" Nonlinear Signal Processing of Electroencephalogram .' Application in the Study of Neurodynamics. "

Interfacings of various subjects generate new field ofstudy and research that help in advancing human knowledge. One of the latest of such fields is Neurotechnology, which is an effective amalgamation of neuroscience, physics, biomedical engineering and computational methods. Neurotechnology provides a platform to interact physicist; neurologist and engineers to break methodology and terminology related barriers. Advancements in Computational capability, wider scope of applications in nonlinear dynamics and chaos in complex systems enhanced study of neurodynamics. However there is a need for an effective dialogue among physicists, neurologists and engineers. Application of computer based technology in the field of medicine through signal and image processing, creation of clinical databases for helping clinicians etc are widely acknowledged. Such synergic effects between widely separated disciplines may help in enhancing the effectiveness of existing diagnostic methods. One of the recent methods in this direction is analysis of electroencephalogram with the help of methods in nonlinear dynamics. This thesis is an effort to understand the functional aspects of human brain by studying electroencephalogram. The algorithms and other related methods developed in the present work can be interfaced with a digital EEG machine to unfold the information hidden in the signal. Ultimately this can be used as a diagnostic tool.

Interface dynamics in planar neural field models

Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves.

Multiple stationary solutions of an irradiated slab

A mathematical model describing the heat budget of an irradiated medium is introduced. The one-dimensional form of the equations and boundary conditions are presented and analysed. Heat transport at one face of the slab occurs by absorption (and reflection) of an incoming beam of short-wave radiation with a fraction of this radiation penetrating into the body of the slab, a diffusive heat flux in the slab and a prescribed incoming heat flux term. The other face of the slab is immersed in its own melt and is considered to be a free surface. Here, temperature continuity is prescribed and evolution of the surface is determined by a Stefan condition. These boundary conditions are flexible enough to describe a range of situations such as a laser shining on an opaque medium, or the natural environment of polar sea ice or lake ice. A two-stream radiation model is used which replaces the simple Beer’s law of radiation attenuation frequently used for semi-infinite domains. The stationary solutions of the governing equations are sought and it is found that there exists two possible stationary solutions for a given set of boundary conditions and a range of parameter choices. It is found that the existence of two stationary solutions is a direct result of the model of radiation absorption, due to its effect on the albedo of the medium. A linear stability analysis and numerical calculations indicate that where two stationary solutions exist, the solution corresponding to a larger thickness is always stable and the solution corresponding to a smaller thickness is unstable. Numerical simulations reveal that when there are two solutions, if the slab is thinner than the smaller stationary thickness it will melt completely, whereas if the slab is thicker than the smaller stationary thickness it will evolve toward the larger stationary thickness. These results indicate that other mechanisms (e.g. wave-induced agglomeration of crystals) are necessary to grow a slab from zero initial thickness in the parameter regime that yields two stationary solutions.

A mathematical model of crystallization in an emulsion

A mathematical model incorporating many of the important processes at work in the crystallization of emulsions is presented. The model describes nucleation within the discontinuous domain of an emulsion, precipitation in the continuous domain, transport of monomers between the two domains, and formation and subsequent growth of crystals in both domains. The model is formulated as an autonomous system of nonlinear, coupled ordinary differential equations. The description of nucleation and precipitation is based upon the Becker–Döring equations of classical nucleation theory. A particular feature of the model is that the number of particles of all species present is explicitly conserved; this differs from work that employs Arrhenius descriptions of nucleation rate. Since the model includes many physical effects, it is analyzed in stages so that the role of each process may be understood. When precipitation occurs in the continuous domain, the concentration of monomers falls below the equilibrium concentration at the surface of the drops of the discontinuous domain. This leads to a transport of monomers from the drops into the continuous domain that are then incorporated into crystals and nuclei. Since the formation of crystals is irreversible and their subsequent growth inevitable, crystals forming in the continuous domain effectively act as a sink for monomers “sucking” monomers from the drops. In this case, numerical calculations are presented which are consistent with experimental observations. In the case in which critical crystal formation does not occur, the stationary solution is found and a linear stability analysis is performed. Bifurcation diagrams describing the loci of stationary solutions, which may be multiple, are numerically calculated.

