Hopf‐pitchfork bifurcation in a two‐neuron system with discrete and distributed delays

Wang, Hongbin; Wang, Jingnan

doi:10.1002/mma.3418

Cited by 11 publications

(5 citation statements)

References 38 publications

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“…From [31] we can obtain Theorem 1. If the assumptions of Lemma 1 are satisfied, σ δ, σδ > 1, and sign(g (0)) sign(β) < 0 hold, then system (2) undergoes a Hopf-pitchfork bifurcation of case I at equilibrium (0, 0, 0, 0), which is shown in Fig.…”

Section: Normal Form For Hopf-zero Bifurcationmentioning

confidence: 91%

Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay

Cai

Zhang

2017

NAMC

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In this paper, the Hopf-pitchfork bifurcation of coupled van der Pol with delay is studied. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly, the normal form is gotten by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly, bifurcation diagrams and phase portraits are given through analyzing the unfolding structure. Finally, numerical simulations are used to support theoretical analysis.

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Section: Normal Form For Hopf-zero Bifurcationmentioning

confidence: 91%

Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay

Cai

Zhang

2017

NAMC

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“…The analysis of coupled delayed differential equations has been amply studied for the analysis of the stability of neural networks. [ 45–47 ] An approach to a system of mutually restrained neurons can be established by using a set of elementary neuron dynamical equations as (1–2) and introducing a coupling function C , as follows [ 17,19,23,48–52 ]

τ_{m} \frac{d u_{i}}{d t} = f false(u_{i}, w_{i}, I_{tot} false) + \sum_{i, j} C_{i j}

τ_{k} \frac{d w_{i}}{d t} = g false(u_{i}, w_{i} false)

where

C_{i j} false(t false) = C_{i j} false(u_{i} false(t false), u_{j} false(t - τ_{normalc} false) false)

is a coupling function with a delay time

τ_{c}

. This system of equations allows us to study not only coupled neurons but also the coupling of neuronal subensembles that operate synchronically.…”

Section: Coupling Of Neuronsmentioning

confidence: 99%

“…The coupling function may include a time‐distributed delay via a memory function

m false(t false)

[ 50,51 ]

C_{j} false(t false) = c_{2} \int_{- \infty}^{t} m false(t - x false) u_{j} false(x - τ_{c} false) d x

…”

Section: Coupling Of Neuronsmentioning

confidence: 99%

The Impedance of Spiking Neurons Coupled by Time‐Delayed Interaction

Bisquert

2022

Physica Status Solidi (a)

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The synchronization of populations of interacting spiking neurons is a topic of interest for understanding the operation of the brain and for developing oscillatory‐based biomimetic computation. The analysis of the operation of neurons by models such as Hodgkin–Huxley (HH) and FitzHugh–Nagumo (FHN) is based on the electrical and electrochemical concepts. The application of small‐signal AC impedance and equivalent circuit methods is a promising tool for the fabrication of artificial neurons and synapses with memristor technologies for neuromorphic computation and sensory‐motor autonomous systems. The time domain and impedance spectroscopy analysis of two FHN neurons coupled with a time delay related to the propagation of the action potentials is developed. Depending on the coupling strength and delay time constant, the two neurons become synchronized, in phase opposition for a weak coupling, and in alternating spike packets for the strong interaction. The delayed interaction introduces a new element in the equivalent circuit. A new oscillatory impedance that causes curling of the basic spectral patterns associated with the onset of a limit cycle in the presence of Hopf bifurcations is found. The new pattern is appealing as an experimental tool to determine experimentally the extent of coupling between neurons.

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“…In this section, we use the center manifold theory and normal form method [6,23] to study Hopf-zero bifurcations. The normal form of a Hopf-zero bifurcation for a general delay-differential equations has been given in the following two papers: one is for a saddlenode-Hopf bifurcation [11], and the other is for a steady-state Hopf bifurcation [20].…”

Section: Normal Form For Hopf-zero Bifurcationmentioning

confidence: 99%

“…In Sect. 3, we use the center manifold theory and normal form method [6,23] to investigate the Hopf-zero bifurcation with original parameters. In Sect.…”

Section: Introductionmentioning

confidence: 99%

Hopf-zero bifurcation of Oregonator oscillator with delay

2018

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In this paper, we study the Hopf-zero bifurcation of Oregonator oscillator with delay. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly, we get the normal form by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly, we obtain a critical value to predict the bifurcation diagrams and phase portraits. Under some conditions, saddle-node bifurcation and pitchfork bifurcation occur along M and N, respectively; Hopf bifurcation and heteroclinic bifurcation occur along H and S, respectively. Finally, we use numerical simulations to support theoretical analysis.

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Hopf‐pitchfork bifurcation in a two‐neuron system with discrete and distributed delays

Cited by 11 publications

References 38 publications

Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay

Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay

The Impedance of Spiking Neurons Coupled by Time‐Delayed Interaction

Hopf-zero bifurcation of Oregonator oscillator with delay

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