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Barać U, Perc M, Gosak M. Determinants of collective failure in excitable networks. CHAOS (WOODBURY, N.Y.) 2023; 33:043120. [PMID: 37097938 DOI: 10.1063/5.0149578] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2023] [Accepted: 03/27/2023] [Indexed: 06/19/2023]
Abstract
We study collective failures in biologically realistic networks that consist of coupled excitable units. The networks have broad-scale degree distribution, high modularity, and small-world properties, while the excitable dynamics is determined by the paradigmatic FitzHugh-Nagumo model. We consider different coupling strengths, bifurcation distances, and various aging scenarios as potential culprits of collective failure. We find that for intermediate coupling strengths, the network remains globally active the longest if the high-degree nodes are first targets for inactivation. This agrees well with previously published results, which showed that oscillatory networks can be highly fragile to the targeted inactivation of low-degree nodes, especially under weak coupling. However, we also show that the most efficient strategy to enact collective failure does not only non-monotonically depend on the coupling strength, but it also depends on the distance from the bifurcation point to the oscillatory behavior of individual excitable units. Altogether, we provide a comprehensive account of determinants of collective failure in excitable networks, and we hope this will prove useful for better understanding breakdowns in systems that are subject to such dynamics.
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Affiliation(s)
- Uroš Barać
- Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
| | - Matjaž Perc
- Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
- Alma Mater Europaea, Slovenska ulica 17, 2000 Maribor, Slovenia
- Department of Medical Research, China Medical University Hospital, China Medical University, 40447 Taichung, Taiwan
- Complexity Science Hub Vienna, Josefstädterstrasse 39, 1080 Vienna, Austria
- Department of Physics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, 02447 Seoul, Republic of Korea
| | - Marko Gosak
- Department of Physics, Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
- Alma Mater Europaea, Slovenska ulica 17, 2000 Maribor, Slovenia
- Institute of Physiology, Faculty of Medicine, University of Maribor, Taborska ulica 8, 2000 Maribor, Slovenia
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Ronge R, Zaks MA. Emergence and stability of periodic two-cluster states for ensembles of excitable units. Phys Rev E 2021; 103:012206. [PMID: 33601632 DOI: 10.1103/physreve.103.012206] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2020] [Accepted: 12/06/2020] [Indexed: 06/12/2023]
Abstract
We study dynamics in ensembles of identical excitable units with global repulsive interaction. Starting from active rotators with additional higher order Fourier modes in on-site dynamics, we observe, at sufficiently strong repulsive coupling, large-scale collective oscillations in which the elements form two separate clusters. Transitions from quiescence to clustered oscillations are caused by global bifurcations involving the unstable clustered steady states. For clusters of equal size, the scenarios evolve either through simultaneous formation of two heteroclinic trajectories or through two simultaneous saddle-node bifurcations on invariant circles. If the sizes of clusters differ, two global bifurcations are separated in the parameter space. Stability of clusters with respect to splitting perturbations depends on the kind of higher order corrections to on-site dynamics; we show that for periodic oscillations of two equal clusters the Watanabe-Strogatz integrability marks a change of stability. By extending our studies to ensembles of voltage-coupled Morris-Lecar neurons, we demonstrate that similar bifurcations and switches in stability occur also for more elaborate models in higher dimensions.
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Affiliation(s)
- Robert Ronge
- Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
| | - Michael A Zaks
- Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
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3
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Orr D, Ermentrout B. Synchronization of oscillators via active media. Phys Rev E 2019; 99:052218. [PMID: 31212450 DOI: 10.1103/physreve.99.052218] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2019] [Indexed: 11/07/2022]
Abstract
In this paper, we study pairs of oscillators that are indirectly coupled via active (excitable) cells. We introduce a scalar phase model for coupled oscillators and excitable cells. We first show that one excitable and one oscillatory cell will exhibit phase locking at a variety of m:n patterns. We next introduce a second oscillatory cell and show that the only attractor is synchrony between the oscillators. We will also study the robustness to heterogeneity when the excitable cell fires or is quiescent. We next examine the dynamics when the oscillators are coupled via two excitable cells. In this case, the dynamics are very complicated with many forms of bistability and, in some cases, chaotic behavior. We also apply weak-coupling analysis to this case and explain some of the degeneracies observed in the bifurcation diagram. Further, we look at pairs of oscillators coupled via long chains of excitable cells and show that small differences in the frequency of the oscillators makes their locking more robust. Finally, we demonstrate many of the same phenomena seen in the phase model for a gap-junction coupled system of Morris-Lecar neurons.
