Pitchfork–Hopf bifurcations in 1D neural field models with transmission delays

Dijkstra, Koen; Gils, Stephan A. van; Janssens, Sebastiaan G.; Kuznetsov, Yu. А.; Visser, Sid

doi:10.1016/j.physd.2015.01.004

Cited by 22 publications

(41 citation statements)

References 32 publications

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“…Similar patterns can also be found in NFMs with spatially dependent delays -modeling the effect of the finite velocity propagation of action potentials [1,15]-as a great deal of theoretical work indicates, see e.g. [16][17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 57%

Synchrony-induced modes of oscillation of a neural field model

et al. 2017

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We investigate the modes of oscillation of heterogeneous ring networks of quadratic integrate-and-fire (QIF) neurons with nonlocal, space-dependent coupling. Perturbations of the equilibrium state with a particular wave number produce transient standing waves with a specific temporal frequency, analogously to those in a tense string. In the neuronal network, the equilibrium corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network's oscillatory modes, which reflect the interplay of episodes of synchronous spiking with the excitatory-inhibitory spatial interactions. In the thermodynamic limit, an exact low-dimensional neural field model describing the macroscopic dynamics of the network is derived. This allows us to obtain formulas for the Turing eigenvalues of the spatially homogeneous state and hence to obtain its stability boundary. We find that the frequency of each Turing mode depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity. The decay rate instead is identical for all oscillation modes as a consequence of the heterogeneity-induced desynchronization of the neurons. Finally, we numerically compute the spectrum of spatially inhomogeneous solutions branching from the Turing bifurcation, showing that similar oscillatory modes operate in neural bump states and are maintained away from onset.

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Section: Introductionmentioning

confidence: 57%

Synchrony-induced modes of oscillation of a neural field model

et al. 2017

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“…As a result we obtain a direct and flexible basic existence proof for a delay neural field equation, which includes a constructive method based on integral equations only. These results have been derived by other authors [ 8 , 10 , 11 , 24 ] with more sophisticated techniques, but it is non-trivial that the arguments used for neural fields without delay are applicable to the delay case, and the approach in our Sect. 2 , based on several relatively simple functional analytic arguments, is of interest by itself.…”

Section: Introductionmentioning

confidence: 56%

“…In addition, Faugeras and Faye [ 10 ], in their Theorem 3.2.1, state the general existence of solutions with a reference to the generic theory of delay equations, based on work such as [ 35 ]. We also point out the work of Van Gils et al [ 8 ] employing the sun–star calculus for their analysis and [ 24 ] in which the local bifurcation theory for delayed neural fields was developed. Here, we develop arguments on how to use the basic functional analytic calculus to work for the delay case as well, with the goal to present a short and elementary approach which is easily accessible.…”

Section: The Mathematical Modelmentioning

confidence: 99%

Kernel Reconstruction for Delayed Neural Field Equations

et al. 2018

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Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with considerable success over the past four decades. However, to make effective use of such models, we need to identify their constituents in practical systems. This includes the determination of model parameters and in particular the reconstruction of the underlying effective connectivity in biological tissues.In this work, we provide an integral equation approach to the reconstruction of the neural connectivity in the case where the neural activity is governed by a delay neural field equation. As preparation, we study the solution of the direct problem based on the Banach fixed-point theorem. Then we reformulate the inverse problem into a family of integral equations of the first kind. This equation will be vector valued when several neural activity trajectories are taken as input for the inverse problem. We employ spectral regularization techniques for its stable solution. A sensitivity analysis of the regularized kernel reconstruction with respect to the input signal u is carried out, investigating the Fréchet differentiability of the kernel with respect to the signal. Finally, we use numerical examples to show the feasibility of the approach for kernel reconstruction, including numerical sensitivity tests, which show that the integral equation approach is a very stable and promising approach for practical computational neuroscience.

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“…Furthermore, in [15,59] the technique was used to calculate the critical normal form coefficients for Hopf and Hopf-Hopf bifurcations occurring in neural field models with propagation delays. For these models sun-reflexivity is lost, which is typical for delay equations in abstract spaces or with infinite delay.…”

Section: Discussionmentioning

confidence: 99%

Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

Bosschaert¹,

Janssens²,

Kuznetsov³

2020

SIAM J. Appl. Dyn. Syst.

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In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models.

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Pitchfork–Hopf bifurcations in 1D neural field models with transmission delays

Cited by 22 publications

References 32 publications

Synchrony-induced modes of oscillation of a neural field model

Synchrony-induced modes of oscillation of a neural field model

Kernel Reconstruction for Delayed Neural Field Equations

Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

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