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The forced oscillations of hair bundle of inner hair cells of auditory nervous system evoked by external force from steady state are related to the fast adaption of hair cells, which are very important for auditory amplification. In the present paper, comprehensive and deep understandings to nonlinear dynamics of forced oscillations are acquired in four aspects. Firstly, the complex dynamics underlying the twitch (fast recoil of displacement X which is fast variable) induced from Case-1 and Case-2 steady states by external pulse force are obtained. With help of vector fields and nullclines, the phase trajectory of forced oscillations is identified to be an evolution process between two equilibrium points corresponding to zero force and pulse force, respectively, and then the twitch is obtained as the behavior running along the nonlinear part of X-nullcline. Especially, twitch observed in experiment are classified into 6 types, which are induced by negative change of force, negative and positive changes of force, and positive change of force, respectively, and further build relationships to three subcases of Case-2 steady state with N-shaped X-nullcline (equilibrium point locates on the left, middle, and right branches of X-nullcline, respectively). Secondly, the experimental observation of fatigue of twitch induced by continual two pulse forces, i.e. the reduced amplitude of the latter twitch when interval between two forces is short, is also explained as a nonlinear behavior beginning from an initial value different from that of the former one. Thirdly, the experimental observation of transition between sustained oscillations and steady state induced by pulse force can be simulated for Case-1 steady state with Z-shaped X-nullcline instead of Case-2, due to that there exists bifurcations with respect to external force for Case-1 while no bifurcations for Case-2. Last, the threshold phenomenon induced by simple pulse stimulation exists for Case-1 steady state rather than Case-2, due to that the upper and lower branches of Z-shaped X-nullcline close to the middle branch exhibit coexisting behaviors of variable X while N-shaped X-nullcline does not. The nonlinear dynamics of forced oscillations are helpful for explanations to the complex experimental observations, which presents potential measures to modulate the functions of twitch such as the fast adaption.

In the traditional viewpoint, inhibitory and excitatory effects always induce opposite responses. In the present study, the enhanced bursting activities induced by excitatory autapses, which are consistent with the recent experimental observations, and those induced by inhibitory autapses, which is a paradoxical phenomenon, were simulated using the Chay model. The same bifurcations and different ionic currents for the same responses were acquired with fast-slow variable dissection and current decomposition, respectively. As the inhibitory or excitatory autaptic conductance increased, the ending phase of the burst related to a homoclinic bifurcation of the fast subsystem changed to widen the burst duration to contain more spikes, which was induced by an elevated minimal potential (Vmin) of spiking of the fast subsystem. Larger inhibitory and excitatory autaptic conductances induced smaller and larger maximal potentials (Vmax) of spiking, respectively. During the downstroke, a weaker potassium current induced by the smaller Vmax played a dominant role for the inhibitory autapse, and the stronger potassium current induced by the larger Vmax became weaker due to the opposite autaptic current of the excitatory autapse, which induced the Vmin elevated. The results present the nonlinear and biophysical mechanisms of the same responses to opposite effects, which extends nonlinear dynamics knowledge and provides potential modulation measures for the nervous system.

Resonance of both subthreshold and suprathreshold membrane potentials has been found in brain or motor neurons involving in the information processing or locomotion control via frequency response, which is mainly related to the hyperpolarization-activated cation current ( $$I_{{\text {h}}}$$ ). In the present paper, the modulations of $$I_{{\text {h}}}$$ current on resonance are acquired in a bursting neuron model, which closely match experimental observations. Firstly, the increase of $$I_{{\text {h}}}$$ current induces stable node changed to stable focus related to Hopf bifurcation, and to bursting via complex bifurcations including the big homoclinic orbit related to type I excitability. Secondly, the subthreshold resonance can be evoked by small stimulation from focus instead of node, resembling the experimental observation that subthreshold resonance appears at strong ( $$I_{{\text {h}}}$$ ) current. Thirdly, the modulation regularity of $$I_{{\text {h}}}$$ current on frequency and amplitude of resonance is simulated, closely matching the experimental observations, which can be well explained with the changing trends of imaginary and real part of eigenvalues of the focus. Last, the suprathreshold bursting is induced by strong stimulation from the stable focus near big homoclinic orbit, resembling that observed in the experiment. The bursting exhibits “suprathreshold resonance” similar to those evoked near Hopf bifurcation corresponding to type II excitability. A novel condition for “suprathreshold resonance” and the underlying codimension-2 bifurcations are acquired. The results present the comprehensive viewpoints on the subthreshold resonance and extend the condition for the “suprathreshold resonance”, which are helpful for deep understanding resonance in the brain or motor neurons with $$I_{{\text {h}}}$$ current.