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Affiliation(s)
- Derek Orr
- Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15213, USA
| | - Bard Ermentrout
- Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15213, USA
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Yuan T, Tanaka G. Robustness of coupled oscillator networks with heterogeneous natural frequencies. CHAOS (WOODBURY, N.Y.) 2017; 27:123105. [PMID: 29289041 DOI: 10.1063/1.4991742] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2017] [Accepted: 11/23/2017] [Indexed: 06/07/2023]
Abstract
Robustness of coupled oscillator networks against local degradation of oscillators has been intensively studied in this decade. The oscillation behavior on the whole network is typically reduced with an increase in the fraction of degraded (inactive) oscillators. The critical fraction of inactive oscillators, at which a transition from an oscillatory to a quiescent state occurs, has been used as a measure for the network robustness. The larger (smaller) this measure is, the more robust (fragile) the oscillatory behavior on the network is. Most previous studies have used oscillators with identical natural frequencies, for which the oscillators are necessarily synchronized and thereby the analysis is simple. In contrast, we focus on the effect of heterogeneity in the natural frequencies on the network robustness. First, we analytically derive the robustness measure for the coupled oscillator models with heterogeneous natural frequencies under some conditions. Then, we show that increasing the heterogeneity in natural frequencies makes the network fragile. Moreover, we discuss the optimal parameter condition to maximize the network robustness.
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Affiliation(s)
- Tianyu Yuan
- Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
| | - Gouhei Tanaka
- Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
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Rahman B, Blyuss KB, Kyrychko YN. Aging transition in systems of oscillators with global distributed-delay coupling. Phys Rev E 2017; 96:032203. [PMID: 29347035 DOI: 10.1103/physreve.96.032203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2016] [Indexed: 06/07/2023]
Abstract
We consider a globally coupled network of active (oscillatory) and inactive (nonoscillatory) oscillators with distributed-delay coupling. Conditions for aging transition, associated with suppression of oscillations, are derived for uniform and gamma delay distributions in terms of coupling parameters and the proportion of inactive oscillators. The results suggest that for the uniform distribution increasing the width of distribution for the same mean delay allows aging transition to happen for a smaller coupling strength and a smaller proportion of inactive elements. For gamma distribution with sufficiently large mean time delay, it may be possible to achieve aging transition for an arbitrary proportion of inactive oscillators, as long as the coupling strength lies in a certain range.
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Affiliation(s)
- B Rahman
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, England, United Kingdom
| | - K B Blyuss
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, England, United Kingdom
| | - Y N Kyrychko
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, England, United Kingdom
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Daido H. Boundary in the dynamic phase of globally coupled oscillatory and excitable units. Phys Rev E 2017; 96:012210. [PMID: 29347184 DOI: 10.1103/physreve.96.012210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2017] [Indexed: 06/07/2023]
Abstract
There is a crucial boundary between dynamic phase 1 and dynamic phase 2 of globally coupled oscillatory and excitable units, where the mean field is constant and oscillates in the former and the latter, respectively. This boundary is theoretically derived here for a large population of dynamical units, each having only a phase variable, where it is assumed that both the coupling strength and the distribution width of bifurcation parameters are equally small. This theory, which is applicable only if all or most of the units are intrinsically oscillatory, is confirmed to agree with simulation results for two different distribution densities.
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Affiliation(s)
- Hiroaki Daido
- Department of Mathematical Sciences, Graduate School of Engineering, Osaka Prefecture University, Sakai 599-8531, Japan
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Ratas I, Pyragas K. Macroscopic self-oscillations and aging transition in a network of synaptically coupled quadratic integrate-and-fire neurons. Phys Rev E 2016; 94:032215. [PMID: 27739712 DOI: 10.1103/physreve.94.032215] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2016] [Indexed: 06/06/2023]
Abstract
We analyze the dynamics of a large network of coupled quadratic integrate-and-fire neurons, which represent the canonical model for class I neurons near the spiking threshold. The network is heterogeneous in that it includes both inherently spiking and excitable neurons. The coupling is global via synapses that take into account the finite width of synaptic pulses. Using a recently developed reduction method based on the Lorentzian ansatz, we derive a closed system of equations for the neuron's firing rate and the mean membrane potential, which are exact in the infinite-size limit. The bifurcation analysis of the reduced equations reveals a rich scenario of asymptotic behavior, the most interesting of which is the macroscopic limit-cycle oscillations. It is shown that the finite width of synaptic pulses is a necessary condition for the existence of such oscillations. The robustness of the oscillations against aging damage, which transforms spiking neurons into nonspiking neurons, is analyzed. The validity of the reduced equations is confirmed by comparing their solutions with the solutions of microscopic equations for the finite-size networks.
