Synchronization of Discontinuous Neural Networks with Delays via Adaptive Control

The drive-response synchronization of delayed neural networks with discontinuous activation functions is investigated via adaptive control. The synchronization of this paper means that the synchronization error approaches to zero for almost all time as time goes to infinity. The discontinuous activation functions are assumed to be monotone increasing which can be unbounded. Due to the mild condition on the discontinuous activations, adaptive control technique is utilized to control the response system. Under the framework of Filippov solution, by using Lyapunov function and chain rule of differential inclusion, rigorous proofs are given to show that adaptive control can realize complete synchronization of the considered model. The results of this paper are also applicable to continuous neural networks, since continuous function is a special case of discontinuous function. Numerical simulations verify the effectiveness of the theoretical results. Moreover, when there are parameter mismatches between drive and response neural networks with discontinuous activations, numerical example is also presented to demonstrate the complete synchronization by using discontinuous adaptive control

Recent Advances and Applications of Fractional-Order Neural Networks

This paper focuses on the growth, development, and future of various forms of fractional-order neural networks. Multiple advances in structure, learning algorithms, and methods have been critically investigated and summarized. This also includes the recent trends in the dynamics of various fractional-order neural networks. The multiple forms of fractional-order neural networks considered in this study are Hopfield, cellular, memristive, complex, and quaternion-valued based networks. Further, the application of fractional-order neural networks in various computational fields such as system identification, control, optimization, and stability have been critically analyzed and discussed

Robust generalized Mittag-Leffler synchronization of fractional order neural networks with discontinuous activation and impulses.

Fractional order system is playing an increasingly important role in terms of both theory and applications. In this paper we investigate the global existence of Filippov solutions and the robust generalized Mittag-Leffler synchronization of fractional order neural networks with discontinuous activation and impulses. By means of growth conditions, differential inclusions and generalized Gronwall inequality, a sufficient condition for the existence of Filippov solution is obtained. Then, sufficient criteria are given for the robust generalized Mittag-Leffler synchronization between discontinuous activation function of impulsive fractional order neural network systems with (or without) parameter uncertainties, via a delayed feedback controller and M-Matrix theory. Finally, four numerical simulations demonstrate the effectiveness of our main results.N/

Fixed-time synchronization problem of coupled delayed discontinuous neural networks via indefinite derivative method

In this brief, we introduce a class of coupled delayed nonautonomous neural networks (CDNNs) with discontinuous activation function. Different from the conventional Lyapunov method, this brief uses the implementation of an indefinite derivative to deal with the nonautonomous system for the case that the topology between neurons is nonlinear coupling, and the system can achieve synchronization in fixed time by selecting the suitable control scheme. The settling time estimation of the system which can get rid of the dependence on the initial value is given. Finally, two examples are given to verify the correctness of the results in this paper

Networks of piecewise linear neural mass models

Neural mass models are ubiquitous in large scale brain modelling. At the node level they are written in terms of a set of ordinary differential equations with a nonlinearity that is typically a sigmoidal shape. Using structural data from brain atlases they may be connected into a network to investigate the emergence of functional dynamic states, such as synchrony. With the simple restriction of the classic sigmoidal nonlinearity to a piecewise linear caricature we show that the famous Wilson-Cowan neural mass model can be explicitly analysed at both the node and network level. The construction of periodic orbits at the node level is achieved by patching together matrix exponential solutions, and stability is determined using Floquet theory. For networks with interactions described by circulant matrices, we show that the stability of the synchronous state can be determined in terms of a low-dimensional Floquet problem parameterised by the eigenvalues of the interaction matrix. Moreover, this network Floquet problem is readily solved using linear algebra, to predict the onset of spatio-temporal network patterns arising from a synchronous instability. We further consider the case of a discontinuous choice for the node nonlinearity, namely the replacement of the sigmoid by a Heaviside nonlinearity. This gives rise to a continuous-time switching network. At the node level this allows for the existence of unstable sliding periodic orbits, which we explicitly construct. The stability of a periodic orbit is now treated with a modification of Floquet theory to treat the evolution of small perturbations through switching manifolds via the use of saltation matrices. At the network level the stability analysis of the synchronous state is considerably more challenging. Here we report on the use of ideas originally developed for the study of Glass networks to treat the stability of periodic network states in neural mass models with discontinuous interactions