Paradoxical enhancement rather than reduction in firing activity induced by inhibitory effect is very important for both nonlinear dynamics and neuroscience. In the present paper, to build a more comprehensive viewpoint related to excitatory effect, a paradoxical phenomenon modulated by the excitatory self-feedback of autapse is extended to bursting activity in a modified Morris–Lecar model. With increasing the strength of excitatory autapse, the bursting patterns exhibit inverse period-adding bifurcations to spiking pattern via chaos, and the burst duration, the spike number per burst, and the firing frequency of bursting decrease, which is different from the traditional view that firing frequency should be enhanced. Furthermore, the paradoxical reduction in bursting activity is well explained with the nonlinear dynamics of the fast subsystem. The burst begins from neighborhood of a fold bifurcation of equilibrium point and terminates at a saddle homoclinic (SH) bifurcation. With increasing conductance of autapse, the fold bifurcation point remains unchanged, which is due to too small autaptic current near the fold point to influence the dynamics, and the SH point shifts left, which follows from the enhanced potassium current at the minimal value of the stable limit cycle. Therefore, the range between the fold and SH point becomes narrower to shorten the burst duration to reduce spike number per burst and firing frequency. The novel example of paradoxical reduction modulated by positive self-feedback and the associated bifurcation mechanism enrich the phenomena of nonlinear science and present the potential functions of excitatory autapse in the brain neurons with bursting behavior.

Coherence resonance (CR), which is typical nonlinear phenomenon that noise plays construct roles, has been widely investigated in single neurons at resting state instead of bursting behavior when noise free. In the present paper, multiple CRs evoked from bursting instead of resting state and the underlying bifurcation mechanism are studied in a theoretical neuron model composed of fast subsystem and slow variable, which is very important for the bursting neurons receiving strong synaptic noise in the central nervous system. With increasing noise intensity within a range, the bursting period decreases to induce the burst number increases firstly and then becomes irregular, which results in the increase firstly and then decrease in the corresponding peak of the power spectrum of membrane potentials, i.e., CR is evoked from bursting instead of resting state. With further increasing noise intensity, two other peaks exhibit CR phenomenon, one at middle frequency and middle noise intensity and the other at large frequency and strong noise intensity, which shows that multiple CRs are evoked from bursting. Furthermore, the bifurcation mechanism of the latter two resonant peaks is acquired with fast–slow variable dissection method. With increasing the value of slow variable, the fast subsystem exhibits a fold bifurcation of limit cycle and a subcritical Hopf bifurcation to a stable focus, and the imaginary part of the characteristic value of the focus increases, which implies that spikes evoked from the focus exhibit increasing frequency. The trajectory of the deterministic bursting in phase space locates between the two bifurcations and alternates between the coexisting limit cycle and focus of the fast subsystem. With increasing noise intensity, the value of slow variable of the stochastic bursting increases and moves to a range of the focus of the fast subsystem, and the spikes of the stochastic bursting are evoked from the focus by noise. Therefore, the stronger the noise intensity, the larger the value of slow variable, and the higher frequency the spikes, due to the changes of characteristic values, which is the cause that the peaks with middle and high frequency appear at middle and strong noise intensity, respectively. The results of the present paper extend the concept of CR to multiple CRs evoked from bursting, provide the bifurcation mechanism underlying the multiple CRs, and imply multiple chances to utilize noise to enhance information of bursting neurons.

Although the bursting patterns with spike undershoot are involved with the achievement of physiological or cognitive functions of brain with synaptic noise, noise induced-coherence resonance (CR) from resting state or subthreshold oscillations instead of bursting has been widely identified to play positive roles in information process. Instead, in the present paper, CR characterized by the increase firstly and then decease of peak value of power spectrum of spike trains is evoked from a bursting pattern with spike undershoot, which means that the minimal membrane potential within burst is lower than that of the subthreshold oscillations between bursts, while CR cannot be evoked from the bursting pattern without spike undershoot. With bifurcations and fast-slow variable dissection method, the bursting patterns with and without spike undershoot are classified into "Sub-Hopf/Fold" bursting and "Fold/Homoclinic" bursting, respectively. For the bursting with spike undershoot, the trajectory of the subthreshold oscillations is very close to that of the spikes within burst. Therefore, noise can induce more spikes from the subthreshold oscillations and modulate the bursting regularity, which leads to the appearance of CR. For the bursting pattern without spike undershoot, the trajectory of the quiescent state is not close to that of the spikes within burst, and noise cannot induce spikes from the quiescent state between bursts, which is cause for non-CR. The result provides a novel case of CR phenomenon and extends the scopes of CR concept, presents that noise can enhance rather than suppress information of the bursting patterns with spike undershoot, which are helpful for understanding the dynamics and the potential physiological or cognitive functions of the nerve fiber or brain neurons with such bursting patterns.

