Chaotic Firing Patterns Research Articles

Dynamics and synchronization in a memristor-coupled discrete heterogeneous neuron network considering noise

Research on discrete memristor-based neural networks has received much attention. However, current research mainly focuses on memristor–based discrete homogeneous neuron networks, while memristor-coupled discrete heterogeneous neuron networks are rarely reported. In this study, a new four-stable discrete locally active memristor is proposed and its nonvolatile and locally active properties are verified by its power-off plot and DC V–I diagram. Based on two-dimensional (2D) discrete Izhikevich neuron and 2D discrete Chialvo neuron, a heterogeneous discrete neuron network is constructed by using the proposed discrete memristor as a coupling synapse connecting the two heterogeneous neurons. Considering the coupling strength as the control parameter, chaotic firing, periodic firing, and hyperchaotic firing patterns are revealed. In particular, multiple coexisting firing patterns are observed, which are induced by different initial values of the memristor. Phase synchronization between the two heterogeneous neurons is discussed and it is found that they can achieve perfect synchronous at large coupling strength. Furthermore, the effect of Gaussian white noise on synchronization behaviors is also explored. We demonstrate that the presence of noise not only leads to the transition of firing patterns, but also achieves the phase synchronization between two heterogeneous neurons under low coupling strength.

Firing pattern in a memristive Hodgkin–Huxley circuit: Numerical simulation and analog circuit validation

Spiking firing patterns and their hardware implementation can assist us in exploring spike-based applications. The classical Hodgkin–Huxley circuit can generate spiking firing patterns, but it is hard to implement on analog platform because of the involvement of complex exponential nonlinearities for characterizing the sodium and potassium ion channels. To solve this issue, this paper proposes a second-order local active memristor (LAM) and a first-order LAM to respectively characterize the sodium and potassium ion channels in the Hodgkin–Huxley circuit, thereby obtains a memristive Hodgkin–Huxley circuit. Numerical simulations demonstrate that the memristive Hodgkin–Huxley circuit can generate abundant spiking firing patterns, i.e. periodic and chaotic spiking firing patterns. These abundant firing patterns can be effectively regulated by memristor- and stimulus-related parameters. Moreover, a PCB-based analog circuit employing off-the-shelf discrete circuit components is manually constructed and hardware experiments are executed. Experimental results captured from hardware measurements satisfactorily verify the numerically simulated spiking firing patterns and effectively exhibit the availability of the memristive Hodgkin–Huxley circuit in generating spiking firing patterns.

Extreme multistability and phase synchronization in a heterogeneous bi-neuron Rulkov network with memristive electromagnetic induction.

Memristive electromagnetic induction effect has been widely explored in bi-neuron network with homogeneous neurons, but rarely in bi-neuron network with heterogeneous ones. This paper builds a bi-neuron network by coupling heterogeneous Rulkov neurons with memristor and investigates the memristive electromagnetic induction effect. Theoretical analysis discloses that the bi-neuron network possesses a line equilibrium state and its stability depends on the memristor coupling strength and initial condition. That is, the stability of the line equilibrium state has a transition between unstable saddle-focus and stable node-focus via Hopf bifurcation. By employing parameters located in the stable node-focus region, dynamical behaviors related to the memristor coupling strength and initial conditions are revealed by Julia- and MATLAB-based multiple numerical tools. Numerical results demonstrate that the proposed heterogeneous bi-neuron Rulkov network can generate point attractor, period, chaos, chaos crisis, and period-doubling bifurcation. Note that extreme multistability are disclosed with respect to initial conditions of memristor and gated ion concentration. Coexisting infinitely multiple firing patterns of periodic firing patterns with different periodicities and chaotic firing patterns for different memristor initial conditions are demonstrated by phase portrait and time-domain waveform. Besides, the phase synchronization related to the memristor coupling strength and its initial condition is explored, which suggests that the two heterogeneous neurons become phase synchronization with large memristor coupling strength and initial condition. This also reflects that the plasticity of memristor synapse enables adaptive regulation in keeping energy balance between the neurons. What's more, MCU-based hardware experiments are executed to further confirm the numerical simulations.

