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Nicola W. Rapid changes in synchronizability in conductance-based neuronal networks with conductance-based coupling. CHAOS (WOODBURY, N.Y.) 2024; 34:023123. [PMID: 38377288 DOI: 10.1063/5.0176956] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/18/2023] [Accepted: 01/14/2024] [Indexed: 02/22/2024]
Abstract
Real neurons connect to each other non-randomly. These connectivity graphs can potentially impact the ability of networks to synchronize, along with the dynamics of neurons and the dynamics of their connections. How the connectivity of networks of conductance-based neuron models like the classical Hodgkin-Huxley model or the Morris-Lecar model impacts synchronizability remains unknown. One powerful tool to resolve the synchronizability of these networks is the master stability function (MSF). Here, we apply and extend the MSF approach to networks of Morris-Lecar neurons with conductance-based coupling to determine under which parameters and for which graphs the synchronous solutions are stable. We consider connectivity graphs with a constant non-zero row sum, where the MSF approach can be readily extended to conductance-based synapses rather than the more well-studied diffusive connectivity case, which primarily applies to gap junction connectivity. In this formulation, the synchronous solution is a single, self-coupled, or "autaptic" neuron. We find that the primary determining parameter for the stability of the synchronous solution is, unsurprisingly, the reversal potential, as it largely dictates the excitatory/inhibitory potential of a synaptic connection. However, the change between "excitatory" and "inhibitory" synapses is rapid, with only a few millivolts separating stability and instability of the synchronous state for most graphs. We also find that for specific coupling strengths (as measured by the global synaptic conductance), islands of synchronizability in the MSF can emerge for inhibitory connectivity. We verified the stability of these islands by direct simulation of pairs of neurons coupled with eigenvalues in the matching spectrum.
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Fatoyinbo HO, Brown RG, Simpson DJW, van Brunt B. Pattern Formation in a Spatially Extended Model of Pacemaker Dynamics in Smooth Muscle Cells. Bull Math Biol 2022; 84:86. [PMID: 35804271 PMCID: PMC9270316 DOI: 10.1007/s11538-022-01043-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2021] [Accepted: 06/16/2022] [Indexed: 11/30/2022]
Abstract
Spatiotemporal patterns are common in biological systems. For electrically coupled cells, previous studies of pattern formation have mainly used applied current as the primary bifurcation parameter. The purpose of this paper is to show that applied current is not needed to generate spatiotemporal patterns for smooth muscle cells. The patterns can be generated solely by external mechanical stimulation (transmural pressure). To do this we study a reaction-diffusion system involving the Morris-Lecar equations and observe a wide range of spatiotemporal patterns for different values of the model parameters. Some aspects of these patterns are explained via a bifurcation analysis of the system without coupling - in particular Type I and Type II excitability both occur. We show the patterns are not due to a Turing instability and that the spatially extended model exhibits spatiotemporal chaos. We also use travelling wave coordinates to analyse travelling waves.
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Affiliation(s)
- H. O. Fatoyinbo
- School of Mathematical and Computational Sciences, Massey University, Palmerston North, New Zealand
| | - R. G. Brown
- School of Mathematical and Computational Sciences, Massey University, Palmerston North, New Zealand
| | - D. J. W. Simpson
- School of Mathematical and Computational Sciences, Massey University, Palmerston North, New Zealand
| | - B. van Brunt
- School of Mathematical and Computational Sciences, Massey University, Palmerston North, New Zealand
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Ge P, Cao H. Intermittent evolution routes to the periodic or the chaotic orbits in Rulkov map. CHAOS (WOODBURY, N.Y.) 2021; 31:093119. [PMID: 34598445 DOI: 10.1063/5.0060205] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2021] [Accepted: 09/07/2021] [Indexed: 06/13/2023]
Abstract
This paper concerns the intermittent evolution routes to the asymptotic regimes in the Rulkov map. That is, the windows with transient approximate periodic and transient chaotic behaviors occur alternatively before the system reaches the periodic or the chaotic orbits. Meanwhile, the evolution routes to chaotic orbits can be classified into different types according to the windows before reaching asymptotic chaotic states. In addition, the initial values can be regarded as a key factor affecting the asymptotic behaviors and the evolution routes. The effects of the initial values are given by parameter planes, bifurcation diagrams, and waveforms. In order to investigate whether the intermittent evolution routes can be learned by machine learning, some experiments are given to understanding the differences between the trajectories of the Rulkov map generated by the numerical simulations and predicted by the neural networks. These results show that there is about 60% accuracy rate of successfully predicting both the evolution routes and the asymptotic period-3 orbits using a three-layer feedforward neural network, while the bifurcation diagrams can be reconstructed using reservoir computing except a few parameter conditions.
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Affiliation(s)
- Penghe Ge
- Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People's Republic of China
| | - Hongjun Cao
- Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People's Republic of China
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Kaminker V, Wackerbauer R. Alternating activity patterns and a chimeralike state in a network of globally coupled excitable Morris-Lecar neurons. CHAOS (WOODBURY, N.Y.) 2019; 29:053121. [PMID: 31154794 DOI: 10.1063/1.5093483] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/21/2019] [Accepted: 04/30/2019] [Indexed: 06/09/2023]
Abstract
Spatiotemporal chaos collapses to either a rest state or a propagating pulse in a ring network of diffusively coupled, excitable Morris-Lecar neurons. Adding global varying synaptic coupling to the ring network reveals complex transient behavior. Spatiotemporal chaos collapses into a transient pulse that reinitiates spatiotemporal chaos to allow sequential pattern switching until a collapse to the rest state. A domain of irregular neuron activity coexists with a domain of inactive neurons forming a transient chimeralike state. Transient spatial localization of the chimeralike state is observed for stronger synapses.
