Modeling Animal Communication Using Multivariate Hawkes Processes with Additive Excitation and Multiplicative Inhibition

This paper proposes a novel multivariate Hawkes process framework combining additive excitation and multiplicative inhibition to effectively model animal acoustic communication, which is validated through simulations and applied to reveal distinct interaction patterns in meerkat and baleen whale datasets.

The Gist

Imagine you are sitting in a busy coffee shop. You hear a conversation start, then someone laughs, which makes another person speak up, but then a loud espresso machine noise makes everyone go quiet for a moment.

Animal communication is a lot like this coffee shop, but instead of words, animals use sounds (calls). Scientists want to understand the "rules" of this conversation: Does a specific call make others talk more? Does it make them shut up? And does one type of animal's call affect a different type of animal?

This paper introduces a new, smarter way to model these conversations using a mathematical tool called a Hawkes Process. Here is the breakdown in simple terms:

1. The Old Way vs. The New Way

The Old Way (The "Additive" Problem):
Imagine trying to explain the coffee shop noise by just adding numbers.

• "The conversation adds +5 decibels."
• "The espresso machine subtracts -5 decibels."
• The Problem: If you add them up and get zero or a negative number, the math breaks because you can't have "negative noise." Also, it's hard to tell if the silence was caused by the machine (inhibition) or if the conversation just naturally died out. It's like trying to figure out if a car stopped because you hit the brakes or because the engine died, when both actions are just "subtracting speed" from the total.

The New Way (The "Additive-Multiplicative" Solution):
The authors propose a clever new formula: Excitation is added, but Inhibition is a dimmer switch.

• Excitation (Adding): When a meerkat makes a "Close Call," it's like adding more people to the conversation. It directly boosts the chance of more calls.
• Inhibition (The Dimmer): When a meerkat makes an "Alarm Call," it doesn't just subtract noise; it turns down the volume knob for everyone. It scales down the entire background noise and the excitement.
• Why it's better: This keeps the math from breaking (you can't have negative volume) and makes it much easier to see exactly how much "excitement" is happening versus how much "suppression" is happening. It's like having a separate dial for "how loud the party is" and a separate switch for "how much the music is turned down."

2. The Two Real-World Tests

The authors tested their new "dimmer switch" model on two very different animal groups.

Case A: The Meerkats (The Busy Coffee Shop)

Meerkats live in groups and have three main types of calls:

1. Close Calls: "I'm here, let's stay together."
2. Alarm Calls: "Danger! Run!"
3. Short Notes: "I'm just chilling/running."

What they found:

• Self-Excitement: If one meerkat makes a "Close Call," others are likely to make one too (like a ripple effect).
• Cross-Excitement: An "Alarm Call" often triggers "Short Notes" (because they are running to hide).
• Cross-Inhibition (The Dimmer): Here is the cool part. "Close Calls" and "Short Notes" actually suppress each other.
  • Analogy: Think of "Close Calls" as a "Work Mode" and "Short Notes" as a "Play Mode." You can't really do both at the exact same time. If a meerkat is busy foraging (Close Call), it stops making the "I'm running" sounds. The model successfully identified that these two behaviors act like a dimmer switch on each other.

Case B: The Whales (The Distant Neighbors)

The authors looked at two types of whales: Humpbacks and North Atlantic Right Whales. They live in the same ocean area but have different diets.

What they found:

• Self-Excitement: If a Humpback sings, other Humpbacks are likely to sing back. If a Right Whale calls, other Right Whales call back.
• No Cross-Talk: Surprisingly, one species' calls did not trigger or suppress the other species.
  • Analogy: Imagine two different radio stations playing in the same room. Station A plays rock, Station B plays jazz. The rock fans get excited when they hear rock, and the jazz fans get excited when they hear jazz. But the rock fans don't care if the jazz station plays, and they don't get quiet because of it. They are just in different "worlds."
• Noise: They also found that for Humpbacks, loud background ocean noise actually turned down their "volume" (they called less), likely because they couldn't hear each other.

3. Why This Matters

This paper isn't just about math; it's about understanding animal behavior without getting confused by the numbers.

• Clarity: By using the "dimmer switch" (multiplicative inhibition) instead of just subtraction, scientists can clearly see how much an animal is being "encouraged" to talk versus "discouraged" to talk.
• Identifiability: It solves a puzzle where old math couldn't tell the difference between "no one is talking because they are bored" and "no one is talking because they are scared."
• Future Use: This model can be used for anything where events trigger or stop other events, from stock market crashes to the spread of viruses on social media.

In a nutshell: The authors built a better mathematical "translator" for animal conversations. They discovered that meerkats have a complex dance of encouraging and suppressing each other based on what they are doing, while whales mostly just talk to their own kind, ignoring the neighbors.

Technical Summary

Here is a detailed technical summary of the paper "Modeling Animal Communication Using Multivariate Hawkes Processes with Additive Excitation and Multiplicative Inhibition."

1. Problem Statement

Animal acoustic communication often exhibits complex temporal dependencies where calls can either trigger (excitation) or suppress (inhibition) subsequent calls within and across individuals, call types, or species. While Hawkes processes are the standard framework for modeling self-excitation (where past events increase future intensity), incorporating inhibition (where past events decrease future intensity) presents significant challenges:

• Identifiability Issues: In standard additive models where both excitation and inhibition are added to the intensity, it is difficult to distinguish whether a lack of events is due to strong inhibition or a low background rate.
• Parameter Interpretation: Additive models often require link functions (e.g., ReLU, sigmoid) to ensure non-negative intensity, which obscures the direct interpretation of parameters (e.g., the "branching" nature of excitation is lost).
• Confounding: In multivariate settings, jointly modeling excitation and inhibition additively can lead to confounding between background activity and interaction effects, making it impossible to separately quantify the expected number of background events versus excitation-driven events.

