Approximating the stability region of a neural network with a general distribution of delays

Jessop, R.; Campbell, Sue Ann

doi:10.1016/j.neunet.2010.06.009

Cited by 38 publications

(23 citation statements)

References 41 publications

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“…(1.4) is stable near λ = d and Hopf bifurcation cannot occur. For a delayed differential equations with a Gamma distribution delay kernel, the effect of β and m on the stability and bifurcations of the equilibrium has been investigated [5,13]. However, when f (s) is given by Eq.…”

Section: Applications and Discussionmentioning

confidence: 99%

Stability Analysis of a Reaction–Diffusion Equation with Spatiotemporal Delay and Dirichlet Boundary Condition

Chen

2014

J Dyn Diff Equat

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In this paper, we concentrate on the study of a reaction-diffusion equation with spatiotemporal delay and homogeneous Dirichlet boundary condition. It is shown that a positive spatially nonhomogeneous equilibrium can bifurcate from the trivial equilibrium. Moreover, the stability of the bifurcated positive equilibrium is investigated. And we prove that, for the given spatiotemporal delay, the bifurcated equilibrium is stable under some conditions, and Hopf bifurcation cannot occur.

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Section: Applications and Discussionmentioning

confidence: 99%

Stability Analysis of a Reaction–Diffusion Equation with Spatiotemporal Delay and Dirichlet Boundary Condition

Chen

2014

J Dyn Diff Equat

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“…This can be argued successfully, for example, in machining when the tool has nearly infinite stiffness perpendicular to the cutting direction [75], or in laser dynamics where light travels over a fixed distance [40]. On the other hand, in many contexts, including in biological systems and in control problems [9,10,11,21,36,38,68,82], the delays one encounters are not actually constant. In particular, they may depend on the state in a significant way, that is, change dynamically during the time-evolution of the system.…”

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confidence: 99%

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

Calleja¹,

Humphries²,

Krauskopf³

2017

SIAM J. Appl. Dyn. Syst.

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Abstract. We study a scalar DDE with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three only contain constant delays. We implemented this expansion and the computation of the normal form coefficients in Matlab using symbolic differentiation, and the resulting code HHnfDDE is supplied as a supplement to this article. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally, and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincaré section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds (when there is a single unstable Floquet multiplier). This allows us to study transitions through resonance tongues and the breakup of a 1 : 4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena.Key words. State-dependent delay differential equations, bifurcation analysis, invariant tori, resonance tongues, Hopf-Hopf bifurcation, normal form computation AMS subject classifications. 34K60, 34K18, 37G05, 37M201. Introduction. Time delays arise naturally in numerous areas of application as an unavoidable phenomenon, for example, in balancing and control [8,19,35,39,64,65,66,67], machining [36], laser physics [40,46,54], agent dynamics [52,53,70,73], neuroscience and biology [1,18,20,42,79], and climate modelling [13,41,48]. Important sources of delays are communication times between components of a system, maturation and reaction times, and the processing time of information received. When they are sufficiently large compared to the relevant internal time scales of the system under consideration, then the delays must be incorporated into its mathematical description. This leads to mathematical models in the form of delay differential equations (DDEs). In many situations the relevant delays can be considered to be fixed; examples are the travel time of light between components of a laser syst...

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“…13 Distributed delay models appear in a wide range of applications such as hematopoiesis, 14 population biology, [15][16][17] or neural networks. [18][19][20] Four-dimensional models that incorporate the positive self-regulation of glucocorticoid receptors (GR) in the pituitary have been investigated in previous studies. [21][22][23][24][25] In particular, in Kaslik and Neamtu, 25 we constructed a 4-dimensional general model with distributed time delays, which represents an extension of the minimal model of Vinther et al 9 In Gupta et al, 21 it has been suggested that positive self-regulation of GR may trigger bistability in the dynamical structure of the HPA model, ie, there exist 2 asymptotically stable equilibrium states: one corresponding to the normal disease-free state with higher cortisol levels and a second one with lower cortisol levels related to a diseased state associated with hypocortisolism.…”

Section: Introductionmentioning

confidence: 99%

Stability and Hopf bifurcation analysis of a 4‐dimensional hypothalamic‐pituitary‐adrenal axis model with distributed delays

Kaslik

Neamţu

2018

Math Methods in App Sciences

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A 4‐dimensional mathematical model of the hypothalamus‐pituitary‐adrenal axis is investigated, incorporating the influence of the glucocorticoid receptors concentration and general feedback functions. The inclusion of distributed time delays provides a more realistic modeling approach, since the whole past history of the variables is taken into account. The positivity of the solutions and the existence of a positively invariant bounded region are proved. It is shown that the considered 4‐dimensional system has at least 1 equilibrium state, and a detailed local stability and Hopf bifurcation analysis is given. Numerical results reveal the fact that an appropriate choice of the system's parameters leads to the coexistence of 2 asymptotically stable equilibria in the nondelayed case. When the total average time delay of the system is large enough, the coexistence of 2 stable limit cycles is revealed, which successfully model the ultradian rhythm of the hypothalamus‐pituitary‐adrenal axis both in a normal disease‐free situation and in a diseased hypocortisolism state, respectively. Numerical simulations reflect the importance of the theoretical results.

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Approximating the stability region of a neural network with a general distribution of delays

Cited by 38 publications

References 41 publications

Stability Analysis of a Reaction–Diffusion Equation with Spatiotemporal Delay and Dirichlet Boundary Condition

Stability Analysis of a Reaction–Diffusion Equation with Spatiotemporal Delay and Dirichlet Boundary Condition

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

Stability and Hopf bifurcation analysis of a 4‐dimensional hypothalamic‐pituitary‐adrenal axis model with distributed delays

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