Chaotic Firing Research Articles

Effect of Higher Order Terms on the Bifurcation Structure of Coupled Rulkov Neurons

Three kinds of bifurcations of two coupled Rulkov neurons with electrical synapses are investigated in this paper. The critical normal forms are derived based on the center manifold theorem and the normal form theory. For the flip and the Neimark–Sacker bifurcation, the quartic terms and above in the normal forms are defined as higher order terms, which originate from the Taylor expansion of the original system. Then the effects of the quartic and quintic terms on the flip and the Neimark–Sacker bifurcation structure are discussed, which verifies that the normal form is locally topologically equivalent to the original system for the infinitesimal 4-sphere of initial conditions and tiny perturbation on the bifurcation curve. By the flip-Neimark–Sacker bifurcation analysis, a novel firing pattern can be found which is that the orbit oscillates between two invariant cycles. Two disconnected cardioid cycles also appear, which makes one, two, three, four, etc. turns happen before closure. Finally, we present a global bifurcation structure in the parameter space and exhibit the distribution of the periodic, quasi-periodic and chaotic firing patterns of the coupled neuron model.

Hyperpolarization-Activated Current Induces Period-Doubling Cascades and Chaos in a Cold Thermoreceptor Model.

In this article, we describe and analyze the chaotic behavior of a conductance-based neuronal bursting model. This is a model with a reduced number of variables, yet it retains biophysical plausibility. Inspired by the activity of cold thermoreceptors, the model contains a persistent Sodium current, a Calcium-activated Potassium current and a hyperpolarization-activated current (Ih) that drive a slow subthreshold oscillation. Driven by this oscillation, a fast subsystem (fast Sodium and Potassium currents) fires action potentials in a periodic fashion. Depending on the parameters, this model can generate a variety of firing patterns that includes bursting, regular tonic and polymodal firing. Here we show that the transitions between different firing patterns are often accompanied by a range of chaotic firing, as suggested by an irregular, non-periodic firing pattern. To confirm this, we measure the maximum Lyapunov exponent of the voltage trajectories, and the Lyapunov exponent and Lempel-Ziv's complexity of the ISI time series. The four-variable slow system (without spiking) also generates chaotic behavior, and bifurcation analysis shows that this is often originated by period doubling cascades. Either with or without spikes, chaos is no longer generated when the Ih is removed from the system. As the model is biologically plausible with biophysically meaningful parameters, we propose it as a useful tool to understand chaotic dynamics in neurons.

Chaotic Resonance in Coupled Inferior Olive Neurons with the Llinás Approach Neuron Model.

It is well known that cerebellar motor control is fine-tuned by the learning process adjusted according to rich error signals from inferior olive (IO) neurons. Schweighofer and colleagues proposed that these signals can be produced by chaotic irregular firing in the IO neuron assembly; such chaotic resonance (CR) was replicated in their computer demonstration of a Hodgkin-Huxley (HH)-type compartment model. In this study, we examined the response of CR to a periodic signal in the IO neuron assembly comprising the Llinás approach IO neuron model. This system involves empirically observed dynamics of the IO membrane potential and is simpler than the HH-type compartment model. We then clarified its dependence on electrical coupling strength, input signal strength, and frequency. Furthermore, we compared the physiological validity for IO neurons such as low firing rate and sustaining subthreshold oscillation between CR and conventional stochastic resonance (SR) and examined the consistency with asynchronous firings indicated by the previous model-based studies in the cerebellar learning process. In addition, the signal response of CR and SR was investigated in a large neuron assembly. As the result, we confirmed that CR was consistent with the above IO neuron's characteristics, but it was not as easy for SR.

