A Curie-Weiss Model with Dissipation

Pra, Paolo Dai; Fischer, Markus; Regoli, Daniele

doi:10.1007/s10955-013-0756-2

Cited by 24 publications

(43 citation statements)

References 18 publications

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“…With this observation, we follow a long tradition, see e.g. [33,34] in the framework of nonlinear diffusion processes, or Dai Pra, Fischer and Regoli (2015) [12] and Collet, Dai Pra and Formentin (2015) [9].…”

Section: Oscillationsmentioning

confidence: 86%

“…This study follows a long tradition, starting probably with [33,34] in the framework of nonlinear diffusion processes, or Dai Pra, Fischer and Regoli (2015) [12], Collet, Dai Pra and Formentin (2015) [9] and Collet, Formentin and Tovazzi (2016) [10]. In all these papers, rhythmic behavior arises in the mean field limit as a consequence of the nonlinear "Mc-Kean-Vlasov"-type dynamics of the limit process.…”

Section: Introductionmentioning

confidence: 99%

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Spiking neurons : interacting Hawkes processes, mean field limits and oscillations

Löcherbach

Coeurjolly²,

Leclercq-Samson³

2017

ESAIM: Procs

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Abstract. This paper gives a short survey of some aspects of the study of Hawkes processes in high dimensions, in view of modeling large systems of interacting neurons. We first present a perfect simulation result allowing for a graphical construction of the process in infinite dimension. Then we turn to the study of mean field limits and show how in some cases the delays present in the Hawkes intensities give rise to oscillatory behavior of the limit process.Résumé. Cet article résume brièvement certains aspects de l'étude de processus de Hawkes en grande dimension, en vue d'une modélisation de systèmes de neurones en interactions. Nous présentons d'abord un résultat de simulation parfaite qui donne lieu à une construction graphique du processus en dimension infinie. Ensuite nous étudions des limites de champ moyen de processus de Hawkes et montrons comment les délais apparaissant dans le processus d'intensité engendrent des oscillations du processus limite.

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Section: Oscillationsmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Spiking neurons : interacting Hawkes processes, mean field limits and oscillations

Löcherbach

Coeurjolly²,

Leclercq-Samson³

2017

ESAIM: Procs

View full text Add to dashboard Cite

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“…Example. If we set Γ (z) = 1 + tanh(z), the analysis of the phase diagram done in [11,Sect. 3] is recovered.…”

Section: Under the Assumptionmentioning

confidence: 99%

“…The process ζ n is the effective potential felt by the i-th particle and it is damped in time according to the equation dζ n = −αζ n dt + βdm n (α, β > 0), where m n = n −1 n i=1 σ i is the empirical average of the spins. Compared to the original version of the dissipative Curie-Weiss model [11], we do not have any external noise in the evolution of ζ n (as in [14]); on the other hand, we work with arbitrary transition rates rather than sticking on the case Γ (x) = 1 + tanh(x). Depending on the choice of the function Γ , the phase structure of the infinite volume system may become very rich.…”

Section: Introductionmentioning

confidence: 99%

Path-space moderate deviations for a class of Curie–Weiss models with dissipation

Collet

Kraaij

2020

Stochastic Processes and their Applications

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We modify the Glauber dynamics of the Curie-Weiss model with dissipation in [11] by considering arbitrary transition rates and we analyze the phase-portrait as well as the dynamics of moderate fluctuations for macroscopic observables. We obtain path-space moderate deviation principles via a general analytic approach based on the convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton-Jacobi equations. The moderate asymptotics depend crucially on the phase we are considering and, moreover, their behavior may be influenced by the choice of the rates.

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“…One of the goals of the mathematical theory in this field is to understand which types of microscopic interactions and mechanisms can lead to or enhance the above self-organization. Among others, we cite noise ( [10], [20], [22]), dissipation in the interaction potential ( [1], [6], [7], [9]), delay in the transmission of information and/or frustration in the interaction network ( [8], [12], [21]). In particular, in [12] the authors consider non-Markovian dynamics, studying systems of interacting nonlinear Hawkes processes for modeling neurons.…”

Section: Introductionmentioning

confidence: 99%

Oscillatory Behavior in a Model of Non-Markovian Mean Field Interacting Spins

2020

Self Cite

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We analyze a non-Markovian mean field interacting spin system, related to the Curie-Weiss model. We relax the Markovianity assumption by replacing the memoryless distribution of the waiting times of a classical spin-flip dynamics with a distribution with memory. The resulting stochastic evolution for a single particle is a spin-valued renewal process, an example of two-state semi-Markov process. We associate to the individual dynamics an equivalent Markovian description, which is the subject of our analysis. We study a corresponding interacting particle system, where a mean field interaction is introduced as a time scaling, depending on the overall magnetization of the system, on the waiting times between two successive particle's jumps. Via linearization arguments on the Fokker-Planck mean field limit equation, we give evidence of emerging periodic behavior. Specifically, numerical analysis on the discrete spectrum of the linearized operator, characterized by the zeros of an explicit holomorphic function, suggests the presence of a Hopf bifurcation for a critical value of the temperature, which is in accordance with the one obtained by simulating the N -particle system.

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A Curie-Weiss Model with Dissipation

Cited by 24 publications

References 18 publications

Spiking neurons : interacting Hawkes processes, mean field limits and oscillations

Spiking neurons : interacting Hawkes processes, mean field limits and oscillations

Path-space moderate deviations for a class of Curie–Weiss models with dissipation

Oscillatory Behavior in a Model of Non-Markovian Mean Field Interacting Spins

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