3,054 research outputs found
Algebraic Analysis of Bifurcation and Limit Cycles for Biological Systems
International audienceIn this paper, we show how to analyze bifurcation and limit cycles for biological systems by using an algebraic approach based on triangular decomposition, Gröbner bases, discriminant varieties, real solution classification, and quantifier elimination by partial CAD. The analysis of bifurcation and limit cycles for a concrete two-dimensional system, the self-assembling micelle system with chemical sinks, is presented in detail. It is proved that this system may have a focus of order 3, from which three limit cycles can be constructed by small perturbation. The applicability of our approach is further illustrated by the construction of limit cycles for a two-dimensional Kolmogorov prey-predator system and a three-dimensional Lotka-Volterra system
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
Transition from homogeneous to inhomogeneous limit cycles : Effect of local filtering in coupled oscillators
10 pages, 8 FigsPeer reviewedPublisher PD
Shaping bursting by electrical coupling and noise
Gap-junctional coupling is an important way of communication between neurons
and other excitable cells. Strong electrical coupling synchronizes activity
across cell ensembles. Surprisingly, in the presence of noise synchronous
oscillations generated by an electrically coupled network may differ
qualitatively from the oscillations produced by uncoupled individual cells
forming the network. A prominent example of such behavior is the synchronized
bursting in islets of Langerhans formed by pancreatic \beta-cells, which in
isolation are known to exhibit irregular spiking. At the heart of this
intriguing phenomenon lies denoising, a remarkable ability of electrical
coupling to diminish the effects of noise acting on individual cells.
In this paper, we derive quantitative estimates characterizing denoising in
electrically coupled networks of conductance-based models of square wave
bursting cells. Our analysis reveals the interplay of the intrinsic properties
of the individual cells and network topology and their respective contributions
to this important effect. In particular, we show that networks on graphs with
large algebraic connectivity or small total effective resistance are better
equipped for implementing denoising. As a by-product of the analysis of
denoising, we analytically estimate the rate with which trajectories converge
to the synchronization subspace and the stability of the latter to random
perturbations. These estimates reveal the role of the network topology in
synchronization. The analysis is complemented by numerical simulations of
electrically coupled conductance-based networks. Taken together, these results
explain the mechanisms underlying synchronization and denoising in an important
class of biological models
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