Francisco Javier Torrealdea

FEEDBACK SYNCHRONIZATION OF CHAOTIC SYSTEMS

Feedback coupling through an interaction term proportional to the difference in the value of some behavioral characteristics of two systems is a very common structural setting that leads to synchronization of the behavior of both systems. The degree of synchronization attained depends on the strength of the interaction term and on the mutual interdependency of the structures of both systems. In this paper, we show that two chaotic systems linked through a feedback coupling interaction term of gain parameter k reach a synchronized regime characterized by a vector of variable errors which tends towards zero with parameter k while the interaction term tends towards a finite nonzero permanent regime. This means that maintaining a certain degree of synchronization has a cost. In the limit, complete synchronization occurs at a finite limit cost. We show that feedback coupling in itself brings about conditions permitting that systems with a degree of structural parameter flexibility evolve close towards each other structures in order to facilitate the maintenance of the synchronized regime. In this paper, we deduce parameter adaptive laws for any family of homochaotic systems provided they are previously forced to work, via feedback coupling, within an appropriate degree of synchronization. The laws are global in the space of parameters and lead eventually to identical synchronization at no interaction cost. We illustrate this point with homochaotic systems from the Lorenz, Rossler and Chua families.

Energy and information in Hodgkin-Huxley neurons.

The generation of spikes by neurons is energetically a costly process and the evaluation of the metabolic energy required to maintain the signaling activity of neurons a challenge of practical interest. Neuron models are frequently used to represent the dynamics of real neurons but hardly ever to evaluate the electrochemical energy required to maintain that dynamics. This paper discusses the interpretation of a Hodgkin-Huxley circuit as an energy model for real biological neurons and uses it to evaluate the consumption of metabolic energy in the transmission of information between neurons coupled by electrical synapses, i.e., gap junctions. We show that for a single postsynaptic neuron maximum energy efficiency, measured in bits of mutual information per molecule of adenosine triphosphate (ATP) consumed, requires maximum energy consumption. For groups of parallel postsynaptic neurons we determine values of the synaptic conductance at which the energy efficiency of the transmission presents clear maxima at relatively very low values of metabolic energy consumption. Contrary to what could be expected, the best performance occurs at a low energy cost.

Hierarchically structured systems

Abstract Obvious reasons of methodological praxis lead to the fact that in our studies of systems these appear as entities situated in an environment without structure. The hierarchical conception of systems induces a structure in the environment in a natural way making its systemic nature evident, as well as the necessity of its role as controll of the system under study. The functions of optimization of the systems goal are shared by the system itself and by its environment and, in this context, the system can interact in competition or in cooperation. Finally, the hierarchical modelling in System Dynamics, that is, the construction of a set of models of progressively decreasing abstraction, is put forward as a method to enhance the structure of the systems.

Cost of synchronizing different chaotic systems

Feedback coupling provides a general scheme for synchronizing two oscillatory chaotic systems through the intervention of a term of interaction that accounts for the difference of behaviors. We define a cost of synchronization based on a measure of the interaction term. Synchronizing different systems is not cost free and the cost increases with the requirements imposed on the synchronized behavior. We prove that many systems can reach a regime of complete synchronization at a limited, and a priory computable, cost. For identical systems, the cost of complete synchronization is zero. Some different systems can also keep a completely synchronized behavior in some of their variables at zero cost. We propose to reserve the name identical synchronization for complete synchronization at zero cost. We compute the cost for different stages of synchronization between two systems as different as the Rossler and Lorenz systems and for homochaotic cases of both families. If the response system is flexible enough to adapt to the structure of the driving system, lower synchronization cost or, eventually, identical synchronization will be possible. In this paper, we deduce adaptation laws to reach identical synchronization for any family of homochaotic systems, and we illustrate their application for the Rossler and Lorenz cases.

Energy efficiency of information transmission by electrically coupled neurons.

