时滞对磁通耦合及化学耦合神经元分岔及同步的影响

以化学突触耦合神经元模型为基础,讨论了抑制性及兴奋性条件下达到同步的区别及同步的类型。并根据磁通耦合对神经元放电的影响,讨论了具有时滞、磁通耦合和化学耦合Morris-Lecar (ML)神经元模型的放电状态、分岔类型及其同步情况。发现具有磁通耦合和化学耦合ML神经元系统在不同参数下会产生丰富的逆倍周期分岔或加周期分岔行为。而时滞的引入,虽然可以增加系统的周期性,但同时也会破环系统同步。相反,适当的耦合强度能够增加同步。     

Stability,coherent spiking and synchronization in noisy excitable systems with coupling and internal delays

We study the onset and the adjustment of different oscillatory modes in a system of excitable units subjected to two forms of noise and delays cast as external or internal according to whether they are associated with inter- or intra-unit activity. Conditions for stability of a single unit are derived in case of the linearized perturbed system, whereas the interplay of noise and internal delay in shaping the oscillatory motion is analyzed by the method of statistical linearization. It is demonstrated that the internal delay, as well as its coaction with external noise, drive the unit away from the bifurcation controlled by the excitability parameter. For the pair of interacting units, it is shown that the external/internal character of noise primarily influences frequency synchronization and the competition between the noise-induced and delay-driven oscillatory modes, while coherence of firing and phase synchronization substantially depend on internal delay. Some of the important effects include: (i) loss of frequency synchronization under external noise; (ii) existence of characteristic regimes of entrainment, where under variation of coupling delay, the optimized unit (noise intensity fixed at resonant value) may be controlled by the adjustable unit (variable noise) and vice versa, or both units may become adjusted to coupling delay; (iii) phase synchronization achieved both for noise-induced and delay-driven modes.     

Modelling gait transition in two-legged animals

The study of locomotor patterns has been a major research goal in the last decades. Understanding how intralimb and interlimb coordination works out so well in animals’ locomotion is a hard and challenging task. Many models have been proposed to model animal’s rhythms. These models have also been applied to the control of rhythmic movements of adaptive legged robots, namely biped, quadruped and other designs.In this paper we study gait transition in a central pattern generator (CPG) model for bipeds, the 4-cells model. This model is proposed by Golubitsky, Stewart, Buono and Collins and is studied further by Pinto and Golubitsky. We briefly resume the work done by Pinto and Golubitsky. We compute numerically gait transition in the 4-cells CPG model for bipeds. We use Morris–Lecar equations and Wilson–Cowan equations as the internal dynamics for each cell. We also consider two types of coupling between the cells: diffusive and synaptic. We obtain secondary gaits by bifurcation of primary gaits, by varying the coupling strengths. Nevertheless, some bifurcating branches could not be obtained, emphasizing the fact that despite analytically those bifurcations exist, finding them is a hard task and requires variation of other parameters of the equations. We note that the type of coupling did not influence the results.     

Existence of weak positive solutions to a nonlinear PDE system around a triple phase boundary, coupling domain and boundary variables

We consider a 2D nonlinear system of PDEs representing a simplified model of processes near a triple-phase boundary (TPB) in cathode catalyst layer of hydrogen fuel cells. The particularity of this system is the coupling of a variable satisfying a PDE in the interior of the domain with another variable satisfying a differential equation (DE) defined only on the boundary, through an adsorption-desorption equilibrium mechanism. The system includes also an isolated singular boundary condition which models the flux continuity at the contact of the TPB with a subdomain. By freezing certain terms we transform the nonlinear PDE system to an equation, which has a variational formulation. We prove several L^∞ and W1,p a priori estimates and then by using Schauder fixed point theorem we prove the existence of a weak positive bounded solution.     

The Theory of Quantum Fermi-Liquid Solutions with a Vector Order Parameter

We construct the thermodynamic theory and derive the hydrodynamic equations for a solution of Fermi superfluids with a vector order parameter. The thermodynamic quantities related to spontaneous breaking of phase symmetries and spin rotations are represented in terms of the order-parameter operator. We predict a new entrainment effect related to the mutual influence of magnetic and orbital degrees of freedom. We find the spectra of collective excitations and discuss the distinctions between the states under consideration and those with scalar and vector order parameters.     

