Dynamics of neural systems with time delays

Complex networks are ubiquitous in nature. Numerous neurological diseases, such as Alzheimer's, Parkinson's, epilepsy are caused by the abnormal collective behaviour of neurons in the brain. In particular, there is a strong evidence that Parkinson's disease is caused by the synchronisation of neurons, and understanding how and why such synchronisation occurs will bring scientists closer to the design and implementation of appropriate control to support desynchronisation required for the normal functioning of the brain. In order to study the emergence of (de)synchronisation, it is necessary first to understand how the dynamical behaviour of the system under consideration depends on the changes in systems parameters. This can be done using a powerful mathematical method, called bifurcation analysis, which allows one to identify and classify different dynamical regimes, such as, for example, stable/unstable steady states, Hopf and fold bifurcations, and find periodic solutions by varying parameters of the nonlinear system. In real-world systems, interactions between elements do not happen instantaneously due to a finite time of signal propagation, reaction times of individual elements, etc. Moreover, time delays are normally non-constant and may vary with time. This means that it is vital to introduce time delays in any realistic model of neural networks. In this thesis, I consider four different models. First, in order to analyse the fundamental properties of neural networks with time-delayed connections, I consider a system of four coupled nonlinear delay differential equations. This model represents a neural network, where one subsystem receives a delayed input from another subsystem. The exciting feature of this model is the combination of both discrete and distributed time delays, where distributed time delays represent the neural feedback between the two sub-systems, and the discrete delays describe neural interactions within each of the two subsystems. Stability properties are investigated for different commonly used distribution kernels, and the results are compared to the corresponding stability results for networks with no distributed delays. It is shown how approximations to the boundary of stability region of an equilibrium point can be obtained analytically for the cases of delta, uniform, and gamma delay distributions. Numerical techniques are used to investigate stability properties of the fully nonlinear system and confirm our analytical findings. In the second part of this thesis, I consider a globally coupled network composed of active (oscillatory) and inactive (non-oscillatory) oscillators with distributed time delayed coupling. Analytical conditions for the amplitude death, where the oscillations are quenched, are obtained in terms of the coupling strength, the ratio of inactive oscillators, the width of the uniformly distributed delay and the mean time delay for gamma distribution. The results show that for uniform distribution, by increasing both the width of the delay distribution and the ratio of inactive oscillators, the amplitude death region increases in the mean time delay and the coupling strength parameter space. In the case of the gamma distribution kernel, we find the amplitude death region in the space of the ratio of inactive oscillators, the mean time delay for gamma distribution, and the coupling strength for both weak and strong gamma distribution kernels. Furthermore, I analyse a model of the subthalamic nucleus (STN)-globus palidus (GP) network with three different transmission delays. A time-shift transformation reduces the model to a system with two time delays, for which the existence of a unique steady state is established. Conditions for stability of the steady state are derived in terms of system parameters and the time delays. Numerical stability analysis is performed using traceDDE and DDE-BIFTOOL in Matlab to investigate different dynamical regimes in the STN-GP model, and to obtain critical stability boundaries separating stable (healthy) and oscillatory (Parkinsonian-like) neural ring. Direct numerical simulations of the fully nonlinear system are performed to confirm analytical findings, and to illustrate different dynamical behaviours of the system. Finally, I consider a ring of n neurons coupled through the discrete and distributed time delays. I show that the amplitude death occurs in the symmetric (asymmetric) region depending on the even (odd) number of neurons in the ring neural system. Analytical conditions for linear stability of the trivial steady state are represented in a parameter space of the synaptic weight of the self-feedback and the coupling strength between the connected neurons, as well as in the space of the delayed self-feedback and the coupling strength between the neurons. It is shown that both Hopf and steady-state bifurcations may occur when the steady state loses its stability. Stability properties are also investigated for different commonly used distribution kernels, such as delta function and weak gamma distributions. Moreover, the obtained analytical results are confirmed by the numerical simulations of the fully nonlinear system

Synchronization and state estimation for discrete-time complex networks with distributed delays