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Affiliation(s)
- Irmantas Ratas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
| | - Kestutis Pyragas
- Center for Physical Sciences and Technology, LT-10257 Vilnius, Lithuania
- Department of Theoretical Physics, Faculty of Physics, Vilnius University, LT-10222 Vilnius, Lithuania
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O'Keeffe KP, Strogatz SH. Dynamics of a population of oscillatory and excitable elements. Phys Rev E 2016; 93:062203. [PMID: 27415251 DOI: 10.1103/physreve.93.062203] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2016] [Indexed: 11/07/2022]
Abstract
We analyze a variant of a model proposed by Kuramoto, Shinomoto, and Sakaguchi for a large population of coupled oscillatory and excitable elements. Using the Ott-Antonsen ansatz, we reduce the behavior of the population to a two-dimensional dynamical system with three parameters. We present the stability diagram and calculate several of its bifurcation curves analytically, for both excitatory and inhibitory coupling. Our main result is that when the coupling function is broad, the system can display bistability between steady states of constant high and low activity, whereas when the coupling function is narrow and inhibitory, one of the states in the bistable regime can show persistent pulsations in activity.
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Affiliation(s)
- Kevin P O'Keeffe
- Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA
| | - Steven H Strogatz
- Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA
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Daido H, Nishio K. Bifurcation and scaling at the aging transition boundary in globally coupled excitable and oscillatory units. Phys Rev E 2016; 93:052226. [PMID: 27300899 DOI: 10.1103/physreve.93.052226] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2016] [Indexed: 06/06/2023]
Abstract
Following a previous paper [Phys. Rev. E 88, 052907 (2013)PLEEE81539-375510.1103/PhysRevE.88.052907], we study in detail the mechanism of aging transition in globally and diffusively coupled excitable and oscillatory units. Here two of the three models taken up in the earlier work are used, each composed of a large number of units with their bifurcation parameters forming a uniform distribution. The control parameters are the coupling strength and the average of the bifurcation parameters. The present work is mostly devoted to a region of the phase diagram near the aging transition boundary with the coupling strength greater than its critical value for the onset of bistability and hysteresis. The bifurcation structure of each system at the aging transition boundary is investigated theoretically as well as numerically. Moreover, we show that critical scaling laws of order parameters S and M used in the previous paper are different depending on which region of the coupling strength to be chosen.
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Affiliation(s)
- Hiroaki Daido
- Department of Mathematical Sciences, Graduate School of Engineering, Osaka Prefecture University, Sakai 599-8531, Japan
| | - Kazuho Nishio
- Department of Mathematical Sciences, Graduate School of Engineering, Osaka Prefecture University, Sakai 599-8531, Japan
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Sasai T, Morino K, Tanaka G, Almendral JA, Aihara K. Robustness of oscillatory behavior in correlated networks. PLoS One 2015; 10:e0123722. [PMID: 25894574 PMCID: PMC4403822 DOI: 10.1371/journal.pone.0123722] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2014] [Accepted: 03/06/2015] [Indexed: 11/23/2022] Open
Abstract
Understanding network robustness against failures of network units is useful for preventing large-scale breakdowns and damages in real-world networked systems. The tolerance of networked systems whose functions are maintained by collective dynamical behavior of the network units has recently been analyzed in the framework called dynamical robustness of complex networks. The effect of network structure on the dynamical robustness has been examined with various types of network topology, but the role of network assortativity, or degree–degree correlations, is still unclear. Here we study the dynamical robustness of correlated (assortative and disassortative) networks consisting of diffusively coupled oscillators. Numerical analyses for the correlated networks with Poisson and power-law degree distributions show that network assortativity enhances the dynamical robustness of the oscillator networks but the impact of network disassortativity depends on the detailed network connectivity. Furthermore, we theoretically analyze the dynamical robustness of correlated bimodal networks with two-peak degree distributions and show the positive impact of the network assortativity.
