Transition to chaos in small-world dynamical network

Yuan, Weihong; Xiao-Shu, Luo; Jiang, Pinqun; Wang, Bing–Hong; Jin-Qing, Fang

doi:10.1016/j.chaos.2006.09.077

Cited by 20 publications

(10 citation statements)

References 21 publications

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“…[72][73][74][75][76][77] In particular, insertion of long-distance shortcuts according to small-world topology leads to chaos even in networks that cannot otherwise be chaotic. [78][79][80][81] At the same time, in-vitro recordings of neuronal cultures on substrate-integrated multi-electrode arrays, indexing mesoscale activity in networks of %10 4 -10 5 neurons, demonstrate the emergence of self-organized small-world functional connectivity during culture maturation. It has separately been shown that the prevalence of chaotic activity in these cultures gradually increases as the number of active sites grows, in turn suggesting that the gradual formation of a dense and complex network promotes the emergence of chaos.…”

Section: Analogy Between Brain and Single-transistor Chaotic Oscilmentioning

confidence: 97%

Synchronization, non-linear dynamics and low-frequency fluctuations: Analogy between spontaneous brain activity and networked single-transistor chaotic oscillators

Minati¹,

Chiesa²,

Tabarelli³

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper, the topographical relationship between functional connectivity (intended as inter-regional synchronization), spectral and non-linear dynamical properties across cortical areas of the healthy human brain is considered. Based upon functional MRI acquisitions of spontaneous activity during wakeful idleness, node degree maps are determined by thresholding the temporal correlation coefficient among all voxel pairs. In addition, for individual voxel time-series, the relative amplitude of low-frequency fluctuations and the correlation dimension (D2), determined with respect to Fourier amplitude and value distribution matched surrogate data, are measured. Across cortical areas, high node degree is associated with a shift towards lower frequency activity and, compared to surrogate data, clearer saturation to a lower correlation dimension, suggesting presence of non-linear structure. An attempt to recapitulate this relationship in a network of single-transistor oscillators is made, based on a diffusive ring (n = 90) with added long-distance links defining four extended hub regions. Similarly to the brain data, it is found that oscillators in the hub regions generate signals with larger low-frequency cycle amplitude fluctuations and clearer saturation to a lower correlation dimension compared to surrogates. The effect emerges more markedly close to criticality. The homology observed between the two systems despite profound differences in scale, coupling mechanism and dynamics appears noteworthy. These experimental results motivate further investigation into the heterogeneity of cortical non-linear dynamics in relation to connectivity and underline the ability for small networks of single-transistor oscillators to recreate collective phenomena arising in much more complex biological systems, potentially representing a future platform for modelling disease-related changes.

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Section: Analogy Between Brain and Single-transistor Chaotic Oscilmentioning

confidence: 97%

Synchronization, non-linear dynamics and low-frequency fluctuations: Analogy between spontaneous brain activity and networked single-transistor chaotic oscillators

Minati¹,

Chiesa²,

Tabarelli³

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The emergence of chaotic behaviour in coupled maps was investigated in Li, Chen, and Ko (2004), later they formalise these results showing that a network of logistic maps can satisfy the conditions for chaos in the sense of Li and Yorke (Ott 2002). The results in Li et al (2004) were extended to the case of dynamical networks where each node is a continuous-time multidimensional system by Zhang, Wu, and Fu (2006) and Yuan, Luo, Jiang, Wang, and Fang (2008). In Femat 2008a, 2008b) the transition to complex behaviour was considered for small-world and scale-free topologies.…”

Section: Introductionmentioning

confidence: 99%

On the emergence of chaos in dynamical networks

Barajas-Ramírez

Femat

2012

International Journal of Systems Science

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We investigate how changes of specific topological features result on transitions among different bounded behaviours in dynamical networks. In particular, we focus on networks with identical dynamical systems, synchronised to a common equilibrium point, then a transition into chaotic behaviour is observed as the number of nodes and the strength of their coupling changes. We analyse the network's transverse Lyapunov exponents (tLes) to derive conditions for the emergence of bounded complex behaviour on different basic network models. We find that, for networks with a given number of nodes, chaotic behaviour emerges when the coupling strength is within a specific bounded interval; this interval is reduced as the number of nodes increases. Furthermore, the endpoints the emergence interval depend on the coupling structure of network. We also find that networks with homogeneous connectivity, such as regular lattices and small-world networks are more conducive to the emergence of chaos than networks with heterogeneous connectivity like scale-free and star-connected graphs. Our results are illustrated with numerical simulations of the chaotic benchmark Lorenz systems, and to underline their potential applicability to real-world systems, our results are used to establish conditions for the chaotic activation of a network of electrically coupled pancreatic -cell models.

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“…On such complex networks, even if in the absence of any external stimuli, the dynamical patterns could be very complicated under the condition of spontaneous communications among the dynamical nodes (Newman, 2003; Boccaletti et al, 2006; Arenas et al, 2008; Yuan et al, 2008). The two simplest patterns are in-phase (zero time lag) and anti-phase oscillations.…”

Section: Introductionmentioning

confidence: 99%

Organization of Anti-Phase Synchronization Pattern in Neural Networks: What are the Key Factors?

Li¹,

Zhou²

2011

Front. Syst. Neurosci.

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Anti-phase oscillation has been widely observed in cortical neural network. Elucidating the mechanism underlying the organization of anti-phase pattern is of significance for better understanding more complicated pattern formations in brain networks. In dynamical systems theory, the organization of anti-phase oscillation pattern has usually been considered to relate to time delay in coupling. This is consistent to conduction delays in real neural networks in the brain due to finite propagation velocity of action potentials. However, other structural factors in cortical neural network, such as modular organization (connection density) and the coupling types (excitatory or inhibitory), could also play an important role. In this work, we investigate the anti-phase oscillation pattern organized on a two-module network of either neuronal cell model or neural mass model, and analyze the impact of the conduction delay times, the connection densities, and coupling types. Our results show that delay times and coupling types can play key roles in this organization. The connection densities may have an influence on the stability if an anti-phase pattern exists due to the other factors. Furthermore, we show that anti-phase synchronization of slow oscillations can be achieved with small delay times if there is interaction between slow and fast oscillations. These results are significant for further understanding more realistic spatiotemporal dynamics of cortico-cortical communications.

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Transition to chaos in small-world dynamical network

Cited by 20 publications

References 21 publications

Synchronization, non-linear dynamics and low-frequency fluctuations: Analogy between spontaneous brain activity and networked single-transistor chaotic oscillators

Synchronization, non-linear dynamics and low-frequency fluctuations: Analogy between spontaneous brain activity and networked single-transistor chaotic oscillators

On the emergence of chaos in dynamical networks

Organization of Anti-Phase Synchronization Pattern in Neural Networks: What are the Key Factors?

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