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arxiv: 2507.22867 · v2 · submitted 2025-07-30 · 📊 stat.ME

Hawkes Processes with Variable Length Memory: Existence, Inference and Application to Neuronal Activity

Sacha Quayle , Anna Bonnet , Maxime Sangnier This is my paper

Pith reviewed 2026-05-19 02:41 UTC · model grok-4.3

classification 📊 stat.ME
keywords Hawkes processesvariable length memorynonlinear point processesexistence and uniquenesslikelihood inferenceneuronal spikingmultivariate point processes
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The pith

Nonlinear Hawkes processes with variable length memory exist and can be inferred by likelihood maximization, generalizing models for reset-prone systems like neurons.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a new class of nonlinear Hawkes processes in which each subprocess can have its memory length reset after its own most recent event. This setup lets the intensity ignore earlier history in some cases while still allowing both excitation and inhibition effects. The authors prove that well-defined versions of these processes exist and supply a practical likelihood maximization procedure that recovers parameters for both the new variable-memory case and ordinary fixed-memory Hawkes models. They test the method on synthetic data and on recordings of neuronal spiking activity. The approach is motivated by neuroscience, where a neuron's firing can effectively clear dependence on events before its last spike.

Core claim

The central claim is that a multivariate nonlinear Hawkes process with variable length memory exists under suitable conditions on the intensity functions, and that a likelihood-based estimation procedure can identify whether a given subprocess follows classical or variable-memory dynamics. In the variable-memory case the intensity of a subprocess depends only on the history after its own last event, while still satisfying the integrability requirements needed for the point process to be well-defined.

What carries the argument

The variable-length-memory intensity function, which renders a subprocess independent of all history before its most recent event while preserving the overall existence conditions of the multivariate point process.

If this is right

  • The model can represent neuronal spiking where each spike resets dependence on earlier activity.
  • The same likelihood procedure distinguishes classical Hawkes behavior from reset behavior on the same dataset.
  • Existence is guaranteed for both excitatory and inhibitory nonlinear intensities that satisfy the variable-memory independence condition.
  • The framework applies directly to any multivariate point process in which some components exhibit event-triggered memory resets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar reset mechanisms could be introduced into other point-process families such as Cox or renewal processes for applications outside neuroscience.
  • The inference method might be adapted to streaming data for online detection of memory-length changes.
  • Connections to marked point processes or self-correcting processes could be explored by treating the last-event time as an explicit mark.
  • Empirical tests on additional datasets would show in which regimes the variable-memory extension improves predictive accuracy over standard Hawkes fits.

Load-bearing premise

The intensity of each subprocess can be defined to ignore history before its last event without violating the conditions required for the existence proof to go through.

What would settle it

Generating data from the proposed variable-memory model and then checking whether the likelihood maximizer recovers the true parameters or whether the simulated process satisfies the existence criteria would confirm or refute the claims.

Figures

Figures reproduced from arXiv: 2507.22867 by Anna Bonnet, Maxime Sangnier, Sacha Quayle.

Figure 1
Figure 1. Figure 1: Test results from estimations under (GVM) for Scenarios (HP) and (VM). Colours [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Boxplots of the relative squared error for each group of parameters ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Heatmaps of true parameters µ, α, β and αr. For each scenario, we display the outcomes from Tests 1 to 3, and the relative squared errors for each group of parameters (µ, α, β and αr) after Step 5, by considering vector norms (unlike the bivariate case, errors after Steps 1 and 3 are omitted). As shown in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Test results from estimations under (GVM) for Scenarios (HP), (VM), (GVM) and [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Boxplots of the relative squared error for each group of parameters ( [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Test results from GVM estimations for 25 resampled trials. On the left: red (0) [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Heatmaps of estimated parameters after Step 5. For [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Transition graph of the Markov chain pZ plq qlPN for d “ 2. Denote P1 :“ Pz p1q 1 . Our goal is to show that P1pTz p1q 1 “ 8q “ 0, meaning that the chain returns to z p1q 1 with probability 1. Examining the graph of the Markov chain, for Tz p1q 1 to be infinite, one of two scenarios must occur: either only event times from N 1 are observed indefinitely, or at some point, an event time from N 2 occurs, afte… view at source ↗
Figure 9
Figure 9. Figure 9: Normalised cumulative spike counts Ni ptq{t for the 10 original trials, shown on the normalised time interval r0, 10s, obtained after Step 1 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Normalised cumulative spike counts Ni ptq{t for the 7 trials retained after Step 3 [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Normalised cumulative spike counts Ni ptq{t for the 25 trials obtained after the resampling procedure in Step 6. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
read the original abstract

