Invariant Circle Bifurcation Research Articles

A bioelectrical phase transition patterns the first vertebrate heartbeats.

A regular heartbeat is essential to vertebrate life. In the mature heart, this function is driven by an anatomically localized pacemaker. By contrast, pacemaking capability is broadly distributed in the early embryonic heart1-3, raising the question of how tissue-scale activity is first established and then maintained during embryonic development. The initial transition of the heart from silent to beating has never been characterized at the timescale of individual electrical events, and the structure in space and time of the early heartbeats remains poorly understood. Using all-optical electrophysiology, we captured the very first heartbeat of a zebrafish and analysed the development of cardiac excitability and conduction around this singular event. The first few beats appeared suddenly, had irregular interbeat intervals, propagated coherently across the primordial heart and emanated from loci that varied between animals and over time. The bioelectrical dynamics were well described by a noisy saddle-node on invariant circle bifurcation with action potential upstroke driven by CaV1.2. Our work shows how gradual and largely asynchronous development of single-cell bioelectrical properties produces a stereotyped and robust tissue-scale transition from quiescence to coordinated beating.

A Reduced Order Modeling Framework for Strongly Perturbed Nonlinear Dynamical Systems Near Arbitrary Trajectory Sets

A reduced order modeling strategy is proposed that can accurately capture the behavior of strongly perturbed nonlinear dynamical systems, i.e., those that are subject to general, large magnitude inputs. In contrast to standard variational approaches which consider dynamics in the neighborhood of a single reference trajectory, the proposed methodology augments the dynamics with an additional variable that continuously selects from a family of reference trajectories in order to limit truncation errors that result from neglected nonlinear terms. Provided the reference trajectories contract sufficiently rapidly in some directions, a reduced order set of equations can be obtained by choosing an appropriate coordinate system. Direct numerical approaches for computation of the necessary terms of the associated reduced order equations are provided. Crucially, because the proposed reduction strategy does not require the existence of a persistent, stable fixed point or periodic orbit, it can be implemented in situations where external inputs cause the dynamics to transition through a bifurcation. Two examples with relevance to neural control are provided. In the first, the proposed reduction strategy allows for the formulation of a numerically tractable control problem to identify energy-optimal inputs that eliminate tonic firing in a neural model. In the second example, the proposed reduction framework accurately captures successive transitions between quiescence and tonic firing across the boundary of a saddle-node on invariant circle bifurcation.

A global bifurcation organizing rhythmic activity in a coupled network

We study a system of coupled phase oscillators near a saddle-node on invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using Ott–Antonsen reduction and geometric techniques for ordinary differential equations, we identify heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before bifurcation from noncontractibile ones after bifurcation. Both families are stable for the model at hand.

Analysis to dynamics of complex electrical activities in Wilson model of brain neocortical neuron using fast-slow variable dissection with two slow variables

