Phase-locking and chaos in a silent Hodgkin–Huxley neuron exposed to sinusoidal electric field

Neuronal firing patterns are related to the information processing in neural system. This paper investigates the response characteristics of a silent Hodgkin–Huxley neuron to the stimulation of externally-applied sinusoidal electric field. The neuron exhibits both p:q phase-locked (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 and amplitudes. In one-parameter space, a rich bifurcation structure including period-adding without chaos and phase-locking alternated with chaos suggests frequency discrimination of the neuronal firing patterns. Furthermore, by mapping out Arnold tongues, we partition the amplitude–frequency parameter space in terms of the qualitative behaviors of the neuron. Thus the neuron’s information (firing patterns) encodes the stimulus information (amplitude and frequency), and vice versa.     

The genesis of period-adding bursting without bursting-chaos in the Chay model

According to the period-adding firing patterns without chaos observed in neuronal experiments, the genesis of the period-adding “fold/homoclinic” bursting sequence without bursting-chaos is explored by numerical simulation, fast/slow dynamics and bifurcation analysis of limit cycle in the neuronal Chay model. It is found that each periodic bursting, from period-1 to period-7, is separately generated by the corresponding periodic spiking pattern through two period-doubling bifurcations, except for the period-1 bursting occurring via a Hopf bifurcation. Consequently, it can be revealed that this period-adding bursting bifurcation without chaos has a compound bifurcation structure with transitions from spiking to bursting, which is closely related to period-doubling bifurcations of periodic spiking in essence.     

The genesis of period-adding bursting without bursting-chaos in the Chay model

According to the period-adding firing patterns without chaos observed in neuronal experiments, the genesis of the period-adding “fold/homoclinic” bursting sequence without bursting-chaos is explored by numerical simulation, fast/slow dynamics and bifurcation analysis of limit cycle in the neuronal Chay model. It is found that each periodic bursting, from period-1 to 7, is separately generated by the corresponding periodic spiking pattern through two period-doubling bifurcations, except for the period-1 bursting occurring via a Hopf bifurcation. Consequently, it can be revealed that this period-adding bursting bifurcation without chaos has a compound bifurcation structure with transitions from spiking to bursting, which is closely related to period-doubling bifurcations of periodic spiking in essence.     

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.     

Complex dynamics in Duffing system with two external forcings

Duffing's equation with two external forcing terms have been discussed. The threshold values of chaotic motion under the periodic and quasi-periodic perturbations are obtained by using second-order averaging method and Melnikov's method. Numerical simulations not only show the consistence with the theoretical analysis but also exhibit the interesting bifurcation diagrams and the more new complex dynamical behaviors, including period-n (n=2,3,6,8) orbits, cascades of period-doubling and reverse period doubling bifurcations, quasi-periodic orbit, period windows, bubble from period-one to period-two, onset of chaos, hopping behavior of chaos, transient chaos, chaotic attractors and strange non-chaotic attractor, crisis which depends on the frequencies, amplitudes and damping. In particular, the second frequency plays a very important role for dynamics of the system, and the system can leave chaotic region to periodic motions by adjusting some parameter which can be considered as an control strategy of chaos. The computation of Lyapunov exponents confirm the dynamical behaviors.     

Exploring action potential initiation in neurons exposed to DC electric fields through dynamical analysis of conductance-based model

Noninvasive direct current (DC) electric stimulation of central nervous system is today a promising therapeutic option to alleviate the symptoms of a number of neurological disorders. Despite widespread use of this noninvasive brain modulation technique, a generalizable explanation of its biophysical basis has not been described which seriously restricts its application and development. This paper investigated the dynamical behaviors of Hodgkin’s three classes of neurons exposed to DC electric field based on a conductance-based neuron model. With phase plane and bifurcation analysis, the different responses of each class of neuron to the same stimulation are shown to derive from distinct spike initiating dynamics. Under the effects of negative DC electric field, class 1 neuron generates repetitive spike through a saddle-node on invariant circle (SNIC) bifurcation, while it ceases this repetitive behavior through a Hopf bifurcation; Class 2 neuron generates repetitive spike through a Hopf bifurcation, meanwhile it ceases this repetitive behavior also by a Hopf bifurcation; Class 3 neuron can generate single spike through a quasi-separatrix-crossing (QSC) at first, then it generates repetitive spike through a Hopf bifurcation, while it ceases this repetitive behavior through a SNIC bifurcation. Furthermore, three classes of neurons’ spiking frequency f–electric field E (f–E) curves all have parabolic shape. Our results highlight the effects of external DC electric field on neuronal activity from the biophysical modeling point of view. It can contribute to the application and development of noninvasive DC brain modulation technique.     

