Spiking patterns of a minimal neuron to ELF sinusoidal electric field

Neuron as the main information carrier in neural systems is able to generate diverse spiking trains in response to different stimuli. Neuronal spiking patterns are related to the information processing in neural system. This paper investigates the dynamical behaviors of a two-dimensional minimal neuron model exposed to externally-applied extremely low frequency (ELF) sinusoidal electric field (EF). By numerical stimulation, it is found that neuron can exhibit different spiking patterns such as p:q mode-locking (i.e. a periodic oscillation defined as p action potentials generated by q cycle stimulations) and chaotic behaviors, depending on the values of stimulus frequencies. Transitions between different spiking patterns during exposure to the external EF are explored by interspike intervals (ISIs) and average firing rate. It is found that frequencies of the external EF can act as a bifurcation parameter, whose small change can cause the transition in neuronal behaviors. It is shown that a rich bifurcation structure including period-adding without chaos and mode-locking alternated with chaos suggests frequency discrimination of the neuronal firing patterns. Our results can provide a useful insight into the organization of similar bifurcation structures in excitable systems such as neurons under periodic forcing. Through detail analysis of the spiking patterns, we have explained how neuron’s information (spiking patterns) encodes the stimulus information (frequency), and vice versa.     

Sparse Functional Dynamical Models—A Big Data Approach

Nonlinear dynamical systems are encountered in many areas of social science, natural science, and engineering, and are of particular interest for complex biological processes like the spiking activity of neural ensembles in the brain. To describe such spiking activity, we adapt the Volterra series expansion of an analytic function to account for the point-process nature of multiple inputs and a single output (MISO) in a neural ensemble. Our model describes the transformed spiking probability for the output as the sum of kernel-weighted integrals of the inputs. The kernel functions need to be identified and estimated, and both local sparsity (kernel functions may be zero on part of their support) and global sparsity (some kernel functions may be identically zero) are of interest. The kernel functions are approximated by B-splines and a penalized likelihood-based approach is proposed for estimation. Even for moderately complex brain functionality, the identification and estimation of this sparse functional dynamical model poses major computational challenges, which we address with big data techniques that can be implemented on a single, multi-core server. The performance of the proposed method is demonstrated using neural recordings from the hippocampus of a rat during open field tasks. Supplementary materials for this article are available online.     

Bifurcations and fast-slow behaviors in a hyperchaotic dynamical system

This paper reports a four-dimension (4D) fast-slow hyperchaotic system with the structure of two time scales by adding a slow state variable w into a three-dimension (3D) chaotic dynamical system, studies the stability and Hopf bifurcation of origin point. Furthermore, based on the fast-slow dynamical bifurcation analysis and the phase planes analysis, different bursting phenomena, symmetric fold/fold bursting, symmetric sub-Hopf/sub-Hopf bursting and chaotic bursting, as well as chaotic and periodic spiking, are observed in the fast-slow hyperchaotic system. Numerical simulations are presented to show these results.     

Bifurcations and Exact Solutions of the Raman Soliton Model in Nanoscale Optical Waveguides with Metamaterials

In this paper, we study Raman soliton model in nanoscale optical waveguides with metamaterials, having polynomial law non-linearity. By using the bifurcation theory method of dynamical systems to the equations of $\phi(\xi)$, under 24 different parameter conditions, we obtain bifurcations of phase portraits and different traveling wave solutions including periodic solutions, homoclinic and heteroclinic solutions for planar dynamical system of the Raman soliton model. Under different parameter conditions, 24 exact explicit parametric representations of the traveling wave solutions are derived. The dynamic behavior of these traveling wave solutions are meaningful and helpful for us to understand the physical structures of the model.     

Synchronization criteria in complex dynamical networks with nonsymmetric coupling and multiple time-varying delays

In this article, we consider a generalized complex dynamical network model with nonsymmetric coupling, and the dynamics of each node has a different time-varying delay. Criteria of exponential synchronization are derived in terms of linear matrix inequalities for the model by constructing suitable Lyapunov functionals. The obtained outcomes are different from those in the current literature, in which the complex dynamical networks are coupling symmetrically and delays are fixed constants. Moreover, the given sufficient conditions extend current available results and are verifiable. A numerical example is provided to illustrate the efficiency of the derived outcome.     