Inhibition-induced enhancement or paradoxical response of firing was related to Hopf bifurcation instead of saddle-node bifurcation on an invariant cycle (SNIC), due to the “negative” threshold and rotated vector fields. In the present letter, by changing multiple parameters to adjust the vector fields, the condition for the paradoxical response and “negative” threshold is extended to SNIC near a codimension-2 bifurcation appearing prior to the Hopf bifurcation, which presents a comprehensive relationship between bifurcations and threshold. Especially, the result for a special current can well explain the enhanced firing along with seizure induced by inhibitory interneuron, implying that SNIC far from the codimension-2 bifurcation of pyramidal neuron is a potential candidate to avoid seizure.

Neuron exhibits nonlinear dynamics such as excitability transition and post-inhibitory rebound (PIR) spike related to bifurcations, which are associated with information processing, locomotor modulation, or brain disease. PIR spike is evoked by inhibitory stimulation instead of excitatory stimulation, which presents a challenge to the threshold concept. In the present paper, 7 codimension-2 or degenerate bifurcations related to 10 codimension-1 bifurcations are acquired in a neuronal model, which presents the bifurcations underlying the excitability transition and PIR spike. Type I excitability corresponds to saddle-node bifurcation on an invariant cycle (SNIC) bifurcation, and type II excitability to saddle-node (SN) bifurcation or sub-critical Hopf (SubH) bifurcation or sup-critical Hopf (SupH) bifurcation. The excitability transition from type I to II corresponds to the codimension-2 bifurcation, Saddle-Node Homoclinic orbit (SNHO) bifurcation, via which SNIC bifurcation terminates and meanwhile big homoclinic orbit (BHom) bifurcation and SN bifurcation emerge. A degenerate bifurcation via which BHom bifurcation terminates and fold limit cycle (LPC) bifurcation emerges is responsible for spiking transition from type I to II, and the roles of other codimension-2 bifurcations (Cusp, Bogdanov-Takens, and Bautin) are discussed. In addition, different from the widely accepted viewpoint that PIR spike is mainly evoked near Hopf bifurcation rather than SNIC bifurcation, PIR spike is identified to be induced near SNIC or BHom or LPC bifurcations, and threshold curves resemble that of Hopf bifurcation. The complex bifurcations present comprehensive and deep understandings of excitability transition and PIR spike, which are helpful for the modulation to neural firing activities and physiological functions.

Identification of dynamics of the mixed-mode oscillations (MMOs), which exhibit transition between oscillations with large and small amplitudes, is very important for nonlinear physics. In this paper, the MMOs with transition between subthreshold oscillations and spikes are investigated in a neuron model. In the absence of noise, the MMOs appear between the resting state and period-1 firing with increasing depolarization current. After introducing white noise, coherence resonance (CR) is evoked from the resting state and non-CR is induced from period-1 firing far from the MMOs, which is consistent with the traditional viewpoint. However, an interesting result that a transition from anti-CR to CR is evoked by noise from both the MMOs and the period-1 firing near the MMOs is acquired, which is characterized by the increase, decrease and increase again of the coefficient of variations of interspike intervals (ISIs) with increasing noise intensity. At small noise intensity, more subthreshold oscillations are evoked by noise to reduce the firing frequency, resulting in faster increase of standard deviation (SD) of ISIs than that of mean value of ISIs, which is the cause for the anti-CR. The decrease of SD is faster for middle noise intensity and is lower for strong noise intensity, which is the cause for the CR. The different stochastic responses of MMOs and period-1 firing nearby at different levels of noise insanity are the dynamical mechanism for the transition from anti-CR to CR. Such results present potential functions of the MMOs and period-1 firing on information processing in the nervous system with noise and extend the conditions for the CR and anti-CR phenomena, which enriches the contents of nonlinear dynamics.