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Affiliation(s)
- Takeyuki Sasai
- Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
| | - Kai Morino
- Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
| | - Gouhei Tanaka
- Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
- Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
- Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
- * E-mail:
| | - Juan A. Almendral
- Complex Systems Group, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
- Center for Biomedical Technology, Univ. Politecnica de Madrid, 28223 Pozuelo de Alarcon, Madrid, Spain
| | - Kazuyuki Aihara
- Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
- Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
- Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
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Mouse hair cycle expression dynamics modeled as coupled mesenchymal and epithelial oscillators. PLoS Comput Biol 2014; 10:e1003914. [PMID: 25375120 PMCID: PMC4222602 DOI: 10.1371/journal.pcbi.1003914] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2014] [Accepted: 09/08/2014] [Indexed: 02/07/2023] Open
Abstract
The hair cycle is a dynamic process where follicles repeatedly move through phases of growth, retraction, and relative quiescence. This process is an example of temporal and spatial biological complexity. Understanding of the hair cycle and its regulation would shed light on many other complex systems relevant to biological and medical research. Currently, a systematic characterization of gene expression and summarization within the context of a mathematical model is not yet available. Given the cyclic nature of the hair cycle, we felt it was important to consider a subset of genes with periodic expression. To this end, we combined several mathematical approaches with high-throughput, whole mouse skin, mRNA expression data to characterize aspects of the dynamics and the possible cell populations corresponding to potentially periodic patterns. In particular two gene clusters, demonstrating properties of out-of-phase synchronized expression, were identified. A mean field, phase coupled oscillator model was shown to quantitatively recapitulate the synchronization observed in the data. Furthermore, we found only one configuration of positive-negative coupling to be dynamically stable, which provided insight on general features of the regulation. Subsequent bifurcation analysis was able to identify and describe alternate states based on perturbation of system parameters. A 2-population mixture model and cell type enrichment was used to associate the two gene clusters to features of background mesenchymal populations and rapidly expanding follicular epithelial cells. Distinct timing and localization of expression was also shown by RNA and protein imaging for representative genes. Taken together, the evidence suggests that synchronization between expanding epithelial and background mesenchymal cells may be maintained, in part, by inhibitory regulation, and potential mediators of this regulation were identified. Furthermore, the model suggests that impairing this negative regulation will drive a bifurcation which may represent transition into a pathological state such as hair miniaturization. The hair cycle represents a complex process of particular interest in the study of regulated proliferation, apoptosis and differentiation. While various modeling strategies are presented in the literature, none attempt to link extensive molecular details, provided by high-throughput experiments, with high-level, system properties. Thus, we re-analyzed a previously published mRNA expression time course study and found that we could readily identify a sizeable subset of genes that was expressed in synchrony with the hair cycle itself. The data is summarized in a dynamic, mathematical model of coupled oscillators. We demonstrate that a particular coupling scheme is sufficient to explain the observed synchronization. Further analysis associated specific expression patterns to general yet distinct cell populations, background mesenchymal and rapidly expanding follicular epithelial cells. Experimental imaging results are presented to show the localization of candidate genes from each population. Taken together, the results describe a possible mechanism for regulation between epithelial and mesenchymal populations. We also described an alternate state similar to hair miniaturization, which is predicted by the oscillator model. This study exemplifies the strengths of combining systems-level analysis with high-throughput experimental data to obtain a novel view of a complex system such as the hair cycle.
