1
|
Erneux T. Strong delayed negative feedback. FRONTIERS IN NETWORK PHYSIOLOGY 2024; 4:1399272. [PMID: 38903729 PMCID: PMC11188390 DOI: 10.3389/fnetp.2024.1399272] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/11/2024] [Accepted: 04/17/2024] [Indexed: 06/22/2024]
Abstract
In this paper, we analyze the strong feedback limit of two negative feedback schemes which have proven to be efficient for many biological processes (protein synthesis, immune responses, breathing disorders). In this limit, the nonlinear delayed feedback function can be reduced to a function with a threshold nonlinearity. This will considerably help analytical and numerical studies of networks exhibiting different topologies. Mathematically, we compare the bifurcation diagrams for both the delayed and non-delayed feedback functions and show that Hopf classical theory needs to be revisited in the strong feedback limit.
Collapse
Affiliation(s)
- Thomas Erneux
- Université Libre de Bruxelles, Optique Nonlinéaire Théorique, Bruxelles, Belgium
| |
Collapse
|
2
|
Papo D, Buldú JM. Does the brain behave like a (complex) network? I. Dynamics. Phys Life Rev 2024; 48:47-98. [PMID: 38145591 DOI: 10.1016/j.plrev.2023.12.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2023] [Accepted: 12/10/2023] [Indexed: 12/27/2023]
Abstract
Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.
Collapse
Affiliation(s)
- D Papo
- Department of Neuroscience and Rehabilitation, Section of Physiology, University of Ferrara, Ferrara, Italy; Center for Translational Neurophysiology, Fondazione Istituto Italiano di Tecnologia, Ferrara, Italy.
| | - J M Buldú
- Complex Systems Group & G.I.S.C., Universidad Rey Juan Carlos, Madrid, Spain
| |
Collapse
|
3
|
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.
Collapse
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
| |
Collapse
|
4
|
Mihara A, Medeiros ES, Zakharova A, Medrano-T RO. Sparsity-driven synchronization in oscillator networks. CHAOS (WOODBURY, N.Y.) 2022; 32:033114. [PMID: 35364836 DOI: 10.1063/5.0074008] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2021] [Accepted: 02/24/2022] [Indexed: 06/14/2023]
Abstract
The emergence of synchronized behavior is a direct consequence of networking dynamical systems. Naturally, strict instances of this phenomenon, such as the states of complete synchronization, are favored or even ensured in networks with a high density of connections. Conversely, in sparse networks, the system state-space is often shared by a variety of coexistent solutions. Consequently, the convergence to complete synchronized states is far from being certain. In this scenario, we report the surprising phenomenon in which completely synchronized states are made the sole attractor of sparse networks by removing network links, the sparsity-driven synchronization. This phenomenon is observed numerically for nonlocally coupled Kuramoto networks and verified analytically for locally coupled ones. In addition, we unravel the bifurcation scenario underlying the network transition to completely synchronized behavior. Furthermore, we present a simple procedure, based on the bifurcations in the thermodynamic limit, that determines the minimum number of links to be removed in order to ensure complete synchronization. Finally, we propose an application of the reported phenomenon as a control scheme to drive complete synchronization in high connectivity networks.
Collapse
Affiliation(s)
- Antonio Mihara
- Departamento de Física, Universidade Federal de São Paulo, UNIFESP, Campus Diadema, 09913-030 São Paulo, Brasil
| | - Everton S Medeiros
- Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky University Oldenburg, 26111 Oldenburg, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Rene O Medrano-T
- Departamento de Física, Universidade Federal de São Paulo, UNIFESP, Campus Diadema, 09913-030 São Paulo, Brasil
| |
Collapse
|
5
|
Ding Y, Ermentrout B. Traveling waves in non-local pulse-coupled networks. J Math Biol 2021; 82:18. [PMID: 33570663 DOI: 10.1007/s00285-021-01572-8] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2020] [Revised: 10/29/2020] [Accepted: 01/19/2021] [Indexed: 11/25/2022]
Abstract
Traveling phase waves are commonly observed in recordings of the cerebral cortex and are believed to organize behavior across different areas of the brain. We use this as motivation to analyze a one-dimensional network of phase oscillators that are nonlocally coupled via the phase response curve (PRC) and the Dirac delta function. Existence of waves is proven and the dispersion relation is computed. Using the theory of distributions enables us to write and solve an associated stability problem. First and second order perturbation theory is applied to get analytic insight and we show that long waves are stable while short waves are unstable. We apply the results to PRCs that come from mitral neurons. We extend the results to smooth pulse-like coupling by reducing the nonlocal equation to a local one and solving the associated boundary value problem.