A switching control for finite-time synchronization of memristor-based BAM neural networks with stochastic disturbances

This paper deals with the finite-time stochastic synchronization for a class of memristorbased bidirectional associative memory neural networks (MBAMNNs) with time-varying delays and stochastic disturbances. Firstly, based on the physical property of memristor and the circuit of MBAMNNs, a MBAMNNs model with more reasonable switching conditions is established. Then, based on the theory of Filippov’s solution, by using Lyapunov–Krasovskii functionals and stochastic analysis technique, a sufficient condition is given to ensure the finite-time stochastic synchronization of MBAMNNs with a certain controller. Next, by a further discussion, an errordependent switching controller is given to shorten the stochastic settling time. Finally, numerical simulations are carried out to illustrate the effectiveness of theoretical results

Robust Observation and Control of Complex Networks

The problem of understanding when individual actions of interacting agents display to a coordinated collective behavior has receiving a considerable attention in many research fields. Especially in control engineering, distributed applications in cooperative environments are achieving resounding success, due to the large number of relevant applications, such as formation control, attitude synchronization tasks and cooperative applications in large-scale systems. Although those problems have been extensively studied in Literature, themost of classic approaches use to consider the unrealistic scenario in which networks always consist of identical, linear, time-invariant entities. It’s clear that this assumption strongly approximates the effective behavior of a network. In fact agents can be subjected to parameter uncertainties, unmodeled dynamics or simply characterized by proper nonlinear dynamics. Therefore, motivated by those practical problems, the present Thesis proposes various approaches for dealing with the problem of observation and control in both the framework of multi-agents and complex interconnected systems. The main contributions of this Thesis consist on the development of several algorithms based on concepts of discontinuous slidingmode control. This techniques can be employed for solving in finite-time problems of robust state estimation and consensus-based synchronization in network of heterogenous nonlinear systems subjected to unknown but bounded disturbances and sudden topological changes. Both directed and undirected topologies have been taken into account. It is worth to mention also the extension of the consensus problem to networks of agents governed by a class parabolic partial differential equation, for which, for the first time, a boundary-based robust local interaction protocol has been presented

Robust Observation and Control of Complex Networks

The problem of understanding when individual actions of interacting agents display to a coordinated collective behavior has receiving a considerable attention in many research fields. Especially in control engineering, distributed applications in cooperative environments are achieving resounding success, due to the large number of relevant applications, such as formation control, attitude synchronization tasks and cooperative applications in large-scale systems. Although those problems have been extensively studied in Literature, themost of classic approaches use to consider the unrealistic scenario in which networks always consist of identical, linear, time-invariant entities. It’s clear that this assumption strongly approximates the effective behavior of a network. In fact agents can be subjected to parameter uncertainties, unmodeled dynamics or simply characterized by proper nonlinear dynamics. Therefore, motivated by those practical problems, the present Thesis proposes various approaches for dealing with the problem of observation and control in both the framework of multi-agents and complex interconnected systems. The main contributions of this Thesis consist on the development of several algorithms based on concepts of discontinuous slidingmode control. This techniques can be employed for solving in finite-time problems of robust state estimation and consensus-based synchronization in network of heterogenous nonlinear systems subjected to unknown but bounded disturbances and sudden topological changes. Both directed and undirected topologies have been taken into account. It is worth to mention also the extension of the consensus problem to networks of agents governed by a class parabolic partial differential equation, for which, for the first time, a boundary-based robust local interaction protocol has been presented

Hierarchy of models: From qualitative to quantitative analysis of circadian rhythms in cyanobacteria

International audienceA hierarchy of models, ranging from high to lower levels of abstraction, is proposed to construct "minimal" but predictive and explanatory models of biological systems. Three hierarchical levels will be considered: Boolean networks, piecewise affine differential (PWA) equations, and a class of continuous, ordinary, differential equations' models derived from the PWA model. This hierarchy provides different levels of approximation of the biological system and, crucially, allows the use of theoretical tools to more exactly analyze and understand the mechanisms of the system. The Kai ABC oscillator, which is at the core of the cyanobacterial circadian rhythm, is analyzed as a case study, showing how several fundamental properties-order of oscillations, synchronization when mixing oscillating samples, structural robustness, and entrainment by external cues-can be obtained from basic mechanisms