The hair bundles of inner hair cells in the auditory nervous exhibit spontaneous oscillations, which is the prerequisite for an important auditory function to enhance the sensitivity of inner ear to weak sounds, otoacoustic emission. In the present paper, the dynamics of spontaneous oscillations and relationships to steady state are acquired in a two-dimensional model with fast variable X (displacement of hair bundles) and slow variable $$X_a$$ . The spontaneous oscillations are derived from negative stiffness modulated by two biological factors (S and D) and are identified to appear in multiple two-dimensional parameter planes. In (S, D) plane, comprehensive bifurcations including 4 types of codimension-2 bifurcation and 5 types of codimension-1 bifurcation related to the spontaneous oscillations are acquired. The spontaneous oscillations are surrounded by supercritical and subcritical Hopf bifurcation curves, and outside of the curves are two cases of steady state. Case-1 and Case-2 steady states exhibit Z-shaped (coexistence of X) and N-shaped (coexistence of $$X_a$$ ) X-nullclines, respectively. In (S, D) plane, left and right to the spontaneous oscillations are two subcases of Case-1, which exhibit the stable equilibrium point locating on the upper and lower branches of X-nullcline, respectively, resembling that of the neuron. Lower to the spontaneous oscillations are 3 subcases of Case-2 from left to right, which manifest stable equilibrium point locating on left, middle, and right branches of X-nullcline, respectively, differing from that of the neuron. The phase plane for steady state is divided into four parts by nullclines, which manifest different vector fields. The phase trajectory of transient behavior beginning from a phase point in the four regions to the stable equilibrium point exhibits different dynamics determined by the vector fields, which is the basis to identify dynamical mechanism of complex forced oscillations induced by external signal. The results present comprehensive viewpoint and deep understanding for dynamics of the spontaneous oscillations and steady states of hair bundles, which can be used to well explain the experimental observations and to modulate functions of spontaneous oscillations.

Neuronal excitability is classified as type I, II, or III, according to the responses of electronic activities, which play different roles. In the present paper, the effect of an excitatory autapse on type III excitability is investigated and compared to type II excitability in the Morris-Lecar model, based on Hopf bifurcation and characteristics of the nullcline. The autaptic current of a fast-decay autapse produces periodic stimulations, and that of a slow-decay autapse highly resembles sustained stimulations. Thus, both fast- and slow-decay autapses can induce a resting state for type II excitability that changes to repetitive firing. However, for type III excitability, a fast-decay autapse can induce a resting state to change to repetitive firing, while a slow-decay autapse can induce a resting state to change to a resting state following a transient spike instead of repetitive spiking, which shows the abnormal phenomenon that a stronger excitatory effect of a slow-decay autapse just induces weaker responses. Our results uncover a novel paradoxical phenomenon of the excitatory effect, and we present potential functions of fast- and slow-decay autapses that are helpful for the alteration and maintenance of type III excitability in the real nervous system related to neuropathic pain or sound localization.

Based on the post-inhibitory rebound (PIR) spike induced by inhibitory current pulse, in the present paper, a novel counterintuitive phenomenon that the inhibitory autapse with time delay can induce the resting state changed to stable spiking pattern is identified near subcritical Hopf bifurcation of Hodgkin–Huxley model. The delayed inhibitory autaptic current pulse induced by the preceded action potential can induce the preceding PIR spike via the hyperpolarization, rebound, and depolarization processes, which is compared with spiking induced by excitatory autapse via only a depolarization process. The threshold of inhibitory or excitatory autaptic conductance to induce spiking with increasing time delay, and the threshold curve of inhibitory or excitatory pulse current to evoke a spike exhibit damping oscillations can be well interpreted with the damping dynamics of focus near subcritical Hopf bifurcation. However, due to PIR mechanism, the threshold conductance of inhibitory autapse is stronger than that of excitatory autapse, and the spiking period for inhibitory autapse, which is composed of time delay and durations of the other three processes, is longer than the one for excitatory autapse, which is composed of time delay and duration of only a depolarization process. Therefore, a linear correlation between spiking period and time delay is identified, which shows that autapse can modulate the spike timing related to temporal coding. The results present a novel viewpoint and a potential function that inhibitory autapse can facilitate spiking like the excitatory autapse and provide effective measures to modulate neuronal spiking pattern, which is related to subcritical Hopf bifurcation.