Productive public expenditures, expectation formations and nonlinear dynamics

In this paper, we investigate the dynamic properties of an overlapping generations’ model with capital accumulation and publicly funded inventions under three different expectations: perfect foresight, myopic expectations and adaptive expectations. We show that considering productive public expenditures in the model will increase the dimension of the dynamical system. To study the dynamic behavior of a high-dimensional dynamical system, we focus on the case when the elasticity of publicly funded invention to output is small and approximate the system by using a one-dimensional dynamical system. This approximation method provides an efficient way to rigorously prove the existence of chaos in high-dimensional dynamical systems. We show that when agents are perfectly foresighted, there exists a unique, nontrivial steady state which is a global attractor. Cycles or even chaos may occur under myopic and adaptive expectations when the inter-temporal elasticity of substitution of consumption is large enough. Furthermore, we find that the impact of fiscal policy is sensible to the expectation formation.     

Predator–prey model with sigmoid functional response

The sigmoid functional response in the predator–prey model was posed in 1977. But its dynamics has not been completely characterized. This paper completes the classification of the global dynamics for the classical predator–prey model with the sigmoid functional response, whose denominator has two different zeros. The dynamical phenomena we obtain here include global stability, the existence of the heteroclinic and homoclinic loops, the consecutive canard explosions via relaxation oscillation, and the canard explosion to a homoclinic loop among others. As we know, the last one is a new dynamical phenomenon, which has never been reported previously. In addition, with the help of geometric singular perturbation theory, we solve the problem of connection between stable and unstable manifolds from different singularities, which has not been well settled in the published literature.     

Optimal homotopy analysis of a chaotic HIV-1 model incorporating AIDS-related cancer cells

The studies of nonlinear models in epidemiology have generated a deep interest in gaining insight into the mechanisms that underlie AIDS-related cancers, providing us with a better understanding of cancer immunity and viral oncogenesis. In this article, we analyze an HIV-1 model incorporating the relations between three dynamical variables: cancer cells, healthy CD4 ⁺ T lymphocytes, and infected CD4 ⁺ T lymphocytes. Recent theoretical investigations indicate that these cells interactions lead to different dynamical outcomes, for instance to periodic or chaotic behavior. Firstly, we analytically prove the boundedness of the trajectories in the system’s attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. Our calculations reveal that the highest observable variable is the population of cancer cells, thus indicating that these cells could be monitored in future experiments in order to obtain time series for attractor’s reconstruction. We identify different dynamical behaviors of the system varying two biologically meaningful parameters: r ₁, representing the uncontrolled proliferation rate of cancer cells, and k ₁, denoting the immune system’s killing rate of cancer cells. The maximum Lyapunov exponent is computed to identify the chaotic regimes. Considering very recent developments in the literature related to the homotopy analysis method (HAM), we calculate the explicit series solutions of the cancer model and focus our analysis on the dynamical variable with the highest observability index. An optimal homotopy analysis approach is used to improve the computational efficiency of HAM by means of appropriate values for the convergence control parameter, which greatly accelerate the convergence of the series solution. The approximated analytical solutions are used to compute density plots, which allow us to discuss additional dynamical features of the model.     

Pullback attractors for asymptotically compact non-autonomous dynamical systems

First, we introduce the concept of pullback asymptotically compact non-autonomous dynamical system as an extension of the similar concept in the autonomous framework. Our definition is different from that of asymptotic compactness already used in the theory of random and non-autonomous dynamical systems (as developed by Crauel, Flandoli, Kloeden, Schmalfuss, amongst others) which means the existence of a (random or time-dependent) family of compact attracting sets. Next, we prove a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. This attractor is minimal and, in most practical applications, it is unique. Finally, we illustrate the theory with a 2D Navier–Stokes model in an unbounded domain.     

