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Klinshov VV, Nekorkin VI. Adaptive myelination causes slow oscillations in recurrent neural loops. CHAOS (WOODBURY, N.Y.) 2024; 34:033101. [PMID: 38427934 DOI: 10.1063/5.0193265] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2023] [Accepted: 02/03/2024] [Indexed: 03/03/2024]
Abstract
The brain is known to be plastic, i.e., capable of changing and reorganizing as it develops and accumulates experience. Recently, a novel form of brain plasticity was described which is activity-dependent myelination of nerve fibers. Since the speed of propagation of action potentials along axons depends significantly on their degree of myelination, this process leads to adaptive change of axonal delays depending on the neural activity. To understand the possible influence of the adaptive delays on the behavior of neural networks, we consider a simple setup, a neuronal oscillator with delayed feedback. We show that introducing the delay plasticity into this circuit can lead to the occurrence of slow oscillations which are impossible with a constant delay.
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Affiliation(s)
- Vladimir V Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, Ulyanova Street 46, 603950, Nizhny Novgorod, Russia
- National Research University Higher School of Economics, 25/12 Bol'shaya Pecherskaya street, Nizhny Novgorod 603155, Russia
| | - Vladimir I Nekorkin
- Institute of Applied Physics of the Russian Academy of Sciences, Ulyanova Street 46, 603950, Nizhny Novgorod, Russia
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2
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Laing CR, Krauskopf B. Theta neuron subject to delayed feedback: a prototypical model for self-sustained pulsing. Proc Math Phys Eng Sci 2022. [DOI: 10.1098/rspa.2022.0292] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022] Open
Abstract
We consider a single theta neuron with delayed self-feedback in the form of a Dirac delta function in time. Because the dynamics of a theta neuron on its own can be solved explicitly—it is either excitable or shows self-pulsations—we are able to derive algebraic expressions for the existence and stability of the periodic solutions that arise in the presence of feedback. These periodic solutions are characterized by one or more equally spaced pulses per delay interval, and there is an increasing amount of multistability with increasing delay time. We present a complete description of where these self-sustained oscillations can be found in parameter space; in particular, we derive explicit expressions for the loci of their saddle-node bifurcations. We conclude that the theta neuron with delayed self-feedback emerges as a prototypical model: it provides an analytical basis for understanding pulsating dynamics observed in other excitable systems subject to delayed self-coupling.
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Affiliation(s)
- Carlo R. Laing
- School of Natural and Computational Sciences Massey University, Private Bag 102-904, North Shore Mail Centre, Auckland 0745, New Zealand
| | - Bernd Krauskopf
- Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
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Klinshov VV, D'Huys O. Noise-induced switching in an oscillator with pulse delayed feedback: A discrete stochastic modeling approach. CHAOS (WOODBURY, N.Y.) 2022; 32:093141. [PMID: 36182395 DOI: 10.1063/5.0100698] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/26/2022] [Accepted: 08/22/2022] [Indexed: 06/16/2023]
Abstract
We study the dynamics of an oscillatory system with pulse delayed feedback and noise of two types: (i) phase noise acting on the oscillator and (ii) stochastic fluctuations of the feedback delay. Using an event-based approach, we reduce the system dynamics to a stochastic discrete map. For weak noise, we find that the oscillator fluctuates around a deterministic state, and we derive an autoregressive model describing the system dynamics. For stronger noise, the oscillator demonstrates noise-induced switching between various deterministic states; our theory provides a good estimate of the switching statistics in the linear limit. We show that the robustness of the system toward this switching is strikingly different depending on the type of noise. We compare the analytical results for linear coupling to numerical simulations of nonlinear coupling and find that the linear model also provides a qualitative explanation for the differences in robustness to both types of noise. Moreover, phase noise drives the system toward higher frequencies, while stochastic delays do not, and we relate this effect to our theoretical results.