A mathematical model of the sterol regulatory element binding protein 2 cholesterol biosynthesis pathway

Cholesterol is one of the key constituents for maintaining the cellular membrane and thus the integrity of the cell itself. In contrast high levels of cholesterol in the blood are known to be a major risk factor in the development of cardiovascular disease. We formulate a deterministic nonlinear ordinary differential equation model of the sterol regulatory element binding protein 2 (SREBP-2) cholesterol genetic regulatory pathway in an hepatocyte. The mathematical model includes a description of genetic transcription by SREBP-2 which is subsequently translated to mRNA leading to the formation of 3-hydroxy-3-methylglutaryl coenzyme A reductase (HMGCR), a main precursor of cholesterol synthesis. Cholesterol synthesis subsequently leads to the regulation of SREBP-2 via a negative feedback formulation. Parameterised with data from the literature, the model is used to understand how SREBP-2 transcription and regulation affects cellular cholesterol concentration. Model stability analysis shows that the only positive steady-state of the system exhibits purely oscillatory, damped oscillatory or monotic behaviour under certain parameter conditions. In light of our findings we postulate how cholesterol homestasis is maintained within the cell and the advantages of our model formulation are discussed with respect to other models of genetic regulation within the literature.

Phytoplankton–zooplankton dynamics in the ‘presence’ or ‘absence’ of toxic phytoplankton

The inhibitory effects of toxin-producing phytoplankton (TPP) on zooplankton modulate the dynamics of marine plankton. In this article, we employ simple mathematical models to compare theoretically the dynamics of phytoplankton–zooplankton interaction in situations where the TPP are present with those where TPP are absent. We consider two sets of three-component interaction models: one that does not include the effect of TPP and the other that does. The negative effects of TPP on zooplankton is described by a non-linear interaction term. Extensive theoretical analyses of the models have been performed to understand the qualitative behaviour of the model systems around every possible equilibria. The results of local-stability analysis and numerical simulations demonstrate that the two model-systems differ qualitatively with regard to oscillations and stability. The model system that does not include TPP is asymptotically stable around the coexisting equilibria, whereas, the system that includes TPP oscillates for a range of parametric values associated with toxin-inhibition rate and competition coefficients. Our analysis suggests that the qualitative dynamics of the plankton–zooplankton interactions are very likely to alter due to the presence of TPP species, and therefore the effects of TPP should be considered carefully while modelling plankton dynamics.

Modeling migratory grazing of zooplankton on toxic and non-toxic phytoplankton

Migratory grazing of zooplankton between non-toxic phytoplankton (NTP) and toxic phytoplankton (TPP) is a realistic phenomena unexplored so far. The present article is a first step in this direction. A mathematical model of NTP–TPP-zooplankton with constant and variable zooplankton migration is proposed and analyzed. The asymptotic dynamics of the model system around the biologically feasible equilibria is explored through local stability analysis. The dynamics of the proposed system is explored and displayed for different combination of migratory parameters and toxin inhibition parameters. Our analysis suggests that the migratory grazing of zooplankton has a significant role in determining the dynamic stability and oscillation of phytoplankton zooplankton systems.

Numerical analyses of Runge–Kutta implicit–explicit schemes for horizontally explicit, vertically implicit solutions of atmospheric models

With the prospect of exascale computing, computational methods requiring only local data become especially attractive. Consequently, the typical domain decomposition of atmospheric models means horizontally-explicit vertically-implicit (HEVI) time-stepping schemes warrant further attention. In this analysis, Runge-Kutta implicit-explicit schemes from the literature are analysed for their stability and accuracy using a von Neumann stability analysis of two linear systems. Attention is paid to the numerical phase to indicate the behaviour of phase and group velocities. Where the analysis is tractable, analytically derived expressions are considered. For more complicated cases, amplification factors have been numerically generated and the associated amplitudes and phase diagnosed. Analysis of a system describing acoustic waves has necessitated attributing the three resultant eigenvalues to the three physical modes of the system. To do so, a series of algorithms has been devised to track the eigenvalues across the frequency space. The result enables analysis of whether the schemes exactly preserve the non-divergent mode; and whether there is evidence of spurious reversal in the direction of group velocities or asymmetry in the damping for the pair of acoustic modes. Frequency ranges that span next-generation high-resolution weather models to coarse-resolution climate models are considered; and a comparison is made of errors accumulated from multiple stability-constrained shorter time-steps from the HEVI scheme with a single integration from a fully implicit scheme over the same time interval. Two schemes, “Trap2(2,3,2)” and “UJ3(1,3,2)”, both already used in atmospheric models, are identified as offering consistently good stability and representation of phase across all the analyses. Furthermore, according to a simple measure of computational cost, “Trap2(2,3,2)” is the least expensive.