Diverse Dynamic Behaviors and Firing Activities of the Modified Fractional-Order Hindmarsh–Rose Neuronal Model Induced by Fractional-Order

It is important to investigate the firing activities of neurons, and previous experimental works have shown that fractional-order neuronal models depict the firing rate of neurons more verifiably. In this study, a modified fractional-order Hindmarsh–Rose neuronal model is proposed, and the dynamics and firing activities are investigated. Some novel phenomenon can be found. First, by analyzing numerically and theoretically, the Hopf bifurcation is found to occur when the external direct current stimulus is chosen appropriately. The effects of fractional-order on the bifurcation are also studied. Second, when injecting a direct current stimulus, compared with the integer-order model, the system has more varying dynamic behaviors and firing pattern transitions. Under different external current stimulus, periodic firing patterns and chaotic firing patterns occur when fractional-order changes, but the regions of chaotic firing patterns are different. In other words, the transition mode of periodic firing and chaotic firing induced by fractional-order is different under different external current stimulus. The two-dimensional colored diagram of firing patterns is also investigated. Finally, when injecting periodic current stimulus, regular/irregular bursting, multiple spiking, regular\irregular square wave bursting, and mixed firing mods are found by setting the appropriate fractional-order, amplitude, and frequency of the external current stimulus. Some firing patterns cannot be found in integer-order models. When the amplitude is chosen at appropriate values, the region of frequency when the system displays the mixed firing modes decreases with increasing fractional-order.

Memristor Initial-Offset Boosting in Memristive HR Neuron Model with Hidden Firing Patterns

A new three-dimensional (3D) memristive HR neuron model is presented, which is improved from an existing memristive HR neuron model using a memristor synapse with sine memductance to substitute the original one. The improved memristive HR neuron model has no equilibrium but hidden firing activities can emerge with discrete memristor initial-offset boosting. Treating the neuron model as a two-dimensional (2D) major subsystem controlled by a magnetic flux variable, fold bifurcations for hidden chaotic and periodic firing patterns are elaborated. The coexistence of hidden firing patterns induced by memristor initial boosting is quantitatively analyzed and numerically simulated by bifurcation plots, phase plots, and basins of attraction. The results demonstrate that the improved memristive HR neuron model can exhibit a discrete memristor initial-offset boosting behavior owning infinitely many disconnected basins of attraction and the generating firing patterns can be boosted to different discrete levels by changing the memristor initial value, differing entirely from various boosting behaviors reported previously. Therefore, infinitely many hidden coexisting offset-boosted firing patterns with the same initial-offsets and attractor types are disclosed along the boosting route, which are homogenous with extreme multistability and are perfectly validated by PSIM circuit simulations based on a physically implementation-oriented analog circuit.

Coexistence of firing patterns and its control in two neurons coupled through an asymmetric electrical synapse.

In this paper, the effects of asymmetry in an electrical synaptic connection between two neuronal oscillators with a small discrepancy are studied in a 2D Hindmarsh-Rose model. We have found that the introduced model possesses a unique unstable equilibrium point. We equally demonstrate that the asymmetric electrical couplings as well as external stimulus induce the coexistence of bifurcations and multiple firing patterns in the coupled neural oscillators. The coexistence of at least two firing patterns including chaotic and periodic ones for some discrete values of coupling strengths and external stimulus is demonstrated using time series, phase portraits, bifurcation diagrams, maximum Lyapunov exponent graphs, and basins of attraction. The PSpice results with an analog electronic circuit are in good agreement with the results of theoretical analyses. Of most/particular interest, multistability observed in the coupled neuronal model is further controlled based on the linear augmentation scheme. Numerical results show the effectiveness of the control strategy through annihilation of the periodic coexisting firing pattern. For higher values of the coupling strength, only a chaotic firing pattern survives. To the best of the authors' knowledge, the results of this work represent the first report on the phenomenon of coexistence of multiple firing patterns and its control ever present in a 2D Hindmarsh-Rose model connected to another one through an asymmetric electrical coupling and, thus, deserves dissemination.