The neocortex of the brain plays a most important role in achieving functions of the brain via the electrical activities of neurons. Understanding the transition regularity of firing patterns and underlying dynamics of firing patterns of neurons can help to identify the brain functions and to treat some brain diseases. Different neocortical neurons exhibit regular spiking (RS), fast spiking (FS), intrinsic bursting (IB), and continuous bursting (CB), which play vital roles and wide range of functions. Fast-slow variable dissection method combined with bifurcation analysis has been an effective method to identify the underlying dynamical mechanism of spiking and bursting modulated by a single slow variable. The spiking is related to the stable limit cycle of the fast subsystem, and the bursting is associated with the transitions or bifurcations between the stable limit cycle and resting state of the fast subsystem. Such underlying dynamics of bursting has been widely used to distinguish different bursting patterns and identify complex dynamics of bursting modulated by various different factors such as synaptic current, autaptic current, and stimulations applied at a suitable phase related to the bifurcations, which play important roles in the real nervous system to regulate neural firing behaviors. Unfortunately, the bursting of neocortical neuronal model (wilson model) is modulated by two slow variables, i.e. the gating variable of calcium-activated potassium channel <i>H</i> and the gating variable of T-type calcium channel <i>T,</i> with <i>H</i> being slower than <i>T</i>. Then, the underlying dynamical mechanism of the IB and CB of the neocortical neurons cannot be acquired by the fast-slow variable dissection method when <i>H</i> is taken as the sole slow variable, due to the fact that the fast-subsystem contains the slow variable <i>T</i>. In the present paper, we use the fast-slow variable dissection method with two slow variables (<i>H</i> and <i>T </i>) to analyze the bursting patterns. The bifurcations of the fast subsystem, and the intersections between the bifurcation curves and the phase trajectory of bursting in the parameter plane (<i>H</i>, <i>T </i>) are acquired. Owing to the fact that neither of the two slow variables of the bursting is sufficiently slow, the bifurcations of only some intersections are related to the bursting behaviors, but others not. Then, the position relationship between the bifurcation curves and bursting trajectory in the three-dimensional space (<i>H</i>, <i>T</i>, <i>V </i>) (<i>V</i> is membrane potential of bursting) is further acquired, from which the bifurcations related to bursting behaviors are acquired and bifurcations unrelated to bursting behaviors are excluded. The start phase and the termination phase of the burst of the IB are related to the saddle-node on invariant circle (SNIC) bifurcation, but not to the saddle-node (SN) bifurcation. The start phase and termination phase of the burst of the CB are related to the SNIC bifurcation and the supercritical Andronov-Hopf (SupHopf) bifurcation, respectively, but not to the SN bifurcation. The results present a comprehensive and in-depth understanding of the underlying dynamics of bursting patterns in the neocortical neurons, thereby laying the foundation for regulating the firing patterns of the neocortical neurons.

Bifurcation Generated Mechanical Frequency Comb.

We demonstrate a novel response of a nonlinear micromechanical resonator when operated in a region of strong, nonlinear mode coupling. The system is excited with a single drive signal and its response is characterized by periodic amplitude modulations that occur at timescales based on system parameters. The periodic amplitude modulations of the resonator are a consequence of nonlinear mode coupling and are responsible for the emergence of a "frequency-comb" regime in the spectral response. By considering a generic model for a 1∶3 internal resonance, we demonstrate that the novel behavior results from a saddle node on an invariant circle (SNIC) bifurcation. The ability to control the operating parameters of the micromechanical structures reported here makes the simple micromechanical resonator an ideal test bed to study the dynamic response of SNIC behavior demonstrated in mechanical, optical, and biological systems.

Bifurcation mechanism for emergence of spontaneous oscillations in coupled heterogeneous excitable units

Biological systems are often composed of various heterogeneous excitable units. It is a fascinating problem to investigate how this heterogeneity affects collective behavior of biological systems. In this paper, to understand the effect of the unit heterogeneity on the dynamical mechanism for the onset of collective oscillatory behavior, we analyze coupled heterogeneous excitable units. We clarify how spontaneous oscillations emerge depending on the degree of heterogeneity of the units. With an increase in the coupling strength, the system undergoes a saddle-node on invariant circle bifurcation and a heteroclinic bifurcation. Based on bifurcation theory, we reveal that the order of the two bifurcations plays key roles in the mechanism of the emergence of spontaneous oscillations. In addition, we analytically show that when the system has a symmetric property, a 5th-order pitchfork bifurcation occurs instead of the two bifurcations. We also find that spontaneous oscillations are more likely to occur when the sizes of subpopulations of excitable units with different parameters are more balanced.

Unraveling the Control of Cell Cycle Periods during Intestinal Stem Cell Differentiation