Multivariate Hawkes processes are past-dependant point processes originally introduced to model excitation effects, later extended to a nonlinear framework to account for the opposite effect, known as inhibition. Motivated by applications in neuroscience, where the memory of a neuron may reset upon firing, we introduce a new class of nonlinear Hawkes processes with variable length memory. Our model generalises classical Hawkes processes, with or without inhibition, describing the situation where the probability of an event occurring within a given subprocess may depend differently on the history before and after its last event. In particular, if the subprocess does not depend on the history before its last event, it is said to have a variable length memory. Our main contributions are to prove existence of such processes, and to derive a workable likelihood maximisation method, capable of identifying both classical and variable memory dynamics. We demonstrate the effectiveness of our approach both on synthetic data, and on a neuronal activity dataset.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a new class of nonlinear Hawkes processes with variable length memory, generalizing classical models to allow the intensity of a subprocess to depend differently on history before and after its last event. The main contributions are a proof of existence for such processes and the derivation of a likelihood maximization method for inference that can identify both classical and variable memory dynamics. The approach is illustrated on synthetic data and a neuronal activity dataset.

Significance. Should the existence result hold under the stated conditions and the inference method prove consistent, this work offers a meaningful extension to Hawkes process modeling, particularly suited for neuroscience applications where neuronal memory may reset upon firing. The capacity to distinguish memory types through likelihood maximization adds practical value for analyzing spiking data.

major comments (2)
  1. [Existence section (likely §3)] The proof of existence needs to address whether the variable-length memory truncation preserves the necessary Lipschitz continuity or boundedness conditions for the intensity function. Standard arguments for nonlinear Hawkes processes rely on these to ensure no finite-time explosion; the paper should explicitly verify that the reset mechanism does not violate them, perhaps by providing a uniform bound independent of the number of events.
  2. [Inference section (likely §4)] The derivation of the likelihood function for the variable memory model should include a discussion of identifiability and consistency of the maximizer, especially when the model can reduce to classical Hawkes under certain parameter choices.
minor comments (2)
  1. [Abstract] Change 'past-dependant' to 'past-dependent'.
  2. [Throughout] Ensure consistent notation for the intensity function and history measures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help strengthen the presentation of our results on variable-length-memory Hawkes processes. We address each major comment below.

read point-by-point responses

  1. Referee: [Existence section (likely §3)] The proof of existence needs to address whether the variable-length memory truncation preserves the necessary Lipschitz continuity or boundedness conditions for the intensity function. Standard arguments for nonlinear Hawkes processes rely on these to ensure no finite-time explosion; the paper should explicitly verify that the reset mechanism does not violate them, perhaps by providing a uniform bound independent of the number of events.

    Authors: We agree that the existence argument should explicitly confirm preservation of the required conditions. In the revised manuscript we will add a short lemma showing that, under the same Lipschitz and boundedness assumptions used for the classical nonlinear Hawkes process, the variable-length truncation yields an intensity that remains Lipschitz continuous with a constant independent of history length and admits a uniform bound independent of the number of events, thereby ruling out finite-time explosion by the standard arguments. revision: yes

  2. Referee: [Inference section (likely §4)] The derivation of the likelihood function for the variable memory model should include a discussion of identifiability and consistency of the maximizer, especially when the model can reduce to classical Hawkes under certain parameter choices.

    Authors: We thank the referee for this suggestion. The revised manuscript will expand the inference section with a paragraph on identifiability, explicitly noting that the variable-memory model nests the classical Hawkes process (recovered when the pre-reset kernel coefficients are set to zero) and that the parameter space remains identifiable under mild separation conditions. We will also sketch consistency of the maximum-likelihood estimator by appealing to standard regularity conditions for point-process likelihoods, with a reference to existing results in the literature. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained; no circular reductions identified

full rationale

The paper defines a new class of nonlinear Hawkes processes with variable-length memory, proves existence under stated conditions, and derives a likelihood maximization procedure for inference. These steps rely on standard point-process theory (compensators, intensity functions) and do not reduce any claimed existence result or estimator to a fitted quantity, self-citation chain, or input by construction. The model specification (truncation at last event) is logically prior to both the existence argument and the subsequent parameter estimation; no equation or theorem is shown to be equivalent to its own inputs. The paper is therefore self-contained against external benchmarks for the purpose of this circularity check.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the paper invokes standard point-process existence conditions and introduces the variable-memory cutoff as a modeling primitive without detailing free parameters or invented entities.

axioms (1)
  • domain assumption The intensity function satisfies conditions that guarantee existence of the point process even when memory is truncated after the last event.
    Existence proof is listed as a main contribution, implying this background regularity is assumed or verified.

pith-pipeline@v0.9.0 · 5692 in / 1150 out tokens · 32064 ms · 2026-05-19T02:41:42.839711+00:00 · methodology

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Reference graph

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