Dynamical behaviour and exact solutions of thirteenth order derivative nonlinear Schr\"{o}dinger equation

In this paper, we considered the model of the thirteenth order derivatives of nonlinear Schr\"{o}dinger equations. It is shown that a wave packet ansatz inserted into these equations leads to an integrable Hamiltonian dynamical sub-system. By using bifurcation theory of planar dynamical systems, in different parametric regions, we determined the phase portraits. In each of these parametric regions we obtain possible exact explicit parametric representation of the traveling wave solutions corresponding to homoclinic, hetroclinic and periodic orbits.     

Reciprocal inhibitory coupling: Measure and control of chaos on a biophysically motivated model of bursting

Bursting activity is an interesting feature of the temporal organization in many cell firing patterns. This complex behavior is characterized by clusters of spikes (action potentials) interspersed with phases of quiescence. As shown in experimental recordings, concerning the electrical activity of real neurons, the analysis of bursting models reveals not only patterned periodic activity but also irregular behavior [1], [2]. The interpretation of experimental results, particularly the study of the influence of coupling on chaotic bursting oscillations, is of great interest from physiological and physical perspectives. The inability to predict the behavior of dynamical systems in presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we focus our attention on a specific class of biophysically motivated maps, proposed in the literature to describe the chaotic activity of spiking–bursting cells [Cazelles B, Courbage M, Rabinovich M. Anti-phase regularization of coupled chaotic maps modelling bursting neurons. Europhys Lett 2001;56:504–9]. More precisely, we study a map that reproduces the behavior of a single cell and a map used to examine the role of reciprocal inhibitory coupling, specially on two symmetrically coupled bursting neurons. Firstly, using results of symbolic dynamics, we characterize the topological entropy associated to the maps, which allows us to quantify and to distinguish different chaotic regimes. In particular, we exhibit numerical results about the effect of the coupling strength on the variation of the topological entropy. Finally, we show that complicated behavior arising from the chaotic coupled maps can be controlled, without changing of its original properties, and turned into a desired attracting time periodic motion (a regular cycle). The control is illustrated by an application of a feedback control technique developed by Romeiras et al. [Romeiras FJ, Grebogi C, Ott E, Dayawansa WP. Controlling chaotic dynamical systems. Physica D 1992;58:165–92]. This work provides an illustration of how our understanding of chaotic bursting models can be enhanced by the theory of dynamical systems.     

一种描述复杂运动的模型的数值研究*

为了研究状态为连续系统的动力学的复杂性,我们根据对无穷维动力系统讨论的结论,提出了一种新的映射模型。利用数值方法,对这种模型进行了研究,发现其动力学行为可能是普适语言(universal language)和上下文有关语言(context-sensitive language).     

Time-reversal symmetry,Poincaré recurrence,irreversibility, and the entropic arrow of time: From mechanics to system thermodynamics

In this paper, we use a large-scale dynamical systems perspective to provide a system-theoretic foundation for thermodynamics. Specifically, using a state space formulation, we develop a nonlinear compartmental dynamical system model characterized by energy conservation laws that is consistent with basic thermodynamic principles. In addition, we establish the existence of a unique, continuously differentiable global entropy function for our large-scale dynamical system, and using Lyapunov stability theory we show that the proposed thermodynamic model has convergent trajectories to Lyapunov stable equilibria determined by the system initial energies. Finally, using the system entropy, we establish the absence of Poincaré recurrence for our thermodynamic model and develop a clear connection between irreversibility, the second law of thermodynamics, and the entropic arrow of time.     

Categorical axiomatics of dynamic programming

In systems theory, we need some general framework to take into account various concepts from different fields (automata theory, dynamical systems, optimization theory, etc.). This work is a proposal for such a framework. A decision process model is given with the help of category theory. Sufficient conditions for the validity of the equation of optimality of Bellman are established in this model. This model also generalizes the concept of finite automaton. Thus we unify the concepts of automaton and decision system. Especially, we propose the new concept of “opt-automaton” (automaton which optimizes).     