A map-based logistic neuron model: an efficient way to obtain many different neural behaviors

The KTz neuron model [1-5] is a map with three dynamic variables: the neuron membrane potential (xt), a recovery variable (yt) and a slow adaptive current (zt), given by the following equations: xt+1=tanhxt-Kyt+zt+ItT, yt+1=xt, zt+1=1-δzt-λxt-xR, where δ is related to the refractory period, xR is the reversal potential, λ controls the burst sizes, It is an external current and K and T are gain parameters for neuron self-interactions. The hyperbolic tangent is biologically plausible, since it is a sigmoidal function that saturates at large absolute inputs. This model exhibits a rich repertoire of dynamical behaviors, especially those of excitable neurons: bursting, fast and regular spiking, spikes with plateau, type I and II excitability, chaotic firing, among others [1-4]. When associated in networks, they exhibit synchronization, power law avalanches, criticality, etc. [4,5]. However, hyperbolic tangent is a computationally expensive sigmoidal function. Considering the need of a good trade-off between computational efficiency and biological relevance [6-8], we then propose the use of the logistic function fu=u/1+u, which displays a similar asymptotic behavior and is computationally faster. We call it the KTz logistic model. A similar model has been recently used to study the unlearning hypothesis in REM sleep [9]. We determine the complete phase diagram of the KTz logistic model with details for zt=δ=λ=0. This phase diagram is similar to the hyperbolic tangent KTz under the same conditions [5], although it is more analytically tractable. We also determine the phase diagram of the logistic KTz model with δ=λ=0 and zt≠0 and compare with the hyperbolic tangent KTz [1,4]. The myriad of dynamical behaviors of the logistic KTz model (δ≠0 and λ≠0) and its T versus xR phase diagram are also described and compared with those of the hyperbolic tangent KTz [1-4], showing the regions of cardiac like behavior, bursting, fast and slow spiking and fixed point. A detailed analysis of the order of the transitions between these regimes is also made. The preliminary results of networks of logistic KTz models connected by map chemical synapses [2,4,5] are displayed and compared with those of hyperbolic KTz model. Finally, we make qualitative and quantitative comparisons between computational efficiency of the hyperbolic tangent and the logistic KTz models, for many different behaviors of the studied neurons.

Difference Between Intermittent Chaotic Bursting and Spiking of Neural Firing Patterns

The chaotic spiking and intermittent chaotic bursting were observed from biological experiment on a neural pacemaker. The intermittent chaotic bursting manifested alternation between a phase of nearly period-3 bursting and another phase of irregular bursting, and exhibited a nonsmooth-like structure in the first return map of interspike interval (ISI) series. The chaotic spiking manifested a smooth structure. The intermittent chaotic bursting and spiking simulated in the Chay model were identified by the dissection of the fast-slow variables. The intermittent chaotic bursting manifested nonsmooth-like structures in the first return map of both ISI and slow variable, while spiking exhibited smooth structures. The intermittent chaotic bursting and spiking manifested different scale laws of the averaged length of periodic phase with respect to the changes of parameter value. The scale law of bursting was similar to those of both type V intermittent chaos generated in a nonsmooth map and type I in a smooth system, while that of spiking resembled type I more than type V. The results revealed the dynamics of intermittent chaotic firing patterns, the differences between the intermittent chaotic bursting and spiking patterns and the relationships to the smooth and nonsmooth-like characteristics, provided a deep understanding of intermittent chaotic firing patterns, and indicated different mechanisms of the adjustment of variables from slow to fast between bursting and spiking patterns.

Balanced neural architecture and the idling brain

A signature feature of cortical spike trains is their trial-to-trial variability. This variability is large in the spontaneous state and is reduced when cortex is driven by a stimulus or task. Models of recurrent cortical networks with unstructured, yet balanced, excitation and inhibition generate variability consistent with evoked conditions. However, these models produce spike trains which lack the long timescale fluctuations and large variability exhibited during spontaneous cortical dynamics. We propose that global network architectures which support a large number of stable states (attractor networks) allow balanced networks to capture key features of neural variability in both spontaneous and evoked conditions. We illustrate this using balanced spiking networks with clustered assembly, feedforward chain, and ring structures. By assuming that global network structure is related to stimulus preference, we show that signal correlations are related to the magnitude of correlations in the spontaneous state. Finally, we contrast the impact of stimulation on the trial-to-trial variability in attractor networks with that of strongly coupled spiking networks with chaotic firing rate instabilities, recently investigated by Ostojic (2014). We find that only attractor networks replicate an experimentally observed stimulus-induced quenching of trial-to-trial variability. In total, the comparison of the trial-variable dynamics of single neurons or neuron pairs during spontaneous and evoked activity can be a window into the global structure of balanced cortical networks.