Effects of extracellular electric fields on electrical activities of two-compartment autaptic-neurons

In our work, we follow a modeling approach to investigate how extracellular electric fields affect the dynamics of two-compartment autaptic-neurons with morphological structures and how those effects translate into changes in electrical activities. Inter-spikes intervals bifurcation diagrams, frequency–input curves as well as phase-locking ratios are used to describe neural firing patterns, input sensitivities as well as mode-locking states. Modulated by extracellular electric fields, the autaptic-neurons exhibit subthreshold oscillation, tonic spiking, bursting and chaotic firing patterns. The input sensitivities and the mode-locking states are enhanced or suppressed depending on the parameters of extracellular electrical fields. Compared with autaptic-neurons with symmetrical structures, the autaptic-neuron with nonsymmetrical morphology have higher input sensitivities to extracellular electric fields, which suggest that neural geometric features play a crucial role in electric field effects on behaviors of neuron. Those results in our works provide a systematic understanding of effects of extracellular electric fields on the electrical activities of autaptic-neurons, which are further steps towards improving the design of electromagnetic brain stimulations.

The dynamic mechanism of a novel stochastic neural firing pattern observed in a real biological system

In the experimental neural pacemaker of a rat, a novel firing pattern has been discovered. This pattern was generated between the period 2 firing pattern and the period 3 firing pattern during the periodic adding bifurcation and inverse periodic adding bifurcation. The pattern was observed and analyzed in the present investigation. The composition of this novel firing pattern could be regarded as a transition between a string of period 2 burst and a string of period 3 burst without single period 2 or period 3 burst, which was different from those chaotic and stochastic neural firing patterns in previous reports. It was identified to be stochastic by the inter-event intervals (IEIs) analysis, although it exhibited chaos-like characteristics with the results of the inter-spike intervals (ISIs) analysis. The numerical simulation suggested that the new pattern observed in the real biological system could be simulated in the stochastic Chay model but not in the deterministic model. With the signal to noise ratio (SNR) analysis and bifurcation analysis, this novel firing pattern was considered to be generated by stochastic resonance under the influence of noise near the periodic adding (inverse) bifurcation point. The probability analysis of transformed binary chain further confirmed that the origin of stochastic and chaos (deterministic)-like characteristics of this novel firing pattern.

Chaotic Dynamics Mediate Brain State Transitions, Driven by Changes in Extracellular Ion Concentrations.

Previous studies have suggested that changes in extracellular ion concentrations initiate the transition from an activity state that characterizes sleep in cortical neurons to states that characterize wakefulness. However, because neuronal activity and extracellular ion concentrations are interdependent, isolating their unique roles during sleep-wake transitions is not possible invivo. Here, we extend the Averaged-Neuron model and demonstrate that, although changes in extracellular ion concentrations occur concurrently, decreasing the conductance of calcium-dependent potassium channels initiates the transition from sleep to wakefulness. We find that sleep is governed by stable, self-sustained oscillations in neuronal firing patterns, whereas the quiet awake state and active awake state are both governed by irregular oscillations and chaotic dynamics; transitions between these separable awake states are prompted by ionic changes. Although waking is indicative of a shift from stable to chaotic neuronal firing patterns, we illustrate that the properties of chaotic dynamics ensure that the transition between states is smooth and robust to noise.

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.

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.

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.

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.

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.

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.

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.

Chaotic decorrelation of Globus Pallidus by periodic forcing: a possible mechanism for the therapeutic effects of deep brain stimulation