Delay-induced diversity of firing behavior and ordered chaotic firing in adaptive neuronal networks

In this paper, we study the effect of time delay on the firing behavior and temporal coherence and synchronization in Newman–Watts thermosensitive neuron networks with adaptive coupling. At beginning, the firing exhibit disordered spiking in absence of time delay. As time delay is increased, the neurons exhibit diversity of firing behaviors including bursting with multiple spikes in a burst, spiking, bursting with four, three and two spikes, firing death, and bursting with increasing amplitude. The spiking is the most ordered, exhibiting coherence resonance (CR)-like behavior, and the firing synchronization becomes enhanced with the increase of time delay. As growth rate of coupling strength or network randomness increases, CR-like behavior shifts to smaller time delay and the synchronization of firing increases. These results show that time delay can induce diversity of firing behaviors in adaptive neuronal networks, and can order the chaotic firing by enhancing and optimizing the temporal coherence and enhancing the synchronization of firing. However, the phenomenon of firing death shows that time delay may inhibit the firing of adaptive neuronal networks. These findings provide new insight into the role of time delay in the firing activity of adaptive neuronal networks, and can help to better understand the complex firing phenomena in neural networks.     

不同手法及频率针刺神经电信号的编码特征提取

神经系统通过时间空间编码的形式表征外部刺激所包含的信息,针刺作为神经系统的一种外部刺激,使神经元产生丰富的放电模式. 为揭示针刺信息传导与作用规律,设计了不同手法以及不同频率的针刺作用足三里时,采集脊髓背根神经电信号的实验.通过对放电波形进行小波特征提取和类选算法把放电波形进行归类,然后提取出不同刺激下的时间编码和空间编码特征. 接着使用平均放电率和编码异质性系数量化了不同针刺刺激下编码的特征. 研究发现捻转法与提插法产生的神经电信号差异较大, 这种差异主要表现在针刺手法对于参与编码神经元的选择性,但对于不同频率的针刺作用, 这种编码选择性并不明显.此外,数据分析中还发现了神经系统对于针刺作用的适应性和饱和特性.     

Control of synchronization and spiking regularity by heterogenous aperiodic high-frequency signal in coupled excitable systems

This paper investigates the synchronization and spiking regularity induced by heterogenous aperiodic (HA) signal in coupled excitable FitzHugh–Nagumo systems. We found new nontrivial effects of couplings and HA signals on the firing regularity and synchronization in coupled excitable systems without a periodic external driving. The phenomenon is similar to array enhanced coherence resonance (AECR), and it is shown that AECR-type behavior is not limited to systems driven by noises. It implies that the HA signal may be beneficial for the brain function, which is similar to the role of noise. Furthermore, it is also found that the mean frequencies, the amplitudes and the heterogeneity of HA stimuli can serve as control parameters in modulating spiking regularity and synchronization in coupled excitable systems. These results may be significant for the control of the synchronized firing of the brain in neural diseases like epilepsy.     

Resonance in a noise-driven excitable neuron model

The effects of variations in bifurcation parameter on coherence resonance in the noisy FitzHugh–Nagumo (FHN) neuron model are studied. We find that the coherence resonance effect depends monotonically on the firing bifurcation parameter, and this result is interpreted analytically. Then an external control method is presented to modulate coherence resonance in the excitable neuron model, and our scheme is based on the result that a weak periodic perturbation can be used to change the critical firing onset value. This method can be used to either enhance the effect of coherence resonance or delay the occurrence of coherence resonance, and the application of the method to another neuron model is also discussed.     