Firing properties and synchronization rate in fractional-order Hindmarsh-Rose model neurons

We find that the fractional-order Hindmarsh-Rose model neuron demonstrates various types of firing behavior as a function of the fractional order in this study. There exists a clear difference in the bifurcation diagram between the fractional-order Hindmarsh-Rose model and the corresponding integer-order model even though the neuron undergoes a Hopf bifurcation to oscillation and then starts a period-doubling cascade to chaos with the decrease of the externally applied current. Interestingly, the discharge frequency of the fractional-order Hindmarsh-Rose model neuron is greater than that of the integer-order counterpart irrespective of whether the neuron exhibits periodic or chaotic firing. Then we demonstrate that the firing behavior of the fractional-order Hindmarsh-Rose model neuron has a higher complexity than that of the integer-order counterpart. Also, the synchronization phenomenon is investigated in the network of two electrically coupled fractional-order model neurons. We show that the synchronization rate increases as the fractional order decreases.

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.

Experimental identification of a comb-shaped chaotic region in multiple parameter spaces simulated by the Hindmarsh—Rose neuron model

A comb-shaped chaotic region has been simulated in multiple two-dimensional parameter spaces using the Hindmarsh—Rose (HR) neuron model in many recent studies, which can interpret almost all of the previously simulated bifurcation processes with chaos in neural firing patterns. In the present paper, a comb-shaped chaotic region in a two-dimensional parameter space was reproduced, which presented different processes of period-adding bifurcations with chaos with changing one parameter and fixed the other parameter at different levels. In the biological experiments, different period-adding bifurcation scenarios with chaos by decreasing the extra-cellular calcium concentration were observed from some neural pacemakers at different levels of extra-cellular 4-aminopyridine concentration and from other pacemakers at different levels of extra-cellular caesium concentration. By using the nonlinear time series analysis method, the deterministic dynamics of the experimental chaotic firings were investigated. The period-adding bifurcations with chaos observed in the experiments resembled those simulated in the comb-shaped chaotic region using the HR model. The experimental results show that period-adding bifurcations with chaos are preserved in different two-dimensional parameter spaces, which provides evidence of the existence of the comb-shaped chaotic region and a demonstration of the simulation results in different two-dimensional parameter spaces in the HR neuron model. The results also present relationships between different firing patterns in two-dimensional parameter spaces.

Biological Experimental Observations of an Unnoticed Chaos as Simulated by the Hindmarsh-Rose Model

An unnoticed chaotic firing pattern, lying between period-1 and period-2 firing patterns, has received little attention over the past 20 years since it was first simulated in the Hindmarsh-Rose (HR) model. In the present study, the rat sciatic nerve model of chronic constriction injury (CCI) was used as an experimental neural pacemaker to investigate the transition regularities of spontaneous firing patterns. Chaotic firing lying between period-1 and period-2 firings was observed located in four bifurcation scenarios in different, isolated neural pacemakers. These bifurcation scenarios were induced by decreasing extracellular calcium concentrations. The behaviors after period-2 firing pattern in the four scenarios were period-doubling bifurcation not to chaos, period-doubling bifurcation to chaos, period-adding sequences with chaotic firings, and period-adding sequences with stochastic firings. The deterministic structure of the chaotic firing pattern was identified by the first return map of interspike intervals and a short-term prediction using nonlinear prediction. The experimental observations closely match those simulated in a two-dimensional parameter space using the HR model, providing strong evidences of the existence of chaotic firing lying between period-1 and period-2 firing patterns in the actual nervous system. The results also present relationships in the parameter space between this chaotic firing and other firing patterns, such as the chaotic firings that appear after period-2 firing pattern located within the well-known comb-shaped region, periodic firing patterns and stochastic firing patterns, as predicted by the HR model. We hope that this study can focus attention on and help to further the understanding of the unnoticed chaotic neural firing pattern.

