Dynamical Analysis of the Hindmarsh-Rose Neuron With Time Delays

Oscillatory Dynamics and Regulatory Mechanisms of the p53-Per2 Network in DNA-Damaged Cells

Circadian rhythm disruptions are linked to increased cancer risk and unfavorable prognosis in patients with cancer, highlighting the critical role of the interplay between the circadian rhythm factor Per2 and the tumor suppressor p53. This brief presents, for the first time, a mathematical model to capture the dynamics of the p53-Per2 network in DNA-damaged cells. The model accurately describes the different stages of the process from unstressed cells to cellular repair and finally to apoptosis as the degree of DNA damage increases. Furthermore, it is found that increasing the inhibition of Per2 by p53 leads to the phase advance of Per2 oscillations, whereas by modulating the inhibition of Mdm2 by Per2, an independent amplitude modulation of active p53 can be achieved, with the range of modulation increasing with the strength of the inhibition. Moreover, the effects of time delays inherent in the transcription, translation, and nuclear translocation of Per2 on the circadian rhythm of DNA-damaged cells are quantitatively investigated by theoretical analyses. It is found that time delays can induce stable oscillations through a supercritical Hopf bifurcation, thereby maintaining the circadian function of DNA-damaged cells and enhancing their DNA-damage repair capacity. This study proposes new insights into cancer prevention and treatment strategies.

Dynamical Modeling and Qualitative Analysis of a Delayed Model for CD8 T Cells in Response to Viral Antigens

Although the immune effector CD8 T cells play a crucial role in clearance of viruses, the mechanisms underlying the dynamics of how CD8 T cells respond to viral infection remain largely unexplored. Here, we develop a delayed model that incorporates CD8 T cells and infected cells to investigate the functional role of CD8 T cells in persistent virus infection. Bifurcation analysis reveals that the model has four steady states that can finely divide the progressions of viral infection into four states, and endows the model with bistability that has ability to achieve the switch from one state to another. Furthermore, analytical and numerical methods find that the time delay resulting from incubation period of virus can induce a stable low-infection steady state to be oscillatory, coexisting with a stable high-infection steady state in phase space. In particular, a novel mechanism to achieve the switch between two stable steady states, time-delay-based switch, is proposed, where the initial conditions and other parameters of the model remain unchanged. Moreover, our model predicts that, for a certain range of initial antigen load: 1) under a longer incubation period, the lower the initial antigen load, the easier the virus infection will evolve into severe state; while the higher the initial antigen load, the easier it is for the virus infection to be effectively controlled and 2) only when the incubation period is small, the lower the initial antigen load, the easier it is to effectively control the infection progression. Our results are consistent with multiple experimental observations, which may facilitate the understanding of the dynamical and physiological mechanisms of CD8 T cells in response to viral infections.

Macroscopic Gamma Oscillation With Bursting Neuron Model Under Stochastic Fluctuation

Gamma oscillations are thought to play a role in information processing in the brain. Bursting neurons, which exhibit periodic clusters of spiking activity, are a type of neuron that are thought to contribute largely to gamma oscillations. However, little is known about how the properties of bursting neurons affect the emergence of gamma oscillation, its waveforms, and its synchronized characteristics, especially when subjected to stochastic fluctuations. In this study, we proposed a bursting neuron model that can analyze the bursting ratio and the phase response function. Then we theoretically analyzed the neuronal population dynamics composed of bursting excitatory neurons, mixed with inhibitory neurons. The bifurcation analysis of the equivalent Fokker-Planck equation exhibits three types of gamma oscillations of unimodal firing, bimodal firing in the inhibitory population, and bimodal firing in the excitatory population under different interaction strengths. The analyses of the macroscopic phase response function by the adjoint method of the Fokker-Planck equation revealed that the inhibitory doublet facilitates synchronization of the high-frequency oscillations. When we keep the strength of interactions constant, decreasing the bursting ratio of the individual neurons increases the relative high-gamma component of the populational phase-coupling functions. This also improves the ability of the neuronal population model to synchronize with faster oscillatory input. The analytical frameworks in this study provide insight into nontrivial dynamics of the population of bursting neurons, which further suggest that bursting neurons have an important role in rhythmic activities.

Anti-control of periodic firing in HR model in the aspects of position, amplitude and frequency

This paper proposes a novel controller to control position, amplitude and frequency of periodic firing activity in Hindmarsh-Rose model based on Hopf bifurcation theory which is composed of linear control gain and nonlinear control gain. First, we select the activation of the fast ion channel as control parameter. Based on explicit criterion of Hopf bifurcation, a series of conditions are obtained to derive the linear gains of controller responsible for control of the location where the periodic firing activity occurs. Then, based on the control parameter, a series of conditions are obtained to derive the nonlinear gains of controller responsible for controlling the amplitude and frequency of periodic firing activity by using center manifold and normal form. Finally, the numerical experiments show that our controller can make the periodic firing activity occur at designed value and control the amplitude and frequency of periodic firing activity by adjusting nonlinear control gain of controller.

Multi-scroll hidden attractor in memristive HR neuron model under electromagnetic radiation and its applications

This paper aims to propose a novel no-equilibrium Hindmarsh-Rose (HR) neuron model with memristive electromagnetic radiation effect. Compared with other memristor-based HR neuron models, the uniqueness of this memristive HR neuron model is that it can generate multi-scroll hidden attractors with sophisticated topological structures and the parity of the scrolls can be controlled conveniently with changing the internal parameters of the memristor. In particular, the number of scrolls of the multi-scroll hidden attractors is also associated with the intensity of external electromagnetic radiation stimuli. The complex dynamics is numerically studied through phase portraits, bifurcation diagrams, Lyapunov exponents, and a two-parameter diagram. Furthermore, hardware circuit experiments are carried out to demonstrate theoretical analyses and numerical simulations. From the perspective of engineering application, a pseudo-random number generator is designed. Besides, an image encryption application and security analysis are also performed. The obtained results show that the memristive HR neuron model possesses excellent randomness and high security, which is suitable for chaos-based real-world applications.