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Hraha TH, Westacott MJ, Pozzoli M, Notary AM, McClatchey PM, Benninger RKP. Phase transitions in the multi-cellular regulatory behavior of pancreatic islet excitability. PLoS Comput Biol 2014; 10:e1003819. [PMID: 25188228 PMCID: PMC4154652 DOI: 10.1371/journal.pcbi.1003819] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2014] [Accepted: 07/16/2014] [Indexed: 12/23/2022] Open
Abstract
The pancreatic islets of Langerhans are multicellular micro-organs integral to maintaining glucose homeostasis through secretion of the hormone insulin. β-cells within the islet exist as a highly coupled electrical network which coordinates electrical activity and insulin release at high glucose, but leads to global suppression at basal glucose. Despite its importance, how network dynamics generate this emergent binary on/off behavior remains to be elucidated. Previous work has suggested that a small threshold of quiescent cells is able to suppress the entire network. By modeling the islet as a Boolean network, we predicted a phase-transition between globally active and inactive states would emerge near this threshold number of cells, indicative of critical behavior. This was tested using islets with an inducible-expression mutation which renders defined numbers of cells electrically inactive, together with pharmacological modulation of electrical activity. This was combined with real-time imaging of intracellular free-calcium activity [Ca2+]i and measurement of physiological parameters in mice. As the number of inexcitable cells was increased beyond ∼15%, a phase-transition in islet activity occurred, switching from globally active wild-type behavior to global quiescence. This phase-transition was also seen in insulin secretion and blood glucose, indicating physiological impact. This behavior was reproduced in a multicellular dynamical model suggesting critical behavior in the islet may obey general properties of coupled heterogeneous networks. This study represents the first detailed explanation for how the islet facilitates inhibitory activity in spite of a heterogeneous cell population, as well as the role this plays in diabetes and its reversal. We further explain how islets utilize this critical behavior to leverage cellular heterogeneity and coordinate a robust insulin response with high dynamic range. These findings also give new insight into emergent multicellular dynamics in general which are applicable to many coupled physiological systems, specifically where inhibitory dynamics result from coupled networks. As science has successfully broken down the elements of many biological systems, the network dynamics of large-scale cellular interactions has emerged as a new frontier. One way to understand how dynamical elements within large networks behave collectively is via mathematical modeling. Diabetes, which is of increasing international concern, is commonly caused by a deterioration of these complex dynamics in a highly coupled micro-organ called the islet of Langerhans. Therefore, if we are to understand diabetes and how to treat it, we must understand how coupling affects ensemble dynamics. While the role of network connectivity in islet excitation under stimulatory conditions has been well studied, how connectivity also suppresses activity under fasting conditions remains to be elucidated. Here we use two network models of islet connectivity to investigate this process. Using genetically altered islets and pharmacological treatments, we show how suppression of islet activity is solely dependent on a threshold number of inactive cells. We found that the islet exhibits critical behavior in the threshold region, rapidly transitioning from global activity to inactivity. We therefore propose how the islet and multicellular systems in general can generate a robust stimulated response from a heterogeneous cell population.
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Affiliation(s)
- Thomas H. Hraha
- Department of Bioengineering, University of Colorado, Anschutz Medical Campus, Aurora, Colorado, United States of America
| | - Matthew J. Westacott
- Department of Bioengineering, University of Colorado, Anschutz Medical Campus, Aurora, Colorado, United States of America
| | - Marina Pozzoli
- Department of Bioengineering, University of Colorado, Anschutz Medical Campus, Aurora, Colorado, United States of America
| | - Aleena M. Notary
- Department of Bioengineering, University of Colorado, Anschutz Medical Campus, Aurora, Colorado, United States of America
| | - P. Mason McClatchey
- Department of Bioengineering, University of Colorado, Anschutz Medical Campus, Aurora, Colorado, United States of America
| | - Richard K. P. Benninger
- Department of Bioengineering, University of Colorado, Anschutz Medical Campus, Aurora, Colorado, United States of America
- Barbara Davis Center for Childhood Diabetes, University of Colorado, Anschutz Medical Campus, Aurora, Colorado, United States of America
- * E-mail:
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Tanaka G, Morino K, Daido H, Aihara K. Dynamical robustness of coupled heterogeneous oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:052906. [PMID: 25353860 DOI: 10.1103/physreve.89.052906] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/03/2013] [Indexed: 05/26/2023]
Abstract
We study tolerance of dynamic behavior in networks of coupled heterogeneous oscillators to deterioration of the individual oscillator components. As the deterioration proceeds with reduction in dynamic behavior of the oscillators, an order parameter evaluating the level of global oscillation decreases and then vanishes at a certain critical point. We present a method to analytically derive a general formula for this critical point and an approximate formula for the order parameter in the vicinity of the critical point in networks of coupled Stuart-Landau oscillators. Using the critical point as a measure for dynamical robustness of oscillator networks, we show that the more heterogeneous the oscillator components are, the more robust the oscillatory behavior of the network is to the component deterioration. This property is confirmed also in networks of Morris-Lecar neuron models coupled through electrical synapses. Our approach could provide a useful framework for theoretically understanding the role of population heterogeneity in robustness of biological networks.
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Affiliation(s)
- Gouhei Tanaka
- Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan and Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan and Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
| | - Kai Morino
- Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
| | - Hiroaki Daido
- Graduate School of Engineering, Osaka Prefecture University, Sakai 599-8531, Japan
| | - Kazuyuki Aihara
- Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan and Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan and Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
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