Dynamic behavior analysis of fractional-order Hindmarsh–Rose neuronal model

Previous experimental work has shown that the firing rate of multiple time-scales of adaptation for single rat neocortical pyramidal neurons is consistent with fractional-order differentiation, and the fractional-order neuronal models depict the firing rate of neurons more verifiably than other models do. For this reason, the dynamic characteristics of the fractional-order Hindmarsh-Rose (HR) neuronal model were here investigated. The results showed several obvious differences in dynamic characteristic between the fractional-order HR neuronal model and an integer-ordered model. First, the fractional-order HR neuronal model displayed different firing modes (chaotic firing and periodic firing) as the fractional order changed when other parameters remained the same as in the integer-order model. However, only one firing mode is displayed in integer-order models with the same parameters. The fractional order is the key to determining the firing mode. Second, the Hopf bifurcation point of this fractional-order model, from the resting state to periodic firing, was found to be larger than that of the integer-order model. Third, for the state of periodically firing of fractional-order and integer-order HR neuron model, the firing frequency of the fractional-order neuronal model was greater than that of the integer-order model, and when the fractional order of the model decreased, the firing frequency increased.

Stimulus-induced transition of clustering firings in neuronal networks with information transmission delay

We study the evolution of spatiotemporal dynamics and transition of clustering firing synchronization on spiking Hodgkin-Huxley neuronal networks as information transmission delay and the periodic stimulus are varied. In particular, it is shown that the tuned information transmission delay can induce a clustering anti-phase synchronization transition with the pacemaker, where two equal clusters can alternatively synchronize in anti-phase firing. More interestingly, we show that the periodic stimulus can drive the delay-induced clustering anti-phase firing synchronization bifurcate to the collective perfect synchronization, which is routed by the complex process including collective chaotic firings and clustering out-of-phase synchronization of the neuronal networks. In addition, the periodic stimulus induced clustering firings of the spiking neuronal networks are robust to the connectivity probability of small world networks. Furthermore, the different stimulus frequency induced complexity is also investigated. We hope that the results of this paper can provide insights that could facilitate the understanding of the joint impact of information transmission delays and periodic stimulus on controlling dynamical behaviors of realistic neuronal networks.

Low dimensional model of bursting neurons

A computationally efficient, biophysically-based model of neuronal behavior is presented; it incorporates ion channel dynamics in its two fast ion channels while preserving simplicity by representing only one slow ion current. The model equations are shown to provide a wide array of physiological dynamics in terms of spiking patterns, bursting, subthreshold oscillations, and chaotic firing. Despite its simplicity, the model is capable of simulating an extensive range of spiking patterns. Several common neuronal behaviors observed in vivo are demonstrated by varying model parameters. These behaviors are classified into dynamical classes using phase diagrams whose boundaries in parameter space prove to be accurately delineated by linear stability analysis. This simple model is suitable for use in large scale simulations involving neural field theory or neuronal networks.

Experimental researches on different complex bifurcation procedures of neural firing patterns

The bifurcation structures of neural firing patterns in parameter space are important in nonlinear dynamics, playing important roles in identifying neural information coding mechanism. The bifurcation structures of neural firing with respect to the changes of concentration of calcium ion are observed in the experiment on an injured sciatic nerve model when concentration of voltage dependent potassium channel blockers is fixed at different values. The simple bifurcation procedure composed of a few kinds of firing patterns and the complex one composed of multiple kinds of periodic and chaotic firing patterns can be transited each other. Two different kinds of complex bifurcation structures can also be transited each other. The difference between chaotic firing patterns near bifurcation points is the key indicator to distinguish the kinds of bifurcation procedures. The results not only enrich the nonlinear phenomenon of neural firing patterns, but also are helpful to identify neural coding mechanism based on the construction of the bifurcation structure of neural firing patterns in parameter space.

Bifurcation Scenarios of Neural Firing Patterns across Two Separated Chaotic Regions as Indicated by Theoretical and Biological Experimental Models

Nonlinear dynamics can be used to identify relationships between different firing patterns, which play important roles in the information processing. The present study provides novel biological experimental findings regarding complex bifurcation scenarios from period-1 bursting to period-1 spiking with chaotic firing patterns. These bifurcations were found to be similar to those simulated using the Hindmarsh-Rose model across two separated chaotic regions. One chaotic region lay between period-1 and period-2 burstings. This region has not attracted much attention. The other region is a well-known comb-shaped chaotic region, and it appears after period-2 bursting. After period-2 bursting, the chaotic firings lay in a period-adding bifurcation scenario or in a period-doubling bifurcation cascade. The deterministic dynamics of the chaotic firing patterns were identified using a nonlinear prediction method. These results provided details regarding the processes and dynamics of bifurcation containing the chaotic bursting between period-1 and period-2 burstings and other chaotic firing patterns within the comb-shaped chaotic region. They also provided details regarding the relationships between different firing patterns in parameter space.

