Brownian motion surviving in the unstable cubic potential and the role of Maxwell's demon

Ornigotti, Luca; Ryabov, Artem; Holubec, Viktor; Filip, Radim

doi:10.1103/physreve.97.032127

Cited by 23 publications

(52 citation statements)

References 83 publications

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“…A rigorous analysis of the complete setting including escape is a problem on its own requiring the introduction of a probability for a path conditioned on the requirement that it has not escaped to infinity. For a leak term equal to zero, this analysis has recently been performed [93]. Another possibility is to consider the time-dependent problem, as it is done for example in the context of laser physics [94].…”

Section: One-loop Correction To the Variance And Higher Order Statisticsmentioning

confidence: 99%

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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: One-loop Correction To the Variance And Higher Order Statisticsmentioning

confidence: 99%

“…93)]. This yields, using the regulator of the form introduced in (36), ∂Γ λ X * ,X * λ X,X exp S λ X,X × exp J T X +J TX…”

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confidence: 99%

Self-consistent formulations for stochastic nonlinear neuronal dynamics

et al. 2020

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“…With increasing ω, the boundary shifts away from maximums ofP 1 andP a . Due to the trajectories trapped in the absorbing state [71,73], the maximum ofP a is slightly farther away from the absorbing boundary than the maximum ofP 1 , and thus, with increasing ω, the tail ofP a with small derivative (small j Da ) reaches the boundary at x = −R before the tail ofP 1 . Hence, for large enough barrier height with respect to the noise strength, the inequality between the rates turns over to κ B (∞) ≥ κ M (∞).…”

Section: Methodsmentioning

confidence: 99%

“…We solve this formula numerically using the method described in Refs. [71,72]. The steady state value of the transition rate predicted using Bullerjahn's method reads…”

Section: Long-time Behavior and Kramers' Methodsmentioning

confidence: 99%

Brownian molecules formed by delayed harmonic interactions

2019

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A time-delayed response of individual living organisms to information exchange within flocks or swarms leads to the formation of complex collective behaviors. A recent experimental setup by Khadka et al., Nat. Commun. 9, 3864 (2018), employing synthetic microswimmers, allows to realize and study such behavior in a controlled way, in the lab. Motivated by these experiments, we study a system of N Brownian particles interacting via a retarded harmonic interaction. For N ≤ 3, we characterize its collective behavior analytically via linear stochastic delay-differential equations, and for N > 3 by Brownian dynamics simulations. The particles form nonequilibrium molecule-like structures which become unstable with increasing number of particles, delay time, and interaction strength. We evaluate the entropy fluxes in the system and develop an approximate time-dependent transition-state theory to characterize transitions between different isomers of the molecules. For completeness, we include a comprehensive discussion of the analytical solution procedure for systems of linear stochastic delay differential equations in finite dimension, and new results for covariance and time-correlation matrices.

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“…We also note that it is possible to construct an effective conservative Markov process that converges towards the distribution Q st (x) in the long-time limit [5]. The process is based on an appropriate return (or resetting) of the diverging trajectories.…”

Section: Quasi-stationary Distributionmentioning

confidence: 99%

Living on the edge of instability

Ryabov¹,

Holubec²,

Berestneva³

2019

J. Stat. Mech.

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Statistical description of stochastic dynamics in highly unstable potentials is strongly affected by properties of divergent trajectories, that quickly leave metastable regions of the potential landscape and never return. Using ideas from theory of Q-processes and quasi-stationary distributions, we analyze position statistics of nondiverging trajectories. We discuss two limit distributions which can be considered as (formal) generalizations of the Gibbs canonical distribution to highly unstable systems. Even though the associated effective potentials differ only slightly, properties of the two distributions are fundamentally different for all highly unstable system. The distribution for trajectories conditioned to diverge in an infinitely distant future is localized and light-tailed. The other distribution, describing trajectories surviving in the meta-stable region at the instant of conditioning, is heavy-tailed. The exponent of the corresponding power-law tail is determined by the leading divergent term of the unstable potential. We discuss different equivalent forms of the two distributions and derive properties of the effective statistical force arising in the ensemble of nondiverging trajectories after the Doob h-transform. The obtained explicit results generically apply to non-linear dynamical models with meta-stable states and fast kinetic transitions.

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Brownian motion surviving in the unstable cubic potential and the role of Maxwell's demon

Cited by 23 publications

References 83 publications

Self-consistent formulations for stochastic nonlinear neuronal dynamics

Self-consistent formulations for stochastic nonlinear neuronal dynamics

Brownian molecules formed by delayed harmonic interactions

Living on the edge of instability

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