During differentiation, intestinal stem cells (ISCs), a prototypical adult stem cell pool, become either secretory transit-amplifying cells, which give rise to all secretory cell types, or absorptive transit-amplifying cells, which give rise to enterocytes. These cells exhibit distinct cell cycle dynamics: ISCs cycle with a period of 24 h and absorptive transit-amplifying cells cycle with a period of ∼12 h, whereas secretory transit-amplifying cells arrest their cycle. The cell cycle dynamics of ISCs and their progeny are a systems-level property that emerges from interactions between the cell cycle control machinery and multiple regulatory pathways. Although many mathematical models have been developed to study the details of the cell cycle and related regulatory pathways, few models have been constructed to unravel the dynamic consequences of their interactions. To fill this gap, we present a simplified model focusing on the interaction between four key regulatory pathways (STAT, Wnt, Notch, and MAPK) and cell cycle control. After experimentally validating a model prediction, which showed that the Notch pathway can fine-tune the cell cycle period, we perform further model analysis that reveals that the change of cell cycle period accompanying ISC differentiation may be controlled by a design principle that has been well studied in dynamical systems theory—a saddle node on invariant circle bifurcation. Given that the mechanisms that control the cell cycle are conserved in most eukaryotic cell types, this general principle potentially controls the interplay between proliferation and differentiation for a broad range of stem cells.

Low temperature nullifies the circadian clock in cyanobacteria through Hopf bifurcation

Cold temperatures lead to nullification of circadian rhythms in many organisms. Two typical scenarios explain the disappearance of rhythmicity: the first is oscillation death, which is the transition from self-sustained oscillation to damped oscillation that occurs at a critical temperature. The second scenario is oscillation arrest, in which oscillation terminates at a certain phase. In the field of nonlinear dynamics, these mechanisms are called the Hopf bifurcation and the saddle-node on an invariant circle bifurcation, respectively. Although these mechanisms lead to distinct dynamical properties near the critical temperature, it is unclear to which scenario the circadian clock belongs. Here we reduced the temperature to dampen the reconstituted circadian rhythm of phosphorylation of the recombinant cyanobacterial clock protein KaiC. The data led us to conclude that Hopf bifurcation occurred at ∼19 °C. Below this critical temperature, the self-sustained rhythms of KaiC phosphorylation transformed to damped oscillations, which are predicted by the Hopf bifurcation theory. Moreover, we detected resonant oscillations below the critical temperature when temperature was periodically varied, which was reproduced by numerical simulations. Our findings suggest that the transition to a damped oscillation through Hopf bifurcation contributes to maintaining the circadian rhythm of cyanobacteria through resonance at cold temperatures.

Rotating waves in simple scalar excitable media: approximations and numerical solutions.

Rotating waves, in a two-dimensional annular region, are studied using a number of semi-analytical methods. For our kinetics, we use a simple one-variable phase model that arises as the normal form in the transition between excitability and oscillation at a saddle-node invariant circle (SNIC) bifurcation. After deriving asymptotic expressions for the scalar dispersion relationship, we use this to compare the approximations to a direct numerical simulation of the governing nonlinear partial differential equation. All of the approximation methods are based on writing the full solution [Formula: see text] as [Formula: see text] + "perturbation" terms where V is some function and [Formula: see text] is the radial phase shift. Finding an expression for the radial phase shift is the main aim of this paper. It is found that the total twist of the spiral is not a monotone function of the excitability. The largest twist occurs in the vicinity of the transition from excitable to oscillatory behavior.

Bursting dynamics in a population of oscillatory and excitable Josephson junctions.

We report an emergent bursting dynamics in a globally coupled network of mixed population of oscillatory and excitable Josephson junctions. The resistive-capacitive shunted junction (RCSJ) model of the superconducting device is considered for this study. We focus on the parameter regime of the junction where its dynamics is governed by the saddle-node on invariant circle (SNIC) bifurcation. For a coupling value above a threshold, the network splits into two clusters when a reductionism approach is applied to reproduce the bursting behavior of the large network. The excitable junctions effectively induce a slow dynamics on the oscillatory units to generate parabolic bursting in a broad parameter space. We reproduce the bursting dynamics in a mixed population of dynamical nodes of the Morris-Lecar model.

Input-output relation and energy efficiency in the neuron with different spike threshold dynamics.

