Discontinuous Activations Research Articles

Quasi-pinning synchronization and stabilization of fractional order BAM neural networks with delays and discontinuous neuron activations

Abstract This manuscript concerns quasi-pinning synchronization and β-exponential pinning stabilization for a class of fractional order BAM neural networks with time-varying delays and discontinuous neuron activations (FBAMNNDDAs). Firstly, under the framework of Filippov solution and fractional-order differential inclusions analysis for the initial value problem of FBAMNNDDAs is presented. Secondly, two kinds of novel pinning controllers according to pinning control technique are designed. By means of fractional order Lyapunov method and designed pinning control strategy, the sufficient criteria is given first to ensure the quasi-synchronization for the dynamic behavior of FBAMNNDDAs. Furthermore, the error bound of pinning synchronization is explicitly evaluated. Thirdly, via Kakutani s fixed point theorem of set-valued map analysis, Razumikhin condition, and a nonlinear pinning controller, the existence and β-exponential stabilization of FBAMNNDDAs equilibrium point is obtained in the voice of linear matrix inequality (LMI) technique. Fourthly, based on as well as Mittag-Leffler function and growth condition, the global existence of a solution in the Filippov sense of such system is guaranteed with detailed proof. At last, a numerical example with computer simulations are performed to illustrate the effectiveness of proposed theoretical consequences.

Periodicity and stabilization control of the delayed Filippov system with perturbation

By employing Leray-Schauder alternative theorem of set-valued maps and non-Lyapunov method (non-smooth analysis, inequality analysis, matrix theory), this paper investigates the problems of periodicity and stabilization for time-delayed Filippov system with perturbation. Several criteria are obtained to ensure the existence of periodic solution of time-delayed Filippov system by using differential inclusion. By designing appropriate switching state-feedback controller, the asymptotic stabilization and exponential stabilization control of Filippov system are realized. Applying these criteria and control design method to a class of time-delayed neural networks with perturbation and discontinuous activation functions under a periodic environment. The developed theorems improve the existing results and their effectiveness are demonstrated by numerical example.

Adaptive synchronization of memristor-based neural networks with discontinuous activations

This paper investigates the adaptive synchronization of memristor-based neural networks (MNNs) with discontinuous activations. First, by using a decoupling strategy, the extensively used models of MNNs are improved, which can describe the real dynamics more accurately. Moreover, we extend the asymptotic synchronization criteria for MNNs with the consideration of discontinuous activation functions. Additionally, a general adaptive controller is devised to synchronize the drive and response systems, and the sufficient stability conditions are established via Lyapunov functional method within the framework of differential inclusions. Finally, numerical simulations are conducted to show the effectiveness of the developed methods.

Discontinuity at fixed points with applications

In this paper, we study new contractive conditions which are strong enough to generate fixed points but which do not force the map to be continuous at fixed points. In this context, we give new results on the fixed-circle problem. We investigate some applications to complex-valued metric spaces and to discontinuous activation functions in real and complex valued neural networks.

Robust synchronisation control of discontinuous CGNNs with time-varying delays

Fixed-time synchronisation requires that the settling time is independent of the initial synchronisation errors and the upper bound of the settling time can be given in advance. From this point of view, the fixed time synchronisation is more favourable and applicable. This paper aims to investigate the fixed-time synchronisation and fixed-time anti-synchronisation of Cohen-Grossberg master-slave neural networks with discontinuous neuron activations and time delays. For the purpose of fixed-time synchronisation, under the framework of Filippov solution, by using the Lyapunov-Krasovskii functional approach and designing a novel discontinuous feedback control procedure for the slave neural networks, some sufficient criteria are firstly derived to design the control parameters for achieving synchronisation within a fixed time for the proposed discontinuous Cohen-Grossberg neural networks. Then, based on the fixed-time synchronisation, another new discontinuous feedback controller is also designed to accomplish the fixed-time anti-synchronisation for the considered neural system. Finally, two numerical examples associated with main theorems and corollaries are presented to illustrate the proposed methodologies.

Stability analysis of almost periodic solutions of discontinuous BAM neural networks with hybrid time-varying delays and D operator

In the paper, we are concerned with a class of discontinuous BAM neural networks with hybrid time-varying delays and D operator. Based on the concept of Filippov solution, by means of the differential inclusions theory and the non-smooth analysis theory with Lyapunov-like approach, some new and novel sufficient conditions are derived to guarantee the existence, uniqueness and global exponential stability of almost-periodic solution of our proposed neural network model. To the authors’ knowledge, the results established in the paper are the only available results on the BAM neural networks, connecting the three main characteristics, i.e., discontinuous activation functions, hybrid time-varying delays and D operator. Some previous works in the literature are significantly extend and complement. Finally, two topical simulation examples are given to show the effectiveness of the established main results.

