The Feed-Forward Chain as a Filter-Amplifier Motif

Golubitsky, Martin; Shiau, LieJune; Postlethwaite, Claire M.; Zhang, Yanyan

doi:10.1007/978-1-4419-0389-1_6

Cited by 19 publications

(25 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This theorem generalizes the results in [8], [14] and [15] to feedforward chains of arbitrary length. We like to point out that, except for the linear algebra, various computational aspects of our proof are the same as those given in [15].…”

Section: Analysis Of the Nonlinear Normal Formsupporting

confidence: 79%

See 1 more Smart Citation

Amplified Hopf Bifurcations in Feed-Forward Networks

Rink¹,

Sanders²

2013

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

In [18] the authors developed a method for computing normal forms of dynamical systems with a coupled cell network structure. We now apply this theory to one-parameter families of homogeneous feed-forward chains with 2-dimensional cells. Our main result is that Hopf bifurcations in such families generically generate branches of periodic solutions with amplitudes growing like ∼ |λ| . We explain here how these bifurcations arise generically in a broader class of feed-forward chains of arbitrary length.

show abstract

Section: Analysis Of the Nonlinear Normal Formsupporting

confidence: 79%

“…is much faster though. One could say that the feed-forward chain acts as an amplifier and it has been conjectured that this is why feed-forward chains occur so often as motifs in larger networks [14].…”

Section: X0mentioning

confidence: 99%

Amplified Hopf Bifurcations in Feed-Forward Networks

Rink¹,

Sanders²

2013

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

show abstract

“…Its important role as the working principle of biological sensors was recognized much later [6,7]; in particular, the mechanism was shown to play a decisive role in insect hearing [8,9]. Regarding mammalian hearing [1,[10][11][12], this principle still is fighting its way against classical engineering hearing solutions. For networks of neurons, which naturally display bifurcation behavior, the corresponding consequences have largely remained unexplored.…”

mentioning

confidence: 99%

“…[12]); a more general study comparable to the superthreshold (i.e., self-oscillatory) case is still missing. In nature, different types of bifurcations occur that mathematics classifies according to the eigenvalues that the system's linearization has when the behavioral change occurs, i.e., at the bifurcation point.…”

mentioning

confidence: 99%

Signal-Coupled Subthreshold Hopf-Type Systems Show a Sharpened Collective Response

2016

View full text Add to dashboard Cite

Astounding properties of biological sensors can often be mapped onto a dynamical system below the occurrence of a bifurcation. For mammalian hearing, a Hopf bifurcation description has been shown to work across a whole range of scales, from individual hair bundles to whole regions of the cochlea. We reveal here the origin of this scale invariance, from a general level, applicable to all dynamics in the vicinity of a Hopf bifurcation (embracing, e.g., neuronal Hodgkin-Huxley equations). When subject to natural "signal coupling," ensembles of Hopf systems below the bifurcation threshold exhibit a collective Hopf bifurcation. This collective Hopf bifurcation occurs at parameter values substantially below where the average of the individual systems would bifurcate, with a frequency profile that is sharpened if compared to the individual systems.

show abstract

“…102 A potential application of this behavior is to the design of a nonlinear filter which selects oscillations close to a specific frequency and amplifies them. 58 Experimental proof-ofconcept can be found in McCullen et al 90 …”

Section: Anomalous Growth In Network Hopf Bifurcationmentioning

confidence: 99%

Recent advances in symmetric and network dynamics

Golubitsky

Stewart

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

View full text Add to dashboard Cite

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 ...

show abstract

The Feed-Forward Chain as a Filter-Amplifier Motif

Cited by 19 publications

References 30 publications

Amplified Hopf Bifurcations in Feed-Forward Networks

Amplified Hopf Bifurcations in Feed-Forward Networks

Signal-Coupled Subthreshold Hopf-Type Systems Show a Sharpened Collective Response

Recent advances in symmetric and network dynamics

Contact Info

Product

Resources

About