2. Methodology

The authors propose a novel class of Multivariate Hawkes Processes that combines additive excitation with multiplicative inhibition.

Model Specification

For a system with $K$ event types (marks), the conditional intensity function $\lambda_k(t)$ for mark $k$ is defined as:
$$\lambda_k(t | H_t) = \underbrace{\left( \mu_k(t) + \sum_{\ell=1}^K \sum_{t_i < t, m_i=\ell} \alpha_{\ell,k} g_{\ell,k}(t-t_i) \right)}_{\text{Additive Excitation Component}} \times \underbrace{\exp\left( - \sum_{\ell=1}^K \sum_{t_i < t, m_i=\ell} \gamma_{\ell,k} h_{\ell,k}(t-t_i) \right)}_{\text{Multiplicative Inhibition Component}}$$

• Additive Excitation: The term in the first parenthesis represents the standard Hawkes structure. $\mu_k(t)$ is the background intensity, and the summation represents the triggering effect of past events (both self and cross-type) via kernels $g$.
• Multiplicative Inhibition: The exponential term acts as a scaling factor. If recent events occur, the exponent becomes more negative, reducing the overall intensity. This preserves the non-negativity of the intensity naturally without requiring a link function.
• Identifiability Constraint: To prevent simultaneous excitation and inhibition for the same pair of marks (which causes identifiability issues), the authors impose a constraint: for any pair $(\ell, k)$, either $\alpha_{\ell,k} > 0$ (excitation) or $\gamma_{\ell,k} > 0$ (inhibition), but not both. This is enforced using latent indicator variables.

Bayesian Inference

• Likelihood Decomposition: The multiplicative structure allows the likelihood to be decomposed using latent branching variables ($z_i$). This indicates whether an event is a background event or triggered by a specific past event.
• Sampling: A Metropolis–Hastings-within-Gibbs sampler is used. The latent branching structure enables efficient updates of parameters by separating the background and excitation components, improving MCMC mixing.
• Model Assessment:
  • Random Time Change Theorem (RTCT): Used to validate model fit by transforming event times into inter-event increments that should follow an Exp(1) distribution.
  • WAIC (Widely Applicable Information Criterion): Used for model comparison and selection.

3. Key Contributions

1. Novel Parameterization: The introduction of an additive-multiplicative formulation that separates background, excitation, and inhibition cleanly.
2. Improved Interpretability: Unlike additive-additive models, this formulation preserves the branching process interpretation of excitation (offspring contributions) and allows for the direct quantification of the expected number of background vs. excitation-driven events via compensators.
3. Solved Identifiability: By enforcing an exclusivity constraint (excitation OR inhibition per pair) and using multiplicative inhibition, the model avoids the parameter confounding common in additive models with negative coefficients.
4. Computational Efficiency: The decomposition of the likelihood facilitates faster and more stable MCMC convergence compared to models requiring full intensity evaluation at every step.

4. Results

Simulation Experiments

• The authors simulated data under three scenarios: (i) Excitation + Inhibition, (ii) Excitation only, and (iii) Inhibition only.
• Findings: The proposed model (i) successfully recovered the true parameters and provided the best fit (lowest WAIC and MSD) across all scenarios. Models lacking the correct component (e.g., fitting an excitation-only model to data with inhibition) showed significant deviations in Q-Q plots and higher WAIC scores, demonstrating the model's ability to distinguish between interaction types.

Real Data Application 1: Meerkat Calls

• Data: Three call types (Close, Alarm, Short Note) from wild meerkats over three days.
• Results: The model revealed a complex interaction structure:
  • Self-Excitation: Strong for all call types (Alarm and Short Note were strongest).
  • Cross-Excitation: Alarm calls excite Short Note calls.
  • Cross-Inhibition: Significant mutual inhibition between Close and Short Note calls.
• Biological Interpretation: The inhibition between Close (foraging) and Short Note (non-foraging contexts like sunning or submission) calls aligns with biological expectations that these behaviors are mutually exclusive. The model successfully captured these distinct temporal dynamics.

Real Data Application 2: Baleen Whale Calls

• Data: Humpback whales (HUWH) and North Atlantic Right whales (RIWH) in Stellwagen Bank.
• Results:
  • Excitation: Strong within-species self-excitation for both species.
  • Inhibition: No significant cross-species or within-species inhibition was detected; the excitation-only model was preferred over the full model.
  • Noise Effect: Ambient noise significantly reduced calling intensity for Humpback whales.
  • Decay Differences: Right whale excitation persisted longer (2.86 mins) compared to Humpback whales (28 secs), likely due to the diversity of Humpback call types (including breeding songs vs. foraging calls) and their transiting behavior in the study area.

5. Significance and Future Work

• Scientific Impact: This work provides a robust statistical framework for decoding animal communication systems, moving beyond simple correlation to quantify causal-like temporal dependencies (excitation/inhibition) in a multivariate setting.
• Ecological Insight: The model successfully differentiated between species that exhibit complex social coordination (meerkats) and those that do not show strong cross-species acoustic interference (whales in this context), offering new tools for behavioral ecology.
• Future Directions:
  • Spatial Dependence: Extending the model to include spatial proximity (spatio-temporal Hawkes processes) to account for the physical location of callers.
  • Individual-Level Analysis: Partitioning data by individual to study within-individual vs. cross-individual dynamics.
  • Broader Applications: The methodology is applicable beyond biology to finance (market shocks), epidemiology (disease spread), and social networks (information diffusion).

In summary, the paper presents a mathematically rigorous and biologically interpretable solution to the problem of modeling competing excitatory and inhibitory forces in temporal point processes, validated through simulation and real-world animal communication datasets.