Asynchronous states and the emergence of synchrony in large networks of interacting excitatory and inhibitory neurons

We investigate theoretically the conditions for the emergence of synchronous activity in large networks, consisting of two populations of extensively connected neurons, one excitatory and one inhibitory. The neurons are modeled with quadratic integrate-and-fire dynamics, which provide a very good approximation for the subthreshold behavior of a large class of neurons. In addition to their synaptic recurrent inputs, the neurons receive a tonic external input that varies from neuron to neuron. Because of its relative simplicity, this model can be studied analytically. We investigate the stability of the asynchronous state (AS) of the network with given average firing rates of the two populations. First, we show that the AS can remain stable even if the synaptic couplings are strong. Then we investigate the conditions under which this state can be destabilized. We show that this can happen in four generic ways. The first is a saddle-node bifurcation, which leads to another state with different average firing rates. This bifurcation, which occurs for strong enough recurrent excitation, does not correspond to the emergence of synchrony. In contrast, in the three other instability mechanisms, Hopf bifurcations, which correspond to the emergence of oscillatory synchronous activity, occur. We show that these mechanisms can be differentiated by the firing patterns they generate and their dependence on the mutual interactions of the inhibitory neurons and cross talk between the two populations. We also show that besides these codimension 1 bifurcations, the system can display several codimension 2 bifurcations: Takens-Bogdanov, Gavrielov-Guckenheimer, and double Hopf bifurcations.     

Bifurcation software in Matlab with applications in neuronal modeling

Many biological phenomena, notably in neuroscience, can be modeled by dynamical systems. We describe a recent improvement of a Matlab software package for dynamical systems with applications to modeling single neurons and all-to-all connected networks of neurons. The new software features consist of an object-oriented approach to bifurcation computations and the partial inclusion of C-code to speed up the computation. As an application, we study the origin of the spiking behaviour of neurons when the equilibrium state is destabilized by an incoming current. We show that Class II behaviour, i.e. firing with a finite frequency, is possible even if the destabilization occurs through a saddle-node bifurcation. Furthermore, we show that synchronization of an all-to-all connected network of such neurons with only excitatory connections is also possible in this case.     

平面时变向量场Hopf分岔的精确定位与可视化

分岔是指动力系统在演化过程中定性行为发生质变的现象．分岔研究对揭示复杂流场的不稳定过程有重要意义．文中提出一个平面时变向量场Hopf分岔的榆出、定位与可视化算法．通过精密跟踪场内全体临界点的变化以及用一个颜色模型图画般地记述这些变化的演化路径,新算法不仅实现了对平面时变向量场中全部Hopf分岔点的检出和定位,还能详尽显示分岔的发展过程,文中方法原则上也适用于其它局部分岔类型的检出,为用数据可视化技术观察和研究分岔现象开辟了新的途径。     

一种自反馈细胞神经网络稳定性分析及在函数优化中的应用

提出一种具有暂态混沌的细胞神经网络,该网络是利用欧拉算法将模型的状态方程转化为离散形式并引入一项负的自反馈而形成的.由对单个神经元的仿真发现,该模型具有分叉和混沌的特性.在函数优化中,该网络首先经过一个倍周期倒分叉过程进行混沌搜索;然后进行类似Hopfield网络的梯度搜索.由于该网络利用了混沌搜索固有的随机性和轨道遍历性,因而具有较强的全局寻优的能力.最后通过2个函数优化的例子验证了该网络的有效性.     

