Biological brains exhibit many interesting and complex behaviours. Understanding of the mechanisms behind brain behaviours is critical for continuing advancement in fields of research such as artificial intelligence and medicine. In particular, synchronisation of neuronal firing is associated with both improvements to and degeneration of the brain's performance; increased synchronisation can lead to enhanced information-processing or neurological disorders such as epilepsy and Parkinson's disease. As a result, it is desirable to research under which conditions synchronisation arises in neural networks and the possibility of controlling its prevalence. Stochastic ensembles of FitzHugh-Nagumo elements are used to model neural networks for numerical simulations and bifurcation analysis. The FitzHugh-Nagumo model is employed because of its realistic representation of the flow of sodium and potassium ions in addition to its advantageous property of allowing phase plane dynamics to be observed. Network characteristics such as connectivity, configuration and size are explored to determine their influences on global synchronisation generation in their respective systems. Oscillations in the mean-field are used to detect the presence of synchronisation over a range of coupling strength values. To ensure simulation efficiency, coupling strengths between neurons that are identical and fixed with time are investigated initially. Such networks where the interaction strengths are fixed are referred to as homogeneously coupled. The capacity of controlling and altering behaviours produced by homogeneously coupled networks is assessed through the application of weak and strong delayed feedback independently with various time delays. To imitate learning, the coupling strengths later deviate from one another and evolve with time in networks that are referred to as heterogeneously coupled. The intensity of coupling strength fluctuations and the rate at which coupling strengths converge to a desired mean value are studied to determine their impact upon synchronisation performance. The stochastic delay differential equations governing the numerically simulated networks are then converted into a finite set of deterministic cumulant equations by virtue of the Gaussian approximation method. Cumulant equations for maximal and sub-maximal connectivity are used to generate two-parameter bifurcation diagrams on the noise intensity and coupling strength plane, which provides qualitative agreement with numerical simulations. Analysis of artificial brain networks, in respect to biological brain networks, are discussed in light of recent research in sleep theory.

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<p>The purpose of this study is to identify an impact on an option pricing within NASDAQ OMX Stockholm Market, if the underlying</p><p>asset prices include jumps. The current financial crisis, when jumps are much more evident than ever, makes this issue very actual and important in the global sense for the portfolio hedging and other risk management applications for example for the banking sector. Therefore, an investigation is based on OMXS30 Index and SEB A Bank. To detect jumps the Barndorff-Nielsen and Shephard non-parametric bipower variation test is used. First it is examined on simulations, to be finally implemented on the real data. An affirmation of a jumps occurrence requires to apply an appropriate model for the option pricing. For this purpose the Kou model, a double exponential jump-diffusion one, is proposed, as it incorporates essential stylized facts not available for another models. Th parameters in the model are estimated by a new approach - a combined cumulant matching with lambda taken from the Barrndorff-Nielsen and Shephard test. To evaluate how the Kou model manages on the option pricing, it is compared to the Black-Scholes model and to the real prices of European call options from the Stockholm Stock Exchange. The results show that the Kou model outperforms the latter.</p>

The purpose of this study is to identify an impact on an option pricing within NASDAQ OMX Stockholm Market, if the underlying asset prices include jumps. The current financial crisis, when jumps are much more evident than ever, makes this issue very actual and important in the global sense for the portfolio hedging and other risk management applications for example for the banking sector. Therefore, an investigation is based on OMXS30 Index and SEB A Bank. To detect jumps the Barndorff-Nielsen and Shephard non-parametric bipower variation test is used. First it is examined on simulations, to be finally implemented on the real data. An affirmation of a jumps occurrence requires to apply an appropriate model for the option pricing. For this purpose the Kou model, a double exponential jump-diffusion one, is proposed, as it incorporates essential stylized facts not available for another models. Th parameters in the model are estimated by a new approach - a combined cumulant matching with lambda taken from the Barrndorff-Nielsen and Shephard test. To evaluate how the Kou model manages on the option pricing, it is compared to the Black-Scholes model and to the real prices of European call options from the Stockholm Stock Exchange. The results show that the Kou model outperforms the latter.