Collapse
Affiliation(s)
- Yujie Ding
- University of Pittsburgh, Pennsylvania, USA
| | | |
Collapse
|
6
|
Chandrasekar VK, Manoranjani M, Gupta S. Kuramoto model in the presence of additional interactions that break rotational symmetry. Phys Rev E 2020; 102:012206. [PMID: 32794959 DOI: 10.1103/physreve.102.012206] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2020] [Accepted: 06/23/2020] [Indexed: 11/07/2022]
Abstract
The Kuramoto model serves as a paradigm to study the phenomenon of spontaneous collective synchronization. We study here a nontrivial generalization of the Kuramoto model by including an interaction that breaks explicitly the rotational symmetry of the model. In an inertial frame (e.g., the laboratory frame), the Kuramoto model does not allow for a stationary state, that is, a state with time-independent value of the so-called Kuramoto (complex) synchronization order parameter z≡re^{iψ}. Note that a time-independent z implies r and ψ are both time independent, with the latter fact corresponding to a state in which ψ rotates at zero frequency (no rotation). In this backdrop, we ask: Does the introduction of the symmetry-breaking term suffice to allow for the existence of a stationary state in the laboratory frame? Compared to the original model, we reveal a rather rich phase diagram of the resulting model, with the existence of both stationary and standing wave phases. While in the former the synchronization order parameter r has a long-time value that is time independent, one has in the latter an oscillatory behavior of the order parameter as a function of time that nevertheless yields a nonzero and time-independent time average. Our results are based on numerical integration of the dynamical equations as well as an exact analysis of the dynamics by invoking the so-called Ott-Antonsen ansatz that allows to derive a reduced set of time-evolution equations for the order parameter.
Collapse
Affiliation(s)
- V K Chandrasekar
- Centre for Nonlinear Science and Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| | - M Manoranjani
- Centre for Nonlinear Science and Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| | - Shamik Gupta
- Department of Physics, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah 711202, India.,Regular Associate, Quantitative Life Sciences Section, ICTP - The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
| |
Collapse
|
7
|
Wüster S, Bhavna R. Spatial correlations in a finite-range Kuramoto model. Phys Rev E 2020; 101:052210. [PMID: 32575303 DOI: 10.1103/physreve.101.052210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2019] [Accepted: 04/21/2020] [Indexed: 11/07/2022]
Abstract
We study spatial correlations between oscillator phases in the steady state of a Kuramoto model, in which phase oscillators that are randomly distributed in space interact with constant strength but within a limited range. Such a model could be relevant, for example, in the synchronization of gene expression oscillations in cells, where only oscillations of neighboring cells are coupled through cell-cell contacts. We analytically infer spatial phase-phase correlation functions from the known steady-state distribution of oscillators for the case of homogenous frequencies and show that these can contain information about the range and strength of interactions, provided the noise in the system can be estimated. We suggest a method for the latter, and also explore when correlations appear to be ergodic in this model, which would enable an experimental measurement of correlation functions to utilize temporal averages. Simulations show that our techniques also give qualitative results for the model with heterogenous frequencies. We illustrate our results by comparison with experimental data on genetic oscillations in the segmentation clock of zebrafish embryos.
Collapse
Affiliation(s)
- Sebastian Wüster
- Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462 023, India
| | - Rajasekaran Bhavna
- Department of Biological Sciences, Tata Institute of Fundamental Research, 400005 Mumbai, India
| |
Collapse
|
8
|
Bick C, Goodfellow M, Laing CR, Martens EA. Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2020; 10:9. [PMID: 32462281 PMCID: PMC7253574 DOI: 10.1186/s13408-020-00086-9] [Citation(s) in RCA: 91] [Impact Index Per Article: 22.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2019] [Accepted: 05/07/2020] [Indexed: 05/03/2023]
Abstract
Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott-Antonsen and Watanabe-Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.
Collapse
Affiliation(s)
- Christian Bick
- Centre for Systems, Dynamics, and Control, University of Exeter, Exeter, UK.
- Department of Mathematics, University of Exeter, Exeter, UK.
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK.
- Mathematical Institute, University of Oxford, Oxford, UK.
- Institute for Advanced Study, Technische Universität München, Garching, Germany.
| | - Marc Goodfellow
- Department of Mathematics, University of Exeter, Exeter, UK
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK
- Living Systems Institute, University of Exeter, Exeter, UK
- Wellcome Trust Centre for Biomedical Modelling and Analysis, University of Exeter, Exeter, UK
| | - Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Auckland, New Zealand
| | - Erik A Martens
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, Denmark.