Reliability of orbital fits for resonant extrasolar planetary systems: the case of HD82943

We perform a statistical study of the process of orbital determination of the HD82943 extrasolar planetary system, using the current observational data set of N = 165 radial velocity (RV) measurements. Our aim is to analyse the dispersion of possible orbital fits leading to residuals compatible with the best solution, and to discuss the sensitivity of the results with respect to both the data set and the error distribution around the best fit. Although some orbital parameters (e.g. semimajor axis) appear well constrained, we show that the best fits for the HD82943 system are not robust, and at present it is not possible to estimate reliable solutions for these bodies. Finally, we discuss the possibility of a third planet, with a mass of 0.35M(Jup) and an orbital period of 900 d. Stability analysis and simulations of planetary migration indicate that such a hypothetical three-planet system could be locked in a double 2/1 mean-motion resonance, similar to the so-called Laplace resonance of the three inner Galilean satellites of Jupiter.

Estabilidade e robustez de um controlador adaptativo indireto por um modelo de referência e estrutura variável

In this thesis, it is developed the robustness and stability analysis of a variable structure model reference adaptive controller considering the presence of disturbances and unmodeled dynamics. The controller is applied to uncertain, monovariable, linear time-invariant plants with relative degree one, and its development is based on the indirect adaptive control. In the direct approach, well known in the literature, the switching laws are designed for the controller parameters. In the indirect one, they are designed for the plant parameters and, thus, the selection of the relays upper bounds becomes more intuitive, whereas they are related to physical parameters, which present uncertainties that can be known easier, such as resistances, capacitances, inertia moments and friction coefficients. Two versions for the controller algorithm with the stability analysis are presented. The global asymptotic stability with respect to a compact set is guaranteed for both cases. Simulation results under adverse operation conditions in order to verify the theoretical results and to show the performance and robustness of the proposed controller are showed. Moreover, for practical purposes, some simplifications on the original algorithm are developed

Contribuições à análise de robustez de sistemas de controle usando redes neurais

This work develops a robustness analysis with respect to the modeling errors, being applied to the strategies of indirect control using Artificial Neural Networks - ANN s, belong to the multilayer feedforward perceptron class with on-line training based on gradient method (backpropagation). The presented schemes are called Indirect Hybrid Control and Indirect Neural Control. They are presented two Robustness Theorems, being one for each proposed indirect control scheme, which allow the computation of the maximum steady-state control error that will occur due to the modeling error what is caused by the neural identifier, either for the closed loop configuration having a conventional controller - Indirect Hybrid Control, or for the closed loop configuration having a neural controller - Indirect Neural Control. Considering that the robustness analysis is restrict only to the steady-state plant behavior, this work also includes a stability analysis transcription that is suitable for multilayer perceptron class of ANN s trained with backpropagation algorithm, to assure the convergence and stability of the used neural systems. By other side, the boundness of the initial transient behavior is assured by the assumption that the plant is BIBO (Bounded Input, Bounded Output) stable. The Robustness Theorems were tested on the proposed indirect control strategies, while applied to regulation control of simulated examples using nonlinear plants, and its results are presented

Estabilidade de agregados do solo após manejo com rotações de culturas e escarificação

