Mathematical modelling and brain dynamical networks

In this thesis, we study the dynamics of the Hindmarsh-Rose (HR) model which studies the spike-bursting behaviour of the membrane potential of a single neuron. We study the stability of the HR system and compute its Lyapunov exponents (LEs). We consider coupled general sections of the HR system to create an undirected brain dynamical network (BDN) of Nn neurons. Then, we study the concepts of upper bound of mutual information rate (MIR) and synchronisation measure and their dependence on the values of electrical and chemical couplings. We analyse the dynamics of neurons in various regions of parameter space plots for two elementary examples of 3 neurons with two different types of electrical and chemical couplings. We plot the upper bound Ic and the order parameter rho (the measure of synchronisation) and the two largest Lyapunov exponents LE1 and LE2 versus the chemical coupling gn and electrical coupling gl. We show that, even for small number of neurons, the dynamics of the system depends on the number of neurons and the type of coupling strength between them. Finally, we evolve a network of Hindmarsh-Rose neurons by increasing the entropy of the system. In particular, we choose the Kolmogorov-Sinai entropy: HKS (Pesin identity) as the evolution rule. First, we compute the HKS for a network of 4 HR neurons connected simultaneously by two undirected electrical and two undirected chemical links. We get different entropies with the use of different values for both the chemical and electrical couplings. If the entropy of the system is positive, the dynamics of the system is chaotic and if it is close to zero, the trajectory of the system converges to one of the fixed points and loses energy. Then, we evolve a network of 6 clusters of 10 neurons each. Neurons in each cluster are connected only by electrical links and their connections form small-world networks. The six clusters connect to each other only by chemical links. We compare between the combined effect of chemical and electrical couplings with the two concepts, the information flow capacity Ic and HKS in evolving the BDNs and show results that the brain networks might evolve based on the principle of the maximisation of their entropies

Extension of Lorenz Unpredictability

It is found that Lorenz systems can be unidirectionally coupled such that the chaos expands from the drive system. This is true if the response system is not chaotic, but admits a global attractor, an equilibrium or a cycle. The extension of sensitivity and period-doubling cascade are theoretically proved, and the appearance of cyclic chaos as well as intermittency in interconnected Lorenz systems are demonstrated. A possible connection of our results with the global weather unpredictability is provided.Comment: 32 pages, 13 figure

Complexity, Emergent Systems and Complex Biological Systems:\ud Complex Systems Theory and Biodynamics. [Edited book by I.C. Baianu, with listed contributors (2011)]

An overview is presented of System dynamics, the study of the behaviour of complex systems, Dynamical system in mathematics Dynamic programming in computer science and control theory, Complex systems biology, Neurodynamics and Psychodynamics.\u

A note on the multi-stage spectral relaxation method for chaos control and synchronization

In this study, we present and apply a new, accurate and easy to implement numerical method to realize and verify the synchronization between two identical chaotic Lorenz, Genesio-Tesi, Rössler, Chen and Rikitake systems. The proposed method is called the multi-stage spectral relaxation method (MSRM). We utilize the active control technique for the synchronization of these systems. To illustrate the effectiveness of the method, simulation results are presented and compared with results obtained using the Runge-Kutta (4, 5) based MATLAB solver, ode45

On new chaotic and hyperchaotic systems: A literature survey

This paper provides a thorough survey of new chaotic and hyperchaotic systems. An analysis of the dynamic behavior of these complex systems is presented by pointing out their originality and elementary characteristics. Recently, such systems have been increasingly used in various fields such as secure communication, encryption and finance and so on. In practice, each field requires specific performances with peculiar complexity. A particular classification is then proposed in this paper based on the Lyapunov exponent, the equilibriums points and the attractor forms

Control of a novel chaotic fractional order system using a state feedback technique

