from the expression for the largest transversal Lyapunov exponent. In particular, it is shown that in the thermodynamical limit, such transition is only possible for sufficiently long-range interactions, namely, for alpha

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.

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.

Chaos synchronization in long-range coupled map lattices

We investigate the synchronization phenomenon in 1D coupled chaotic map lattices where the couplings decay with distance following a power-law. Depending on the number of maps, the coupling strength and the range of the interactions, complete chaos synchronization may be attained. The synchronization domain in the coupling parameter space can be analytically determined by means of the condition of negativity of the largest transversal Lyapunov exponent. In this Letter we use previously found analytical expressions for the synchronization frontier to analyze in detail the role of all the system parameters in the ability of the lattice to achieve complete synchronization. Analytical predictions are shown to be in accord with the outcomes of numerical experiments.

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.

Bursting synchronization in networks with long-range coupling mediated by a diffusing chemical substance

Abstract Many networks of physical and biological interest are characterized by a long-range coupling mediated by a chemical which diffuses through a medium in which oscillators are embedded. We considered a one-dimensional model for this effect for which the diffusion is fast enough so as to be implemented through a coupling whose intensity decays exponentially with the lattice distance. In particular, we analyzed the bursting synchronization of neurons described by two timescales (spiking and bursting activity), and coupled through such a long-range interaction network. One of the advantages of the model is that one can pass from a local (Laplacian) type of coupling to a global (all-to-all) one by varying a single parameter in the interaction term. We characterized bursting synchronization using an order parameter which undergoes a transition as the coupling parameters are changed through a critical value. We also investigated the role of an external time-periodic signal on the bursting synchronization properties of the network. We show potential applications in the control of pathological rhythms in biological neural networks.

Chimera-like states in a neuronal network model of the cat brain

Abstract Neuronal systems have been modelled by complex networks in different description levels. Recently, it has been verified that the networks can simultaneously exhibit one coherent and other incoherent domain, known as chimera states. In this work, we study the existence of chimera-like states in a network considering the connectivity matrix based on the cat cerebral cortex. The cerebral cortex of the cat can be separated in 65 cortical areas organised into the four cognitive regions: visual, auditory, somatosensory-motor and frontolimbic. We consider a network where the local dynamics is given by the Hindmarsh–Rose model. The Hindmarsh–Rose equations are a well known model of the neuronal activity that has been considered to simulate the membrane potential in neuron. Here, we analyse under which conditions chimera-like states are present, as well as the effects induced by intensity of coupling on them. We identify two different kinds of chimera-like states: spiking chimera-like state with desynchronised spikes, and bursting chimera-like state with desynchronised bursts. Moreover, we find that chimera-like states with desynchronised bursts are more robust to neuronal noise than with desynchronised spikes.

Suppression of phase synchronisation in network based on cat's brain

We have studied the effects of perturbations on the cats cerebral cortex. According to the literature, this cortex structure can be described by a clustered network. This way, we construct a clustered network with the same number of areas as in the cat matrix, where each area is described as a sub-network with a small-world property. We focus on the suppression of neuronal phase synchronisation considering different kinds of perturbations. Among the various controlling interventions, we choose three methods: delayed feedback control, external time-periodic driving, and activation of selected neurons. We simulate these interventions to provide a procedure to suppress undesired and pathological abnormal rhythms that can be associated with many forms of synchronisation. In our simulations, we have verified that the efficiency of synchronisation suppression by delayed feedback control is higher than external time-periodic driving and activation of selected neurons of the cats cerebral cortex with the same coupling strengths.

Synchronization threshold in coupled logistic map lattices

We consider regular lattices of coupled chaotic maps. Depending on lattice size, there may exist a window in parameter space where complete synchronization is eventually attained after a transient regime. Close outside this window, an intermittent transition to synchronization occurs. While asymptotic transversal Lyapunov exponents allow to determine the synchronization threshold, the distribution of finite-time Lyapunov exponents, in the vicinity of the critical frontier, is expected to provide relevant information on phenomena such as intermittency. In this work we scrutinize the distribution of finite-time exponents when the local dynamics is ruled by the logistic map