Noise-Induced Order in Extended Systems: A Tutorial

Sancho, J. M.; García‐Ojalvo, Jordi

doi:10.1007/3-540-45396-2_22

Cited by 13 publications

(13 citation statements)

References 42 publications

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“…This is mainly because it has been impossible thus far to find a nonequilibrium model whose steady-state probability distribution and the associated effective potential are known. In particular, noise-induced phase transitions have been systematically explained in terms of a short-time instability of the local dynamics, which becomes "frozen" at larger times by the spatial coupling [7,8].In this Letter, we introduce a class of systems exhibiting NIOPTs for which the steady-state probability distribution can be obtained exactly, so that one can define the corresponding nonequilibrium free energy or potential. As a consequence, the NIOPT can be studied in the steady state, with no dynamical reference.…”

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Noise-Induced Scenario for Inverted Phase Diagrams

et al. 2001

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We introduce a class of exactly solvable models exhibiting an ordering noise-induced phase transition in which order arises as a result of a balance between the relaxing deterministic dynamics and the randomizing character of the fluctuations. A finite-size scaling analysis of the phase transition reveals that it belongs to the universality class of the equilibrium Ising model. All these results are analyzed in the light of the nonequilibrium probability distribution of the system, which can be obtained analytically. Our results could constitute a possible scenario of inverted phase diagrams in the so-called lower critical solution temperature transitions. From a fundamental point of view, however, the most interesting example is that of noise-induced ordering phase transitions (NIOPTs) [7]. In this case, transitions between true extended phases (in a thermodynamic sense) are produced when the noise intensity is used as a control parameter. These nonequilibrium phase transitions can therefore be characterized using standard techniques of critical phenomena, such as dynamic renormalization group, mean-field approximations, as well as finite-size scaling analyses using, for instance, results obtained from numerical simulations of stochastic partial differential equations. Until now, all the arguments presented to account for the occurrence of NIOPTs have been dynamical ones. This is mainly because it has been impossible thus far to find a nonequilibrium model whose steady-state probability distribution and the associated effective potential are known. In particular, noise-induced phase transitions have been systematically explained in terms of a short-time instability of the local dynamics, which becomes "frozen" at larger times by the spatial coupling [7,8].In this Letter, we introduce a class of systems exhibiting NIOPTs for which the steady-state probability distribution can be obtained exactly, so that one can define the corresponding nonequilibrium free energy or potential. As a consequence, the NIOPT can be studied in the steady state, with no dynamical reference. It turns out that the noiseinduced phase transition that appears in these model systems is not a consequence of an instability of the disordered homogeneous state. Rather, the situation is similar to what happens for noise-induced transitions in zero-dimensional systems, where the disordered state is linearly stable, but the effective nonequilibrium potential in the steady state is bimodal [9,10]. A phase transition is obtained by coupling adequately many of those zero-dimensional systems. For a strong enough spatial coupling, the neighboring variables tend to a common value and a macroscopic ordered phase appears as a result of ergodicity breaking. This situation contrasts vividly with NIOPTs reported thus far, where the need of a short-time dynamical instability prevents systems that undergo noise-induced transitions in 0D from exhibiting NIOPTs in the presence of spatial coupling [7].The phase transition that we report in what follows shares som...

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Noise-Induced Scenario for Inverted Phase Diagrams

et al. 2001

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“…As the fraction δ of removed connections increases both maxima of C(D) appear at lower noise intensity. The network size was set to N = 800 and a factor N instead of k is considered in the definition (4). Each curve has been obtained with a single realization of the corresponding network.…”

Section: Discussionmentioning

confidence: 99%

“…Nevertheless, these fluctuations may play a fundamental role in natural systems. For instance, they may optimize signals propagation by turning the medium into an excitable one -e.g., the case of ionic channel stochasticity in neurons that can affect the first spike latency [1,2] or enhance signal propagation through different neuronal layers [3] -, originate order at macroscopic and mesoscopic levels [4,5] or induce coherence between the intrinsic dynamics of a system and some weak stimuli it receives, a phenomenon known as stochastic resonance (SR) (see for instance [6] for a review). Precisely, this intriguing phenomenon has attracted the interest of the computational neuroscience community for its possible implications in the complex processing of information in the brain [7][8][9][10][11], or as a way to control specific brain states [12].…”