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Affiliation(s)
- Vladimir V Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, Nizhny Novgorod 603950, Russia
| | - Otti D'Huys
- Department of Applied Computing Sciences, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands
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Klinshov V, Lücken L, Feketa P. On the interpretation of Dirac δ pulses in differential equations for phase oscillators. CHAOS (WOODBURY, N.Y.) 2021; 31:031102. [PMID: 33810720 DOI: 10.1063/5.0040995] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2020] [Accepted: 02/05/2021] [Indexed: 06/12/2023]
Abstract
In this note, we discuss the usage of the Dirac δ function in models of phase oscillators with pulsatile inputs. Many authors use a product of the delta function and the phase response curve in the right-hand side of an ordinary differential equation to describe the discontinuous phase dynamics in such systems. We point out that this notation has to be treated with care as it is ambiguous. We argue that the presumably most canonical interpretation does not lead to the intended behavior in many cases.
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Affiliation(s)
- Vladimir Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950 Nizhny Novgorod, Russia
| | - Leonhard Lücken
- Institute for Chemistry and Biology of the Marine Environment, University of Oldenburg, 26111 Oldenburg, Germany
| | - Petro Feketa
- Chair of Automatic Control, Kiel University, Kaiserstraße 2, 24143 Kiel, Germany
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Terrien S, Pammi VA, Krauskopf B, Broderick NGR, Barbay S. Pulse-timing symmetry breaking in an excitable optical system with delay. Phys Rev E 2021; 103:012210. [PMID: 33601571 DOI: 10.1103/physreve.103.012210] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/11/2020] [Accepted: 11/10/2020] [Indexed: 06/12/2023]
Abstract
Excitable systems with delayed feedback are important in areas from biology to neuroscience and optics. They sustain multistable pulsing regimes with different numbers of equidistant pulses in the feedback loop. Experimentally and theoretically, we report on the pulse-timing symmetry breaking of these regimes in an optical system. A bifurcation analysis unveils that this originates in a resonance phenomenon and that symmetry-broken states are stable in large regions of the parameter space. These results have impact in photonics for, e.g., optical computing and versatile sources of optical pulses.
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Affiliation(s)
- Soizic Terrien
- The Dodd-Walls Centre for Photonic and Quantum Technologies, The University of Auckland, New Zealand
| | - Venkata A Pammi
- Université Paris-Saclay, Centre National de la Recherche Scientifique, Centre de Nanosciences et de Nanotechnologies, Palaiseau, France
| | - Bernd Krauskopf
- The Dodd-Walls Centre for Photonic and Quantum Technologies, The University of Auckland, New Zealand
| | - Neil G R Broderick
- The Dodd-Walls Centre for Photonic and Quantum Technologies, The University of Auckland, New Zealand
| | - Sylvain Barbay
- Université Paris-Saclay, Centre National de la Recherche Scientifique, Centre de Nanosciences et de Nanotechnologies, Palaiseau, France
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Ruschel S, Krauskopf B, Broderick NGR. The limits of sustained self-excitation and stable periodic pulse trains in the Yamada model with delayed optical feedback. CHAOS (WOODBURY, N.Y.) 2020; 30:093101. [PMID: 33003905 DOI: 10.1063/5.0007758] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2020] [Accepted: 08/11/2020] [Indexed: 06/11/2023]
Abstract
We consider the Yamada model for an excitable or self-pulsating laser with saturable absorber and study the effects of delayed optical self-feedback in the excitable case. More specifically, we are concerned with the generation of stable periodic pulse trains via repeated self-excitation after passage through the delayed feedback loop and their bifurcations. We show that onset and termination of such pulse trains correspond to the simultaneous bifurcation of countably many fold periodic orbits with infinite period in this delay differential equation. We employ numerical continuation and the concept of reappearance of periodic solutions to show that these bifurcations coincide with codimension-two points along families of connecting orbits and fold periodic orbits in a related advanced differential equation. These points include heteroclinic connections between steady states and homoclinic bifurcations with non-hyperbolic equilibria. Tracking these codimension-two points in parameter space reveals the critical parameter values for the existence of periodic pulse trains. We use the recently developed theory of temporal dissipative solitons to infer necessary conditions for the stability of such pulse trains.