- Department of Biomedical Science, University of Copenhagen, Copenhagen N, Denmark.
- Centre for Translational Neuroscience, University of Copenhagen, Copenhagen N, Denmark.
| |
Collapse
|
9
|
Xie Y, Zhang L, Guo S, Dai Q, Yang J. Twisted states in nonlocally coupled phase oscillators with frequency distribution consisting of two Lorentzian distributions with the same mean frequency and different widths. PLoS One 2019; 14:e0213471. [PMID: 30861016 PMCID: PMC6413906 DOI: 10.1371/journal.pone.0213471] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2018] [Accepted: 02/21/2019] [Indexed: 11/19/2022] Open
Abstract
In globally coupled phase oscillators, the distribution of natural frequency has strong effects on both synchronization transition and synchronous dynamics. In this work, we study a ring of nonlocally coupled phase oscillators with the frequency distribution made up of two Lorentzians with the same center frequency but with different half widths. Using the Ott-Antonsen ansatz, we derive a reduced model in the continuum limit. Based on the reduced model, we analyze the stability of the incoherent state and find the existence of multiple stability islands for the incoherent state depending on the parameters. Furthermore, we numerically simulate the reduced model and find a large number of twisted states resulting from the instabilities of the incoherent state with respect to different spatial modes. For some winding numbers, the stability region of the corresponding twisted state consists of two disjoint parameter regions, one for the intermediate coupling strength and the other for the strong coupling strength.
Collapse
Affiliation(s)
- Yuan Xie
- School of Science, Beijing University of Posts and Telecommunications, Beijing, China
| | - Lan Zhang
- School of Science, Beijing University of Posts and Telecommunications, Beijing, China
| | - Shuangjian Guo
- School of Science, Beijing University of Posts and Telecommunications, Beijing, China
| | - Qionglin Dai
- School of Science, Beijing University of Posts and Telecommunications, Beijing, China
- * E-mail: (QD); (JY)
| | - Junzhong Yang
- School of Science, Beijing University of Posts and Telecommunications, Beijing, China
- * E-mail: (QD); (JY)
| |
Collapse
|
10
|
Rule ME, Vargas-Irwin C, Donoghue JP, Truccolo W. Phase reorganization leads to transient β-LFP spatial wave patterns in motor cortex during steady-state movement preparation. J Neurophysiol 2018; 119:2212-2228. [PMID: 29442553 DOI: 10.1152/jn.00525.2017] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023] Open
Abstract
Previous studies on the origin and properties of spatial patterns in motor cortex β-local field potential (β-LFP) oscillations have focused on planar traveling waves. However, it is unclear 1) whether β-LFP waves are limited to plane waves, or even 2) whether they are propagating waves of excito-excitatory activity, i.e., primarily traveling waves in excitable media; they could reflect, instead, reorganization in the relative phases of transient oscillations at different spatial sites. We addressed these two problems in β-LFPs recorded via microelectrode arrays implanted in three adjacent motor cortex areas of nonhuman primates during steady-state movement preparation. Our findings are fourfold: 1) β-LFP wave patterns emerged as transient events, despite stable firing rates of single neurons concurrently recorded during the same periods. 2) β-LFP waves showed a richer variety of spatial dynamics, including rotating and complex waves. 3) β-LFP wave patterns showed no characteristic wavelength, presenting instead a range of scales with global zero-lag phase synchrony as a limiting case, features surprising for purely excito-excitatory waves but consistent with waves in coupled oscillator systems. 4) Furthermore, excito-excitatory traveling waves induced by optogenetic stimulation in motor cortex showed, in contrast, a characteristic wavelength and reduced phase synchrony. Overall, β-LFP wave statistics differed from those of induced traveling waves in excitable media recorded under the same microelectrode array setup. Our findings suggest phase reorganization in neural coupled oscillators contribute significantly to the origin of transient β-LFP spatial dynamics during preparatory steady states and outline important constraints for spatially extended models of β-LFP dynamics in motor cortex. NEW & NOTEWORTHY We show that a rich variety of transient β-local field potential (β-LFP) wave patterns emerge in motor cortex during preparatory steady states, despite stable neuronal firing rates. Furthermore, unlike optogenetically induced traveling waves, β-LFP waves showed no characteristic wavelength, presenting instead a range of scales with global phase synchrony as a limiting case. Overall, our statistical analyses suggest that transient phase reorganization in neural coupled oscillators, beyond purely excito-excitatory traveling waves, contribute significantly to the origin of motor cortex β-LFP wave patterns.