Dynamics of chaotic maps for modelling the multifractal spectrum of human brain Diffusion Tensor Images

The multifractal spectra of 3d Diffusion Tensor Images (DTI) obtained by magnetic resonance imaging of the human brain are studied. They are shown to deviate substantially from artificial brain images with the same white matter intensity. All spectra, obtained from 12 healthy subjects, show common characteristics indicating non-trivial moments of the intensity. To model the spectra the dynamics of the chaotic Ikeda map are used. The DTI multifractal spectra for positive q are best approximated by 3d coupled Ikeda maps in the fully developed chaotic regime. The coupling constants are as small as α = 0.01. These results reflect not only the white tissue non-trivial architectural complexity in the human brain, but also demonstrate the presence and importance of coupling between neuron axons. The architectural complexity is also mirrored by the deviations in the negative q-spectra, where the rare events dominate. To obtain a good agreement in the DTI negative q-spectrum of the brain with the Ikeda dynamics, it is enough to slightly modify the most rare events of the coupled Ikeda distributions. The representation of Diffusion Tensor Images with coupled Ikeda maps is not unique: similar conclusions are drawn when other chaotic maps (Tent, Logistic or Henon maps) are employed in the modelling of the neuron axons network.     

The effects of time delay on the stochastic resonance in feed-forward-loop neuronal network motifs

The dependence of stochastic resonance in the feed-forward-loop neuronal network motifs on the noise and time delay are studied in this paper. By computational modeling, Izhikevich neuron model with the chemical coupling is used to build the triple-neuron feed-forward-loop motifs with all possible motif types. Numerical results show that the correlation between the periodic subthreshold signal’s frequency and the dynamical response of the network motifs is resonantly dependent on the intensity of additive spatiotemporal noise. Interestingly, the excitatory intermediate neuron could induce intermittent stochastic resonance, whereas the inhibitory one weakens its influence on the intermittent mode. More importantly, it is found that the increasing delays can induce the intermittent appearance of regions of stochastic resonance. Based on the effects of the time delay on the stochastic resonance, the reasons and conditions of such intermittent resonance phenomenon are analyzed.     

Spatio-temporal patterns of Hopf bifurcating periodic oscillations in a pair of identical tri-neuron network loops

Recently, the coupling time delay has been considered as the source of the occurrence of the phase-flip bifurcation in time-delay coupled system. But the analytical results of how the coupling time delay affects this phenomenon is still lacking. In this paper, we consider a pair of identical tri-neuron network coupled with time delay. By using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups, we investigate the spatio-temporal patterns of Hopf bifurcating periodic oscillations induced by the coupling time delay. The explicit intervals of delay and the regions in the plane of the coupling strength and the gain of the inherent response function for the existence of synchronized in-phase or anti-phase oscillation are obtained. Our study show that the coupling time delay does not affect the spatio-temporal patterns of the individual neural loop but it has the significant impact on the spatio-temporal patterns between the two loops. These analytic results are then verified by numerical simulations.     

Driven response of time delay coupled limit cycle oscillators

We study the periodic forced response of a system of two limit cycle oscillators that interact with each other via a time delayed coupling. Detailed bifurcation diagrams in the parameter space of the forcing amplitude and forcing frequency are obtained for various interesting limits using numerical and analytical means. In particular, the effects of the coupling strength, the natural frequency spread of the two oscillators and the time delay parameter on the size and nature of the entrainment domain are delineated. For an appropriate choice of time delay, synchronization can occur with infinitesimal forcing amplitudes even at off-resonant driving. The system is also found to display a nonlinear response on certain critical contours in the space of the coupling strength and time delay. Numerical simulations with a large number of coupled driven oscillators display similar behavior. Time delay offers a novel tuning knob for controlling the system response over a wide range of frequencies and this may have important practical applications.     