Coherent Swing Instability of Power Grids

We interpret and explain a phenomenon in short-term swing dynamics of multi-machine power grids that we term the Coherent Swing Instability (CSI). This is an undesirable and emergent phenomenon of synchronous machines in a power grid, in which most of the machines in a sub-grid coherently lose synchronism with the rest of the grid after being subjected to a finite disturbance. We develop a minimal mathematical model of CSI for synchronous machines that are strongly coupled in a loop transmission network and weakly connected to the infinite bus. This model provides a dynamical origin of CSI: it is related to the escape from a potential well, or, more precisely, to exit across a separatrix in the dynamical system for the amplitude of the weak nonlinear mode that governs the collective motion of the machines. The linear oscillations between strongly coupled machines then act as perturbations on the nonlinear mode. Thus we reveal how the three different mode oscillations??local plant, inter-machine, and inter-area modes??interact to destabilize a power grid. Furthermore, we present a phenomenon of short-term swing dynamics in the New England (NE) 39-bus test system, which is a well-known benchmark model for power grid stability studies. Using a partial linearization of the nonlinear swing equations and the proper orthonormal decomposition, we show that CSI occurs in the NE test system, because it is a dynamical system with a nonlinear mode that is weak relative to the linear oscillatory modes.     

Bifurcations of phase portraits of a Singular Nonlinear Equation of the Second Class

The soliton dynamics is studied using the Frenkel Kontorova (FK) model with non-convex interparticle interactions immersed in a parameterized on-site substrate potential. The case of a deformable substrate potential allows theoretical adaptation of the model to various physical situations. Non-convex interactions in lattice systems lead to a number of interesting phenomena that cannot be produced with linear coupling alone. In the continuum limit for such a model, the particles are governed by a Singular Nonlinear Equation of the Second Class. The dynamical behavior of traveling wave solutions is studied by using the theory of bifurcations of dynamical systems. Under different parametric situations, we give various sufficient conditions leading to the existence of propagating wave solutions or dislocation threshold, highlighting namely that the deformability of the substrate potential plays only a minor role.     

具有内生人口转变的Ramsey模型与仿真(Ⅰ)

本文将内生人口转变函数引入经典的Ramsey模型.通过解家庭效用最大化问题得到一个二维的动力系统,文中证明描述模型的动力系统至少存在一个非零平衡点并着重讨论非零平衡点惟一的情形.通过数值仿真得出在最优经济增长轨道上人口转变时期的人口变化对人均消费有本质的影响,而对人均资本影响不大.     

Dynamical analysis of chemotherapy models with time-dependent infusion

Classical mathematical models for chemotherapy assume a constant infusion rate of the chemotherapy agent. However in reality the infusion rate usually varies with respect to time, due to the natural (temporal or random) fluctuation of environments or clinical needs. In this work we study a non-autonomous chemotherapy model where the injection rate and injection concentration of the chemotherapy agent are time-dependent. In particular, we prove that the non-autonomous dynamical system generated by solutions to the non-autonomous chemotherapy system possesses a pullback attractor. In addition, we investigate the detailed interior structures of the pullback attractor to provide crucial information on the effectiveness of the treatment. The main analytical tool used is the theory of non-autonomous dynamical systems. Numerical experiments are carried out to supplement the analysis and illustrate the effectiveness of different types of infusions.     

Piecewise linear approximation for the dynamical Φ_3~4 model

We construct a piecewise linear approximation for the dynamicalΦ3~4 model on T^3.The approximation is based on the theory of regularity structures developed by Hairer(2014).They proved that renormalization in a dynamicalΦ3~4 model is necessary for defining the nonlinear term.In contrast to Hairer(2014),we apply piecewise linear approximations to space-time white noise,and prove that the solutions of the approximating equations converge to the solution of the dynamicalΦ_3~4 model.In this case,the renormalization corresponds to multiplying the solution by a t-dependent function,and adding it to the approximating equation.     

Asymptotic justification of dynamical equations for generalized Marguerre–von Kármán anisotropic shallow shells

In this paper, using technics from asymptotic analysis, we show that the three-dimensional dynamical model for a non-linearly elastic shallow shell made of anisotropic material, with boundary conditions of generalized Marguerre–von Kármán’s type, reduces to two-dimensional dynamical model.