Murilo S. Baptista

Combined effect of chemical and electrical synapses in Hindmarsh-Rose neural networks on synchronization and the rate of information.

In this work we studied the combined action of chemical and electrical synapses in small networks of Hindmarsh-Rose (HR) neurons on the synchronous behavior and on the rate of information produced (per time unit) by the networks. We show that if the chemical synapse is excitatory, the larger the chemical synapse strength used the smaller the electrical synapse strength needed to achieve complete synchronization, and for moderate synaptic strengths one should expect to find desynchronous behavior. Otherwise, if the chemical synapse is inhibitory, the larger the chemical synapse strength used the larger the electrical synapse strength needed to achieve complete synchronization, and for moderate synaptic strengths one should expect to find synchronous behaviors. Finally, we show how to calculate semianalytically an upper bound for the rate of information produced per time unit (Kolmogorov-Sinai entropy) in larger networks. As an application, we show that this upper bound is linearly proportional to the number of neurons in a network whose neurons are highly connected.

Integrated chaos-based communication

Abstract We characterize a recently proposed new integrated communication scheme fully based on chaos theory (Phys. Rev. E 62 (2000) 4835). We show that this method is very efficient, i.e., it sends large amounts of information with low bit-error rate. This is achieved because we explore the deterministic characteristic of a chaotic system, using it to predict non-transmitted signals.

Models for the modern power grid

This article reviews different kinds of models for the electric power grid that can be used to understand the modern power system, the smart grid. From the physical network to abstract energy markets, we identify in the literature different aspects that co-determine the spatio-temporal multilayer dynamics of power system. We start our review by showing how the generation, transmission and distribution characteristics of the traditional power grids are already subject to complex behaviour appearing as a result of the the interplay between dynamics of the nodes and topology, namely synchronisation and cascade effects. When dealing with smart grids, the system complexity increases even more: on top of the physical network of power lines and controllable sources of electricity, the modernisation brings information networks, renewable intermittent generation, market liberalisation, prosumers, among other aspects. In this case, we forecast a dynamical co-evolution of the smart grid and other kind of networked systems that cannot be understood isolated. This review compiles recent results that model electric power grids as complex systems, going beyond pure technological aspects. From this perspective, we then indicate possible ways to incorporate the diverse co-evolving systems into the smart grid model using, for example, network theory and multi-agent simulation.

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.

Mutual Information Rate and Bounds for It

The amount of information exchanged per unit of time between two nodes in a dynamical network or between two data sets is a powerful concept for analysing complex systems. This quantity, known as the mutual information rate (MIR), is calculated from the mutual information, which is rigorously defined only for random systems. Moreover, the definition of mutual information is based on probabilities of significant events. This work offers a simple alternative way to calculate the MIR in dynamical (deterministic) networks or between two time series (not fully deterministic), and to calculate its upper and lower bounds without having to calculate probabilities, but rather in terms of well known and well defined quantities in dynamical systems. As possible applications of our bounds, we study the relationship between synchronisation and the exchange of information in a system of two coupled maps and in experimental networks of coupled oscillators.

Finding quasi-optimal network topologies for information transmission in active networks

This work clarifies the relation between network circuit (topology) and behaviour (information transmission and synchronization) in active networks, e.g. neural networks. As an application, we show how one can find network topologies that are able to transmit a large amount of information, possess a large number of communication channels, and are robust under large variations of the network coupling configuration. This theoretical approach is general and does not depend on the particular dynamic of the elements forming the network, since the network topology can be determined by finding a Laplacian matrix (the matrix that describes the connections and the coupling strengths among the elements) whose eigenvalues satisfy some special conditions. To illustrate our ideas and theoretical approaches, we use neural networks of electrically connected chaotic Hindmarsh-Rose neurons.

Detecting phase synchronization by localized maps: Application to neural networks

We present an approach which enables to state about the existence of phase synchronization in coupled chaotic oscillators without having to measure the phase. This is done by observing the oscillators at special times, and analyzing whether this set of points is localized. In particular, we show that this approach is fruitful to analyze the onset of phase synchronization in chaotic attractors whose phases are not well defined, as well as, in networks of non-identical spiking/bursting neurons connected by chemical synapses.

Collective almost synchronisation in complex networks

This work introduces the phenomenon of Collective Almost Synchronisation (CAS), which describes a universal way of how patterns can appear in complex networks for small coupling strengths. The CAS phenomenon appears due to the existence of an approximately constant local mean field and is characterised by having nodes with trajectories evolving around periodic stable orbits. Common notion based on statistical knowledge would lead one to interpret the appearance of a local constant mean field as a consequence of the fact that the behaviour of each node is not correlated to the behaviours of the others. Contrary to this common notion, we show that various well known weaker forms of synchronisation (almost, time-lag, phase synchronisation, and generalised synchronisation) appear as a result of the onset of an almost constant local mean field. If the memory is formed in a brain by minimising the coupling strength among neurons and maximising the number of possible patterns, then the CAS phenomenon is a plausible explanation for it.

Dynamics of the kicked logistic map

Abstract The modulation of the logistic map by a sequence of periodic kicks brings up a three-parameter kicked logistic map (klm) with new distinct dynamic features. Thus, its parameter space structure exhibits highly interleaved sets with different attractors, and complex basins of attraction are created. Additional roots to chaos and abrupt attractor changes are identified in the parameter space. The observed intermittency route to chaos is distinct to those typical of spatially discontinuous unidimensional maps, with a characteristic power-law dependence of the average laminar length on the control parameters. This behaviour is verified for both the internal and the transfer crisis-induced intermittency.

Simple and robust solver for the Poisson-Boltzmann equation

A variational approach is used to develop a robust numerical procedure for solving the nonlinear Poisson-Boltzmann equation. Following Maggs and Rossetto [Phys. Rev. Lett. 88, 196402 (2002)], we construct an appropriate constrained free energy functional such that its Euler-Lagrange equations are equivalent to the Poisson-Boltzmann equation. This is a formulation that searches for a true minimum in function space, in contrast to previous variational approaches that rather searched for a saddle point. We then develop, implement, and test an algorithm for its numerical minimization, which is quite simple and unconditionally stable. The analytic solution for planar geometry is used for validation. Some results are presented for a charged colloidal sphere surrounded by counterions and optimizations based upon fast Fourier transforms and hierarchical preconditioning are briefly discussed.