Nonlinear dynamics of networks:  the groupoid formalism

Golubitsky, Martin; Stewart, Ian

doi:10.1090/s0273-0979-06-01108-6

Cited by 354 publications

(381 citation statements)

References 84 publications

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“…16;Golubitsky and Stewart 2006). Again, simultaneous occurrence of several such 'phase condensations' would account for observed patterns of comorbidity, albeit with distinct cultural convolutions.…”

Section: Discussionmentioning

confidence: 97%

Environmental Induction of Neurodevelopmental Disorders

Wallace

2016

Bull Math Biol

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Chemical exposures, pre-and neonatal infections, psychosocial stress, and the cross-generational cultural and epigenetic impacts of these and other toxicants become an integrated, sometimes synergistic, signal that can overwhelm essential neurodevelopmental regulation. We characterize that dynamic through statistical models based on the asymptotic limit theorems of control and information theories. Schizophrenia and autism emerge as two different 'phases' of pathological neurodevelopmental 'condensations' that impair the dynamic, shifting global workspace of normal mental function.

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Section: Discussionmentioning

confidence: 97%

Environmental Induction of Neurodevelopmental Disorders

Wallace

2016

Bull Math Biol

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“…The name is derived from the fact that this network is a chain of coupled neurons, without feedback to neurons earlier in the chain. Feed-forward chains occur often as 'motifs' in larger networks, because a feed-forward chain can act as an amplifier [9]. This behaviour is proved mathematically in Rink and Sanders (2013) as they found faster amplitude growth for state variables further in the chain [15,16].…”

Section: 4mentioning

confidence: 93%

“…This system of equations is a special case of a general feed-forward network [16]. The system is studied in Rink and Sanders [16] and is often encountered in the literature on coupled cell networks [3,7,8,9]. …”

Section: 4mentioning

confidence: 99%

Synchrony Breaking Bifurcations in Small Neuronal Networks

Hemminga¹

2017

SIURO

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Abstract. We investigate synchrony breaking bifurcations in neuronal networks. These bifurcations occur from synchronous steady-states. In the mutual dyad and a three-neuron feed-forward chain we show that the generic bifurcation behaviour can be derived from the physical modelling parameters, in particular from the sign of the interaction between neurons. Each neuron is equipped with a simplified FitzHugh-Nagumo model and the coupling is based on synaptic coupling. An inhibitory or excitatory coupling can determine if the bifurcation is 'soft' (supercritical) or 'hard' (subcritical). For the analysis of the three-neuron feed-forward chain we follow the work of Rink and Sanders (2013): we can relate excitatory and inhibitory coupling to a 'soft' and a 'hard' transition, respectively. For the mutual dyad system we make use of a centre manifold reduction to find the type of pitchfork bifurcation. As we find an expression in terms of physical parameters, we can state whether the bifurcation is subcritical or supercritical in the weak coupling limit, and for slow and fast input.1. Introduction. We will consider neuronal networks as dynamical systems with a network structure. The nodes of the network model the neurons. Real life neuronal networks are immensely complex but in this paper we will consider small networks. Such small networks are called 'motifs' and are often present as sub-networks of large neuronal networks. In this paper we consider the mutual dyad and a feedforward chain. The mutual dyad contains two identical neurons, mutually coupled by identical coupling. The feed-forward chain contains three identical neurons with a specific feed-forward coupling. These two motifs are shown in figures 1.1 and 1.2.

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“…65,71,115 Here, there is no assumption about the existence of a global group of symmetries. Instead, the network architecture (topology) plays a role analogous to symmetry.…”

Section: Introductionmentioning

confidence: 99%

Recent advances in symmetric and network dynamics

Golubitsky

Stewart

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We summarize some of the main results discovered over the past three decades concerning symmetric dynamical systems and networks of dynamical systems, with a focus on pattern formation. In both of these contexts, extra constraints on the dynamical system are imposed, and the generic phenomena can change. The main areas discussed are time-periodic states, mode interactions, and non-compact symmetry groups such as the Euclidean group. We consider both dynamics and bifurcations. We summarize applications of these ideas to pattern formation in a variety of physical and biological systems, and explain how the methods were motivated by transferring to new contexts Rene Thom's general viewpoint, one version of which became known as "catastrophe theory." We emphasize the role of symmetry-breaking in the creation of patterns. Topics include equivariant Hopf bifurcation, which gives conditions for a periodic state to bifurcate from an equilibrium, and the H/K theorem, which classifies the pairs of setwise and pointwise symmetries of periodic states in equivariant dynamics. We discuss mode interactions, which organize multiple bifurcations into a single degenerate bifurcation, and systems with non-compact symmetry groups, where new technical issues arise. We transfer many of the ideas to the context of networks of coupled dynamical systems, and interpret synchrony and phase relations in network dynamics as a type of pattern, in which space is discretized into finitely many nodes, while time remains continuous. We also describe a variety of applications including animal locomotion, Couette-Taylor flow, flames, the Belousov-Zhabotinskii reaction, binocular rivalry, and a nonlinear filter based on anomalous growth rates for the amplitude of periodic oscillations in a feed-forward network. Symmetry is a feature of many systems of interest in applied science. Mathematically, a symmetry is a transformation that preserves structure; for example, a square looks unchanged if it is rotated through any multiple of a right angle, or reflected in a diagonal or a line joining the midpoints of oppose edges. These symmetries also appear in mathematical models of real-world systems, and their effect is often extensive. The last thirty to forty years has seen considerable advances in the mathematical understanding of the effects of symmetry on dynamical systems-systems of ordinary differential equations. In general, symmetry leads to pattern formation, via a mechanism called symmetry-breaking. For example, if a system with circular symmetry has a time-periodic state, repeating the same behavior indefinitely, then typically this state is either a standing wave or a rotating wave. This observation applies to the movement of a flexible hosepipe as water passes through it: in the standing wave, the hosepipe moves to and fro like a pendulum; in the rotating wave, it goes round and round with its end describing a circle. This observation also applies to how flame fronts move on a circular burner. We survey some basic mathematical ideas that have ...

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Nonlinear dynamics of networks: the groupoid formalism

Cited by 354 publications

References 84 publications

Environmental Induction of Neurodevelopmental Disorders

Environmental Induction of Neurodevelopmental Disorders

Synchrony Breaking Bifurcations in Small Neuronal Networks

Recent advances in symmetric and network dynamics

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