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Cooray GK, Rosch RE, Friston KJ. Modelling cortical network dynamics. SN APPLIED SCIENCES 2024; 6:36. [PMID: 38299095 PMCID: PMC10824794 DOI: 10.1007/s42452-024-05624-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2023] [Accepted: 12/21/2023] [Indexed: 02/02/2024] Open
Abstract
We have investigated the theoretical constraints of the interactions between coupled cortical columns. Each cortical column consists of a set of neural populations where each population is modelled as a neural mass. The existence of semi-stable states within a cortical column is dependent on the type of interaction between the neuronal populations, i.e., the form of the synaptic kernels. Current-to-current coupling has been shown, in contrast to potential-to-current coupling, to create semi-stable states within a cortical column. The interaction between semi-stable states of the cortical columns is studied where we derive the dynamics for the collected activity. For small excitations the dynamics follow the Kuramoto model; however, in contrast to previous work we derive coupled equations between phase and amplitude dynamics with the possibility of defining connectivity as a stationary and dynamic variable. The turbulent flow of phase dynamics which occurs in networks of Kuramoto oscillators would indicate turbulent changes in dynamic connectivity for coupled cortical columns which is something that has been recorded in epileptic seizures. We used the results we derived to estimate a seizure propagation model which allowed for inversions using the Laplace assumption (Dynamic Causal Modelling). The seizure propagation model was trialed on simulated data, and future work will investigate the estimation of the connectivity matrix from empirical data. This model can be used to predict changes in seizure evolution after virtual changes in the connectivity network, something that could be of clinical use when applied to epilepsy surgical cases.
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Affiliation(s)
- Gerald Kaushallye Cooray
- Clinical Neuroscience, Karolinska Institutet, Stockholm, Sweden
- GOS-UCL Institute of Child Health, University College London, London, UK
| | - Richard Ewald Rosch
- The Wellcome Centre for Human Neuroimaging, Queen Square Institute of Neurology, University College London, London, UK
- MRC Centre for Neurodevelopmental Disorders, Institute of Psychiatry, Psychology and Neuroscience, King’s College London, London, UK
| | - Karl John Friston
- The Wellcome Centre for Human Neuroimaging, Queen Square Institute of Neurology, University College London, London, UK
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Gogoi PB, Kumarasamy S, Prasad A, Ramaswamy R. Phase slips in coupled oscillator systems. Phys Rev E 2023; 108:014209. [PMID: 37583223 DOI: 10.1103/physreve.108.014209] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2023] [Accepted: 07/07/2023] [Indexed: 08/17/2023]
Abstract
Phase slips are a typical dynamical behavior in coupled oscillator systems: the route to phase synchrony is characterized by intervals of constant phase difference interrupted by abrupt changes in the phase difference. Qualitatively similar to stick-slip phenomena, analysis of phase slip has mainly relied on identifying remnants of saddle-nodes or "ghosts." We study sets of phase oscillators and by examining the dynamics in detail, offer a more precise, quantitative description of the phenomenon. Phase shifts and phase sticks, namely, the temporary locking of phases required for phase slips, occur at stationary points of phase velocities. In networks of coupled phase oscillators, we show that phase slips between pairs of individual oscillators do not occur simultaneously, in general. We consider additional systems that show phase synchrony: one where saddle-node ghosts are absent, one where the coupling is similarity dependent, and two cases of coupled chaotic oscillators.
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Affiliation(s)
| | - Suresh Kumarasamy
- Centre for Computational Modelling, Chennai Institute of Technology, Chennai 600069, India
| | - Awadhesh Prasad
- Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
| | - Ram Ramaswamy
- Department of Chemistry, Indian Institute of Technology Delhi, New Delhi 110016, India
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Castro MA, Mancilla-Almonacid D, Dieny B, Allende S, Buda-Prejbeanu LD, Ebels U. Mutual synchronization of spin-torque oscillators within a ring array. Sci Rep 2022; 12:12030. [PMID: 35835771 PMCID: PMC9283394 DOI: 10.1038/s41598-022-15483-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2021] [Accepted: 05/12/2022] [Indexed: 11/09/2022] Open
Abstract
An array of spin torque nano-oscillators (STNOs), coupled by dipolar interaction and arranged on a ring, has been studied numerically and analytically. The phase patterns and locking ranges are extracted as a function of the number N, their separation, and the current density mismatch between selected subgroups of STNOs. If \documentclass[12pt]{minimal}
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\begin{document}$$N\ge 6$$\end{document}N≥6 for identical current densities through all STNOs, two degenerated modes are identified an in-phase mode (all STNOs have the same phase) and a splay mode (the phase makes a 2\documentclass[12pt]{minimal}
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\begin{document}$$\pi$$\end{document}π turn along the ring). When inducing a current density mismatch between two subgroups, additional phase shifts occur. The locking range (maximum current density mismatch) of the in-phase mode is larger than the one for the splay mode and depends on the number N of STNOs on the ring as well as on the separation. These results can be used for the development of magnetic devices that are based on STNO arrays.