We consider a new fractional order chaotic system displaying an interesting behavior. A necessary condition for the system to remain chaotic is derived. It is found that chaos exists in the system with order less than three. Using the Routh-Hurwitz and the Matignon stability criteria, we analyze the novel chaotic fractional order system and propose a control methodology that is better than the nonlinear counterparts available in the literature, in the sense of simplicity of implementation and analysis. A scalar control input that excites only one of the states is proposed, and sufficient conditions for the controller gain to stabilize the unstable equilibrium points derived. Numerical simulations confirm the theoretical analysis. © 2013 Elsevier Ltd. All rights reserved

The zero-Hopf bifurcations of a four-dimensional hyperchaotic system

We consider the four-dimensional hyperchaotic system ẋ=a(y-x), y˙=bx+u-y-xz, ż=xy-cz, and u˙=-du-jx+exz, where a, b, c, d, j, and e are real parameters. This system extends the famous Lorenz system to four dimensions and was introduced in Zhou et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 27, 1750021 (2017). We characterize the values of the parameters for which their equilibrium points are zero-Hopf points. Using the averaging theory, we obtain sufficient conditions for the existence of periodic orbits bifurcating from these zero-Hopf equilibria and give some examples to illustrate the conclusions. Moreover, the stability conditions of these periodic orbits are given using the Routh-Hurwitz criterion

Strain and Inflation-Unemployment Relationship in Transitional Economies: A theoretical and empirical investigation

Standard economic theory tells that a command system, like the former eastern economies, allocates resources poorly due to the impossibility of accurate calculation. Therefore, once prices are freed and start to operate at quasi-equilibrium (market-clearing) levels, the hidden inefficiencies come into the open and a possible massive resource reallocation would have to take place. More precisely, the issue refers to the possible and probable intensity of resource reallocation in view of constraints like the balance between exit and entry in the labour market, the size of the budget deficit and the means for its non-inflationary financing, social and political stability, etc. This paper tries to conceptualise the fact that the dynamics of unemployment and inflation are correlated not in a classical sense but in a very complicated mode that suggests the occurrence of some attractors when certain slow parameters are evolving in the neighbourhood of special threshold-values. The start is made with simple models that are based on empirical data and can show to us the traces to discover the steps of transition in eastern economies on the inflation-unemployment relationship space. Then, using economic theory combined with non-linear modelling more refined information is extracted from the statistical standardised data for to evaluate other faces of the eastern transition.natural rate of unemployment, potential function, Hurst exponent, pitchfork bifurcation, modified Phillips curve, Rössler attractor

A direct method to detect deterministic and stochastic properties of data

AbstractA fundamental question of data analysis is how to distinguish noise corrupted deterministic chaotic dynamics from time-(un)correlated stochastic fluctuations when just short length data is available. Despite its importance, direct tests of chaos vs stochasticity in finite time series still lack of a definitive quantification. Here we present a novel approach based on recurrence analysis, a nonlinear approach to deal with data. The main idea is the identification of how recurrence microstates and permutation patterns are affected by time reversibility of data, and how its behavior can be used to distinguish stochastic and deterministic data. We demonstrate the efficiency of the method for a bunch of paradigmatic systems under strong noise influence, as well as for real-world data, covering electronic circuit, sound vocalization and human speeches, neuronal activity, heart beat data, and geomagnetic indexes. Our results support the conclusion that the method distinguishes well deterministic from stochastic fluctuations in simulated and empirical data even under strong noise corruption, finding applications involving various areas of science and technology. In particular, for deterministic signals, the quantification of chaotic behavior may be of fundamental importance because it is believed that chaotic properties of some systems play important functional roles, opening doors to a better understanding and/or control of the physical mechanisms behind the generation of the signals.Financiadora de Estudos e Projetoshttps://doi.org/10.13039/501100004809Coordenação de Aperfeiçoamento de Pessoal de Nível Superiorhttps://doi.org/10.13039/501100002322Conselho Nacional de Desenvolvimento Científico e Tecnológicohttps://doi.org/10.13039/501100003593Peer Reviewe