Computation in a single neuron: Hodgkin and Huxley revisited

A spiking neuron "computes" by transforming a complex dynamical input into a train of action potentials, or spikes. The computation performed by the neuron can be formulated as dimensional reduction, or feature detection, followed by a nonlinear decision function over the low-dimensional space. Generalizations of the reverse correlation technique with white noise input provide a numerical strategy for extracting the relevant low-dimensional features from experimental data, and information theory can be used to evaluate the quality of the low-dimensional approximation. We apply these methods to analyze the simplest biophysically realistic model neuron, the Hodgkin-Huxley (HH) model, using this system to illustrate the general methodological issues. We focus on the features in the stimulus that trigger a spike, explicitly eliminating the effects of interactions between spikes. One can approximate this triggering "feature space" as a two-dimensional linear subspace in the high-dimensional space of input histories, capturing in this way a substantial fraction of the mutual information between inputs and spike time. We find that an even better approximation, however, is to describe the relevant subspace as two dimensional but curved; in this way, we can capture 90% of the mutual information even at high time resolution. Our analysis provides a new understanding of the computational properties of the HH model. While it is common to approximate neural behavior as "integrate and fire," the HH model is not an integrator nor is it well described by a single threshold.     

神经元周期放电模式的分岔

利用一种可以计算自治非线性系统周期解及周期的改进打靶法,求解了神经元电活动Rose—Hind-marsh（R-H）模型自发放电的周期解和周期;计算了周期放电的Floquet乘子并分析了周期解的分岔,如倍周期分岔,鞍-结分岔．研究结果有助于进一步理解神经放电模式转迁的动力学和生物学意义．     

A novel delayed chaotic neural model and its circuitry implementation

In this paper, we investigate a novel delayed chaotic neural model, in which a non-monotonously increasing transfer function is employed as activation function. Local stability and existence of Hopf bifurcation are analyzed in details. Chaos behavior of the neuron model is observed in computer simulations. An electronic implementation of the neuron is also considered. The dynamical behavior of the designed circuits is closely similar to the results simulated by numerical experiments.     

Coding of temporally varying signals in networks of spiking neurons with global delayed feedback

Oscillatory and synchronized neural activities are commonly found in the brain, and evidence suggests that many of them are caused by global feedback. Their mechanisms and roles in information processing have been discussed often using purely feedforward networks or recurrent networks with constant inputs. On the other hand, real recurrent neural networks are abundant and continually receive information-rich inputs from the outside environment or other parts of the brain. We examine how feedforward networks of spiking neurons with delayed global feedback process information about temporally changing inputs. We show that the network behavior is more synchronous as well as more correlated with and phase-locked to the stimulus when the stimulus frequency is resonant with the inherent frequency of the neuron or that of the network oscillation generated by the feedback architecture. The two eigenmodes have distinct dynamical characteristics, which are supported by numerical simulations and by analytical arguments based on frequency response and bifurcation theory. This distinction is similar to the class I versus class II classification of single neurons according to the bifurcation from quiescence to periodic firing, and the two modes depend differently on system parameters. These two mechanisms may be associated with different types of information processing.     

Mathematical-model-based design of silicon burst neurons

Conventionally, silicon neurons have been designed based on two major principles, namely phenomenological and conductance-based principles. In previous studies [T. Kohno, K. Aihara, Parameter tuning of a MOSFET-based nerve membrane, in: Proceedings of the 10th International Symposium on Artificial Life and Robotics 2005, 2005, pp. 91–94; T. Kohno, K. Aihara, A MOSFET-based model of a Class 2 Nerve membrane, IEEE Trans. Neural Networks 16 (3) (2005) 754–773; T. Kohno, K. Aihara, Bottom-up design of Class 2 silicon nerve membrane, J. Intell. Fuzzy Syst., in press], we proposed a mathematical-model-based design principle that is based on phase plane and bifurcation analyses. It reproduces the mathematical structures of biological neuron models, thus making the silicon neurons simple and biologically realistic. In this study, we demonstrate that square-wave and another type of silicon bursters can be constructed by adding simple circuitries and tuning the system parameters for the silicon nerve membrane designed in our previous studies. Our simple square-wave burster exhibits various firing patterns, including chaotic spiking and bursting.     