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Affiliation(s)
- M A Castro
- Universidad de Santiago de Chile (USACH), Departamento de Física, CEDENNA, Avda. V. Jara 3493, Estación Central, Santiago, Chile
| | - D Mancilla-Almonacid
- Universidad de Santiago de Chile (USACH), Departamento de Física, CEDENNA, Avda. V. Jara 3493, Estación Central, Santiago, Chile
| | - B Dieny
- Univ. Grenoble Alpes, CEA, CNRS, Grenoble INP, SPINTEC, 38000, Grenoble, France
| | - S Allende
- Universidad de Santiago de Chile (USACH), Departamento de Física, CEDENNA, Avda. V. Jara 3493, Estación Central, Santiago, Chile.
| | - L D Buda-Prejbeanu
- Univ. Grenoble Alpes, CEA, CNRS, Grenoble INP, SPINTEC, 38000, Grenoble, France
| | - U Ebels
- Univ. Grenoble Alpes, CEA, CNRS, Grenoble INP, SPINTEC, 38000, Grenoble, France
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Goebel M, Mizuhara MS, Stepanoff S. Stability of twisted states on lattices of Kuramoto oscillators. CHAOS (WOODBURY, N.Y.) 2021; 31:103106. [PMID: 34717314 DOI: 10.1063/5.0060095] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2021] [Accepted: 09/14/2021] [Indexed: 06/13/2023]
Abstract
Real world systems comprised of coupled oscillators have the ability to exhibit spontaneous synchronization and other complex behaviors. The interplay between the underlying network topology and the emergent dynamics remains a rich area of investigation for both theory and experiment. In this work, we study lattices of coupled Kuramoto oscillators with non-local interactions. Our focus is on the stability of twisted states. These are equilibrium solutions with constant phase shifts between oscillators resulting in spatially linear profiles. Linear stability analysis follows from studying the quadratic form associated with the Jacobian matrix. Novel estimates on both stable and unstable regimes of twisted states are obtained in several cases. Moreover, exploiting the "almost circulant" nature of the Jacobian obtains a surprisingly accurate numerical test for stability. While our focus is on 2D square lattices, we show how our results can be extended to higher dimensions.
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Affiliation(s)
- Monica Goebel
- Department of Mathematics and Statistics, The College of New Jersey, Ewing, New Jersey 08618, USA
| | - Matthew S Mizuhara
- Department of Mathematics and Statistics, The College of New Jersey, Ewing, New Jersey 08618, USA
| | - Sofia Stepanoff
- Department of Mathematics and Statistics, The College of New Jersey, Ewing, New Jersey 08618, USA
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Rouzaire Y, Levis D. Defect Superdiffusion and Unbinding in a 2D XY Model of Self-Driven Rotors. PHYSICAL REVIEW LETTERS 2021; 127:088004. [PMID: 34477446 DOI: 10.1103/physrevlett.127.088004] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2021] [Accepted: 07/15/2021] [Indexed: 06/13/2023]
Abstract
We consider a nonequilibrium extension of the 2D XY model, equivalent to the noisy Kuramoto model of synchronization with short-range coupling, where rotors sitting on a square lattice are self-driven by random intrinsic frequencies. We study the static and dynamic properties of topological defects (vortices) and establish how self-spinning affects the Berezenskii-Kosterlitz-Thouless phase transition scenario. The nonequilibrium drive breaks the quasi-long-range ordered phase of the 2D XY model into a mosaic of ordered domains of controllable size and results in self-propelled vortices that generically unbind at any temperature, featuring superdiffusion ⟨r^{2}(t)⟩∼t^{3/2} with a Gaussian distribution of displacements. Our work provides a simple framework to investigate topological defects in nonequilibrium matter and sheds new light on the problem of synchronization of locally coupled oscillators.
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Affiliation(s)
- Ylann Rouzaire
- Institute of Physics, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
- Departament de Física de la Materia Condensada, Universitat de Barcelona, Martí i Franquès 1, E08028 Barcelona, Spain
| | - Demian Levis
- Departament de Física de la Materia Condensada, Universitat de Barcelona, Martí i Franquès 1, E08028 Barcelona, Spain
- UBICS University of Barcelona Institute of Complex Systems, Martí i Franquès 1, E08028 Barcelona, Spain
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