The generation of spikes by neurons is energetically a costly process. This paper studies the consumption of energy and the information entropy in the signalling activity of a model neuron both when it is supposed isolated and when it is coupled to another neuron by an electrical synapse. The neuron has been modelled by a four-dimensional Hindmarsh-Rose type kinetic model for which an energy function has been deduced. For the isolated neuron values of energy consumption and information entropy at different signalling regimes have been computed. For two neurons coupled by a gap junction we have analyzed the roles of the membrane and synapse in the contribution of the energy that is required for their organized signalling. Computational results are provided for cases of identical and nonidentical neurons coupled by unidirectional and bidirectional gap junctions. One relevant result is that there are values of the coupling strength at which the organized signalling of two neurons induced by the gap junction takes place at relatively low values of energy consumption and the ratio of mutual information to energy consumption is relatively high. Therefore, communicating at these coupling values could be energetically the most efficient option.

A parametric gradient descent MRI intensity inhomogeneity correction algorithm

Given an appropriate imaging resolution, a common Magnetic Resonance Imaging (MRI) model assumes that the object under study is composed of homogeneous tissues whose imaging intensity is constant, so that MRI produces piecewise constant images. The intensity inhomogeneity (IIH) is modeled by a multiplicative inhomogeneity field. It is due to the spatial inhomogeneity in the excitatory Radio Frequency (RF) signal and other effects. It has been acknowledged as a greater source of error for automatic segmentation algorithms than additive noise. We propose a parametric IIH correction algorithm for MRI that consists of the gradient descent of an error function related to the classification error of the IIH corrected image. The inhomogeneity field is modeled as a linear combination of 3D products of Legendre polynomials. In this letter we test both the image restoration capabilities and the classification accuracy of the algorithm. In restoration processes the adaptive algorithm is used only to estimate the inhomogeneity field. Test images to be restored are IIH corrupted versions of the BrainWeb site simulations. The algorithm image restoration is evaluated by the correlation of the restored image with the known clean image. In classification processes the algorithm is used to estimate both the inhomogeneity field and the intensity class means. The algorithm classification accuracy is tested over the images from the IBSR site. The proposed algorithm is compared with Maximum A Posteriori (MAP) and Fuzzy Clustering algorithms.

Efficient synchronization of structurally adaptive coupled Hindmarsh–Rose neurons

Abstract The use of spikes to carry information between brain areas implies complete or partial synchronization of the neurons involved. The degree of synchronization reached by two coupled systems and the energy cost of maintaining their synchronized behavior is highly dependent on the nature of the systems. For non-identical systems the maintenance of a synchronized regime is energetically a costly process. In this work, we study conditions under which two non-identical electrically coupled neurons can reach an efficient regime of synchronization at low energy cost. We show that the energy consumption required to keep the synchronized regime can be spontaneously reduced if the receiving neuron has adaptive mechanisms able to bring its biological parameters closer in value to the corresponding ones in the sending neuron.

Solving satisfiability via Boltzmann machines

Boltzmann machines (BMs) are proposed as a computational model for the solution of the satisfiability (SAT) problem in the propositional calculus setting. Conditions that guarantee consensus function maxima for configurations of the BM associated with solutions to the satisfaction problem are given. Experimental results that show a linear behavior of BMs solving the satisfiability problem are presented and discussed. >

Experiments of Fast Learning with High Order Boltzmann Machines

This work reports the results obtained with the application of High Order Boltzmann Machines without hidden units to construct classifiers for some problems that represent different learning paradigms. The Boltzmann Machine weight updating algorithm remains the same even when some of the units can take values in a discrete set or in a continuous interval. The absence of hidden units and the restriction to classification problems allows for the estimation of the connection statistics, without the computational cost involved in the application of simulated annealing. In this setting, the learning process can be sped up several orders of magnitude with no appreciable loss of quality of the results obtained.

ENERGY-LIKE FUNCTIONS FOR SOME DISSIPATIVE CHAOTIC SYSTEMS

Functions of the phase space variables that can considered as possible energy functions for a given family of dissipative chaotic systems are discussed. This kind of functions are interesting due to their use as an energy-like quantitative measure to characterize different aspects of dynamic behavior of associated chaotic systems. We have calculated quadratic energy-like functions for the cases of Lorenz, Chen, Lu–Chen and Chua, and show the patterns of dissipation of energy on their respective attractors. We also show that in the case of the Rossler system at least a fourth-order polynomial is required to properly represent its energy.