Memristor has been identified to modulate electronic activities of the nervous system, and conduction failure of action potentials along axon has been identified to play important roles in information transition related to pathological pain. In the present paper, the influences of the memristor on the conduction failure induced by the external stimulation with high frequency are investigated. After introducing electromagnetic induction mediated by memristor into Hodgkin-Hexley (HH) model, the Hopf bifurcation advances with increasing depolarization current, and the membrane potential of the resting state near the bifurcation increases. Such two changes mainly lead to the decrease of conduction failure rate of action potentials along axon which is described with a network model composed of HH neurons. Moreover, the reduction of conduction failure rate induced by memristor is well interpreted with the nonlinear dynamics near the bifurcation point, for example, the dynamics of the current threshold to evoke the second action potential from the after-potential following the first one. The results present that memristor can promote the conduction ability of action potentials along axon, which presents a novel function of memristor and contributes to the information transmission related to pathological pain in the nervous system.

Action potentials can be induced by external electronic impulsive stimulations applied at one end of the unmyelinated fibers (C-fibers), while some action potentials fail to conduct to the other end of the fiber when the stimulation frequency becomes high. Such a phenomenon is called as conduction failure, which was observed in the biological experiments and related to the painful diabetic neuropathy, inflammation, and trauma in the previous studies. On-off firing pattern was recorded from the fiber when conduction failure happened. In the present study, the diffusion Hodgkin-Huxley (HH) model with resting state near a Hopf bifurcation is adopted to simulate the experimental observations. When the periodic electrical pulses with high frequency are applied to one end of the fiber described by the deterministic HH model, conduction failure and the corresponding firing patterns different from the on-off firing pattern are simulated. When noise is introduced to form the stochastic HH model, the firing pattern corresponding to conduction failure becomes the on-off firing pattern, which is characterised bytransition behaviors between on-phase (continuous action potentials) and off-phase (a long quiescent state) and large variations in the durations of both phases. Furthermore, the increase of potassium conductance can enhance the conduction failure degree, which closely matches those observed in the experiment and is suggested to be related to the reduction of pain signals. The results show that noise is an important factor to evoke the on-off firing pattern, reveal the functional capability in the pain signals propagation along C-fiber, and present a possible measure for the treatment of chronic pain.

In contrast to the traditional viewpoint that inhibitory effect can often induce the reduction of neural firing activities, negative self-feedback current mediated by inhibitory autapse is identified to enhance neuronal bursting activities in the Chay model composed of fast and slow subsystems. With increasing autaptic strength of the autapse, bursting patterns exhibit period-adding bifurcations, which leads to the increase in the mean firing frequency. Such a phenomenon can be well interpreted with fast–slow variable dissection and bifurcation analysis to the bursting patterns. The initial and termination phases of the burst correspond to a saddle-node bifurcation and a saddle-homoclinic (SH) bifurcation of the fast subsystem, respectively. With increasing the autaptic strength, the termination phase of the burst delays and the initial phase remains nearly unchanged. Therefore, the width of the burst becomes wider to contain more spikes, which leads to the period-adding bifurcations and the increases of mean firing frequency. The detailed relationship between the delay of termination phase (SH point) and the characteristic of the inhibitory autaptic current is identified. Moreover, the distinction between the present paper and the previous investigations of the enhancement of bursting activities induced by inhibitory current with time delay is discussed. Based on the results that inhibitory autapse without time delay can enhance bursting activities via bifurcation mechanism, a novel case of nonlinear dynamics in contrast to the traditional viewpoint of the inhibitory effect, a possible function of the autapse, a novel potential measure to modulate bursting activities are present.

Different regions in the human brain functionally connect with each other forming a brain functional network, and the time evolution of functional connectivity between different brain regions exhibits complex nonlinear dynamics. This study intends to characterize the nonlinear properties of dynamic functional connectivity and to explore how schizophrenia influences such nonlinear properties. The dynamic functional connectivity is constructed by analyzing resting-state functional magnetic resonance imaging data, and its nonlinear properties are characterized by sample entropy (SampEn), with larger SampEn values corresponding to more complexity. To identify the influence of schizophrenia on SampEn, the difference in SampEn between patients with schizophrenia and healthy controls is analyzed at different levels of the brain. It is shown that the patients exhibit significantly higher SampEn at different levels of the brain, and such phenomenon is mainly caused by a significantly higher SampEn in the visual cortex of the patients. Furthermore, it is also shown that SampEn of the visual cortex is significantly and positively correlated with the illness duration or the symptom severity scores. Because the visual cortex is implicated in the visual information processing, these results can shed light on abnormal visual functions of patients with schizophrenia, and also are consistent with the notion that the nonlinearity underlies the irregularity in psychotic symptoms of schizophrenia. This study extends the application of nonlinear dynamics in brain sciences and suggests that nonlinear properties are effective biomarkers in characterizing the brain functional networks of patients with brain diseases.