Efficient estimation of phase-resetting curves in real neurons and its significance for neural-network modeling

The phase-resetting curve (PRC) of a neural oscillator describes the effect of a perturbation on its periodic motion and is therefore useful to study how the neuron responds to stimuli and whether it phase locks to other neurons in a network. Combining theory, computer simulations and electrophysiological experiments we present a simple method for estimating the PRC of real neurons. This allows us to simplify the complex dynamics of a single neuron to a phase model. We also illustrate how to infer the existence of coherent network activity from the estimated PRC.     

Dynamical origin of independent spiking and bursting activity in neural microcircuits

The relationship between spiking and bursting dynamics is a key question in neuroscience, particularly in understanding the origins of different neural coding strategies and the mechanisms of motor command generation and neural circuit coordination. Experiments indicate that spiking and bursting dynamics can be independent. We hypothesize that different mechanisms for spike and burst generation, intrinsic neuron dynamics for spiking and a modulational network instability for bursting, are the origin of this independence. We tested the hypothesis in a detailed dynamical analysis of a minimal inhibitory neural microcircuit (motif) of three reciprocally connected Hodgkin-Huxley neurons. We reduced this high-dimensional dynamical system to a rate model and showed that both systems have identical bifurcations from tonic spiking to burst generation, which, therefore, does not depend on the details of spiking activity.     

Dynamics of simple electronic neural networks

We examine systems of one and two nonlinear threshold switching elements (“neurons”), of the kind used in electronic neural networks. Characteristics of these systems which deviate from standard ideal models are found to induce complex dynamics. When the neurons possess a finite frequency response or a transfer characteristic with a time delay, underdamped transients and instability leading to oscillation can occur. Inertia in the neuron connections is found to cause ringing about fixed points, convoluted basin boundaries, instability and spontaneous oscillation, and chaotic behavior when driven. Furthermore, the collective behavior of a network of multiple neurons can be underdamped even when the individual connections are overdamped. These results imply that care should be exercised in implementing networks with electronic devices or when adding inertia to enhance the performance of optimizing networks.     

Interaction function of coupled bursting neurons

The interaction functions of electrically coupled Hindmarsh–Rose(HR) neurons for different firing patterns are investigated in this paper.By applying the phase reduction technique,the phase response curve(PRC) of the spiking neuron and burst phase response curve(BPRC) of the bursting neuron are derived.Then the interaction function of two coupled neurons can be calculated numerically according to the PRC(or BPRC) and the voltage time course of the neurons.Results show that the BPRC is more and more complicated with the increase of the spike number within a burst,and the curve of the interaction function oscillates more and more frequently with it.However,two certain things are unchanged:Φ = 0,which corresponds to the in-phase synchronization state,is always the stable equilibrium,while the anti-phase synchronization state with Φ = 0.5 is an unstable equilibrium.     

Google matrix analysis of C.elegans neural network

We study the structural properties of the neural network of the C.elegans (worm) from a directed graph point of view. The Google matrix analysis is used to characterize the neuron connectivity structure and node classifications are discussed and compared with physiological properties of the cells. Our results are obtained by a proper definition of neural directed network and subsequent eigenvector analysis which recovers some results of previous studies. Our analysis highlights particular sets of important neurons constituting the core of the neural system. The applications of PageRank, CheiRank and ImpactRank to characterization of interdependency of neurons are discussed.     

Phase synchronization of bursting neural networks with electrical and delayed dynamic chemical couplings

Diffusive electrical connections in neuronal networks are instantaneous, while excitatoryor inhibitory couplings through chemical synapses contain a transmission time-delay.Moreover, chemical synapses are nonlinear dynamical systems whose behavior can bedescribed by nonlinear differential equations. In this work, neuronal networks withdiffusive electrical couplings and time-delayed dynamic chemical couplings are considered.We investigate the effects of distributed time delays on phase synchronization of burstingneurons. We observe that in both excitatory and Inhibitory chemical connections, the phasesynchronization might be enhanced when time-delay is taken into account. This distributedtime delay can induce a variety of phase-coherent dynamical behaviors. We also study thecollective dynamics of network of bursting neurons. The network model presents theso-called Small-World property, encompassing neurons whose dynamics have two time scales(fast and slow time scales). The neuron parameters in such Small-World network, aresupposed to be slightly different such that, there may be synchronization of the bursting(slow) activity if the coupling strengths are large enough. Bounds for the criticalcoupling strengths to obtain burst synchronization in terms of the network structure aregiven. Our studies show that the network synchronizability is improved, as itsheterogeneity is reduced. The roles of synaptic parameters, more precisely those of thecoupling strengths and the network size are also investigated.     