Bifurcations and Chaos in Duffing Equation

The Duffing equation with even-odd asymmetrical nonlinear-restoring force and one external forcingis investigated.The conditions of existence of primary resonance,second-order,third-order subharmonics,m-order subharmonics and chaos are given by using the second-averaging method,the Melnikov method andbifurcation theory.Numerical simulations including bifurcation diagram,bifurcation surfaces and phase portraitsshow the consistence with the theoretical analysis.The numerical results also exhibit new dynamical behaviorsincluding onset of chaos,chaos suddenly disappearing to periodic orbit,cascades of inverse period-doublingbifurcations,period-doubling bifurcation,symmetry period-doubling bifurcations of period-3 orbit,symmetry-breaking of periodic orbits,interleaving occurrence of chaotic behaviors and period-one orbit,a great abundanceof periodic windows in transient chaotic regions with interior crises and boundary crisis and varied chaoticattractors.Our results show that many dynamical behaviors are strictly departure from the behaviors of theDuffing equation with odd-nonlinear restoring force.     

Frequency transitions in synchronized neural networks

Temporal organization of events can emerge in complex systems, like neural networks. Here, random graph and cellular automaton are used to represent coupled neural structures, in order to investigate the occurrence of synchronization. The connectivity pattern of this toy model of neural system is of Newman–Watts type, formed from a regular lattice with additional random connections. Two networks with this coupling topology are connected by extra random links and an impulse stimulus is either constantly or periodically applied to a unique neuron. Numerical simulations reveal that this model can exhibit a variety of dynamic behaviors. Usually, the whole system achieves synchronization; however, the oscillation frequencies of the stimulus and of each network can be different. The dynamics is evaluated in function of the network size, the amount of the randomly added edges and the number of time steps in which a neuron can remain firing. The biological relevance of these results is discussed.     

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。     

Effect of an autapse on the firing pattern transition in a bursting neuron

On the basis of the Hindmarsh–Rose (HR) neuron model, the dynamics of electrical activity and the transition of firing patterns induced by three types of autapses have been investigated in detail. The dynamic effect of an autapse is detected by imposing a feedback term with a specific time-delay and autaptic intensity. We found that the delayed autaptic feedback connection switches the electrical activities of the HR neuron among quiescent, periodic and chaotic firing patterns. In the case of an electrical autapse, the transition from a periodic to a chaotic state occurs depending on the specific autaptic intensity and the time-delay. The excitatory chemical autapse plays a positive role in generating and enhancing the chaotic state. A time delay could decrease and suppress the chaotic state in the case of inhibitory chemical self-connections with a proper autaptic intensity. The bifurcation diagram vs. time-delay and autaptic intensity has been extensively studied, and the time series of membrane potentials and the distribution of information entropy have also been calculated to confirm the bifurcation analysis.     

周期激励下的前包钦格呼吸神经元的簇振荡现象研究

研究周期激励作用下的非自治前包钦格呼吸神经元模型,结果表明当外界激励频率与系统固有频率存在着量级差距时,系统可以产生典型的簇放电模式.由于激励频率远小于系统的固有频率,因此将整个周期激励项视为慢变参数,从而可以利用稳定性分析理论研究慢变参数变化下的平衡点的分岔类型,进一步应用快慢动力学分析方法给出簇模式产生的动力学机理.本文的结果说明外界激励对神经元的动力学行为有着重要影响,为进一步揭示呼吸节律的产生机制提供了重要帮助.     