Bifurcation analysis in a discrete-time single-directional network with delays

In this paper, we consider a simple discrete-time single-directional network of four neurons. The characteristics equation of the linearized system at the zero solution is a polynomial equation involving very high-order terms. We first derive some sufficient and necessary conditions ensuring that all the characteristic roots have modulus less than 1. Hence, the zero solution of the model is asymptotically stable. Then, we study the existence of three types of bifurcations, such as fold bifurcations, flip bifurcations, and Neimark–Sacker (NS) bifurcations. Based on the normal form theory and the center manifold theorem, we discuss their bifurcation directions and the stability of bifurcated solutions. In addition, several codimension two bifurcations can be met in the system when curves of codimension one bifurcations intersect or meet tangentially. We proceed through listing smooth normal forms for all the possible codimension 2 bifurcations.     

Dynamical Analysis of a Tri‐Neuron Fractional Network

The present paper concerns with the dynamics of a fractional neural network involving three neurons. Firstly, the bifurcation point is identified for which Hopf bifurcations may occur by taking the system parameter as a bifurcation parameter via the stability analysis of fractional systems. It is indicated that the system parameter can significantly affect the dynamical properties of such network. Secondly, the impact of the order on the bifurcation point is carefully examined. It is found that the occurrence of bifurcation is delayed as the order increases as long as the other system parameters are established. Finally, a numerical example is exploited to verify the efficiency of theoretical results.     

高斯白噪声对神经元映射模型动力学的影响*

研究了噪声引发的神经元相干共振现象。基于神经元二维映射模型,用高斯白噪声模拟生物神经系统中的环境噪声,通过数字仿真研究了噪声对神经元动力学特性的影响。研究发现,噪声可以提高静态神经元的可兴奋性,诱导其产生动作电位,当噪声达到某一适中强度时,动作电位的有序性可以达到最佳,产生相干共振现象。研究了系统参数的选取对神经元共振特性的影响,发现控制参数越接近分岔点,相干共振性越好。结果表明合适强度的噪声可以促进神经元的信息传递。     

不可压缩流中二元机翼运动的Hopf分岔

机翼的颤振是一种典型的自激振动,它是由气动力、弹性力和惯性力的相互作用引起的一种气动弹性现象.本文研究了具有结构非线性刚度恢复力的机翼颤振的Hopf分岔问题.首先,利用连续时间的Hopf分岔显式临界准则分析了机翼颤振Hopf分岔的存在性,推导了第一李雅普诺夫系数的通项公式,为判定机翼Hopf分岔的稳定性提供了依据.其次,分析了机翼颤振退化的余维二Hopf分岔的存在性条件,得到了满足条件的双参数分岔区域.然后,推导了第二李雅普诺夫系数的通项公式并结合中心流形降阶原理和同构变换进一步分析了余维二Hopf分岔的稳定性以及其局部开折问题.最后,通过推导第三李雅普诺夫系数分析了余维三Hopf分岔中心的稳定性.     

Two computational regimes of a single-compartment neuron separated by a planar boundary in conductance space

Recent in vitro data show that neurons respond to input variance with varying sensitivities. Here we demonstrate that Hodgkin-Huxley (HH) neurons can operate in two computational regimes: one that is more sensitive to input variance (differentiating) and one that is less sensitive (integrating). A boundary plane in the 3D conductance space separates these two regimes. For a reduced HH model, this plane can be derived analytically from the V nullcline, thus suggesting a means of relating biophysical parameters to neural computation by analyzing the neuron's dynamical system.     

Bifurcation analysis of a two-dimensional simplified Hodgkin–Huxley model exposed to external electric fields

In this paper, the dynamical behaviors of a two-dimensional simplified Hodgkin–Huxley (H–H) model exposed to external electric fields are investigated through qualitative analysis and numerical simulation. A necessary and sufficient condition is proposed for the existence of the Hopf bifurcation. Saddle-node bifurcations and canards of the simplified model with the coefficients of different linear forms are also discussed. Finally, the bifurcation curves with the coefficients of different linear forms are shown. The numerical results demonstrate that some linear forms can retain the bifurcation characteristics of the original model, which is of great use to simplify the H–H model for the real-world applications.     