Copyright [2008] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this paper, a synchronization problem is investigated for an array of coupled complex discrete-time networks with the simultaneous presence of both the discrete and distributed time delays. The complex networks addressed which include neural and social networks as special cases are quite general. Rather than the commonly used Lipschitz-type function, a more general sector-like nonlinear function is employed to describe the nonlinearities existing in the network. The distributed infinite time delays in the discrete-time domain are first defined. By utilizing a novel Lyapunov-Krasovskii functional and the Kronecker product, it is shown that the addressed discrete-time complex network with distributed delays is synchronized if certain linear matrix inequalities (LMIs) are feasible. The state estimation problem is then studied for the same complex network, where the purpose is to design a state estimator to estimate the network states through available output measurements such that, for all admissible discrete and distributed delays, the dynamics of the estimation error is guaranteed to be globally asymptotically stable. Again, an LMI approach is developed for the state estimation problem. Two simulation examples are provided to show the usefulness of the proposed global synchronization and state estimation conditions. It is worth pointing out that our main results are valid even if the nominal subsystems within the network are unstable

Time-domain analysis of RF and microwave autonomous circuits by vector fitting-based approach

This work presents a new method for the analysis of RF and microwave autonomous circuits directly in the time-domain, which is the most effective approach at simulation level to evaluate nonlinear phenomena. For RF and microwave autonomous circuits, time-domain simulations usually experiment convergence problems or numerical inaccuracies due to the presence of distributed elements, preventing de-facto their use. The proposed solution is based on the Vector Fitting algorithm applied directly at circuit level. A case study relative to a RF hybrid oscillator is presented for practical demonstration and evaluation of performance reliability of the proposed method

Relevance of Dynamic Clustering to Biological Networks

Network of nonlinear dynamical elements often show clustering of synchronization by chaotic instability. Relevance of the clustering to ecological, immune, neural, and cellular networks is discussed, with the emphasis of partially ordered states with chaotic itinerancy. First, clustering with bit structures in a hypercubic lattice is studied. Spontaneous formation and destruction of relevant bits are found, which give self-organizing, and chaotic genetic algorithms. When spontaneous changes of effective couplings are introduced, chaotic itinerancy of clusterings is widely seen through a feedback mechanism, which supports dynamic stability allowing for complexity and diversity, known as homeochaos. Second, synaptic dynamics of couplings is studied in relation with neural dynamics. The clustering structure is formed with a balance between external inputs and internal dynamics. Last, an extension allowing for the growth of the number of elements is given, in connection with cell differentiation. Effective time sharing system of resources is formed in partially ordered states.Comment: submitted to Physica D, no figures include

Improved Distributed Estimation Method for Environmental\ud time-variant Physical variables in Static Sensor Networks

In this paper, an improved distributed estimation scheme for static sensor networks is developed. The scheme is developed for environmental time-variant physical variables. The main contribution of this work is that the algorithm in [1]-[3] has been extended, and a filter has been designed with weights, such that the variance of the estimation errors is minimized, thereby improving the filter design considerably\ud and characterizing the performance limit of the filter, and thereby tracking a time-varying signal. Moreover, certain parameter optimization is alleviated with the application of a particular finite impulse response (FIR) filter. Simulation results are showing the effectiveness of the developed estimation algorithm

Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time delays

Copyright [2009] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this paper, we introduce a new class of discrete-time neural networks (DNNs) with Markovian jumping parameters as well as mode-dependent mixed time delays (both discrete and distributed time delays). Specifically, the parameters of the DNNs are subject to the switching from one to another at different times according to a Markov chain, and the mixed time delays consist of both discrete and distributed delays that are dependent on the Markovian jumping mode. We first deal with the stability analysis problem of the addressed neural networks. A special inequality is developed to account for the mixed time delays in the discrete-time setting, and a novel Lyapunov-Krasovskii functional is put forward to reflect the mode-dependent time delays. Sufficient conditions are established in terms of linear matrix inequalities (LMIs) that guarantee the stochastic stability. We then turn to the synchronization problem among an array of identical coupled Markovian jumping neural networks with mixed mode-dependent time delays. By utilizing the Lyapunov stability theory and the Kronecker product, it is shown that the addressed synchronization problem is solvable if several LMIs are feasible. Hence, different from the commonly used matrix norm theories (such as the M-matrix method), a unified LMI approach is developed to solve the stability analysis and synchronization problems of the class of neural networks under investigation, where the LMIs can be easily solved by using the available Matlab LMI toolbox. Two numerical examples are presented to illustrate the usefulness and effectiveness of the main results obtained