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Affiliation(s)
- Vitaliy Kaminker
- Department of Physics, University of Alaska, Fairbanks, Alaska 99775-5920, USA
| | - Renate Wackerbauer
- Department of Physics, University of Alaska, Fairbanks, Alaska 99775-5920, USA
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Lilienkamp T, Parlitz U. Terminal Transient Phase of Chaotic Transients. PHYSICAL REVIEW LETTERS 2018; 120:094101. [PMID: 29547310 DOI: 10.1103/physrevlett.120.094101] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2017] [Revised: 12/16/2017] [Indexed: 06/08/2023]
Abstract
Transient chaos in spatially extended systems can be characterized by the length of the transient phase, which typically grows quickly with the system size (supertransients). For a large class of these systems, the chaotic phase terminates abruptly, without any obvious precursors in commonly used observables. Here we investigate transient spatiotemporal chaos in two different models of this class. By probing the state space using perturbed trajectories we show the existence of a "terminal transient phase," which occurs prior to the abrupt collapse of chaotic dynamics. During this phase the impact of perturbations is significantly different from the earlier transient and particular patterns of (non)susceptible regions in state space occur close to the chaotic trajectories. We therefore hypothesize that even without perturbations proper precursors for the collapse of chaotic transients exist, which might be highly relevant for coping with spatiotemporal chaos in cardiac arrhythmias or brain functionality, for example.
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Affiliation(s)
- Thomas Lilienkamp
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
- Institute for Nonlinear Dynamics, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
| | - Ulrich Parlitz
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
- Institute for Nonlinear Dynamics, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
- German Center for Cardiovascular Research (DZHK), Partner Site Göttingen, Robert-Koch-Straße 42a, 37075 Göttingen, Germany
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Hartle H, Wackerbauer R. Transient chaos and associated system-intrinsic switching of spacetime patterns in two synaptically coupled layers of Morris-Lecar neurons. Phys Rev E 2018; 96:032223. [PMID: 29347029 DOI: 10.1103/physreve.96.032223] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2017] [Indexed: 11/07/2022]
Abstract
Spatiotemporal chaos collapses to either a rest state or a propagating pulse solution in a single layer of diffusively coupled, excitable Morris-Lecar neurons. Weak synaptic coupling of two such layers reveals system intrinsic switching of spatiotemporal activity patterns within and between the layers at irregular times. Within a layer, switching sequences include spatiotemporal chaos, erratic and regular pulse propagation, spontaneous network wide neuron activity, and rest state. A momentary substantial reduction in neuron activity in one layer can reinitiate transient spatiotemporal chaos in the other layer, which can induce a swap of spatiotemporal chaos with a pulse state between the layers. Presynaptic input maximizes the distance between propagating pulses, in contrast to pulse merging in the absence of synapses.
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Affiliation(s)
- Harrison Hartle
- Department of Physics, University of Alaska, Fairbanks, Alaska 99775-5920, USA
| | - Renate Wackerbauer
- Department of Physics, University of Alaska, Fairbanks, Alaska 99775-5920, USA
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Zou HL, Katori Y, Deng ZC, Aihara K, Lai YC. Controlled generation of switching dynamics among metastable states in pulse-coupled oscillator networks. CHAOS (WOODBURY, N.Y.) 2015; 25:103109. [PMID: 26520075 DOI: 10.1063/1.4930840] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Switching dynamics among saddles in a network of nonlinear oscillators can be exploited for information encoding and processing (hence computing), but stable attractors in the system can terminate the switching behavior. An effective control strategy is presented to sustain switching dynamics in networks of pulse-coupled oscillators. The support for the switching behavior is a set of saddles, or unstable invariant sets in the phase space. We thus identify saddles with a common property, localize the system in the vicinity of them, and then guide the system from one metastable state to another to generate desired switching dynamics. We demonstrate that the control method successfully generates persistent switching trajectories and prevents the system from entering stable attractors. In addition, there exists correspondence between the network structure and the switching dynamics, providing fundamental insights on the development of a computing paradigm based on the switching dynamics.
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Affiliation(s)
- Hai-Lin Zou
- School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an 710072, China
| | - Yuichi Katori
- Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
| | - Zi-Chen Deng
- School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an 710072, China
| | - Kazuyuki Aihara
- Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
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Wu K, Luo T, Lu H, Wang Y. Bifurcation study of neuron firing activity of the modified Hindmarsh–Rose model. Neural Comput Appl 2015. [DOI: 10.1007/s00521-015-1892-1] [Citation(s) in RCA: 33] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Lafranceschina J, Wackerbauer R. Impact of weak excitatory synapses on chaotic transients in a diffusively coupled Morris-Lecar neuronal network. CHAOS (WOODBURY, N.Y.) 2015; 25:013119. [PMID: 25637930 DOI: 10.1063/1.4907193] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Spatiotemporal chaos collapses to either a rest state or a propagating pulse solution in a ring network of diffusively coupled, excitable Morris-Lecar neurons. Weak excitatory synapses can increase the Lyapunov exponent, expedite the collapse, and promote the collapse to the rest state rather than the pulse state. A single traveling pulse solution may no longer be asymptotic for certain combinations of network topology and (weak) coupling strengths, and initiate spatiotemporal chaos. Multiple pulses can cause chaos initiation due to diffusive and synaptic pulse-pulse interaction. In the presence of chaos initiation, intermittent spatiotemporal chaos exists until typically a collapse to the rest state.
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Affiliation(s)
| | - Renate Wackerbauer
- Department of Physics, University of Alaska, Fairbanks, Alaska 99775-5920, USA
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