Section: Introductionmentioning

confidence: 99%

Efficient Transmission of Subthreshold Signals in Complex Networks of Spiking Neurons

2015

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We investigate the efficient transmission and processing of weak, subthreshold signals in a realistic neural medium in the presence of different levels of the underlying noise. Assuming Hebbian weights for maximal synaptic conductances—that naturally balances the network with excitatory and inhibitory synapses—and considering short-term synaptic plasticity affecting such conductances, we found different dynamic phases in the system. This includes a memory phase where population of neurons remain synchronized, an oscillatory phase where transitions between different synchronized populations of neurons appears and an asynchronous or noisy phase. When a weak stimulus input is applied to each neuron, increasing the level of noise in the medium we found an efficient transmission of such stimuli around the transition and critical points separating different phases for well-defined different levels of stochasticity in the system. We proved that this intriguing phenomenon is quite robust, as it occurs in different situations including several types of synaptic plasticity, different type and number of stored patterns and diverse network topologies, namely, diluted networks and complex topologies such as scale-free and small-world networks. We conclude that the robustness of the phenomenon in different realistic scenarios, including spiking neurons, short-term synaptic plasticity and complex networks topologies, make very likely that it could also occur in actual neural systems as recent psycho-physical experiments suggest.

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“…Under the influence of multiplicative noises, the extended-system correlative of the phenomenon of noise-induced transitions (NIT) [1], namely a purely noise-induced phase transition (NIPT), may occur. A comprehensive account of the many ways the phenomenon may take place can be found in [2,3]. However, for consistency with our previous work [4,5], we shall restrict here to the 1994 model by Van den Broeck, Parrondo and Toral (VPT) [6,7], in which they proposed the following mechanism for the NIPT:…”

Section: Introductionmentioning

confidence: 95%

Noise-Induced Phase Transitions: Effects of the Noises’ Statistics and Spectrum

Deza

Wio

Fuentes

2007

AIP Conference Proceedings

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Abstract. The local, uncorrelated multiplicative noises driving a second-order, purely noise-induced, ordering phase transition (NIPT) were assumed to be Gaussian and white in the model of [Phys. Rev. Lett. 73, 3395 (1994)]. The potential scientific and technological interest of this phenomenon calls for a study of the effects of the noises' statistics and spectrum. This task is facilitated if these noises are dynamically generated by means of stochastic differential equations (SDE) driven by white noises. One such case is that of Ornstein-Uhlenbeck noises which are stationary, with Gaussian pdf and a variance reduced by the self-correlation time τ, and whose effect on the NIPT phase diagram has been studied some time ago. Another such case is when the stationary pdf is a (colored) Tsallis' q-Gaussian which, being a fat-tail distribution for q > 1 and a compact-support one for q < 1, allows for a controlled exploration of the effects of the departure from Gaussian statistics. As done before with stochastic resonance and other phenomena, we now exploit this tool to study-within a simple mean-field approximation and with an emphasis on the order parameter and the "susceptibility"-the combined effect on NIPT of the noises' statistics and spectrum. Even for relatively small τ, it is shown that whereas fat-tail noise distributions (q > 1) counteract the effect of self-correlation, compact-support ones (q < 1) enhance it. Also, an interesting effect on the susceptibility is seen in the last case.

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Noise-Induced Order in Extended Systems: A Tutorial

Cited by 13 publications

References 42 publications

Noise-Induced Scenario for Inverted Phase Diagrams

Noise-Induced Scenario for Inverted Phase Diagrams

Efficient Transmission of Subthreshold Signals in Complex Networks of Spiking Neurons

Noise-Induced Phase Transitions: Effects of the Noises’ Statistics and Spectrum

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