Control of an anaerobic digester through normal form of fold bifurcation

Nonlinear dynamics is ubiquitous in engineering systems. As some parameters are varied bifurcations arise in the state variables. Generically, when one parameter changes, Hopf and fold bifurcations are found. Other ones can also be present due to special systems characteristics, such as symmetries. Knowing in advance the significant bifurcation scenario, a novel approach to control can be considered. We compute the normal form corresponding to such a bifurcation and we take this model as the nominal model of the plant. Then we design a nonlinear control which takes advantage of the precise bifurcation scenario. This general method is applied, in this paper, to an anaerobic digester. We will control the process with an adaptive controller.Specifically, we want to compare with the case that the nominal plant is considered as a linear model, such as it is typical in adaptive control techniques. Our proposed method has more benefits in signal control effort, faster convergence rate and low error.This paper shows how the combination of appropriated nonlinear dynamic techniques such as bifurcations and normal forms, and nonlinear control, can give rise to an improvement of the traditional methodology.     

Bifurcation study of neuron firing activity of the modified Hindmarsh–Rose model

In this paper, the effects of different parameters on the dynamic behavior of the nonlinear dynamical system are investigated based on modified Hindmarsh–Rose neural nonlinear dynamical system model. We have calculated and analyzed dynamic characteristics of the model under different parameters by using single parameter bifurcation diagram, time response diagram and two parameter bifurcation diagram. The results show that the period-adding bifurcation (with or without chaos), period-doubling bifurcation and intermittent chaos phenomenon (periodic and intermittent chaotic) can be observed more clearly and directly from the two parameter bifurcation diagram, and the optimal parameters matching interval can also be found easily.     

皮层锥体神经元模型的动力学分析

簇发放是锥体神经元的一种典型特性,在确定性的信号传递和突触可塑性方面有着很重要的功能作用,本文通过对一类可产生复杂簇发放的皮层锥体神经元房室模型的研究,从非线性动力学角度对模型所产生的复杂簇发放做了详细的分析,讨论了不同电生理参数条件下,模型簇发放中所蕴含着的丰富的动力学性质,如:峰峰间距(InterSpike Intervals,ISIs)的加周期分岔和倍周期分岔等,通过模型分析结果可进一步理解皮层锥体神经元动作电位簇发放中所蕴含的丰富的发放模式和节律编码.     

Numerical and analytical study of bifurcations in a model of electrochemical reactions in fuel cells

The bifurcations in a three-variable ODE model describing the oxygen reduction reaction on a platinum surface is studied. The investigation is motivated by the fact that this reaction plays an important role in fuel cells. The goal of this paper is to determine the dynamical behaviour of the ODE system, with emphasis on the number and type of the stationary points, and to find the possible bifurcations. It is shown that a non-trivial steady state can appear through a transcritical bifurcation, or a stable and an unstable steady state can arise as a result of saddle-node bifurcation. The saddle-node bifurcation curve is determined by using the parametric representation method, and this enables us to determine numerically the parameter domain where bistability occurs, which is important from the chemical point of view.     

神经元模型的复杂动力学：分岔与编码

研究了改进的Morris—Lecar（ML）神经元模型的放电节律模式和模式转化的峰峰间期（interspike intervals,ISIs）分岔结构,通过调节模型中的两个重要参数μ和Vk,发现对于固定的μ,改变Vk,神经元呈现出从倍周期级联分岔到加周期分岔的复杂结构,放电模式从静息态转化为周期、混沌簇放电状态;若选取此分岔过程中的某一Vk值,对μ进行调节,呈现出的ISIs分岔结构在很大程度上取决于单个神经元的放电节律模式,且单个神经元处于混沌簇放电时,肛带来的分岔动力学行为较丰富．由于神经元能够通过动作电位对信息进行编码,所以我们推测,研究神经元的放电节律模式和动作电位的ISIs分岔结构能为理解神经信息编码机制提供线索．