Dynamical selection of critical exponents

Wiese, Kay Joerg

doi:10.1103/physreve.93.042105

Cited by 12 publications

(22 citation statements)

References 39 publications

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“…Thus it would enable experimental predictions, for example with regard to the spatial structure of correlated activity. For the Manna sandpile model [139] such a continuous theory has been formulated and it has been found to belong to the universality class of directed percolation with a conserved quantity (C-DP) [140][141][142][143]; the conservation of the number of sand grains here gives rise to the conserved quantity. Belonging to the C-DP universality class, the Manna sandpile model in particular features an absorbing state.…”

Section: Discussionmentioning

confidence: 99%

Self-consistent formulations for stochastic nonlinear neuronal dynamics

et al. 2020

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Neuronal networks are interesting physical systems in various respects: they operate outside thermodynamic equilibrium [1], a consequence of directed synaptic connections that prohibit detailed balance [2]; they show relaxational dynamics and hence do not conserve but rather constantly dissipate energy; and they show collective behavior that self-organizes as a result of exposure to structured, correlated inputs and the interaction among their constituents. But their analysis is complicated by three fundamental properties: Neuronal activity is stochastic, the input-output transfer function of single neurons is non-linear, and networks show massive recurrence [3] that gives rise to strong interaction effects. They hence bear similarity with systems that are investigated in the field of (quantum) many particle systems. Here, as well, (quantum) fluctuations need to be taken into account and the challenge is to understand collective phenomena that arise from the non-linear interaction of their constituents. Not surprisingly, similar methods can in principle be used to study these two a priori distinct system classes [4][5][6][7][8].But so far, the techniques employed within theoretical neuroscience just begin to harvest this potential. Here we will take essential steps towards this goal. Concretely, we adapt methods from statistical field theory and functional renormalization group techniques to the study of neuronal dynamics.A central motivation for this work is a coherent presentation of the technical machinery, which is well-developed in other fields of physics [9], to study the statistics and in particular phase transitions in stochastic neuronal systems and to provide a bridge between the stochastic dynamics and effective descriptions with reduced complexity.The large number of synaptic inputs to each neuron in a network allows the application of mean-field theory [10-12] to explain many dynamical phenomena, among them first order phase transitions. The transition from a quiescent to a highly active state in a bistable neuronal network is a prime example of a first order phase transition in neuronal networks [12]. The activation of attractors embedded into the connectivity of a Hopfield network is a second [13]. Combined with linear response theory, network fluctuations can quantitatively be described in binary [14][15][16] and in spiking networks [17][18][19][20][21][22]. Also transitions into oscillatory states by Andronov-Hopf bifurcations are in the realm of linear response theory around a mean-field solution [23][24][25][26].Second order phase transitions in neuronal networks are more challenging because the behavior of the system is dominated by fluctuations on all length scales, so that mean-field theory and its systematic correction by loopwise expansion break down [4]. But understanding these transitions is highly interesting from a neuroscientific point of view, because networks then show large susceptibility to signals. Moreover, signatures of critical states are found ubiquitously in experiments: Par...

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Section: Discussionmentioning

confidence: 99%

Self-consistent formulations for stochastic nonlinear neuronal dynamics

et al. 2020

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“…The eigenvalue λ = 0 corresponds to the fact that one can redefine ζ as ζ = r/[b(T − t) δc ] with arbitrary b and there is a freedom in the choice of b. For a BSS of first kind (δ > δ c ) the boundary condition at ζ = 0 has to be imposed using additional physical arguments [58]. The natural choice is considering the perturbations which resemble the presence of higher order cumulants in the bare disorder distribution.…”

Section: F Stability Of the Nonanalytic Fixed Pointmentioning

confidence: 99%

Disorder-Driven Quantum Transition in Relativistic Semimetals: Functional Renormalization via the Porous Medium Equation

2018

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In the presence of randomness, a relativistic semimetal undergoes a quantum transition towards a diffusive phase. A standard approach relates this transition to the U (N ) Gross-Neveu model in the limit of N → 0. We show that the corresponding fixed point is infinitely unstable, demonstrating the necessity to include fluctuations beyond the usual Gaussian approximation. We develop a functional renormalization group method amenable to include these effects and show that the disorder distribution renormalizes following the so-called porous medium equation. We find that the transition is controlled by a nonanalytic fixed point drastically different from that of the U (N ) Gross-Neveu model. Our approach provides a unique mechanism of spontaneous generation of a finite density of states and also characterizes the scaling behavior of the broad distribution of fluctuations close to the transition. It can be applied to other problems where nonanalytic effects may play a role, such as the Anderson localization transition.Introduction. -The interplay between disorder and quantum fluctuations leads to unique phenomena, the most remarkable being the Anderson localization. After more than half a century of intensive efforts, it remains a topical subject of research with applications to various domains of physics ranging from condensed matter to cold atoms and light propagation [1]. Remarkably, a different type of disorder-driven quantum phase transition was discovered recently when considering waves with a quantum relativistic dispersion relation [2]. This transition happens between a pseudoballistic phase and a diffusive metal as a function of the disorder strength (or the energy). It is predicted to occur in particular in the recently discovered three-dimensional (3D) Weyl [3,4] and Dirac semimetals [5][6][7] in which, respectively, two and four electronic bands cross linearly at isolated points. However we expect these phenomena to be relevant to other relativistic waves beyond condensed matter, such as ultracold atoms [8].In spite of numerous efforts, the understanding of this transition remains elusive. In this Letter we show that the fluctuations of the randomness beyond the standard Gaussian approximation invalidate previous fieldtheoretic descriptions of this transition. A very similar mechanism occurs in the context of the Anderson transition: there discrepancies between the results obtained using renormalization group (RG) and numerical simulations grow with the number of loops [9]. One may attribute them to the existence of infinitely many relevant operators of the associated field theory [10] which destabilize the fixed point (FP) usually considered to describe the transition [11][12][13]. We find that the same problem appears at the new semimetal-diffusive metal transition. We demonstrate how to overcome this obstacle by deriving and solving a functional renormalization group (FRG) for the whole (non-Gaussian) disorder distribution. To our knowledge this solution consti-tutes the only example of an analytical de...

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“…This again leads to the full set of equations (2.1,2.2), and finally to the EGEP response functional (2.9). Or, as some readers might prefer, one can recast the master-equation corresponding to the reaction scheme of the SWIR (1.1) as a coherent state path integral (CSPI)-action [18][19][20][21][22][23][24][25] and then integrate out the coherent fields corresponding to S, W , and R followed by a naïve continuum approximation. After switching to number densities as field variables via the Grassberger transformation and deletion of irrelevant operators, see e.g.…”

Section: Response Functionalmentioning

confidence: 99%

Attacks and infections in percolation processes

Janssen

Stenull

2017

J. Phys. A: Math. Theor.

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We discuss attacks and infections at propagating fronts of percolation processes based on the extended general epidemic process. The scaling behavior of the number of the attacked and infected sites in the long time limit at the ordinary and tricritical percolation transitions is governed by specific composite operators of the field-theoretic representation of this process. We calculate corresponding critical exponents for tricritical percolation in mean-field theory and for ordinary percolation to 1-loop order. Our results agree well with the available numerical data.

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Dynamical selection of critical exponents

Cited by 12 publications

References 39 publications

Self-consistent formulations for stochastic nonlinear neuronal dynamics

Self-consistent formulations for stochastic nonlinear neuronal dynamics

Disorder-Driven Quantum Transition in Relativistic Semimetals: Functional Renormalization via the Porous Medium Equation

Attacks and infections in percolation processes

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