Stability and Hopf-Bifurcation Analysis of Delayed BAM Neural Network under Dynamic Thresholds

In this paper the dynamics of a three neuron model with self-connection and distributed delay under dynamical threshold is investigated. With the help of topological degree theory and Homotopy invariance principle existence and uniqueness of equilibrium point are established. The conditions for which the Hopf-bifurcation occurs at the equilibrium are obtained for the weak kernel of the distributed delay. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and central manifold theorem. Lastly global bifurcation aspect of such periodic solutions is studied. Some numerical simulations for justifying the theoretical analysis are also presented

Dynamics in a Delayed Neural Network Model of Two Neurons with Inertial Coupling

A delayed neural network model of two neurons with inertial coupling is dealt with in this paper. The stability is investigated and Hopf bifurcation is demonstrated. Applying the normal form theory and the center manifold argument, we derive the explicit formulas for determining the properties of the bifurcating periodic solutions. An illustrative example is given to demonstrate the effectiveness of the obtained results

Nonlinear dynamics of full-range CNNs with time-varying delays and variable coefficients

In the article, the dynamical behaviours of the full-range cellular neural networks (FRCNNs) with variable coefficients and time-varying delays are considered. Firstly, the improved model of the FRCNNs is proposed, and the existence and uniqueness of the solution are studied by means of differential inclusions and set-valued analysis. Secondly, by using the Hardy inequality, the matrix analysis, and the Lyapunov functional method, we get some criteria for achieving the globally exponential stability (GES). Finally, some examples are provided to verify the correctness of the theoretical results

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

Dynamics of Macrosystems; Proceedings of a Workshop, September 3-7, 1984

There is an increasing awareness of the important and persuasive role that instability and random, chaotic motion play in the dynamics of macrosystems. Further research in the field should aim at providing useful tools, and therefore the motivation should come from important questions arising in specific macrosystems. Such systems include biochemical networks, genetic mechanisms, biological communities, neutral networks, cognitive processes and economic structures. This list may seem heterogeneous, but there are similarities between evolution in the different fields. It is not surprising that mathematical methods devised in one field can also be used to describe the dynamics of another. IIASA is attempting to make progress in this direction. With this aim in view this workshop was held at Laxenburg over the period 3-7 September 1984. These Proceedings cover a broad canvas, ranging from specific biological and economic problems to general aspects of dynamical systems and evolutionary theory