Neuron encodes and transmits information through generating sequences of output spikes, which is a high energy-consuming process. The spike is initiated when membrane depolarization reaches a threshold voltage. In many neurons, threshold is dynamic and depends on the rate of membrane depolarization (dV/dt) preceding a spike. Identifying the metabolic energy involved in neural coding and their relationship to threshold dynamic is critical to understanding neuronal function and evolution. Here, we use a modified Morris-Lecar model to investigate neuronal input-output property and energy efficiency associated with different spike threshold dynamics. We find that the neurons with dynamic threshold sensitive to dV/dt generate discontinuous frequency-current curve and type II phase response curve (PRC) through Hopf bifurcation, and weak noise could prohibit spiking when bifurcation just occurs. The threshold that is insensitive to dV/dt, instead, results in a continuous frequency-current curve, a type I PRC and a saddle-node on invariant circle bifurcation, and simultaneously weak noise cannot inhibit spiking. It is also shown that the bifurcation, frequency-current curve and PRC type associated with different threshold dynamics arise from the distinct subthreshold interactions of membrane currents. Further, we observe that the energy consumption of the neuron is related to its firing characteristics. The depolarization of spike threshold improves neuronal energy efficiency by reducing the overlap of Na+ and K+ currents during an action potential. The high energy efficiency is achieved at more depolarized spike threshold and high stimulus current. These results provide a fundamental biophysical connection that links spike threshold dynamics, input-output relation, energetics and spike initiation, which could contribute to uncover neural encoding mechanism.

A Mathematical Model of a Midbrain Dopamine Neuron Identifies Two Slow Variables Likely Responsible for Bursts Evoked by SK Channel Antagonists and Terminated by Depolarization Block.

Midbrain dopamine neurons exhibit a novel type of bursting that we call “inverted square wave bursting” when exposed to Ca2+-activated small conductance (SK) K+ channel blockers in vitro. This type of bursting has three phases: hyperpolarized silence, spiking, and depolarization block. We find that two slow variables are required for this type of bursting, and we show that the three-dimensional bifurcation diagram for inverted square wave bursting is a folded surface with upper (depolarized) and lower (hyperpolarized) branches. The activation of the L-type Ca2+ channel largely supports the separation between these branches. Spiking is initiated at a saddle node on an invariant circle bifurcation at the folded edge of the lower branch and the trajectory spirals around the unstable fixed points on the upper branch. Spiking is terminated at a supercritical Hopf bifurcation, but the trajectory remains on the upper branch until it hits a saddle node on the upper folded edge and drops to the lower branch. The two slow variables contribute as follows. A second, slow component of sodium channel inactivation is largely responsible for the initiation and termination of spiking. The slow activation of the ether-a-go-go-related (ERG) K+ current is largely responsible for termination of the depolarized plateau. The mechanisms and slow processes identified herein may contribute to bursting as well as entry into and recovery from the depolarization block to different degrees in different subpopulations of dopamine neurons in vivo.

Changes of firing rate induced by changes of phase response curve in bifurcation transitions.

We study dynamical mechanisms responsible for changes of the firing rate during four different bifurcation transitions in the two-dimensional Hindmarsh-Rose (2DHR) neuron model: the saddle node on an invariant circle (SNIC) bifurcation to the supercritical Andronov-Hopf (AH) one, the SNIC bifurcation to the saddle-separatrix loop (SSL) one, the AH bifurcation to the subcritical AH (SAH) one, and the SSL bifurcation to the AH one. For this purpose, we study slopes of the firing rate curve with respect to not only an external input current but also temperature that can be interpreted as a timescale in the 2DHR neuron model. These slopes are mathematically formulated with phase response curves (PRCs), expanding the firing rate with perturbations of the temperature and external input current on the one-dimensional space of the phase [Formula: see text] in the 2DHR oscillator. By analyzing the two different slopes of the firing rate curve with respect to the temperature and external input current, we find that during changes of the firing rate in all of the bifurcation transitions, the calculated slope with respect to the temperature also changes. This is largely dependent on changes in the PRC size that is also related to the slope with respect to the external input current. Furthermore, we find phase transition-like switches of the firing rate with a possible increase of the temperature during the SSL-to-AH bifurcation transition.

