Collective states in a ring network of theta neurons

Omel’chenko, Oleh; Laing, Carlo R.

doi:10.1098/rspa.2021.0817

Cited by 10 publications

(4 citation statements)

References 68 publications

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“…The model we consider first is that in Omel’chenko and Laing ( 2022 ), Laing ( 2014a ) and Laing and Omel’chenko ( 2020 ), which we briefly present here. The discrete network consists of N synaptically coupled theta neurons described by where each is an angular variable.…”

Section: Theta Neuron Network Modelmentioning

confidence: 99%

“…Such a network can support solutions which are — at a macroscopic level — periodic in time; see Fig. 3 in Omel’chenko and Laing ( 2022 ). Such solutions are unlikely to be true periodic solutions of ( 1 ), since for a typical realisation of the , one or more neurons will have extreme values of this parameter, resulting in them not frequency-locking to other neurons.…”

Section: Theta Neuron Network Modelmentioning

confidence: 99%

“…( 17 ) coincides with Eq. (5.1) from Omel’chenko and Laing ( 2022 ), except that the coefficients a ( x , t ) and are now time-dependent. Since Eq.…”

Section: Periodic Statesmentioning

confidence: 99%

“…We studied such a model in the past (Omel’chenko and Laing 2022 ), concentrating on describing spatially-uniform states and also stationary “bump” states in which there is a spatially-localised group of active neurons while the remainder of the network is quiescent. We determined the existence and stability of such states and also found regions of parameter space in which neither of these types of states were stable.…”

Section: Introductionmentioning

confidence: 99%

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Periodic solutions in next generation neural field models

Laing

Omel’chenko

2023

Biol Cybern

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We consider a next generation neural field model which describes the dynamics of a network of theta neurons on a ring. For some parameters the network supports stable time-periodic solutions. Using the fact that the dynamics at each spatial location are described by a complex-valued Riccati equation we derive a self-consistency equation that such periodic solutions must satisfy. We determine the stability of these solutions, and present numerical results to illustrate the usefulness of this technique. The generality of this approach is demonstrated through its application to several other systems involving delays, two-population architecture and networks of Winfree oscillators.

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Section: Theta Neuron Network Modelmentioning

confidence: 99%

Section: Theta Neuron Network Modelmentioning

confidence: 99%

“…( 17 ) coincides with Eq. (5.1) from Omel’chenko and Laing ( 2022 ), except that the coefficients a ( x , t ) and are now time-dependent. Since Eq.…”

Section: Periodic Statesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Periodic solutions in next generation neural field models

Laing

Omel’chenko

2023

Biol Cybern

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Chimeras on annuli

Laing

2022

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Chimeras occur in networks of coupled oscillators and are characterized by the coexistence of synchronous and asynchronous groups of oscillators in different parts of the network. We consider a network of nonlocally coupled phase oscillators on an annular domain. The Ott/Antonsen ansatz is used to derive a continuum level description of the oscillators’ expected dynamics in terms of a complex-valued order parameter. The equations for this order parameter are numerically analyzed in order to investigate solutions with the same symmetry as the domain and chimeras which are analogous to the “multi-headed” chimeras observed on one-dimensional domains. Such solutions are stable only for domains with widths that are neither too large nor too small. We also study rotating waves with different winding numbers, which are similar to spiral wave chimeras seen in two-dimensional domains. We determine ranges of parameters, such as the size of the domain for which such solutions exist and are stable, and the bifurcations by which they lose stability. All of these bifurcations appear subcritical.

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Patched patterns and emergence of chaotic interfaces in arrays of nonlocally coupled excitable systems

Franović

Eydam

2022

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We disclose a new class of patterns, called patched patterns, in arrays of non-locally coupled excitable units with attractive and repulsive interactions. The self-organization process involves the formation of two types of patches, majority and minority ones, characterized by uniform average spiking frequencies. Patched patterns may be temporally periodic, quasiperiodic, or chaotic, whereby chaotic patterns may further develop interfaces comprised of units with average frequencies in between those of majority and minority patches. Using chaos and bifurcation theory, we demonstrate that chaos typically emerges via a torus breakup and identify the secondary bifurcation that gives rise to chaotic interfaces. It is shown that the maximal Lyapunov exponent of chaotic patched patterns does not decay, but rather converges to a finite value with system size. Patched patterns with a smaller wavenumber may exhibit diffusive motion of chaotic interfaces, similar to that of the incoherent part of chimeras.

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Collective states in a ring network of theta neurons

Cited by 10 publications

References 68 publications

Periodic solutions in next generation neural field models

Periodic solutions in next generation neural field models

Chimeras on annuli

Patched patterns and emergence of chaotic interfaces in arrays of nonlocally coupled excitable systems

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