1,189 research outputs found
Delay-induced patterns in a two-dimensional lattice of coupled oscillators
We show how a variety of stable spatio-temporal periodic patterns can be
created in 2D-lattices of coupled oscillators with non-homogeneous coupling
delays. A "hybrid dispersion relation" is introduced, which allows studying the
stability of time-periodic patterns analytically in the limit of large delay.
The results are illustrated using the FitzHugh-Nagumo coupled neurons as well
as coupled limit cycle (Stuart-Landau) oscillators
Stability and Hopf bifurcation of controlled complex networks model with two delays
none3siopenJinde Cao, Luca Guerrini, Zunshui ChengCao, Jinde; Guerrini, Luca; Cheng, Zunshu
Time-delayed models of gene regulatory networks
We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternativemodelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems
Amplitude Death: The emergence of stationarity in coupled nonlinear systems
When nonlinear dynamical systems are coupled, depending on the intrinsic
dynamics and the manner in which the coupling is organized, a host of novel
phenomena can arise. In this context, an important emergent phenomenon is the
complete suppression of oscillations, formally termed amplitude death (AD).
Oscillations of the entire system cease as a consequence of the interaction,
leading to stationary behavior. The fixed points that the coupling stabilizes
can be the otherwise unstable fixed points of the uncoupled system or can
correspond to novel stationary points. Such behaviour is of relevance in areas
ranging from laser physics to the dynamics of biological systems. In this
review we discuss the characteristics of the different coupling strategies and
scenarios that lead to AD in a variety of different situations, and draw
attention to several open issues and challenging problems for further study.Comment: Physics Reports (2012
The Dynamics of Hybrid Metabolic-Genetic Oscillators
The synthetic construction of intracellular circuits is frequently hindered
by a poor knowledge of appropriate kinetics and precise rate parameters. Here,
we use generalized modeling (GM) to study the dynamical behavior of topological
models of a family of hybrid metabolic-genetic circuits known as
"metabolators." Under mild assumptions on the kinetics, we use GM to
analytically prove that all explicit kinetic models which are topologically
analogous to one such circuit, the "core metabolator," cannot undergo Hopf
bifurcations. Then, we examine more detailed models of the metabolator.
Inspired by the experimental observation of a Hopf bifurcation in a
synthetically constructed circuit related to the core metabolator, we apply GM
to identify the critical components of the synthetically constructed
metabolator which must be reintroduced in order to recover the Hopf
bifurcation. Next, we study the dynamics of a re-wired version of the core
metabolator, dubbed the "reverse" metabolator, and show that it exhibits a
substantially richer set of dynamical behaviors, including both local and
global oscillations. Prompted by the observation of relaxation oscillations in
the reverse metabolator, we study the role that a separation of genetic and
metabolic time scales may play in its dynamics, and find that widely separated
time scales promote stability in the circuit. Our results illustrate a generic
pipeline for vetting the potential success of a potential circuit design,
simply by studying the dynamics of the corresponding generalized model
Comparative exploration on bifurcation behavior for integer-order and fractional-order delayed BAM neural networks
In the present study, we deal with the stability and the onset of Hopf bifurcation of two type delayed BAM neural networks (integer-order case and fractional-order case). By virtue of the characteristic equation of the integer-order delayed BAM neural networks and regarding time delay as critical parameter, a novel delay-independent condition ensuring the stability and the onset of Hopf bifurcation for the involved integer-order delayed BAM neural networks is built. Taking advantage of Laplace transform, stability theory and Hopf bifurcation knowledge of fractional-order differential equations, a novel delay-independent criterion to maintain the stability and the appearance of Hopf bifurcation for the addressed fractional-order BAM neural networks is established. The investigation indicates the important role of time delay in controlling the stability and Hopf bifurcation of the both type delayed BAM neural networks. By adjusting the value of time delay, we can effectively amplify the stability region and postpone the time of onset of Hopf bifurcation for the fractional-order BAM neural networks. Matlab simulation results are clearly presented to sustain the correctness of analytical results. The derived fruits of this study provide an important theoretical basis in regulating networks
The complexity of dynamics in small neural circuits
Mean-field theory is a powerful tool for studying large neural networks.
However, when the system is composed of a few neurons, macroscopic differences
between the mean-field approximation and the real behavior of the network can
arise. Here we introduce a study of the dynamics of a small firing-rate network
with excitatory and inhibitory populations, in terms of local and global
bifurcations of the neural activity. Our approach is analytically tractable in
many respects, and sheds new light on the finite-size effects of the system. In
particular, we focus on the formation of multiple branching solutions of the
neural equations through spontaneous symmetry-breaking, since this phenomenon
increases considerably the complexity of the dynamical behavior of the network.
For these reasons, branching points may reveal important mechanisms through
which neurons interact and process information, which are not accounted for by
the mean-field approximation.Comment: 34 pages, 11 figures. Supplementary materials added, colors of
figures 8 and 9 fixed, results unchange
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