Coordination of a three-level supply chain (supplier–manufacturer–retailer) with permissible delay in payments and price discounts

Different coordination mechanisms are used to manage supply chains. This paper investigates a coordination of a three-level supply chain (supplier–manufacturer–retailer) by coupling two well-known trade credit mechanisms that are widely used in practice, permissible delay in payments and price discounts, where the length of the delay period and discounts offered along the supply chain are decision variables. The paper investigates nine different cases of delay in payments along with eight cases of price discounts among the three players in the supply chain. A numerical example was presented to compare between the cases considered. Also, extensive sensitivity analyses were performed to study the effect of changing the model parameters on the optimal decisions. In addition, we point out the limitations of each model developed in this paper. The numerical examples and the sensitivity analyses conclude that the coupling of delay in payments and price discounts increases the profit of the supply chain more than using only a single mechanism at a time.     

The applications of a higher-dimensional Lie algebra and its decomposed subalgebras

With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings.     

Chaotic neuron clock

A chaotic model of spontaneous (without external stimulus) neuron firing has been analyzed by mapping the irregular spiking time-series into telegraph signals. In this model the fundamental frequency of chaotic Rössler attractor provides (with a period doubling) the strong periodic component of the generated irregular signal. The exponentially decaying broad-band part of the spectrum of the Rössler attractor has been transformed by the threshold firing mechanism into a scaling tale. These results are compared with irregular spiking time-series obtained in vitro from a spontaneous activity of hippocampal (CA3) singular neurons (rat’s brain slice culture). The comparison shows good agreement between the model and experimentally obtained spectra.     

大脑皮层信号作用下人体步态节律运动的探讨

中枢模式发生器可产生节律性运动．目前的中枢模式发生器（CPG）建模研究可以很好地表现CPG的自激行为,但对于人脑信号的调节作用没有讨论．为了体现大脑皮层信号对于CPG网络的调控性,基于Matsuoka神经振荡器的CPG模型,对原有模型中输入刺激与网络内部参数的关联进行了复杂构建,使得模型本身各参数随输入信号的变化而变化,增强了输入信号对于网络自身的影响,令CPG网络不仅仅产生自激状态,同时能够产生自我调节的运动形式,从而体现出大脑信号的调控作用．数值模拟计算结果表明,修正后的模型随着输入刺激的变化可以产生不同模式及不同频率的运动形式,且各不同形式之间可以相互转换,从而在理论上很好地反映出大脑信号在步态节律运动过程中对步态的模式和频率起到了一定的调节作用,实现了各种步态运动之间的行为转换及恢复的功能,从理论上实现了自发节律与大脑调节性节律运动的共存性,做到大脑信号与CPG模型的统一．     

The dynamics of a mutual interference age structured predator-prey model with time delay and impulsive perturbations on predators

In this paper, we introduce a mutual interference age structured predator-prey (natural enemy-pest) model with constant maturation time delay for the prey, and then propose a pest management strategy by constant periodic releasing for the predator. We show that there exists a global attractive pest-eradication periodic solution when the periodic releasing amount μ₁ and μ₂ are lager than some critical value. Further, to obtain a more effective pest control strategy, we give the conditions (involving the estimate of μ₁ and μ₂) in which the model is uniformly permanent and the pest population is under the economic threshold level. We believe that the results will provide reliable tactic basis for the practical pest management.     

The effects of synaptic time delay on motifs of chemically coupled Rulkov model neurons

Given the importance of the network motifs, we consider a pair of Rulkov chaotic map neurons, reciprocally coupled via symmetrical chemical synapses with the time delay τ. For the inhibitory and excitatory synapses, the system dynamics is determined by the synaptic weight g_c, synaptic gain parameter k, time delay τ and the external excitation σ. Due to chaotic nature of the map and synaptic model complexity, the appropriately averaged cross-correlation of membrane potentials represents a suitable numerical diagnostics to quantify mutual synchronization. Along with the expected phase and anti-phase synchronization regimes, we find the emergent phenomena that significantly influence the synchronization behavior.     

