Frustration, Stability, and Delay-Induced Oscillations in a Neural Network Model

Bélair, Jacques; Campbell, Sue Ann; Driessche, P. van den

doi:10.1137/s0036139994274526

Cited by 197 publications

(84 citation statements)

References 14 publications

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“…It should be mentioned that the constant r here is not the absolute size of the time lag required for the communication and response among neurons. In fact, system (1) is obtained after some rescaling and reparametrization, and the constant r represents the ratio of the absolute size of the delay over the relaxation time of the system (see, for example, Belair, Campbell and van den Driessche [1], Marcus and Westervelt [9] and Wu [11]). Hence, this constant can be relatively large, and in such a case the dynamics of system (1) can be significantly different from that of the corresponding ordinary differential equation model.…”

Section: Introductionmentioning

confidence: 99%

Desynchronization of large scale delayed neural networks

Chen

Huang

2000

Proc. Amer. Math. Soc.

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Abstract. We consider a ring of identical neurons with delayed nearest neighborhood inhibitory interaction. Under general conditions, such a network has a slowly oscillatory synchronous periodic solution which is completely characterized by a scalar delay differential equation with negative feedback. Despite the fact that the slowly oscillatory periodic solution of the scalar equation is stable, we show that the associated synchronous solution is unstable if the size of the network is large.

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Section: Introductionmentioning

confidence: 99%

Desynchronization of large scale delayed neural networks

Chen

Huang

2000

Proc. Amer. Math. Soc.

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“…There are two secondary bifurcation branches emanating from each codimension two point: a Hopf bifurcation of limit cycles surrounding the nontrivial equilibria and a pitchfork bifurcation of these limit cycles. This agreement could be checked quantitatively by using a centre manifold reduction of the delay differential equation as was done for a standard Hopf pitchfork interaction in Bélair et al [1996]. In the remaining three cases, one or more of the bifurcations is equivariant.…”

Section: Discussionmentioning

confidence: 93%

Patterns of Oscillation in a Ring of Identical Cells With Delayed Coupling

Bungay

Campbell

2007

Int. J. Bifurcation Chaos

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We investigate the behaviour of a neural network model consisting of three neurons with delayed self and nearest-neighbour connections. We give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the system. We show the existence of codimension two bifurcation points involving both standard and D 3 -equivariant, Hopf and pitchfork bifurcation points. We use numerical simulation and numerical bifurcation analysis to investigate the dynamics near the pitchfork-Hopf interaction points. Our numerical investigations reveal that multiple secondary Hopf bifurcations and pitchfork bifurcations of limit cycles may emanate from the pitchforkHopf points. Further, these secondary bifurcations give rise to 10 different types of periodic solutions. In addition, the secondary bifurcations can lead to multistability between equilibrium points and periodic solutions in some regions of parameter space. We conclude by generalizing our results into conjectures about the secondary bifurcations that emanate from codimension two pitchfork-Hopf bifurcation points in systems with D n symmetry.

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“…Consider the neural-network model discussed in [12], described by In region 2), the system has a stable limit cycle due to a Hopf bifurcation at = 0:96, = 4:265, and !0 = 0:3898. In this case, an eigenvalue of the linearized system crosses the point 01+0i when the frequency is sweeping onto the Nyquist contour.…”

Section: Example 2: Multiple Oscillations Of a Neural-network Modelmentioning

confidence: 99%

“…The TDCC is used to test and verify the new algorithm. In addition, the capacity of the proposed graphical-analysis method is applied to the detection of multiple oscillations in a time-delayed neural-network model frequently discussed in the literature [12], [13].…”

Section: Introductionmentioning

confidence: 99%

Predicting period-doubling bifurcations and multiple oscillations in nonlinear time-delayed feedback systems

Berns

Moiola

Chen

1998

IEEE Trans. Circuits Syst. I

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Abstract-In this brief, a graphical approach is developed from an engineering frequency-domain approach enabling prediction of perioddoubling bifurcations (PDB's) starting from a small neighborhood of Hopf bifurcation points useful for analysis of multiple oscillations of periodic solutions for time-delayed feedback systems. The proposed algorithm employs higher order harmonic-balance approximations (HBA's) for the predicted periodic solutions of the time-delayed systems. As compared to the same study of feedback systems without time delays, the HBA's used in the new algorithm include only some simple modifications. Two examples are used to verify the graphical algorithm for prediction: one is the well-known time-delayed Chua's circuit (TDCC) and the other is a time-delayed neural-network model.

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Frustration, Stability, and Delay-Induced Oscillations in a Neural Network Model

Cited by 197 publications

References 14 publications

Desynchronization of large scale delayed neural networks

Desynchronization of large scale delayed neural networks

Patterns of Oscillation in a Ring of Identical Cells With Delayed Coupling

Predicting period-doubling bifurcations and multiple oscillations in nonlinear time-delayed feedback systems

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