Rhythmic behavior of an Ising Model
with dissipation at low temperature

Cerf, Raphaël; Pra, Paolo Dai; Formentin, Marco; Tovazzi, Daniele

doi:10.30757/alea.v18-20

“…Here z, q i := zq i (z; t)dz, with i = 1, 2. The regularizing effect of the second-order partial derivatives guarantees that, for t ∈ [0, T ], the laws of x(t) and y(t) have respective densities q 1 (•; t) and q 2 (•; t) solving (5). By using the finite element method [23], we performed numerical simulations of system (5) starting from the initial distributions q 1 (x; 0) = q 2 (x; 0) = δ 0.8 (x).…”

Section: The Fokker-planck Equationmentioning

confidence: 99%

Noise-induced periodicity in a frustrated network of interacting diffusions

Marini¹,

Andreis²,

Collet³

et al. 2022

Preprint

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We investigate the emergence of a collective periodic behavior in a frustrated network of interacting diffusions. Particles are divided into two communities depending on their mutual couplings. On the one hand, both intra-population interactions are positive; each particle wants to conform to the average position of the particles in its own community. On the other hand, inter-population interactions have different signs: the particles of one population want to conform to the average position of the particles of the other community, while the particles in the latter want to do the opposite. We show that this system features the phenomenon of noise-induced periodicity: in the infinite volume limit, in a certain range of interaction strengths, although the system has no periodic behavior in the zero-noise limit, a moderate amount of noise may generate an attractive periodic law.

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“…For example, neurons neither have any tendency to behave periodically on their own, nor are subject to any periodic forcing; nevertheless, they organize to produce a regular motion perceived at the macroscopic scale [28]. Various models of large families of interacting particles showing self-sustained oscillations have been proposed; we refer the reader to [1,2,4,5,7,[9][10][11]13,15,16,19,20], where possible mechanisms leading to a rhythmic behavior are discussed and many related references are given.…”

Section: Introductionmentioning

confidence: 99%

Noise-induced periodicity in a frustrated network of interacting diffusions

Marini

¹

,

Andreis

²

,

Collet

³

et al. 2023

Nonlinear Differ. Equ. Appl.

Self Cite

0

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We investigate the emergence of a collective periodic behavior in a frustrated network of interacting diffusions. Particles are divided into two communities depending on their mutual couplings. On the one hand, both intra-population interactions are positive; each particle wants to conform to the average position of the particles in its own community. On the other hand, inter-population interactions have different signs: the particles of one population want to conform to the average position of the particles of the other community, while the particles in the latter want to do the opposite. We show that this system features the phenomenon of noise-induced periodicity: in the infinite volume limit, in a certain range of interaction strengths, although the system has no periodic behavior in the zero-noise limit, a moderate amount of noise may generate an attractive periodic law.

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“…Self-organized rhythmic oscillation in stochastic systems has been studied in different contexts in biology and physics. Cerf, Dai Pra, Formentin and Tovazzi [3] study spin systems with nearest neighbour interaction along a circuit which show the following behaviour. Starting from magnetisation (all spins equal to 1, say) a rather long waiting time is needed to observe flipping of a first spin, rapidly followed -in virtue of the structure of the interaction-by spins flipping at successive neighbouring sites along the circuit which leads to magnetisation of opposite sign (all spins equal to −1, say).…”

Section: Introductionmentioning

confidence: 99%

On the long time behaviour of single stochastic Hodgkin-Huxley neurons with constant signal, and a construction of circuits of interacting neurons showing self-organized rhythmic oscillations

Höpfner

¹

2023

Mathematical Neuroscience and Applications

0

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The stochastic Hodgkin-Huxley neurons considered in this paper replace time-constant deterministic input $a dt$ of the classical deterministic model by increments $\vartheta dt + dX_t$ of a stochastic process: $X$ is Ornstein-Uhlenbeck with volatility $\sigma>0$ and backdriving force $\tau>0$, and we call $\vartheta>0$ the signal. We have ergodicity and strong laws of large numbers for various functionals of the process, and characterize 'quiet behaviour' and 'regular spiking' as events whose probability depends on the parameters $(\tau,\sigma)$ and on the signal $\vartheta$. The notions of quiet behaviour and regular spiking allow for a construction of circuits of interacting stochastic Hodgkin-Huxley neurons, combining excitation with inhibition according to a bloc structure along the circuit, on which self-organized rhythmic oscillations can be observed.

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Rhythmic behavior of an Ising Model with dissipation at low temperature

Cited by 5 publications

References 21 publications

Noise-induced periodicity in a frustrated network of interacting diffusions

Noise-induced periodicity in a frustrated network of interacting diffusions

Noise-induced periodicity in a frustrated network of interacting diffusions

On the long time behaviour of single stochastic Hodgkin-Huxley neurons with constant signal, and a construction of circuits of interacting neurons showing self-organized rhythmic oscillations

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