Geometric properties of discontinuous fixed point set of ($$\varvec{\epsilon -\delta }$$) contractions and applications to neural networks

In this paper, we prove some fixed point theorems under a convex combination of generalized ( $$\epsilon -\delta $$ ) type rational contractions in which the fixed point may or may not be a point of discontinuity. As a by-product we explore some new answers to the open question posed by Rhoades (Contemp Math 72:233–245, 1988). Furthermore, we consider geometric properties of the fixed point set of a self-mapping on a metric space. We define a new kind of contractive mapping and prove that the fixed point set of this kind of contraction contains a circle (resp. a disc). Several non-trivial examples are given to illustrate our results. Apart from these, an application of discontinuous activation functions, frequently used in neural networks is also given.

Global quasi-synchronization and global anti-synchronization of delayed neural networks with discontinuous activations via non-fragile control strategy

This paper deals with the global quasi-synchronization and global anti-synchronization issue of delayed neural networks with discontinuous activations (DNNDA) by means of the non-fragile control strategy. In this paper, a class of DNNDA model is presented. Furthermore, applying the Lyapunov-Krasovskii function with matrix measure, the sufficient conditions are put forward to ensure that such DNNDA drive system and DNNDA response system are global quasi-synchronized. In addition, based on a novel Lur′e-Postnikov Lyapunov function, differential inclusion theory and inequality analysis technique, the sufficient conditions to achieve the global anti-synchronization of the DNNDA drive system and DNNDA response system are proposed in term of the Linear matrix inequalities (LMIs). Finally, two typical numerical simulations are provided to show the effectiveness of the obtained results.

Robust fixed-time synchronization of discontinuous Cohen–Grossberg neural networks with mixed time delays

This paper aims to investigate the fixed-time synchronization (i.e., synchronization in fixed-time sense) of Cohen–Grossberg drive-response neural networks with discontinuous neuron activations and mixed time delays (both time-varying discrete delay and distributed delay). To accomplish the target of fixed-time synchronization, a novel discontinuous feedback control procedure is firstly designed for the response neural networks. Then, under the framework of Filippov solutions, by means of functional differential inclusions theory, inequality technique and the nonsmooth analysis theory with Lyapunov-like approach, some sufficient criteria are derived to design the control parameters for achieving fixed-time synchronization of the proposed drive-response systems. Finally, two numerical examples are presented to illustrate the proposed methodologies.

Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects

Usually, it is very difficult to overcome the effects of time delay and impulse simultaneously in finite-time control area. The present paper solves this problem by considering finite-time synchronization (FTS) and control in an array of competitive neural networks (CNNs) with discontinuous activations, proportional delays, as well as impulsive effects. Since the proportional delays are unbounded and the discontinuous activations usually induce uncertain Filippov solutions, new analytical techniques have to be developed. Quantized control scheme is designed to reduce the control cost and save channel resources. By constructing new Lyapunov functionals and comparison system, several sufficient conditions are obtained to ensure that the coupled CNNs achieve synchronization in a finite settling time which is explicitly estimated. Finally, numerical simulations are provided to illustrate the effectiveness of theoretical analysis.

Anti-Synchronization in Fixed Time for Discontinuous Reaction-Diffusion Neural Networks With Time-Varying Coefficients and Time Delay.

This paper studies the fixed-time anti-synchronization (FTAS) of discontinuous reaction-diffusion neural networks (DRDNNs) with both time-varying coefficients and time delay. First, differential inclusion theory is used to deal with the influence caused by discontinuous activations. In addition, a new fixed-time convergence theorem is used to handle the time-varying coefficients. Second, a novel state-feedback control algorithm and integral state-feedback control algorithm are proposed to realize FTAS of DRDNNs. During the generalized (adaptive) pinning control strategy, a guideline is proposed to select neurons to pin the designed controller. Furthermore, we present several criteria on FTAS by using the generalized Lyapunov function method. Different from the traditional Lyapunov function with negative definite derivative, the derivative of the Lyapunov function can be positive in this paper. Finally, we give two numerical simulations to substantiate the merits of the obtained results.