Evaluating the effectiveness of several synchronization control methods applying to the electrically and the chemically coupled hindmarsh-rose neurons

This study focuses on the synchronization control between the coupled neurons. The achievements of several synchronization control methods have been checked by evaluating the effects of the synaptic coupling weight alteration on the synchronization. Here, a neural ensemble has been constructed by utilizing the Hindmarsh Rose (HR) Neuron Model. The HR neurons have been linked to each other with the bidirectional coupling. The synchrony or the asynchrony states between these coupled neurons have been observed by using the standard deviation results. Here, firstly, the electrically and the chemically coupled HR neurons have been handled without using any control method, separately and the effects of the synaptic coupling weight alteration on the synchronic firing have been assessed by considering the features of the coupling types. Then, while the electrically coupled HR neurons are generally preferred in the available synchronization control studies; the Lyapunov, the back-stepping, and the feedback synchronization control methods have been adapted to both the electrically and the chemically coupled HR neurons. Thus, a remarkable contribution has been provided to the limited number of studies, which are about the synchronization control of the chemically coupled HR neurons. Also, the synchronization control between the electrically or the chemically coupled HR neurons has been provided by the back-stepping method for the first time. Finally, the differences between the membrane potentials of the coupled neurons have been calculated by utilizing an alternative error function. Since this function calculates the amplitude and the phase errors, separately; the effectiveness of these methods can be evaluated correctly in terms of the performing the minimum differences between the neural dynamics.

Adaptive Synchronization of Reaction-Diffusion Neural Networks and Its Application to Secure Communication

This paper is mainly concerned with the synchronization problem of reaction-diffusion neural networks (RDNNs) with delays and its direct application in image secure communications. An adaptive control is designed without a sign function in which the controller gain matrix is a function of time. The synchronization criteria are established for an error model derived from master-slave models through solving the set of linear matrix inequalities derived by constructing the suitable novel Lyapunov-Krasovskii functional candidate, Green's formula, and Wirtinger's inequality. If the proposed sufficient conditions are satisfied, then the global asymptotic synchronization of the error model is guaranteed. The numerical illustrations are provided to demonstrate the validity of the derived synchronization criteria. In addition, the role of system parameters is picturized through the chaotic nature of RDNNs and those unprecedented solutions is utilized to promote better security of image transactions. As is evident, the enhancement of image encryption algorithm is designed with two levels, namely, image watermarking and diffusion process. The contributions of this paper are discussed as concluding remarks.

Hidden Bursting Firings and Bifurcation Mechanisms in Memristive Neuron Model With Threshold Electromagnetic Induction

Memristors can be employed to mimic biological neural synapses or to describe electromagnetic induction effects. To exhibit the threshold effect of electromagnetic induction, this paper presents a threshold flux-controlled memristor and examines its frequency-dependent pinched hysteresis loops. Using an electromagnetic induction current generated by the threshold memristor to replace the external current in 2-D Hindmarsh-Rose (HR) neuron model, a 3-D memristive HR (mHR) neuron model with global hidden oscillations is established and the corresponding numerical simulations are performed. It is found that due to no equilibrium point, the obtained mHR neuron model always operates in hidden bursting firing patterns, including coexisting hidden bursting firing patterns with bistability also. In addition, the model exhibits complex dynamics of the actual neuron electrical activities, which acts like the 3-D HR neuron model, indicating its feasibility. In particular, by constructing the fold and Hopf bifurcation sets of the fast-scale subsystem, the bifurcation mechanisms of hidden bursting firings are expounded. Finally, circuit experiments on hardware breadboards are deployed and the captured results well match with the numerical results, validating the physical mechanism of biological neuron and the reliability of electronic neuron.

Filippov Hindmarsh-Rose Neuronal Model With Threshold Policy Control

A Filippov system of Hindmarsh-Rose (HR) neuronal model with threshold policy control is proposed, membrane potential has been taken as the threshold and the corresponding switching function is also established. We first discuss the existence and stability of the equilibria for the two Filippov subsystems based on the 2-D HR model. Subsequently, the sliding dynamics of HR model including the sliding segments, sliding regions, and various equilibria under the Filippov framework are studied. Then, we further consider the equilibria and the sliding bifurcation set of the Filippov system, and find there exist the bistable equilibria and several sliding bifurcation phenomena, such as boundary-node bifurcation, pseudosaddle-node bifurcation, the emergence and disappearance of limit cycles on the sliding line, and so on. Finally, we study the Filippov system of the 3-D HR model, and provide a phase diagram of the system that generates the sliding spiking and the sliding bursting, which lie on the sliding line.

Nonlinear optimal control for the synchronization of biological neurons under time-delays

The article proposes a nonlinear optimal control method for synchronization of neurons that exhibit nonlinear dynamics and are subject to time-delays. The model of the Hindmarsh-Rose (HR) neurons is used as a case study. The dynamic model of the coupled HR neurons undergoes approximate linearization around a temporary operating point which is recomputed at each iteration of the control method. The linearization procedure relies on Taylor series expansion of the model and on computation of the associated Jacobian matrices. For the approximately linearized model of the coupled HR neurons an H-infinity controller is designed. For the selection of the controller's feedback gain an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. The stability properties of the control loop are proven through Lyapunov analysis. First, it is shown that the H-infinity tracking performance criterion is satisfied. Moreover, it is proven that the control loop is globally asymptotically stable.