Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

Ashwin, Peter; Coombes, Stephen; Nicks, Rachel

doi:10.1186/s13408-015-0033-6

Cited by 256 publications

(268 citation statements)

References 366 publications

(619 reference statements)

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“…Note that (37) (grey dashed) has a singularity at PF, whilst there is none for (38) (black dashed). The predicted asymptotic escape times T (1) , T (2) and T (4) for each network in the limit β → ∞ are shown. Note that in the limit β → 0, all times limit to T (2) .…”

Section: Discussionmentioning

confidence: 99%

“…For low noise amplitude (2) shows similar behavior to the underlying deterministic system, whereas for large noise the dynamics are dominated by large stochastic fluctuations. The dynamics' realisation shown in Figure 1 is computed in Matlab using the Heun method for stochastic differential equations [25] with the initial condition at the origin and step size h = 10 −5 .…”

Section: Single Node Escape Timesmentioning

confidence: 91%

“…More precisely, for the given value of ν and α, constants A, B are chosen to be the unique solution such that the one-node estimates for T (1) and T (2) are exact, namely…”

Section: 2mentioning

confidence: 99%

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Sequential Noise-Induced Escapes for Oscillatory Network Dynamics

Creaser¹,

Tsaneva-Atanasova²,

Ashwin³

2018

SIAM J. Appl. Dyn. Syst.

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It is well known that the addition of noise in a multistable system can induce random transitions between stable states. The rate of transition can be characterised in terms of the noise-free system's dynamics and the added noise: for potential systems in the presence of asymptotically low noise the well-known Kramers' escape time gives an expression for the mean escape time. This paper examines some general properties and examples of transitions between local steady and oscillatory attractors within networks: the transition rates at each node may be affected by the dynamics at other nodes. We use first passage time theory to explain some properties of scalings noted in the literature for an idealised model of initiation of epileptic seizures in small systems of coupled bistable systems with both steady and oscillatory attractors. We focus on the case of sequential escapes where a steady attractor is only marginally stable but all nodes start in this state. As the nodes escape to the oscillatory regime, we assume that the transitions back are very infrequent in comparison. We quantify and characterise the resulting sequences of noise-induced escapes. For weak enough coupling we show that a master equation approach gives a good quantitative understanding of sequential escapes, but for strong coupling this description breaks down.

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Section: Discussionmentioning

confidence: 99%

Section: Single Node Escape Timesmentioning

confidence: 91%

See 1 more Smart Citation

Sequential Noise-Induced Escapes for Oscillatory Network Dynamics

Creaser¹,

Tsaneva-Atanasova²,

Ashwin³

2018

SIAM J. Appl. Dyn. Syst.

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“…The synchronous state is a very special network solution, especially when compared to more exotic ones, such as phase-locked clusters, chimeras (mixing synchronous and asynchronous states), chaos and death (where the dynamics of some nodes is silenced), as recently reviewed in [42] from a neuroscience perspective. Although the tools of weakly coupled oscillator theory have shed some light on these states, as for example described in the book by Hoppensteadt and Izhikevich, [43], we currently lack a general theory of network dynamics valid for strong coupling.…”

Section: Discussionmentioning

confidence: 99%

Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

Coombes

Thul

2016

Eur. J. Appl. Math

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The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov-Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.

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“…For example, network structure can represent social contacts, and structural features can profoundly impact the propagation dynamics of diseases and memes [25]. Network architecture also has a huge effect on collective behaviour in networks of coupled oscillators [1,2] and many other phenomena. In the special issue's third paper, Do et al [8] examine (as in the first two papers) how the interplay of microscale constituents leads to the emergence of macroscale properties.…”

Section: The Articles In This Issuementioning

confidence: 99%