Stability and pinning synchronization analysis of fractional order delayed Cohen–Grossberg neural networks with discontinuous activations

This article, we explore the asymptotic stability and asymptotic synchronization analysis of fractional order delayed Cohen–Grossberg neural networks with discontinuous neuron activation functions (FCGNNDDs). First, under the framework of Filippov theory and differential inclusion theoretical analysis, the global existence of Filippov solution for FCGNNDDs is studied by means of the given growth condition. Second, by virtue of suitable Lyapunov functional, Young inequality and comparison theorem for fractional order delayed linear system, some global asymptotic stability conditions for such system is derived by limiting discontinuous neuron activations. Third, the global asymptotic synchronization condition for FCGNNDDs is obtained based on the pinning control. At last, two numerical simulations are given to verify the theoretical findings.

Event-triggered H∞ anti-synchronisation for delayed neural networks with discontinuous neuron activations via non-fragile control strategy

This paper treats of the global event-triggered anti-synchronisation issue for discontinuous neural networks with the mixed time-varying delays and random feedback gain fluctuation via non-fragile control strategy. The random gain uncertainties are described by stochastic variables satisfying the Bernoulli distribution. Firstly, the novel hybrid controllers, which are composed of the non-fragile controller and the event-triggered controller, are designed. Then, based on Clarke's non-smooth analysis theory, general free-weighting matrix method, the Lyapunov-Krasovskii functional approach and Wirtinger-based multiple integral inequality analysis technology, the global event-triggered non-fragile anti-synchronisation conditions are established in terms of linear matrix inequalities (LMIs). In addition, under the considered external disturbance, the conditions with respect to the global event-triggered non-fragile anti-synchronisation are also addressed in forms of LMIs. Finally, two illustrative examples are provided to verify the effectiveness of the designed event-triggered non-fragile control scheme and the validity of theoretical results.

Fixed-time pinning-controlled synchronization for coupled delayed neural networks with discontinuous activations

This paper deals with the fixed-time synchronization problem of coupled delayed neural networks with discontinuous activations. Based on pinning control, a discontinuous controller is firstly proposed to guarantee that coupled neural networks achieve synchronization with a desired trajectory in finite time. Then, a discontinuous fixed-time controller is designed. With the fixed-time controller, the settling time can be estimated regardless of initial conditions. By providing a topology-dependent Lyapunov function, some criteria of finite-/fixed-time synchronization are derived. Finally, two numerical examples are given to show the effectiveness of the proposed controllers.

Use of Novel Electrogram "Lumipoint" Algorithm to Detect Critical Isthmus and Abnormal Potentials for Ablation in Ventricular Tachycardia.

This study reports the use of a novel "Lumipoint" algorithm in ventricular tachycardia (VT) ablation. Automatic mapping systems aid rapid acquisition of activation maps. However, they may annotate farfield rather than nearfield signal in low voltage areas, making maps difficult to interpret. The Lumipoint algorithm analyzes the complete electrogram tracing and therefore includes nearfield signals in its analysis. Twenty-two patients with ischemic cardiomyopathy and 5 with dilated cardiomyopathy underwent mapping using the ultra-high density Rhythmia system. Lumipoint algorithms were applied retrospectively. In all left ventricular substrate maps, changing the window of interest to the post-QRS phase automatically identified late potentials. In 25 of 27 left ventricular VT activation maps, a minimum spatial window of interest correctly identified the VT isthmus as seen by the manually annotated map, entrainment, and response to ablation. In 6 maps, the algorithm identified the isthmus where the standard automatically annotated map did not. The Lumipoint algorithm automatically highlights areas with electrograms having specific characteristics or timings. This can identify late and fractionated potentials and regions that exhibit discontinuous activation, as well as the isthmus of a VT circuit. These features may enhance human interpretation of the electrogram signals during a case, particularly where the circuit lies in partial scar with low amplitude nearfield signals and potentially allow a more targeted ablation strategy.

Coexistence and local Mittag–Leffler stability of fractional-order recurrent neural networks with discontinuous activation functions**Project supported by the Natural Science Foundation of Zhejiang Province, China (Grant Nos. LY18F030023, LY17F030016, and LY18F020028) and the National Natural Science Foundation of China (Grant Nos. 61503338, 61502422, and 61773348).