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Affiliation(s)
- Stefan Ruschel
- Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand
| | - Bernd Krauskopf
- Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand
| | - Neil G R Broderick
- Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin 9054, New Zealand
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Klinshov V, Shchapin D, D'Huys O. Mode Hopping in Oscillating Systems with Stochastic Delays. PHYSICAL REVIEW LETTERS 2020; 125:034101. [PMID: 32745403 DOI: 10.1103/physrevlett.125.034101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/24/2020] [Revised: 05/06/2020] [Accepted: 06/08/2020] [Indexed: 06/11/2023]
Abstract
We study a noisy oscillator with pulse delayed feedback, theoretically and in an electronic experimental implementation. Without noise, this system has multiple stable periodic regimes. We consider two types of noise: (i) phase noise acting on the oscillator state variable and (ii) stochastic fluctuations of the coupling delay. For both types of stochastic perturbations the system hops between the deterministic regimes, but it shows dramatically different scaling properties for different types of noise. The robustness to conventional phase noise increases with coupling strength. However for stochastic variations in the coupling delay, the lifetimes decrease exponentially with the coupling strength. We provide an analytic explanation for these scaling properties in a linearized model. Our findings thus indicate that the robustness of a system to stochastic perturbations strongly depends on the nature of these perturbations.
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Affiliation(s)
- Vladimir Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia
| | - Dmitry Shchapin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia
| | - Otti D'Huys
- Department of Mathematics, Aston University, B4 7ET Birmingham, United Kingdom
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Klinshov V, Shchapin D, Yanchuk S, Wolfrum M, D'Huys O, Nekorkin V. Embedding the dynamics of a single delay system into a feed-forward ring. Phys Rev E 2017; 96:042217. [PMID: 29347517 DOI: 10.1103/physreve.96.042217] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2017] [Indexed: 11/07/2022]
Abstract
We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example, we demonstrate how the complex bifurcation scenario of simultaneously emerging multijittering solutions can be transferred from a single oscillator with delayed pulse feedback to multijittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type.
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Affiliation(s)
- Vladimir Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950 Nizhny Novgorod, Russia
| | - Dmitry Shchapin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950 Nizhny Novgorod, Russia
| | - Serhiy Yanchuk
- Technical University of Berlin, Institute of Mathematics, Straße des 17. Juni 136, 10623 Berlin, Germany
| | - Matthias Wolfrum
- Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany
| | - Otti D'Huys
- Aston University, Department of Mathematics, B4 7ET Birmingham, United Kingdom
| | - Vladimir Nekorkin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950 Nizhny Novgorod, Russia
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Klinshov V, Shchapin D, Yanchuk S, Nekorkin V. Jittering waves in rings of pulse oscillators. Phys Rev E 2016; 94:012206. [PMID: 27575122 DOI: 10.1103/physreve.94.012206] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2016] [Indexed: 06/06/2023]
Abstract
Rings of oscillators with delayed pulse coupling are studied analytically, numerically, and experimentally. The basic regimes observed in such rings are rotating waves with constant interspike intervals and phase lags between the neighbors. We show that these rotating waves may destabilize leading to the so-called jittering waves. For these regimes, the interspike intervals are no more equal but form a periodic sequence in time. Analytic criterion for the emergence of jittering waves is derived and confirmed by the numerical and experimental data. The obtained results contribute to the hypothesis that the multijitter instability is universal in systems with pulse coupling.
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Affiliation(s)
- Vladimir Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanova Street, 603950, Nizhny Novgorod, Russia
| | - Dmitry Shchapin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanova Street, 603950, Nizhny Novgorod, Russia
| | - Serhiy Yanchuk
- Technical University of Berlin, Institute of Mathematics, Straße des 17. Juni 136, 10623 Berlin
| | - Vladimir Nekorkin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanova Street, 603950, Nizhny Novgorod, Russia
- University of Nizhny Novgorod, 23 Prospekt Gagarina, 603950, Nizhny Novgorod, Russia
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