Random recurrent neural networks dynamics

This paper is a review dealing with the study of large size random recurrent neural networks. The connection weights are varying according to a probability law and it is possible to predict the network dynamics at a macroscopic scale using an averaging principle. After a first introductory section, the section 2 reviews the various models from the points of view of the single neuron dynamics and of the global network dynamics. A summary of notations is presented, which is quite helpful for the sequel. In section 3, mean-field dynamics is developed. The probability distribution characterizing global dynamics is computed. In section 4, some applications of mean-field theory to the prediction of chaotic regime for Analog Formal Random Recurrent Neural Networks (AFRRNN) are displayed. The case of AFRRNN with an homogeneous population of neurons is studied in section 4.1. Then, a two-population model is studied in section 4.2. The occurrence of a cyclo-stationary chaos is displayed using the results of [16]. In section 5, an insight of the application of mean-field theory to IF networks is given using the results of [9].     

Chaos in neural systems

The occurence of chaos in continuous-time neural network models is demonstrated through two examples, (i) a single neuron with an unusual kind of periodic input and (ii) a randomly connected network of 26 neurons with 7 incoming synapses per neuron, half the neurons being subject to a steady external stimulus. The former case is susceptible to exhaustive analysis, while the latter, discovered by computer simulation, is more attuned to neurobiological reality.     

Correlation detection and resonance in neural systems with distributed noise sources

We investigated the resonance behavior in model neurons receiving a large number of random synaptic inputs, whose distributed nature permits one to introduce correlations between them and investigate its effect on cellular responsiveness. A change in the strength of this background led to enhanced responsiveness, consistent with stochastic resonance. Altering the correlation revealed a type of resonance behavior in which the neuron is sensitive to statistical properties rather than the strength of the noise. Remarkably, the neuron could detect weak correlations among the distributed inputs within millisecond time scales.     

Speed of synchronization in complex networks of neural oscillators: analytic results based on Random Matrix Theory

We analyze the dynamics of networks of spiking neural oscillators. First, we present an exact linear stability theory of the synchronous state for networks of arbitrary connectivity. For general neuron rise functions, stability is determined by multiple operators, for which standard analysis is not suitable. We describe a general nonstandard solution to the multioperator problem. Subsequently, we derive a class of neuronal rise functions for which all stability operators become degenerate and standard eigenvalue analysis becomes a suitable tool. Interestingly, this class is found to consist of networks of leaky integrate-and-fire neurons. For random networks of inhibitory integrate-and-fire neurons, we then develop an analytical approach, based on the theory of random matrices, to precisely determine the eigenvalue distributions of the stability operators. This yields the asymptotic relaxation time for perturbations to the synchronous state which provides the characteristic time scale on which neurons can coordinate their activity in such networks. For networks with finite in-degree, i.e., finite number of presynaptic inputs per neuron, we find a speed limit to coordinating spiking activity. Even with arbitrarily strong interaction strengths neurons cannot synchronize faster than at a certain maximal speed determined by the typical in-degree.     

双耳幅值差确定声源方向的神经信息处理机理研究

神经信息系统实质上是定量系统, 应引起足够重视. 关于神经系统的定量研究方面的报道比较少见. 这一问题将会影响进一步的研究, 如双耳声音定向. 双耳定向是定量测量, 用定性分析的方法无法满足要求. 已有的生理实验发现声音输入信号强度与听觉神经的输出频率存在单调递增关系, 所以本文中声音强度的变化被简化成神经脉冲频率的变化. 本文基于圆映射和符号动力学原理, 建立了神经回路定量模型, 模型中对同侧输入回路采用兴奋性耦合, 对侧输入回路采用抑制性耦合, 并考虑神经元间突触连接的量子释放特征, 采用化学耦合模型实现连接, 用耦合系数表示神经元间的耦合程度. 采用Hodgkin-Huxley模型仿真研究听觉神经回路的输入/输出脉冲序列关系. 在已经仿真过的参数范围, 模型在输入信号变化与输出脉冲频率变化间存在单调递增/递减的关系. 对于单输入单输出的神经元, 采用符号动力学方法进行符号化; 对于多输入单输出的神经元, 采用分析各输出脉冲的产生时间, 判断其变化位置, 从神经脉冲序列中得到对应的两耳声音幅值差变化, 以此定位声源. 随着输出脉冲数的增加, 符号序列的长度增加, 符号序列对输入信号变化敏感, 能够得到较高的测量精度. 仿真结果表明这个模型是定量的, 神经脉冲序列能够区分信号的大小.     

Structured information in sparse-code metric neural networks

Sparse-code networks have retrieval abilities which are strongly dependent on the firing threshold for the neurons. If the connections are spatially uniform, the macroscopic properties of the network can be measured by the overlap between neurons and learned patterns, and by the global activity. However, for nonuniform networks, for instance small-world networks, the neurons can retrieve fragments of patterns without performing global retrieval. Local overlaps are needed to describe the network. We characterize the structure type of the neural states using a parameter that is related to fluctuations of the local overlaps, with distinction between bump and block phases. Simulation of neural dynamics shows a competition between localized (bump), structured (block) and global retrieval. When the network topology randomness increases, the phase-diagram shows a transition from local to global retrieval. Furthermore, the local phase splits into a bump phase for low activity and a block phase for high activity. A theoretical approach solves the asymptotic limit of the model, and confirms the simulation results which predicts the change of stability from bumps to blocks when the storage ratio increases.     