In this paper, coexistence and local Mittag–Leffler stability of fractional-order recurrent neural networks with discontinuous activation functions are addressed. Because of the discontinuity of the activation function, Filippov solution of the neural network is defined. Based on Brouwerʼs fixed point theorem and definition of Mittag–Leffler stability, sufficient criteria are established to ensure the existence of equilibrium points, among which equilibrium points are locally Mittag–Leffler stable. Compared with the existing results, the derived results cover local Mittag–Leffler stability of both fractional-order and integral-order recurrent neural networks. Meanwhile discontinuous networks might have higher storage capacity than the continuous ones. Two numerical examples are elaborated to substantiate the effective of the theoretical results.

Spectral Content of Electroencephalographic Burst-Suppression Patterns May Reflect Neuronal Recovery in Comatose Post-Cardiac Arrest Patients.

To assess the potential biologic significance of variations in burst-suppression patterns (BSPs) after cardiac arrest in relation to recovery of consciousness. In the context of recent theoretical models of BSP, bursting frequency may be representative of underlying network dynamics; discontinuous activation of membrane potential during impaired cellular energetics may promote neuronal rescue. We reviewed a database of 73 comatose post-cardiac arrest patients who underwent therapeutic hypothermia to assess for the presence of BSP and clinical outcomes. In a subsample of patients with BSP (n = 14), spectral content of burst and suppression periods were quantified using multitaper method. Burst-suppression pattern was seen in 45/73 (61%) patients. Comparable numbers of patients with (31.1%) and without (35.7%) BSP regained consciousness by the time of hospital discharge. In addition, in two unique cases, BSP initially resolved and then spontaneously reemerged after completion of therapeutic hypothermia and cessation of sedative medications. Both patients recovered consciousness. Spectral analysis of bursts in all patients regaining consciousness (n = 6) showed a prominent theta frequency (5-7 Hz) feature, but not in age-matched patients with induced BSP who did not recover consciousness (n = 8). The prognostic implications of BSP after hypoxic brain injury may vary based on the intrinsic properties of the underlying brain state itself. The presence of theta activity within bursts may index potential viability of neuronal networks underlying recovery of consciousness; emergence of spontaneous BSP in some cases may indicate an innate neuroprotective mechanism. This study highlights the need for better characterization of various BSP patterns after cardiac arrest.

Finite-time synchronization of memristive neural networks with discontinuous activation functions and mixed time-varying delays

This paper is concerned with the issue of the finite-time adaptive synchronization and finite-time synchronization of memristive neural networks with discontinuous activation functions and mixed time-varying delays. For synchronizing the drive-response memristive neural networks in finite time, an adaptive state-feedback controller and a state-feedback controller are proposed, respectively. Then by using the theories of differential inclusions and set-valued map, the synchronization issue of drive-response memristive neural networks with discontinuous activation functions and mixed time-varying delays is transformed into the stabilization issue of the error system. Moreover, based on the stability theory, Forti Lemma and Hardy inequality, some novel algebraic synchronization criteria are deduced to ensure the finite-time adaptive synchronization and finite-time synchronization of memristive neural networks with discontinuous activation functions and mixed time-varying delays under the adaptive state-feedback controller and the state-feedback controller. And the settling times for finite-time adaptive synchronization and finite-time synchronization are given. Furthermore, it is hard to estimate the initial conditions for a large system, so the settling times in this paper are not dependent on initial conditions of system. Finally, an example is provided to demonstrate the effectiveness of the obtained results.

Finite-time stabilization of memristor-based inertial neural networks with discontinuous activations and distributed delays

In this paper, the finite-time stabilization problem for memristor-based inertial neural networks (MINNs) with discontinuous activations (DAs) and distributed delays is investigated. To deal with the discontinuous property of the MINNs, the nonsmooth analysis theory is invoked. Furthermore, to simplify the MINNs with second-order state derivative, an order-reduced method is adopted. Then the second-order MINNs is transformed into a simpler first-order differential system. Moreover, the verifiable algebraic criteria are derived for the finite-time stabilization of MINNs with DAs and distributed delays under the designed control approach. Finally, the effectiveness of the obtained results are illustrated via numerical simulations.

Global Exponential Synchronization of Delayed Complex-Valued Recurrent Neural Networks with Discontinuous Activations

In this paper, we are concerned with the exponential synchronization for a class of two delayed complex-valued recurrent neural networks (CVRNNs) with discontinuous neuron activations. By separating CVRNNs into real and imaginary parts, forming an equivalent real-valued subsystems, under the framework of differential inclusions, novel state feedback controllers are designed and novel criteria are established to ensure the exponential stability of error system, and thus the drive system exponentially synchronize with the response system. The obtained results are essentially new and complement previously known ones. The practicability of theoretical results is also supported via a numerical example.