Collapse
Affiliation(s)
- Michael E Rule
- Department of Neuroscience, Brown University , Providence, Rhode Island
| | | | - John P Donoghue
- Department of Neuroscience, Brown University , Providence, Rhode Island.,Institute for Brain Science, Brown University , Providence, Rhode Island.,Center for Neurorestoration and Neurotechnology, U.S. Department of Veterans Affairs , Providence, Rhode Island
| | - Wilson Truccolo
- Department of Neuroscience, Brown University , Providence, Rhode Island.,Institute for Brain Science, Brown University , Providence, Rhode Island.,Center for Neurorestoration and Neurotechnology, U.S. Department of Veterans Affairs , Providence, Rhode Island
| |
Collapse
|
11
|
Devalle F, Roxin A, Montbrió E. Firing rate equations require a spike synchrony mechanism to correctly describe fast oscillations in inhibitory networks. PLoS Comput Biol 2017; 13:e1005881. [PMID: 29287081 PMCID: PMC5764488 DOI: 10.1371/journal.pcbi.1005881] [Citation(s) in RCA: 43] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2017] [Revised: 01/11/2018] [Accepted: 11/15/2017] [Indexed: 12/25/2022] Open
Abstract
Recurrently coupled networks of inhibitory neurons robustly generate oscillations in the gamma band. Nonetheless, the corresponding Wilson-Cowan type firing rate equation for such an inhibitory population does not generate such oscillations without an explicit time delay. We show that this discrepancy is due to a voltage-dependent spike-synchronization mechanism inherent in networks of spiking neurons which is not captured by standard firing rate equations. Here we investigate an exact low-dimensional description for a network of heterogeneous canonical Class 1 inhibitory neurons which includes the sub-threshold dynamics crucial for generating synchronous states. In the limit of slow synaptic kinetics the spike-synchrony mechanism is suppressed and the standard Wilson-Cowan equations are formally recovered as long as external inputs are also slow. However, even in this limit synchronous spiking can be elicited by inputs which fluctuate on a time-scale of the membrane time-constant of the neurons. Our meanfield equations therefore represent an extension of the standard Wilson-Cowan equations in which spike synchrony is also correctly described. Population models describing the average activity of large neuronal ensembles are a powerful mathematical tool to investigate the principles underlying cooperative function of large neuronal systems. However, these models do not properly describe the phenomenon of spike synchrony in networks of neurons. In particular, they fail to capture the onset of synchronous oscillations in networks of inhibitory neurons. We show that this limitation is due to a voltage-dependent synchronization mechanism which is naturally present in spiking neuron models but not captured by traditional firing rate equations. Here we investigate a novel set of macroscopic equations which incorporate both firing rate and membrane potential dynamics, and that correctly generate fast inhibition-based synchronous oscillations. In the limit of slow-synaptic processing oscillations are suppressed, and the model reduces to an equation formally equivalent to the Wilson-Cowan model.
Collapse
Affiliation(s)
- Federico Devalle
- Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
- Department of Physics, Lancaster University, Lancaster, United Kingdom
| | - Alex Roxin
- Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, Bellaterra, Barcelona, Spain
| | - Ernest Montbrió
- Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
- * E-mail:
| |
Collapse
|
12
|
Esnaola-Acebes JM, Roxin A, Avitabile D, Montbrió E. Synchrony-induced modes of oscillation of a neural field model. Phys Rev E 2017; 96:052407. [PMID: 29347806 DOI: 10.1103/physreve.96.052407] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2017] [Indexed: 05/01/2023]
Abstract
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.
Collapse
Affiliation(s)
- Jose M Esnaola-Acebes
- Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain
| | - Alex Roxin
- Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain
| | - Daniele Avitabile
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham NG2 7RD, United Kingdom
| | - Ernest Montbrió
- Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain
| |
Collapse
|
13
|
English LQ, Zampetaki A, Kevrekidis PG, Skowronski K, Fritz CB, Abdoulkary S. Analysis and observation of moving domain fronts in a ring of coupled electronic self-oscillators. CHAOS (WOODBURY, N.Y.) 2017; 27:103125. [PMID: 29092454 DOI: 10.1063/1.5009088] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
In this work, we consider a ring of coupled electronic (Wien-bridge) oscillators from a perspective combining modeling, simulation, and experimental observation. Following up on earlier work characterizing the pairwise interaction of Wien-bridge oscillators by Kuramoto-Sakaguchi phase dynamics, we develop a lattice model for a chain thereof, featuring an exponentially decaying spatial kernel. We find that for certain values of the Sakaguchi parameter α, states of traveling phase-domain fronts involving the coexistence of two clearly separated regions of distinct dynamical behavior, can establish themselves in the ring lattice. Experiments and simulations show that stationary coexistence domains of synchronization only manifest themselves with the introduction of a local impurity; here an incoherent cluster of oscillators can arise reminiscent of the chimera states in a range of systems with homogeneous oscillators and suitable nonlocal interactions between them.