Adaptive synchronization in an array of chaotic neural networks with mixed delays and jumping stochastically hybrid coupling

In this paper, a general model of an array of N linearly coupled delayed neural networks with Markovian jumping hybrid coupling is introduced. The hybrid coupling consists of constant coupling, discrete and distributed time-varying delay coupling. The complex dynamical network jumps from one mode to another according to a Markovian chain, where all the coupling configurations are also dependent on mode switching. Meanwhile, all the coupling terms are subjected to stochastic disturbances which are described in terms of a Brownian motion. By adaptive approach, some sufficient criteria have been derived to ensure the synchronization in an array of jump neural networks with mixed delays and hybrid coupling in mean square. Surprisingly, it is found that complex networks with two different structure can also be synchronized according to known probability matrix. Finally, an example illustrated by switching between small-world networks and nearest-neighbor networks is given to show the effectiveness of the proposed criteria.     

Single or multiple synchronization transitions in scale-free neuronal networks with electrical or chemical coupling

In this paper, we have studied time delay- and coupling strength-induced synchronization transitions in scale-free modified Hodgkin–Huxley (MHH) neuron networks with gap-junctions and chemical synaptic coupling. It is shown that the synchronization transitions are much different for these two coupling types. For gap-junctions, the neurons exhibit a single synchronization transition with time delay and coupling strength, while for chemical synapses, there are multiple synchronization transitions with time delay, and the synchronization transition with coupling strength is dependent on the time delay lengths. For short delays we observe a single synchronization transition, whereas for long delays the neurons exhibit multiple synchronization transitions as the coupling strength is varied. These results show that gap junctions and chemical synapses have different impacts on the pattern formation and synchronization transitions of the scale-free MHH neuronal networks, and chemical synapses, compared to gap junctions, may play a dominant and more active function in the firing activity of the networks. These findings would be helpful for further understanding the roles of gap junctions and chemical synapses in the firing dynamics of neuronal networks.     

Ordering chaos and synchronization transitions by chemical delay and coupling on scale-free neuronal networks

Chemical synaptic connections are more common than electric ones in neurons, and information transmission delay is especially significant for the synapses of chemical type. In this paper, we report a phenomenon of ordering spatiotemporal chaos and synchronization transitions by the delays and coupling through chemical synapses of modified Hodgkin–Huxley (MHH) neurons on scale-free networks. As the delay τ is increased, the neurons exhibit transitions from bursting synchronization (BS) to intermittent multiple spiking synchronizations (SS). As the coupling g_syn is increased, the neurons exhibit different types of firing transitions, depending on the values of τ. For a smaller τ, there are transitions from spatiotemporal chaotic bursting (SCB) to BS or SS; while for a larger τ, there are transitions from SCB to intermittent multiple SS. These findings show that the delays and coupling through chemical synapses can tame the chaotic firings and repeatedly enhance the firing synchronization of neurons, and hence could play important roles in the firing activity of the neurons on scale-free networks.     

Traveling wave solutions of diffusive Hindmarsh–Rose-type equations with recurrent neural feedback

From the perspective of bifurcation theory, this study investigates the existence of traveling wave solutions for diffusive Hindmarsh–Rose-type (dHR-type) equations with recurrent neural feedback (RNF). The applied model comprises two additional terms: 1) a diffusion term for the conduction process of action potentials and 2) a delay term. The delay term is introduced because if a neuron excites a second neuron, the second neuron, in turn, excites or inhibits the first neuron. To probe the existence of traveling wave solutions, this study applies center manifold reduction and a normal form method, and the results demonstrate the existence of a heteroclinic orbit of a three-dimensional vector for dHR-type equations with RNF near a fold–Hopf bifurcation. Finally, numerical simulations are presented.     

货币供给动态调控模型及其稳定性

根据我国货币政策传导机制并结合目前经济金融形势,建立了货币供给系统的一般动态模型,为了便于分析系统的稳定性,在此基础上建立前向离散货币供给系统简化模型并用经济控制论的方法分析了系统的稳定性。