Spike initiating dynamics of the neuron with different adaptation mechanisms to extracellular electric fields

In this paper, we address how adaptation mediated by different biophysical mechanisms modulates neuronal spike initiating dynamics to extracellular electric fields. We incorporate two adaptation currents, i.e., voltage-sensitive potassium current (IM) and calcium-sensitive potassium current (IAHP), into a reduced two-compartment neuron model, and extensively investigate the modeling behavior to a range of electric fields. With phase plane analysis, it is shown whether neuron continues to spike depends on whether adaptation currents could be sufficiently activated to stabilize membrane potential at subthreshold voltages. With stability and bifurcation analysis, we find the steady-state spiking in the neuron with IM occurs through a Hopf bifurcation, whereas it is generated through a saddle-node on invariant circle (SNIC) bifurcation in the cases of IAHP or no adaptation. By identifying the biophysical basis for these dynamics, we observe that IM could alter the competitive outcomes between kinetically mismatched opposite currents to result in a Hopf bifurcation, while IAHP cannot alter these competitive outcomes. From this, we conclude that different modulations of spike initiating dynamics derive from the biophysical mechanism responsible for distinct adaptation currents. Our study suggests that the adaptation mediated by different mechanisms indeed has different effects on neuronal dynamics to electric field stimulus. It could contribute to uncover the underlying mechanism of how neuron encodes electric field signals.

Event-triggered feedback in noise-driven phase oscillators.

Using a stochastic nonlinear phase oscillator model, we study the effect of event-triggered feedback on the statistics of interevent intervals. Events are associated with the entering of a new cycle. The feedback is modeled by an instantaneous increase (positive feedback) or decrease (negative feedback) of the oscillator frequency whenever an event occurs followed by an exponential decay on a slow time scale. In addition to the known excitable and oscillatory regimes, which are separated by a saddle node on invariant circle bifurcation, positive feedback can lead to bistable dynamics and a change of the system's excitability. The feedback has also a strong effect on noise-induced phenomena like coherence resonance or anticoherence resonance. Both positive and negative feedback can lead to more regular output for particular noise strengths. Finally, we investigate serial correlations in the sequence of interevent intervals that occur due to the additional slow dynamics. We derive approximations for the serial correlation coefficient and show that positive feedback results in extended positive interval correlations, whereas negative feedback yields short-ranging negative correlations. Investigating the interplay of feedback and the nonlinear phase dynamics close to the bifurcation, we find that correlations are most pronounced for optimal feedback strengths.

Transient spatiotemporal chaos in the Morris-Lecar neuronal ring network

Transient behavior is thought to play an integral role in brain functionality. Numerical simulations of the firing activity of diffusively coupled, excitable Morris-Lecar neurons reveal transient spatiotemporal chaos in the parameter regime below the saddle-node on invariant circle bifurcation point. The neighborhood of the chaotic saddle is reached through perturbations of the rest state, in which few initially active neurons at an effective spatial distance can initiate spatiotemporal chaos. The system escapes from the neighborhood of the chaotic saddle to either the rest state or to a state of pulse propagation. The lifetime of the chaotic transients is manipulated in a statistical sense through a singular application of a synchronous perturbation to a group of neurons.

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.

Exploring how extracellular electric field modulates neuron activity through dynamical analysis of a two-compartment neuron model

To investigate how extracellular electric field modulates neuron activity, a reduced two-compartment neuron model in the presence of electric field is introduced in this study. Depending on neuronal geometric and internal coupling parameters, the behaviors of the model have been studied extensively. The neuron model can exist in quiescent state or repetitive spiking state in response to electric field stimulus. Negative electric field mainly acts as inhibitory stimulus to the neuron, positive weak electric field could modulate spiking frequency and spike timing when the neuron is already active, and positive electric fields with sufficient intensity could directly trigger neuronal spiking in the absence of other stimulations. By bifurcation analysis, it is observed that there is saddle-node on invariant circle bifurcation, supercritical Hopf bifurcation and subcritical Hopf bifurcation appearing in the obtained two parameter bifurcation diagrams. The bifurcation structures and electric field thresholds for triggering neuron firing are determined by neuronal geometric and coupling parameters. The model predicts that the neurons with a nonsymmetric morphology between soma and dendrite, are more sensitive to electric field stimulus than those with the spherical structure. These findings suggest that neuronal geometric features play a crucial role in electric field effects on the polarization of neuronal compartments. Moreover, by determining the electric field threshold of our biophysical model, we could accurately distinguish between suprathreshold and subthreshold electric fields. Our study highlights the effects of extracellular electric field on neuronal activity from the biophysical modeling point of view. These insights into the dynamical mechanism of electric field may contribute to the investigation and development of electromagnetic therapies, and the model in our study could be further extended to a neuronal network in which the effects of electric fields on network activity may be investigated.