Collapse
Affiliation(s)
- L Q English
- Department of Physics and Astronomy, Dickinson College, Carlisle, Pennsylvania 17013, USA
| | - A Zampetaki
- Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003, USA
| | - K Skowronski
- Department of Physics and Astronomy, Dickinson College, Carlisle, Pennsylvania 17013, USA
| | - C B Fritz
- Department of Physics and Astronomy, Dickinson College, Carlisle, Pennsylvania 17013, USA
| | - Saidou Abdoulkary
- Département des Sciences Fondamentales, IMIP University of Maroua, P.O. Box 46, Maroua, Cameroon
| |
Collapse
|
14
|
Bertalan T, Wu Y, Laing C, Gear CW, Kevrekidis IG. Coarse-Grained Descriptions of Dynamics for Networks with Both Intrinsic and Structural Heterogeneities. Front Comput Neurosci 2017; 11:43. [PMID: 28659781 PMCID: PMC5467358 DOI: 10.3389/fncom.2017.00043] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2017] [Accepted: 05/12/2017] [Indexed: 11/23/2022] Open
Abstract
Finding accurate reduced descriptions for large, complex, dynamically evolving networks is a crucial enabler to their simulation, analysis, and ultimately design. Here, we propose and illustrate a systematic and powerful approach to obtaining good collective coarse-grained observables—variables successfully summarizing the detailed state of such networks. Finding such variables can naturally lead to successful reduced dynamic models for the networks. The main premise enabling our approach is the assumption that the behavior of a node in the network depends (after a short initial transient) on the node identity: a set of descriptors that quantify the node properties, whether intrinsic (e.g., parameters in the node evolution equations) or structural (imparted to the node by its connectivity in the particular network structure). The approach creates a natural link with modeling and “computational enabling technology” developed in the context of Uncertainty Quantification. In our case, however, we will not focus on ensembles of different realizations of a problem, each with parameters randomly selected from a distribution. We will instead study many coupled heterogeneous units, each characterized by randomly assigned (heterogeneous) parameter value(s). One could then coin the term Heterogeneity Quantification for this approach, which we illustrate through a model dynamic network consisting of coupled oscillators with one intrinsic heterogeneity (oscillator individual frequency) and one structural heterogeneity (oscillator degree in the undirected network). The computational implementation of the approach, its shortcomings and possible extensions are also discussed.
Collapse
Affiliation(s)
- Tom Bertalan
- Department of Chemical and Biological Engineering, Princeton UniversityPrinceton, NJ, United States
| | - Yan Wu
- Department of Chemical and Biological Engineering, Princeton UniversityPrinceton, NJ, United States.,Department of Bioengineering, University of California, La JollaLa Jolla, CA, United States
| | - Carlo Laing
- Institute of Natural and Mathematical Sciences, Massey UniversityAuckland, New Zealand
| | - C William Gear
- Department of Chemical and Biological Engineering, Princeton UniversityPrinceton, NJ, United States
| | - Ioannis G Kevrekidis
- Department of Chemical and Biological Engineering, Princeton UniversityPrinceton, NJ, United States.,Program in Applied and Computational Mathematics, Princeton UniversityPrinceton, NJ, United States
| |
Collapse
|
15
|
Abrams DM, Pecora LM, Motter AE. Introduction to focus issue: Patterns of network synchronization. CHAOS (WOODBURY, N.Y.) 2016; 26:094601. [PMID: 27781481 DOI: 10.1063/1.4962970] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
The study of synchronization of coupled systems is currently undergoing a major surge fueled by recent discoveries of new forms of collective dynamics and the development of techniques to characterize a myriad of new patterns of network synchronization. This includes chimera states, phenomena determined by symmetry, remote synchronization, and asymmetry-induced synchronization. This Focus Issue presents a selection of contributions at the forefront of these developments, to which this introduction is intended to offer an up-to-date foundation.
Collapse
Affiliation(s)
- Daniel M Abrams
- Department of Engineering Sciences and Applied Mathematics and Northwestern Institute on Complex Systems (NICO), Northwestern University, Evanston, Illinois 60208, USA
| | - Louis M Pecora
- U.S. Naval Research Laboratory, Washington, DC 20375, USA
| | - Adilson E Motter
- Department of Physics and Astronomy and Northwestern Institute on Complex Systems (NICO), Northwestern University, Evanston, Illinois 60208, USA
| |
Collapse
|