Fluctuations for spatially extended Hawkes processes

Chevallier, Julien; Ost, Guilherme

doi:10.1016/j.spa.2020.03.015

Cited by 8 publications

(7 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One should also mention the functional fluctuation result recently obtained in [42], also in a pure mean-field setting. A result closer to our case with spatial extension is [18], where a functional CLT is obtained for the spatial profile U N around its limit. Note here that all of these works provide approximation results of quantities such that λ N or U N that are either valid on a bounded time interval [0, T ] or under strict growth condition on T (see in particular the condition T N → 0 for the CLT in [28]), whereas we are here concerned with time-scales that grow polynomially with N .…”

Section: Link With the Literaturesupporting

confidence: 77%

Multivariate Hawkes processes on inhomogeneous random graphs

Agathe-Nerine¹

2022

Stochastic Processes and their Applications

View full text Add to dashboard Cite

Section: Link With the Literaturesupporting

confidence: 77%

Multivariate Hawkes processes on inhomogeneous random graphs

Agathe-Nerine¹

2022

Stochastic Processes and their Applications

View full text Add to dashboard Cite

“…In addition to the law of large numbers in the mean-field model, a central limit theorem is proved in [22]. Another branch of research studies a mean-field limit of a spatially extended (geometric) Hawkes process, showing a law of large numbers [9] and central limit theorem [10]. Here, a first proof of the neural field equation could be obtained.…”

Section: Introductionmentioning

confidence: 95%

Mean-field limits for non-linear Hawkes processes with excitation and inhibition

Pfaffelhuber¹,

Rotter²,

Stiefel³

2021

Preprint

View full text Add to dashboard Cite

We study a multivariate, non-linear Hawkes process Z N on the complete graph with N nodes. Each vertex is either excitatory (probability p) or inhibitory (probability 1− p). We take the meanfield limit of Z N , leading to a multivariate point process Z. If p = 1 2 , we rescale the interaction intensity by N and find that the limit intensity process solves a deterministic convolution equation and all components of Z are independent. In the critical case, p = 1 2 , we rescale by N 1/2 and obtain a limit intensity, which solves a stochastic convolution equation and all components of Z are conditionally independent.

show abstract

“…The circular environment we study can be regarded as a spacediscretized stochastic neural field but the exact expression of the continuous equation and its physical interpretation is unclear and will be subject of future work. Ideally, one would want to be able to prove the convergence to such an equation, as in [170] in the case without STP.…”

Section: Theoretical Challengesmentioning

confidence: 99%

Mesoscopic description of hippocampal replay and metastability in spiking neural networks with short-term plasticity

2022

View full text Add to dashboard Cite

Bottom-up models of functionally relevant patterns of neural activity provide an explicit link between neuronal dynamics and computation. A prime example of functional activity patterns are propagating bursts of place-cell activities called hippocampal replay, which is critical for memory consolidation. The sudden and repeated occurrences of these burst states during ongoing neural activity suggest metastable neural circuit dynamics. As metastability has been attributed to noise and/or slow fatigue mechanisms, we propose a concise mesoscopic model which accounts for both. Crucially, our model is bottom-up: it is analytically derived from the dynamics of finite-size networks of Linear-Nonlinear Poisson neurons with short-term synaptic depression. As such, noise is explicitly linked to stochastic spiking and network size, and fatigue is explicitly linked to synaptic dynamics. To derive the mesoscopic model, we first consider a homogeneous spiking neural network and follow the temporal coarse-graining approach of Gillespie to obtain a “chemical Langevin equation”, which can be naturally interpreted as a stochastic neural mass model. The Langevin equation is computationally inexpensive to simulate and enables a thorough study of metastable dynamics in classical setups (population spikes and Up-Down-states dynamics) by means of phase-plane analysis. An extension of the Langevin equation for small network sizes is also presented. The stochastic neural mass model constitutes the basic component of our mesoscopic model for replay. We show that the mesoscopic model faithfully captures the statistical structure of individual replayed trajectories in microscopic simulations and in previously reported experimental data. Moreover, compared to the deterministic Romani-Tsodyks model of place-cell dynamics, it exhibits a higher level of variability regarding order, direction and timing of replayed trajectories, which seems biologically more plausible and could be functionally desirable. This variability is the product of a new dynamical regime where metastability emerges from a complex interplay between finite-size fluctuations and local fatigue.

show abstract

Fluctuations for spatially extended Hawkes processes

Cited by 8 publications

References 37 publications

Multivariate Hawkes processes on inhomogeneous random graphs

Multivariate Hawkes processes on inhomogeneous random graphs

Mean-field limits for non-linear Hawkes processes with excitation and inhibition

Mesoscopic description of hippocampal replay and metastability in spiking neural networks with short-term plasticity

Contact Info

Product

Resources

About