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

2025-07-30

AI Summary

Conventional Hawkes processes assume fixed memory lengths, failing to capture the dynamic resetting of neuronal memory following spiking events. Method: We propose a novel nonlinear multivariate Hawkes process with adaptive memory length—where the memory horizon evolves endogenously based on event history. We rigorously establish existence and develop a unified maximum-likelihood inference framework jointly estimating kernel functions and memory-length parameters, accommodating both classical fixed-memory and adaptive-memory settings. Our approach integrates point-process theory, nonlinear kernel modeling, and optimization techniques to jointly identify excitatory/inhibitory effects and complex history-dependent dependencies. Results: Extensive experiments on synthetic data and real calcium imaging recordings from neurons demonstrate the model’s effectiveness and robustness, significantly improving accuracy and interpretability in modeling intricate neurodynamics.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems

Abstract

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. Multivariate Hawkes processes are past-dependant point processes originally introduced tomodel excitation effects, later extended to a nonlinear framework to account for the opposite effect, known as inhibition. Our model generalises classical Hawkes processes, with or without inhibition, focusing on the situation where the probability of an event occurring within a given subprocess depends solely on the history since its last event. 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.

Problem

• Modeling neuron memory reset with variable-length memory Hawkes processes
• Generalizing classical Hawkes processes to include event-dependent memory dynamics
• Developing likelihood maximization for identifying variable memory in neural activity

Innovation

• Variable length memory Hawkes processes
• Likelihood maximization for dynamics identification
• Application to neuronal activity modeling