Information processing,memories, and synchronization in chaotic neural network with the time delay

Information processing and two types of memory in an analog neural network model with time delay that produces chaos similar to the human and animal EEGs are considered. There are two levels of information processing in this neural network: the level of individual neurons and the level of the neural network. Similar to the state of brain, the state of chaotic neural network is defined. It is characterized by two types of memories (memory I and memory II) and correlation structure between the neurons. In normal (unperturbed) state, the neural network generates chaotic patterns of averaged neuronal activities (memory I) and patterns of oscillation amplitudes (memory II). In the presence of external stimulation, the activity patterns change, showing changes in both types of memory. As in experiments on stimulation of the brain, the neural network model shows synchronization of neuronal activities due to stimulus measured by Pearson's correlation coefficient. An increase in neural network asymmetry (increase of the neural network excitability) leads to the phenomenon similar to the epilepsy. Modeling of brain injury, Parkinson's disease, and dementia is performed by removing and weakening interneuron connections. In all cases, the chaotic neural network shows a decrease of the degree of chaos and changes in both types of memory similar to those observed in experiments with healthy human subjects and patients with Parkinson's disease and dementia. © 2005 Wiley Periodicals, Inc. Complexity 11:39–52, 2005     

Crisis of interspike intervals in Hodgkin–Huxley model

The bifurcations of the chaotic attractor in a Hodgkin–Huxley (H–H) model under stimulation of periodic signal is presented in this work, where the frequency of signal is taken as the controlling parameter. The chaotic behavior is realized over a wide range of frequency and is visualized by using interspike intervals (ISIs). Many kinds of abrupt undergoing changes of the ISIs are observed in different frequency regions, such as boundary crisis, interior crisis and merging crisis displaying alternately along with the changes of external signal frequency. And there are logistic-like bifurcation behaviors, e.g., periodic windows and fractal structures in ISIs dynamics. The saddle-node bifurcations resulting in collapses of chaos to period-6 orbit in dynamics of ISIs are identified.     

Stochastic resonance induced by novel random transitions of motion of FitzHugh–Nagumo neuron model

In contrast to the previous studies which have dealt with stochastic resonance induced by random transitions of system motion between two coexisting limit cycle attractors in the FitzHugh–Nagumo (FHN) neuron model after Hopf bifurcation and which have dealt with the phenomenon of stochastic resonance induced by external noise when the model with periodic input has only one attractor before Hopf bifurcation, in this paper we have focused our attention on stochastic resonance (SR) induced by a novel transition behavior, the transitions of motion of the model among one attractor on the left side of bifurcation point and two attractors on the right side of bifurcation point under the perturbation of noise. The results of research show: since one bifurcation of transition from one to two limit cycle attractors and the other bifurcation of transition from two to one limit cycle attractors occur in turn besides Hopf bifurcation, the novel transitions of motion of the model occur when bifurcation parameter is perturbed by weak internal noise; the bifurcation point of the model may stochastically slightly shift to the left or right when FHN neuron model is perturbed by external Gaussian distributed white noise, and then the novel transitions of system motion also occur under the perturbation of external noise; the novel transitions could induce SR alone, and when the novel transitions of motion of the model and the traditional transitions between two coexisting limit cycle attractors after bifurcation occur in the same process the SR also may occur with complicated behaviors types; the mechanism of SR induced by external noise when FHN neuron model with periodic input has only one attractor before Hopf bifurcation is related to this kind of novel transition mentioned above.     

Complex dynamics in Duffing–Van der Pol equation

Duffing–Van der Pol equation with fifth nonlinear-restoring force and two external forcing terms is investigated. The threshold values of existence of chaotic motion are obtained under the periodic perturbation. By second-order averaging method and Melnikov method, we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 1, 3, 5, and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 2, 4, 6, 7, 8, 9, 10, where σ is not rational to ω₁, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent, phase portraits and Poincaré map, not only show the consistence with the theoretical analysis but also exhibit the more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations from period-2 to -4 and -6 orbits, interleaving occurrence of chaotic behaviors and quasi-periodic orbits, transient chaos with a great abundance of period windows, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos which occurs more than one, chaos suddenly disappearing to period orbits, interior crisis, strange non-chaotic attractor, non-attracting chaotic set and nice chaotic attractors. Our results show many dynamical behaviors and some of them are strictly departure from the behaviors of Duffing–Van der Pol equation with a cubic nonlinear-restoring force and one external forcing.