Hierarchical excitatory processes for modelling event-time data in the presence of exogenous stimuli

Technical Summary: Hierarchical Excitatory Processes for Modelling Event-Time Data

Problem Statement
Event-time data frequently arise in scientific domains such as neuroscience, epidemiology, and finance, where events occur in the presence of exogenous interventions. A central modelling challenge is characterizing how these external stimuli affect event intensities over time while accounting for non-stationarity, adaptation (e.g., habituation), and heterogeneity across experimental units. Classical point process models, such as inhomogeneous Poisson processes or self-exciting Hawkes processes, are often insufficient because they are not explicitly designed to accommodate systematic changes based on known exogenous stimuli or to capture the evolution of excitation dynamics across repeated interventions. While previous work has attempted to incorporate exogenous effects (e.g., in social media or disease spread models), there is a need for a flexible framework that models the dynamic modulation of excitation strength and allows for the identification of latent subgroups with similar response patterns.

Methodology
The authors propose the Hierarchical Excitatory Process (HEP), a point process model designed for event-time data observed under repeated external stimuli. The framework operates on two levels:

1. Conditional Intensity Modeling: The model assumes events follow an inhomogeneous Poisson process with a conditional intensity function $\lambda(t)$. This intensity is defined as a baseline rate $\lambda_0$ plus a superposition of excitation effects induced by external stimuli occurring at known changepoints $\tau_j$. The excitation is modeled via a kernel function $\phi_{\theta(\tau_j)}(t - \tau_j)$, where the parameters $\theta$ (e.g., jump magnitude $\alpha$, decay rate $\beta$, and delay $\delta$) are not static but evolve dynamically over time.
2. Dynamic Parameter Evolution: The parameters governing the excitation kernels are themselves modeled as functions of time, driven by the history of stimuli. Specifically, the authors employ a hierarchical structure where parameters like $\alpha(s)$ and $\beta(s)$ are modulated by inhibition functions (e.g., scaled exponential decay) that reduce the parameter values following successive stimuli. This captures phenomena such as habituation, where the neuronal response attenuates over repeated stimulation. The model supports various kernel forms, including delayed scaled exponential functions to account for latency between stimulus and response.
3. Inference and Clustering:
   • Parameter Estimation: Inference is conducted via maximum likelihood estimation. The log-likelihood is derived analytically, with exact integration of the intensity function over the observation window, including provisions for censored data (periods with no observation).
   • Model-Based Clustering: To address heterogeneity, the HEP framework is extended to a mixture model. Neurons are assumed to belong to one of $K$ latent groups, each characterized by a distinct set of excitation-inhibition parameters. An Expectation-Maximisation (EM) algorithm is used to estimate group memberships and parameters simultaneously. The optimal number of clusters is selected using the Integrated Completed Likelihood (ICL) criterion.

Key Contributions

• The HEP Framework: Introduction of a flexible point process model that explicitly links event intensities to exogenous stimuli through time-varying excitation parameters, allowing for the modeling of adaptation and habituation.
• Hierarchical Structure: A novel construction where the intensity depends on changepoints via an excitation structure, and the parameters of that excitation are themselves dynamic and history-dependent via latent inhibition or excitation functions.
• Likelihood-Based Inference: Development of a principled likelihood-based approach for parameter estimation, including handling of censored observation intervals common in experimental designs.
• Clustering Capability: Integration of HEPs into a model-based clustering framework to identify latent groups of experimental units (e.g., neurons) sharing similar response dynamics to external stimuli.

Results

• Simulation Studies: Extensive simulations demonstrate the model's ability to accurately recover true model parameters, including the underlying excitation and inhibition functions, even with a limited number of external stimuli. The inferential procedure successfully tracks the true intensity function and predicts intensity for unseen future stimuli. In clustering simulations, the EM algorithm effectively recovers latent group structures, achieving high Adjusted Rand Index (ARI) values after observing just a few stimuli, and the ICL criterion correctly identifies the true number of groups in the majority of cases.
• Application to Aplysia Data: The methodology was applied to optical recordings of neuronal activity from the dorsal pedal ganglia of the sea slug Aplysia californica under four different experimental conditions involving repeated electrical stimuli.
  • The HEP model provided an excellent fit to the observed spike trains, capturing the stimulus-driven increases in firing intensity and the subsequent gradual decay (habituation).
  • Model-based clustering successfully identified distinct groups of neurons (e.g., 9 groups in the June 2023 dataset) exhibiting heterogeneous response patterns. These groups ranged from neurons with strong habituation to those with sustained responses or minimal reaction to stimuli, demonstrating the model's capacity to uncover latent subpopulations that a single set of parameters could not describe.

Significance and Claims
The paper claims that HEPs provide a simple yet flexible framework for practitioners to:

1. Quantify the effect of past exogenous interventions on event rates.
2. Estimate the impact of additional future stimuli.
3. Identify groups of units (such as neurons) exhibiting similar response patterns.
4. Assess evidence of habituation or adaptation in individual units.

The authors note that while the current implementation focuses on inhomogeneous Poisson processes for mathematical tractability, the framework offers a foundation for understanding how external interventions modulate point process data. They acknowledge limitations, such as the independence assumption between neurons and the exclusion of self-exciting or mutually-exciting multivariate structures, suggesting these as potential future extensions. The work emphasizes interpretability, allowing for the characterization of excitability, persistence, and recovery dynamics in a hierarchical manner.