Avalanche dynamics of idealized neuron function in the brain on an uncorrelated random scale-free network

We study a simple model for a neuron function in a collective brain system. The neural network is composed of an uncorrelated configuration model (UCM) for eliminating the degree correlation of dynamical processes. The interaction of neurons is assumed to be isotropic and idealized. These neuron dynamics are similar to biological evolution in extremal dynamics with locally isotropic interaction but has a different time scale. The functioning of neurons takes place as punctuated patterns based on avalanche dynamics. In our model, the avalanche dynamics of neurons exhibit self-organized criticality which shows power-law behavior of the avalanche sizes. For a given network, the avalanche dynamic behavior is not changed with different degree exponents of networks, γ≥2.4 and various refractory periods referred to the memory effect, T_r. Furthermore, the avalanche size distributions exhibit power-law behavior in a single scaling region in contrast to other networks. However, return time distributions displaying spatiotemporal complexity have three characteristic time scaling regimes Thus, we find that UCM may be inefficient for holding a memory.     

Spontaneous dynamics and response properties of a Hodgkin-Huxley-type neuron model driven by harmonic synaptic noise

We study statistical properties, response dynamics, and information transmission in a Hodgkin-Huxley–type neuron system, modeling peripheral electroreceptors in paddlefish. In addition to sodium and potassium currents, the neuron model includes fast calcium and slow afterhyperpolarization (AHP) potassium currents. The synaptic transmission from sensory epithelium is modeled by a Poission process with a rate modulated by narrow-band noise, mimicking stochastic epithelial oscillations observed experimentally. We study how the interplay of parameters of AHP current and synaptic noise affects the statistics of spontaneous dynamics and response properties of the system. In particular, we confirm predictions made earlier with perfect integrate and fire and phase neuron models that epithelial oscillations enhance stimulus–response coherence and thus information transmission in electroreceptor system. In addition, we consider a strong stimulus regime and show that coherent epithelial oscillations may reduce variability of electroreceptor responses to time-varying stimuli.     

Bifurcation in neuronal networks with hub structure

We investigate bifurcations in neuronal networks with a hub structure. It is known that hubs play a leading role in characterizing the network dynamical behavior. However, the dynamics of hubs or star-coupled systems is not well understood. Here, we study rather subnetworks with a star-like configuration. This coupled system is an important motif in complex networks. Thus, our study is a basic step for understanding structure formation in large networks. We use the Morris-Lecar neuron with class I and class II excitabilities as a node. Homogeneous (coupling the same class neurons) and heterogeneous (coupling different class neurons) cases are considered for both excitatory and inhibitory coupling. For the homogeneous system class II neurons are suitable for achieving both complete and cluster synchronization in excitatory and inhibitory coupling, respectively. For the heterogeneous system with inhibitory coupling, the class I hub neuron has a wider parameter region of synchronous firings than the class II hub. Moreover, the class I hub neuron with the excitatory synapse gives rise to bifurcations of synchronized states and multi-stability (coexistence of a few different states) is observed.     

State generators and complex neural memories

The mechanism of self-indexing for feedback neural networks that generates memories from short subsequences is generalized so that a single bit together with an appropriate update order suffices for each memory. This mechanism explains how stimulating an appropriate neuron can recall a memory. Although information is distributed in this model, yet our self-indexing mechanism makes it appear localized. Also a new complex valued neuron model is presented to generalize McCulloch-Pitts neurons.     

Equalization of synaptic efficacy by synchronous neural activity

It is commonly believed that spike timings of a postsynaptic neuron tend to follow those of the presynaptic neuron. Such orthodromic firing may, however, cause a conflict with the functional integrity of complex neuronal networks due to asymmetric temporal Hebbian plasticity. We argue that reversed spike timing in a synapse is a typical phenomenon in the cortex, which has a stabilizing effect on the neuronal network structure. We further demonstrate how the firing causality in a synapse is perturbed by synchronous neural activity and how the equilibrium property of spike-timing dependent plasticity is determined principally by the degree of synchronization. Remarkably, even noise-induced activity and synchrony of neurons can result in equalization of synaptic efficacy.     

Irregular collective behavior of heterogeneous neural networks

We investigate a network of integrate-and-fire neurons characterized by a distribution of spiking frequencies. Upon increasing the coupling strength, the model exhibits a transition from an asynchronous regime to a nontrivial collective behavior. Numerical simulations of large systems indicate that, at variance with the Kuramoto model, (i) the macroscopic dynamics stays irregular and (ii) the microscopic (single-neuron) evolution is linearly stable.