Stochastic dynamics of conduction failure of action potential along nerve fiber with Hopf bifurcation

Zhang, XinJing; Gu, Huaguang; Guan, Lei

doi:10.1007/s11431-018-9515-4

Cited by 14 publications

(4 citation statements)

References 55 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Introduction. Nonlinear dynamics has played very important roles in identifying dynamics of neural electrical activities [10,29,32], which are involved in the information processing, locomotion control, cognitive functions, and brain disease [8,19,31]. These activities mainly include the steady state such as the resting state and the firing behavior composed of action potentials or spikes, for example, the spiking and bursting, which has been widely investigated with the conceptions of bifurcations and chaos [6,16,18,23,33].…”

mentioning

confidence: 99%

Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron

Wang

2021

era

Self Cite

View full text Add to dashboard Cite

Post-inhibitory rebound (PIR) spike induced by the negative stimulation, which plays important roles and presents counterintuitive nonlinear phenomenon in the nervous system, is mainly related to the Hopf bifurcation and hyperpolarization-active caution (I h) current. In the present paper, the emerging condition for the PIR spike is extended to the bifurcation of the big homoclinic (BHom) orbit in a model without I h current. The threshold curve for a spike evoked from a mono-stable or coexisting steady state surrounds the steady state from left, to below, and to right, because the BHom orbit is big enough to surround the steady state. The right part of the threshold curve coincides with the stable manifold of the saddle and acts the threshold for the spike induced by the positive stimulation, resembling that of the saddle-node bifurcation on an invariant cycle, and the left part acts the threshold for the PIR spike, resembling that of the Hopf bifurcation. The bifurcation curve and a codimension-2 bifurcation point related to the BHom orbit are acquired in the two-parameter plane. The results present a comprehensive viewpoint to the dynamics near the BHom orbit bifurcation, which presents a novel threshold curve and extends the conditions for the PIR spike.

show abstract

mentioning

confidence: 99%

Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron

Wang

2021

era

Self Cite

View full text Add to dashboard Cite

show abstract

“…Therefore, reduction of firing frequency appears for "Circle/Fold cycle" bursting, which can be explained with the relation between the trajectory of bursting and bifurcation points of fast subsystem. In future, the influences of noise [48][49][50][51] on the paradoxical phenomenon for various bursting patterns should be investigated and extended to networks models. [16] As suggested in Refs.…”

Section: Discussionmentioning

confidence: 99%

Delayed excitatory self-feedback-induced negative responses of complex neuronal bursting patterns*

Ben

2021

Chinese Phys. B

Self Cite

View full text Add to dashboard Cite

In traditional viewpoint, excitatory modulation always promotes neural firing activities. On contrary, the negative responses of complex bursting behaviors to excitatory self-feedback mediated by autapse with time delay are acquired in the present paper. Two representative bursting patterns which are identified respectively to be “Fold/Big Homoclinic” bursting and “Circle/Fold cycle” bursting with bifurcations are studied. For both burstings, excitatory modulation can induce less spikes per burst for suitable time delay and strength of the self-feedback/autapse, because the modulation can change the initial or termination phases of the burst. For the former bursting composed of quiescent state and burst, the mean firing frequency exhibits increase, due to that the quiescent state becomes much shorter than the burst. However, for the latter bursting pattern with more complex behavior which is depolarization block lying between burst and quiescent state, the firing frequency manifests decrease in a wide range of time delay and strength, because the duration of both depolarization block and quiescent state becomes long. Therefore, the decrease degree of spike number per burst is larger than that of the bursting period, which is the cause for the decrease of firing frequency. Such reduced bursting activity is explained with the relations between the bifurcation points of the fast subsystem and the bursting trajectory. The present paper provides novel examples of paradoxical phenomenon that the excitatory effect induces negative responses, which presents possible novel modulation measures and potential functions of excitatory self-feedback/autapse to reduce bursting activities.

show abstract

“…The bifurcations have played important roles in identifying the nonlinear dynamics of neuronal electrical activities, which are involved in the physiological functions of the neural system such as information processing and locomotor control and pathology or diseases of the brain [1][2][3][4][5][6][7][8][9]. The neuronal electrical activities include the steady state and firing, and the transition between steady state and firing or between different firing patterns can be induced by various modulations via different bifurcations [10][11][12][13][14][15][16][17][18][19][20]. For example, the steady state changes to firing via the sub-critical Hopf (SubH) bifurcation, or the sup-critical Hopf (SupH) bifurcation, or the saddle-node bifurcation on an invariant cycle (SNIC), or the saddle-node (SN) bifurcation [21,22].…”

Section: Introductionmentioning

confidence: 99%

Multiple and Complex Codimension-2 Bifurcations underlying Post-inhibitory Rebound Spike and Excitability Transition

Wang

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Stochastic dynamics of conduction failure of action potential along nerve fiber with Hopf bifurcation

Cited by 14 publications

References 55 publications

Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron

Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron

Delayed excitatory self-feedback-induced negative responses of complex neuronal bursting patterns*

Multiple and Complex Codimension-2 Bifurcations underlying Post-inhibitory Rebound Spike and Excitability Transition

Contact Info

Product

Resources

About