740 research outputs found
Does dynamics reflect topology in directed networks?
We present and analyze a topologically induced transition from ordered,
synchronized to disordered dynamics in directed networks of oscillators. The
analysis reveals where in the space of networks this transition occurs and its
underlying mechanisms. If disordered, the dynamics of the units is precisely
determined by the topology of the network and thus characteristic for it. We
develop a method to predict the disordered dynamics from topology. The results
suggest a new route towards understanding how the precise dynamics of the units
of a directed network may encode information about its topology.Comment: 7 pages, 4 figures, Europhysics Letters, accepte
Unstable Attractors: Existence and Robustness in Networks of Oscillators With Delayed Pulse Coupling
We consider unstable attractors; Milnor attractors such that, for some
neighbourhood of , almost all initial conditions leave . Previous
research strongly suggests that unstable attractors exist and even occur
robustly (i.e. for open sets of parameter values) in a system modelling
biological phenomena, namely in globally coupled oscillators with delayed pulse
interactions.
In the first part of this paper we give a rigorous definition of unstable
attractors for general dynamical systems. We classify unstable attractors into
two types, depending on whether or not there is a neighbourhood of the
attractor that intersects the basin in a set of positive measure. We give
examples of both types of unstable attractor; these examples have
non-invertible dynamics that collapse certain open sets onto stable manifolds
of saddle orbits.
In the second part we give the first rigorous demonstration of existence and
robust occurrence of unstable attractors in a network of oscillators with
delayed pulse coupling. Although such systems are technically hybrid systems of
delay differential equations with discontinuous `firing' events, we show that
their dynamics reduces to a finite dimensional hybrid system system after a
finite time and hence we can discuss Milnor attractors for this reduced finite
dimensional system. We prove that for an open set of phase resetting functions
there are saddle periodic orbits that are unstable attractors.Comment: 29 pages, 8 figures,submitted to Nonlinearit
Being Critical of Criticality in the Brain
Relatively recent work has reported that networks of neurons can produce avalanches of activity whose sizes follow a power law distribution. This suggests that these networks may be operating near a critical point, poised between a phase where activity rapidly dies out and a phase where activity is amplified over time. The hypothesis that the electrical activity of neural networks in the brain is critical is potentially important, as many simulations suggest that information processing functions would be optimized at the critical point. This hypothesis, however, is still controversial. Here we will explain the concept of criticality and review the substantial objections to the criticality hypothesis raised by skeptics. Points and counter points are presented in dialog form
A Tutorial for Information Theory in Neuroscience
Understanding how neural systems integrate, encode, and compute information is central to understanding brain function. Frequently, data from neuroscience experiments are multivariate, the interactions between the variables are nonlinear, and the landscape of hypothesized or possible interactions between variables is extremely broad. Information theory is well suited to address these types of data, as it possesses multivariate analysis tools, it can be applied to many different types of data, it can capture nonlinear interactions, and it does not require assumptions about the structure of the underlying data (i.e., it is model independent). In this article, we walk through the mathematics of information theory along with common logistical problems associated with data type, data binning, data quantity requirements, bias, and significance testing. Next, we analyze models inspired by canonical neuroscience experiments to improve understanding and demonstrate the strengths of information theory analyses. To facilitate the use of information theory analyses, and an understanding of how these analyses are implemented, we also provide a free MATLAB software package that can be applied to a wide range of data from neuroscience experiments, as well as from other fields of study
Speed of synchronization in complex networks of neural oscillators Analytic results based on Random Matrix Theory
We analyze the dynamics of networks of spiking neural oscillators. First, we
present an exact linear stability theory of the synchronous state for networks
of arbitrary connectivity. For general neuron rise functions, stability is
determined by multiple operators, for which standard analysis is not suitable.
We describe a general non-standard solution to the multi-operator problem.
Subsequently, we derive a class of rise functions for which all stability
operators become degenerate and standard eigenvalue analysis becomes a suitable
tool. Interestingly, this class is found to consist of networks of leaky
integrate and fire neurons. For random networks of inhibitory
integrate-and-fire neurons, we then develop an analytical approach, based on
the theory of random matrices, to precisely determine the eigenvalue
distribution. This yields the asymptotic relaxation time for perturbations to
the synchronous state which provides the characteristic time scale on which
neurons can coordinate their activity in such networks. For networks with
finite in-degree, i.e. finite number of presynaptic inputs per neuron, we find
a speed limit to coordinating spiking activity: Even with arbitrarily strong
interaction strengths neurons cannot synchronize faster than at a certain
maximal speed determined by the typical in-degree.Comment: 17 pages, 12 figures, submitted to Chao
Chaos in Symmetric Phase Oscillator Networks
Phase-coupled oscillators serve as paradigmatic models of networks of weakly
interacting oscillatory units in physics and biology. The order parameter which
quantifies synchronization was so far found to be chaotic only in systems with
inhomogeneities. Here we show that even symmetric systems of identical
oscillators may not only exhibit chaotic dynamics, but also chaotically
fluctuating order parameters. Our findings imply that neither inhomogeneities
nor amplitude variations are necessary to obtain chaos, i.e., nonlinear
interactions of phases give rise to the necessary instabilities.Comment: 4 pages; Accepted by Physical Review Letter
Stochastic facilitation in heteroclinic communication channels
Biological neural systems encode and transmit information as patterns of activity tracing complex trajectories in high-dimensional state spaces, inspiring alternative paradigms of information processing. Heteroclinic networks, naturally emerging in artificial neural systems, are networks of saddles in state space that provide a transparent approach to generate complex trajectories via controlled switches among interconnected saddles. External signals induce specific switching sequences, thus dynamically encoding inputs as trajectories. Recent works have focused either on computational aspects of heteroclinic networks, i.e., Heteroclinic Computing, or their stochastic properties under noise. Yet, how well such systems may transmit information remains an open question. Here, we investigate the information transmission properties of heteroclinic networks, studying them as communication channels. Choosing a tractable but representative system exhibiting a heteroclinic network, we investigate the mutual information rate (MIR) between input signals and the resulting sequences of states as the level of noise varies. Intriguingly, MIR does not decrease monotonically with increasing noise. Intermediate noise levels indeed maximize the information transmission capacity by promoting an increased yet controlled exploration of the underlying network of states. Complementing standard stochastic resonance, these results highlight the constructive effect of stochastic facilitation (i.e., noise-enhanced information transfer) on heteroclinic communication channels and possibly on more general dynamical systems exhibiting complex trajectories in state space
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