Šiame darbe nagrinėjama atsitiktinio dėmenų (atsitiktinių dydžių) skaičiaus skirstinio funkcijos aproksimacijos normaliuoju dėsniu tolygusis ir netolygusis įverčiai. Atsitiktinių dydžių atsitiktinio dėmenų skaičiaus suma traktuojama kaip draudėjo suminės išmokos. Pilnai išnagrinėtas atvejis, kai atsitiktinis dėmenų skaičius yra homogeninis Puasono procesas. Kita darbo dalis skirta atvejui, kai nėra Puasono procesas. Darbą sudaro 8 dalys: įvadas, analitinė – metodinė dalis, rezultatai, kuriais remsimės darbe, darbo tikslas, darbo rezultatai, publikuotas straipsnis, išvados, literatūros sąrašas. Darbo apimtis – 49 p. teksto be priedų, 2 iliustr., 4 bibliografiniai šaltiniai. / This work analyses the even and not even evaluations of distribution function approximation of random amount of random variables using normal law. The sum of random amount of random variables is treated as assurers total payout. In this paper there is fully explored the case when (random amount of random values) is homogeneous Poisson process. The other part of this work explores the case when is not a Poisson process. Structure: introduction, analytical – methodical part, results, are used in work, woprk point, work results, conclusions, references. Thesis consist of: 49 p. text without appendixes, 2 pictures, 4 bibliographical entries.

The research object of this thesis is the sum of a random number of summands of independent identically distributed random variables with positive weights. Such sums appear as models, for example, in insurance, finance mathematics. Throughout the thesis, it is assumed that the random number of summands is independent of the summands, the summands satisfy S. N. Bernstein's condition, and the random number of summands together with weights satisfy some compatibility conditions. The aim of this dissertation is a normal approximation to a distribution of the sum of a random number of summands of independent identically distributed random variables with positive weights that takes into consideration large deviations in both the Cramer and the power Linnik zones. / Disertacinio darbo tyrimo objektas yra atsitiktinio dėmenų skaičiaus nepriklausomų vienodai pasiskirsčiusių atsitiktinių dydžių su teigiamais svoriniais koeficientais sumos, kurios kaip modelis sutinkamos, pavyzdžiui, finansų, draudos matematikose. Daromos prielaidos, kad atsitiktinis dėmenų skaičius yra nepriklausomas nuo sumos dėmenų, atsitiktiniai dėmenys tenkina apibendrintą S. N. Bernšteino sąlygą, o atsitiktinis dėmenų skaičius kartu su svoriais tenkina tam tikras suderinamumo sąlygas. Disertacijos tikslas yra standartizuotos (centruotos ir normuotos) minėtos atsitiktinės sumos skirstinio aproksimacija standartiniu normaliuoju dėsniu didžiųjų nuokrypių tiek Kramero, tiek ir laipsninėse Liniko zonose.

Disertacinio darbo tyrimo objektas yra atsitiktinio dėmenų skaičiaus nepriklausomų vienodai pasiskirsčiusių atsitiktinių dydžių su teigiamais svoriniais koeficientais sumos, kurios kaip modelis sutinkamos, pavyzdžiui, finansų, draudos matematikose. Daromos prielaidos, kad atsitiktinis dėmenų skaičius yra nepriklausomas nuo sumos dėmenų, atsitiktiniai dėmenys tenkina apibendrintą S. N. Bernšteino sąlygą, o atsitiktinis dėmenų skaičius kartu su svoriais tenkina tam tikras suderinamumo sąlygas. Disertacijos tikslas yra standartizuotos (centruotos ir normuotos) minėtos atsitiktinės sumos skirstinio aproksimacija standartiniu normaliuoju dėsniu didžiųjų nuokrypių tiek Kramero, tiek ir laipsninėse Liniko zonose. / The research object of this thesis is the sum of a random number of summands of independent identically distributed random variables with positive weights. Such sums appear as models, for example, in insurance, finance mathematics. Throughout the thesis, it is assumed that the random number of summands is independent of the summands, the summands satisfy S. N. Bernstein's condition, and the random number of summands together with weights satisfy some compatibility conditions. The aim of this dissertation is a normal approximation to a distribution of the sum of a random number of summands of independent identically distributed random variables with positive weights that takes into consideration large deviations in both the Cramer and the power Linnik zones.

Game theory is the study of tactical interactions involving conflicts and cooperations among multiple decision makers called players with applications in diverse disciplines such as economics, biology, management, communication networks, electric power systems and control. This dissertation studies a statistical differential game problem where finite N players optimize their system performance by shaping the distribution of their cost function through cost cumulants. This research integrates game theory with statistical optimal control theory and considers a statistical Nash non-cooperative nonzero-sum game for a nonlinear dynamic system with nonquadratic cost functions. The objective of the statistical Nash game is to find the equilibrium solution where no player has the incentive to deviate once other players maintain their equilibrium strategy. The necessary condition for the existence of the Nash equilibrium solution is given for the m-th cumulant cost optimization using the Hamilton-Jacobi-Bellman (HJB) equations. In addition, the sufficient condition which is the verification theorem for the existence of Nash equilibrium solution is given for the m-th cumulant cost optimization using the Hamilton-Jacobi-Bellman (HJB) equations. However, solving the HJB equations even for relatively low dimensional game problem is not trivial, we propose to use neural network approximate method to find the solution of the HJB partial differential equations for the statistical game problem. Convergence proof of the neural network approximate method solution to exact solution is given. In addition, numerical examples are provided for the statistical game to demonstrate the applicability of the proposed theoretical developments. / Electrical and Computer Engineering