High frequency (~100 Hz) stimulation of the subthalamic nucleus is an effective therapy for the symptoms of Parkinson's disease. Presumably, stimulation partly normalizes a pathological change in rate or pattern of neuronal activity in the output neurons of the globus pallidus. The rate effects of subthalamic stimulation are paradoxical, because firing rate in the globus pallidus is increased in Parkinson's disease, and subthalamic input is excitatory. Parkinson's disease is also accompanied by changes in globus pallidus cell firing pattern. Pallidal neurons are autonomous single-spiking oscillators that fire continuously and usually show no periodic bursting. They fire independently of each other, having flat cross-correlations. In Parkinson's disease the cells burst at 5-10/s and are highly correlated [1]. Rubin and Terman [2] have proposed that subthalamic stimulation entrains globus pallidus cells interfering with the low frequency bursting. In their model, the synchrony of globus pallidus cells should be high, because all cells are entrained by the same stimulus. The periodic nature of deep brain stimulation is inconsequential in their model. A recent clinical study has shown that the periodicity of deep brain stimulation is critical to its effectiveness [3]. Poisson stimulation at the same rate was much less beneficial. This observation, coupled with the frequency dependence of DBS effectiveness, suggests that disruption of low frequency bursting may not be the only mechanism of action. Globus pallidus cells are oscillators, so it is possible that periodic stimulation may not only suppress low frequency bursting, but also disrupt synchrony by driving the cells into chaotic firing patterns. We examined the range of stimulus frequencies and amplitudes required to chaotically desynchronize pallidal neurons. Oscillating pallidal cells were represented by their phase resetting curves. We used a type 1 phase resetting curve similar to that measured from model pallidal neurons [4]. The variance of the phase resetting curve was estimated as in Ermentrout et al [5] and used to form a stochastic phase map. The stationary distribution of latencies and the phase resetting curve were used to calculate Lyapunov exponents for both hyperpolarizing and depolarizing stimulation, and compared to the synchronizing or desynchronizing effect of stimulation on a population of simulated pallidal neurons that were weakly synchronized by common noisy input. Powerful synchronization and desynchronization were evoked at different frequencies as predicted by the sign of the Lyapunov exponents. Chaotic decorrelation in response to depolarizing stimulation occurred at frequencies similar to those known to be most effective for deep brain stimulation clinically. We suggest that in addition to its effects on low frequency oscillations in the pallido-thalamic projection, deep brain stimulation decorrelates the activity of pallidal neurons by driving them into a chaotic firing pattern. When the stimulus is randomized the chaotic response is destroyed, explaining why it is less effective. This chaotic effect may be essential to the clinical benefit of stimulation, being responsible for its frequency sensitivity and the requirement for periodicity in the stimulus pattern.

Dynamics of a novel chaotic neural firing pattern discovered in experiment and simulated in mathematical model

A special chaotic firing pattern lying between period-1 and period-2 firing pattern simulated in theoretical neuronal firing model, Hindmarsh-Rose (HR) model, has not been adequately understood for a long time. The non-periodic neural firing patterns lying between period-1 and period-2 firing pattern discovered in the biological experiments on neural pacemakers of rats are identified to be chaotic bursting and divided into three styles in appearance, according to the deterministic structures of the first return map and the short-term predictability of nonlinear predication. One style of the experimental chaos exhibits characteristics similar to the numerical simulations of the theoretical model, verifying the theoretical participation of HR model, while other styles display different characteristics. The characteristics of the three styles and the relationship and distinction among 3 styles of the chaotic rhythms are identified, and compared with those lying between period-2 and period-3 firing pattern, and between period-3 and period-4 firing pattern. In addition, the distinction between the transition procedure from period-1 to chaos and then to period-2 and other bifurcation scenarios from period-1 to period-2 firing pattern is also identified. The results confirm the novel chaos lying between period-1 and period-2 and the corresponding novel bifurcation scenario, enriching the kinds of the chaotic rhythms and bifurcation scenarios of neural firing. Finally simulations of the diversity and non-smooth characteristics of the chaotic rhythms discovered in the experiment and identification of the routine to chaos are also discussed.

Intermittent Chaotic Neural Firing Characterized by Non-Smooth Features

A chaotic firing pattern, characterized by non-smooth features and generated through the routine of intermittency from period 3, is observed in biological experiments on a neural firing pacemaker and reproduced in simulations by using a theoretical neuronal model with multiple time scales. This chaotic activity exhibits a scale law very similar to those of both the type-I intermittency generated in smooth systems and the type-V intermittency in non-smooth systems.