The Influence of Parameters on an Individual Neuron Firing Patterns

The influence of parameters to the firing patterns of time-delayed linear and compound Hindmarsh-Rose(HR) model neuron are studied in this work. For an individual neuron, it can be found from different Intervalspike Interval (ISI) maps that the firing patterns transform among resting state, busting firing, tonic firing, and chaotic spiking, especially, the transfer of the firing patters are speeded up under feedback mechanism, it is easier to get into the chaotic firing state compared with linear feedback.

Firing patterns in a conductance-based neuron model: bifurcation, phase diagram, and chaos

Responding to various stimuli, some neurons either remain resting or can fire several distinct patterns of action potentials, such as spiking, bursting, subthreshold oscillations, and chaotic firing. In particular, Wilson's conductance-based neocortical neuron model, derived from the Hodgkin-Huxley model, is explored to understand underlying mechanisms of the firing patterns. Phase diagrams describing boundaries between the domains of different firing patterns are obtained via extensive numerical computations. The boundaries are further studied by standard instability analyses, which demonstrates that the chaotic neural firing could develop via period-doubling and/or period- adding cascades. Sequences of the firing patterns often observed in many neural experiments are also discussed in the phase diagram framework developed. Our results lay the groundwork for wider use of the model, especially for incorporating it into neural field modeling of the brain.

Chaotic phase synchronization in bursting-neuron models driven by a weak periodic force

We investigate the entrainment of a neuron model exhibiting a chaotic spiking-bursting behavior in response to a weak periodic force. This model exhibits two types of oscillations with different characteristic time scales, namely, long and short time scales. Several types of phase synchronization are observed, such as 1:1 phase locking between a single spike and one period of the force and 1:l phase locking between the period of slow oscillation underlying bursts and l periods of the force. Moreover, spiking-bursting oscillations with chaotic firing patterns can be synchronized with the periodic force. Such a type of phase synchronization is detected from the position of a set of points on a unit circle, which is determined by the phase of the periodic force at each spiking time. We show that this detection method is effective for a system with multiple time scales. Owing to the existence of both the short and the long time scales, two characteristic phenomena are found around the transition point to chaotic phase synchronization. One phenomenon shows that the average time interval between successive phase slips exhibits a power-law scaling against the driving force strength and that the scaling exponent has an unsmooth dependence on the changes in the driving force strength. The other phenomenon shows that Kuramoto's order parameter before the transition exhibits stepwise behavior as a function of the driving force strength, contrary to the smooth transition in a model with a single time scale.

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.

Identification of a stochastic neural firing rhythm lying in period-adding bifurcation and resembling chaos

To identify non-periodic neural rhythm to be chaos or stochasticity has been an important scientific thesis. A kind of non-periodic spontaneous firing pattern, whose behavior is transition between period-k burst in a string and period-k+1 burst in a string (k=1,2), lying between period-k bursting pattern and period-k+1 bursting pattern, is found in the experimental neural pacemaker. The deterministic structures of the firing are identified by nonlinear prediction and first return map of the interspike intervals (ISIs) series. The co-existence of the period-k bursting and period-k+1 bursting is manifested in the deterministic theoretical neuronal model, Chay model. Non-periodic firing patterns similar to the experimental observation are simulated in the co-existing parameter region, implying that the firing pattern is transition between two kinds of bursts induced by noise. A binary series can be acquired by transforming two kinds of bursts to symbols 0 and 1, respectively. The stochastic dynamics within the transitions between two kinds of bursts are detected by probability analysis on the binary series. It not only shows that the rhythm is stochastic firing with deterministic structures instead of chaos, but also provides the typical examples and effective methods to intensively identify the chaotic and stochastic firing patterns in a real nervous system.