Higher Order Statistics (HOS) and Second Order Statistics (SOS) approaches have certain advantages and disadvantages in signal processing applications. HOS approach provides more statistical information for non-Gaussian signals. On the other hand, SOS approach is more robust to the estimation errors than the HOS approach, especially when the number of observations is small. In this thesis, HOS and SOS approaches are jointly used in order to take advantage of both methods. In this respect, the joint use of HOS and SOS approaches are introduced for online calibration of sensor arrays with arbitrary geometries. Three different problems in online array calibration are considered and new algorithms for each of these problems are proposed. In the first problem, the positions of the randomly deployed sensors are completely unknown except the two reference sensors and HOS and SOS approaches are used iteratively for the joint Direction of Arrival (DOA) and sensor position estimation. Iterative HOS-SOS algorithm (IHOSS) solves the ambiguity problem in sensor position estimation by observing the source signals at least in two different frequencies and hence it is applicable for wideband signals. The conditions on these frequencies are presented. IHOSS is the first algorithm in the literature which finds the DOA and sensor position estimations in case of randomly deployed sensors with unknown coordinates. In the second problem, narrowband signals are considered and it is assumed that the nominal sensor positions are known. Modified IHOSS (MIHOSS) algorithm uses the nominal sensor positions to solve the ambiguity problem in sensor position estimation. This algorithm can handle both small and large errors in sensor positions. The upper bound of perturbations for unambiguous sensor position estimation is presented. In the last problem, an online array calibration method is proposed for sensor arrays where the sensors have unknown gain/phase mismatches and mutual coupling coefficients. In this case, sensor positions are assumed to be known. The mutual coupling matrix is unstructured. The two reference sensors are assumed to be perfectly calibrated. IHOSS algorithm is adapted for online calibration and parameter estimation, and hence CIHOSS algorithm is obtained. While CIHOSS originates from IHOSS, it is fundamentally different in many aspects. CIHOSS uses multiple virtual ESPRIT structures and employs an alignment technique to order the elements of rows of the actual array steering matrix. In this thesis, a new cumulant matrix estimation technique is proposed for the HOS approach by converting the multi-source problem into a single source one. The proposed algorithms perform well even in the case of correlated source signals due to the effectiveness of the proposed cumulant matrix estimate. The iterative procedure in all the proposed algorithms is guaranteed to converge. Closed form expressions are derived for the deterministic Cram&acute / er-Rao bound (CRB) for DOA and unknown calibration parameters for non-circular complex Gaussian noise with unknown covariance matrix. Simulation results show that the performances of the proposed methods approach to the CRB for both DOA and unknown calibration parameter estimations for high SNR.

Abstract This thesis reports simulations of direct observables in electron and nuclear spin relaxation experiments in an example paramagnetic system, as well as polarization transfer occurring in a spin-exchange optical pumping (SEOP) experiment. Studies of paramagnetic relaxation are important, e.g., in the development of agents used for enhanced contrast in magnetic resonance imaging. SEOP is used to produce hyperpolarized noble gases, which are then used to, e.g., enhance sensitivity in structural studies of matter with nuclear magnetic resonance. Presently the theory, available software and hardware for such computational modeling have reached a state in which quantitative reproduction of the experimentally observed magnetization decay is possible from first principles. The present multiscale computations are carried out from first principles combining molecular dynamics simulations of atomistic motion and quantum-chemical electronic structure calculations of the spin interaction parameters that enter the effective spin Hamiltonian. A time series of the spin Hamiltonian is then explicitly used to propagate spin dynamics in the system, and dynamical time constants of the magnetization are obtained through ensemble averaging. The complete decay of electron spin magnetization could be followed directly within the duration of the simulation, whereas the nuclear spin relaxation rates were extracted using Kubo’s theory regarding generalized cumulant expansion and stochastic processes. The extracted electron and nuclear spin relaxation rates for the chosen prototypic system, the aqueous solution of Ni²⁺, are in quantitative and semi-quantitative agreement, respectively, with the available experimental results. The simulations of polarization transfer corroborate the empirical observations on the importance of van der Waals complexes and binary collisions in the spin-exchange process. Long van der Waals complexes represent the overwhelmingly most significant kind of individual events, but the short binary collisions can also give a relatively important contribution due to their vast abundance. This thesis represents a first study in which first principles-calculated trajectories of individual events could be followed. The simulations reported in this thesis were run without any empirical parametrization and thus represent a significant step in first-principles computational modeling of magnetization dynamics.