Why are all phase resetting curves bimodal?

Neurons are excitable cells, which means that they operate close to a bifurcation points that allows them to switch back and forth between stable steady states and periodic attractors [1]. Repetitive firing neurons belong to either Class 1, i.e., characterized by a continuous frequency versus bias current (f-I) curve and arbitrarily low frequency of oscillation, or Class 2, i.e., discontinuous f-I curve and firing starts with a relatively high frequency at threshold stimulation [2]. The bifurcation mechanism leading to repetitive firing is related to excitability classes, i.e., saddle node on invariant circle (SNIC) bifurcation is associated to Class 1 and Andronov-Hopf (HB) bifurcation to Class 2 excitability [3]. The complexity of model neurons can be significantly reduced near bifurcation points, e.g., by using normal forms. Although such dramatic reductions oversimplify biophysical mechanisms they allow a better understanding of the phenomenology of phase locked modes and synchrony in neural networks. A phase resetting curve (PRC) is not concerned with the biophysical details of neurons, but rather tabulates the transient changes in the firing period due to an external perturbation, usually a brief current or synaptic conductance pulse. Despite its limitations, PRC-based approaches to treating Parkinson's and epilepsy are at the forefront of practical applications [4], or predicting the shape and timing of necessary stimuli that destabilize a synchronous state of a large population of neurons [5]. It was long advocated that the PRCs are either of Type I when the bifurcation mechanism leading to oscillatory behavior is a SNIC, or of Type II when the bifurcation is of HB type [3,6]. Type II PRC has an almost sinusoidal shape and comparable sizes of the positive and negative lobes, and synchronize more readily due to their ability to both slow down and speed up [3,6]. Type I PRC has a disproportionate ratio of the two PRC lobes, can predominantly speed up (or only slow down) their rhythm when coupled with other neurons. We showed that any stable limit cycle oscillator has two remarkable points, which we called neutral points, at which brief perturbations do not induce any phase resetting. Furthermore, for a planar system there are only two such neutral points and we derived an analytical method of localizing them based on the Floquet multipliers. The PRC changes its sign at neutral points. Since there are two neutral points for any planar system it results that the corresponding PRCs must be all bimodal. Additionally, we derived analytical criteria for the degree of bimodality and were able to smoothly control the transition of the PRC from a Type I to a Type II. The ability to smoothly change the PRC type through experimentally available control parameters, such as bias currents or pharmacologically controlled ionic channels, has potential application in shaping the activity of neural cells and networks.

A phase plane analysis of neuron–astrocyte interactions

Intensive experimental studies have shown that astrocytes are active partners in modulation of synaptic transmission. In the present research, we study neuron–astrocyte signaling using a biologically inspired model of one neuron synapsing one astrocyte. In this model, the firing dynamics of the neuron is described by the Morris–Lecar model and the Ca2+ dynamics of a single astrocyte explained by a functional model introduced by Postnov and colleagues. Using the coupled neuron–astrocyte model and based on the results of the phase plane analyses, it is demonstrated that the astrocyte is able to activate the silent neuron or change the neuron spiking frequency through bidirectional communication. This suggests that astrocyte feedback signaling is capable of modulating spike transmission frequency by changing neuron spiking frequency. This effect is described by a saddle–node on invariant circle bifurcation in the coupled neuron–astrocyte model. In this way, our results suggest that the neuron–astrocyte crosstalk has a fundamental role in producing diverse neuronal activities and therefore enhances the information processing capabilities of the brain.