Comparative exploration on bifurcation behavior for integer-order and fractional-order delayed BAM neural networks

In the present study, we deal with the stability and the onset of Hopf bifurcation of two type delayed BAM neural networks (integer-order case and fractional-order case). By virtue of the characteristic equation of the integer-order delayed BAM neural networks and regarding time delay as critical parameter, a novel delay-independent condition ensuring the stability and the onset of Hopf bifurcation for the involved integer-order delayed BAM neural networks is built. Taking advantage of Laplace transform, stability theory and Hopf bifurcation knowledge of fractional-order differential equations, a novel delay-independent criterion to maintain the stability and the appearance of Hopf bifurcation for the addressed fractional-order BAM neural networks is established. The investigation indicates the important role of time delay in controlling the stability and Hopf bifurcation of the both type delayed BAM neural networks. By adjusting the value of time delay, we can effectively amplify the stability region and postpone the time of onset of Hopf bifurcation for the fractional-order BAM neural networks. Matlab simulation results are clearly presented to sustain the correctness of analytical results. The derived fruits of this study provide an important theoretical basis in regulating networks

Periodic Solution for a Complex-valued Network Model with Discrete Delay

For a tridiagonal two-layer real six-neuron model, the Hopf bifurcation was investigated by studying the eigenvalue equations of the related linear system in the literature. In the present paper, we extend this two-layer real six-neuron network model into a complex-valued delayed network model. Based on the mathematical analysis method, some sufficient conditions to guarantee the existence of periodic oscillatory solutions are established under the assumption that the activation function can be separated into its real and imaginary parts. Our sufficient conditions obtained by the mathematical analysis method in this paper are simpler than those obtained by the Hopf bifurcation method. Computer simulation is provided to illustrate the correctness of the theoretical results

Stability and Hopf bifurcation of controlled complex networks model with two delays

none3siopenJinde Cao, Luca Guerrini, Zunshui ChengCao, Jinde; Guerrini, Luca; Cheng, Zunshu

Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay

In this paper, the Hopf-pitchfork bifurcation of coupled van der Pol with delay is studied. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly, the normal form is gotten by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly, bifurcation diagrams and phase portraits are given through analyzing the unfolding structure. Finally, numerical simulations are used to support theoretical analysis

Bifurcation Analysis in a Two-Dimensional Neutral Differential Equation

The dynamics of a 2-dimensional neural network model in neutral form are investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. Finally, some numerical simulations are carried out to support the analytic results

Dynamical systems and their applications in neuroscience

This thesis deals with dynamical systems, numerical software for the continuation study of dynamical systems, and some important neurobiological applications. First there are two introductory chapters, in which a background is given in dynamical systems and neuroscience. We elucidate what the problems are with some existing classifications of neural models, and suggest an improved version. We introduce the Phase Response Curve (PRC), which is a curve that describes the effect of an input on a periodic orbit. We derive an efficient method to compute this PRC. The extended functionalities of MatCont, a software package for the study of dynamical systems and their bifurcations, are explained: the user can compute the PRC of a limit cycle and its derivative, he can detect and continue homoclinic bifurcations, initiate these curves from different bifurcations and detect many codim 2 bifurcations on these curves. The speed of the software was improved by introducing C-code among the matlab-routines. We have for the first time made a complete bifurcation diagram of the Morris-Lecar neural model. We show that PRCs can be used to determine the synchronizing and/or phase-locking abilities of neural networks, and how the connection delay plays a role in this, and demonstrate some phenomena to do with PRCs and bifurcations. In collaboration with biologists at the University of Bristol, we have built detailed models of the neurons in the spinal cord of the hatchling Xenopus laevis. The biological background and the equations and parameters for the models of individual neurons and synapses are listed elaborately. These models are used to construct biologically realistic networks of neurons. The first network was used to simulate the swimming behaviour of the tadpole and to show that to disregard some important differences in the models for different neurons, will result in breakdown of the good network output. Then we have used the individual models to study a hypothesis regarding synaptogenesis, which states that the specificity in connection between neurons could be purely based on the anatomical organization of the neurons, instead of the ability of growing synapses to make a distinction between the different neurons

Temporal dynamics of a two-neuron continuous network model with time delay

AbstractIn this paper we consider a simple analog neural network model consisting of two continuous nonlinear neurons with delay in signal transmission under appropriate restrictions on internal parameters. We derive conditions for the existence of single steady-state conditions for asymptotic stability, stability switches about the steady state, and bifurcation of the linearized system

metastability and its dynamical cortical core

In the human brain, spontaneous activity during resting state consists of rapid transitions between functional network states over time but the underlying mechanisms are not understood. We use connectome based computational brain network modeling to reveal fundamental principles of how the human brain generates large-scale activity observable by noninvasive neuroimaging. We used structural and functional neuroimaging data to construct whole- brain models. With this novel approach, we reveal that the human brain during resting state operates at maximum metastability, i.e. in a state of maximum network switching. In addition, we investigate cortical heterogeneity across areas. Optimization of the spectral characteristics of each local brain region revealed the dynamical cortical core of the human brain, which is driving the activity of the rest of the whole brain. Brain network modelling goes beyond correlational neuroimaging analysis and reveals non-trivial network mechanisms underlying non-invasive observations. Our novel findings significantly pertain to the important role of computational connectomics in understanding principles of brain function