O objetivo desse trabalho foi avaliar, em solo compactado, a estabilidade dos agregados influenciada pelo cultivo de espécies de cobertura em esquema de rotações de culturas, em sistema de semeadura direta, mediante o efeito da escarificação mecânica. As rotações de culturas repetidas por três anos consecutivos envolveram o cultivo de triticale e girassol, no outono-inverno, associados ao cultivo de milheto, de sorgo forrageiro e de Crotalária júncea como plantas de cobertura, antecedendo o da soja (cultura de verão). No tratamento envolvendo a escarificação mecânica, a área permaneceu em pousio entre os cultivos de outono-inverno e de verão. O experimento foi realizado na Fazenda Experimental Lageado (Botucatu-SP), nos anos agrícolas de 2003/2004, 2004/2005 e 2005/2006. O delineamento experimental foi o de blocos ao acaso, com quatro repetições, em esquema de parcelas subdivididas. As amostras para a análise da estabilidade de agregados foram coletadas nas profundidades de 0 a 0,05 m e de 0,05 a 0,10 m após o manejo das plantas de cobertura em dezembro de 2003 e de 2005. Logo no primeiro ano de instalação do experimento, o cultivo de triticale resultou em maior porcentagem de agregados com mais de 2 mm, maior DMG e maior DMP na camada de 0 a 5 cm, além de maior DMP na camada de 0,05 a 0,10 m. Já a escarificação do solo e a ausência do cultivo de plantas de cobertura proporcionaram menor porcentagem de agregados maiores que 2 mm e menor DMP na camada de 0,05 a 0,10 m. A estabilidade dos agregados foi influenciada pela rotação de culturas, sendo maior na camada de 0 a 0,05 m e de 0,05 a 0,10 m quando o triticale foi introduzido como espécie de outono-inverno.

Prospective study for the development of emulsion systems containing natural oil products

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Efeitos de acoplamentos biquadráticos e campos aleatórios em vidros de spins

In this work we have studied the effects of random biquadratic and random fields in spin-glass models using the replica method. The effect of a random biquadratic coupling was studied in two spin-1 spin-glass models: in one case the interactions occur between pairs of spins, whereas in the second one the interactions occur between p spins and the limit p > oo is considered. Both couplings (spin glass and biquadratic) have zero-mean Gaussian probability distributions. In the first model, the replica-symmetric assumption reveals that the system presents two pha¬ses, namely, paramagnetic and spin-glass, separated by a continuous transition line. The stability analysis of the replica-symmetric solution yields, besides the usual instability associated with the spin-glass ordering, a new phase due to the random biquadratic cou¬plings between the spins. For the case p oo, the replica-symmetric assumption yields again only two phases, namely, paramagnetic and quadrupolar. In both these phases the spin-glass parameter is zero. Besides, it is shown that they are stable under the Almeida-Thouless stability analysis. One of them presents negative entropy at low temperatures. We developed one step of replica simmetry breaking and noticed that a new phase, the biquadratic glass phase, emerge. In this way we have obtained the correct phase diagram, with.three first-order transition lines. These lines merges in a common triple point. The effects of random fields were studied in the Sherrington-Kirkpatrick model consi¬dered in the presence of an external random magnetic field following a trimodal distribu¬tion {P{hi) = p+S(hi - h0) +Po${hi) +pS(hi + h0))- It is shown that the border of the ferromagnetic phase may present, for conveniently chosen values of p0 and hQ, first-order phase transitions, as well as tricritical points at finite temperatures. It is verified that the first-order phase transitions are directly related to the dilution in the fields: the extensions of these transitions are reduced for increasing values of po- In fact, the threshold value pg, above which all phase transitions are continuous, is calculated analytically. The stability analysis of the replica-symmetric solution is performed and the regions of validity of such a solution are identified

On relaxed LMI-based design for fuzzy controllers

Relaxed conditions for stability of nonlinear continuous-time systems given by fuzzy models axe presented. A theoretical analysis shows that the proposed method provides better or at least the same results of the methods presented in the literature. Digital simulations exemplify this fact. This result is also used for fuzzy regulators design. The nonlinear systems are represented by fuzzy models proposed by Takagi and Sugeno. The stability analysis and the design of controllers axe described by LMIs (Linear Matrix Inequalities), that can be solved efficiently using convex programming techniques.