Dynamic Event Units in Adaptive Systems

• Dynamic Event Units (DEUs) are formal abstractions of discrete, adaptive events that encode precise temporal, semantic, and causal information.
• They are applied in event-centric retrieval, control systems, neuromorphic hardware, and process modeling to enhance precision and efficiency.
• Their design enables adaptive causality, exact temporal anchoring, and scalable event-driven computation across diverse applications.

A Dynamic Event Unit (DEU) is a formalization of a discrete event, state change, or process step that possesses adaptive, context-aware properties with respect to time, causality, or operational conditions. Across fields including event-centric retrieval-augmented generation, dynamical systems control, neuromorphic hardware, event-based sensing, process modeling, and causal analysis, DEUs serve as atomic or minimal units of computation, detection, or knowledge, explicitly encoding when and how events occur and how their dependencies or processing should evolve in response to context.

1. Formal Definitions and Scope

DEUs are constructed to capture the precise semantics, temporal location, and adaptive behavior of discrete events:

• Retrieval and knowledge representation: In frameworks such as DyG-RAG, a DEU is defined as a minimal, self-contained factual statement describing a discrete event or a stable state at a specific time point or interval. It is formally represented as

$$\text{DEU} = \{ s_i,\, t_i,\, \text{ID}_{\text{event}},\, \text{ID}_{\text{source}} \}$$

where $s_i$ is the event statement, $t_i$ is a normalized timestamp, and $$\text{ID}_{\text{event}}, \text{ID}_{\text{source}}$$ are unique identifiers (Sun et al., 16 Jul 2025).

• Dynamic causal models: In Dynamic Causality Event Structures (DCESs), DEUs are events augmented with rules for dynamically adding or dropping causal dependencies between events as execution progresses. A DCES is characterized as

$$\Delta = \left( E,\, \rightarrow,\, \mathcal{S},\, \mathcal{G} \right)$$

where $E$ is the event set, $\rightarrow$ is the initial causality, $\mathcal{S}$ (shrinking) and $\mathcal{G}$ (growing) are dynamic relations encoding when dependencies are removed or introduced (Arbach et al., 2018).

• Control and trigger systems: DEUs are embodied as internal dynamic variables (e.g., timers or thresholds) that regulate when a system triggers communication, measurement, or feedback, according to the evolving state and system dynamics (George et al., 2018, Wang et al., 2023, Xu et al., 7 Jan 2024).
• Neuromorphic/event-based hardware: DEUs refer to event-based processing units—such as TDEs (Time–Difference Encoders)—which encode, accumulate, and relay spatiotemporal differences between input events, allowing parallel low-latency computation (Greatorex et al., 17 Jan 2025).

These formulations share the notion that a DEU is not a static timestamped event, but a construct with temporal and operational logic that adapts to system or knowledge state.

2. Temporal Anchoring and Semantic Encoding

A defining property of DEUs is precise temporal locatability combined with semantic granularity:

• Temporal anchoring: Each DEU is linked to an explicit timestamp or time interval, extracted via temporal parsing and normalization. In textual knowledge domains, relative dates (e.g., "last year") are resolved in the context of absolute references to provide exact time anchors (Sun et al., 16 Jul 2025). In control systems or hardware, DEUs track system time via internal dynamic variables (e.g., integration voltage, clock-like timers) that regulate event processing.
• Semantic encoding: DEU content is encoded using representations suitable for downstream processing, such as contextualized embeddings for textual events or explicit mapping to domain ontologies for sensor events (Zhou et al., 2013, Sun et al., 16 Jul 2025). In retrieval tasks, DEUs are represented as concatenated vectors:

$$z_i = \text{Concat}(h_i^{(\text{text})},\, h_i^{(\text{time})})$$

with $h_i^{(\text{text})}$ as the semantic embedding and $h_i^{(\text{time})} = \varphi(t_i)$ as the time embedding (Sun et al., 16 Jul 2025).

• Precision methods: Pruning via eventivity predicates, presence of entities, and numeric indicators ensures that only well-localized, semantically cohesive statements are registered as DEUs. In hardware, calibration routines guarantee accurate mapping from analog signals to event encoding (Greatorex et al., 17 Jan 2025).

The explicit linkage of every DEU to both its meaning and its occurrence time distinguishes it from broader event modeling techniques that either lack semantic precision or fail to capture temporal specificity.

3. Causality and Adaptivity in Event Modeling

DEUs are central in frameworks where the causality or operational rules governing event sequencing can change at runtime:

• Dynamic Causality: In DCES theory, DEUs can modify the causal structure by activating “dropping” or “adding” relations:

$\drops{d}{c}{t} \text{ (“event $d$drops the$c \to t$ dependency”)}, ~~ \addcause{a}{c}{t} \text{ (“event $a$adds$c$as a precondition to$t$”) }$

This enables compact representation of systems with exceptions, alternative flows, and evolving prerequisites, directly supporting workflow, process management, and control system modeling without event duplication (Arbach et al., 2018).

• Adaptivity in control/consensus: In multi-agent and control settings, DEUs take the form of dynamically evolving internal thresholds, such as

$$\dot{\eta}_i(t) = -\alpha_i \eta_i(t) - \delta_i (\beta_i 1_r^T|\epsilon_i(t)| - w_i^T(t) \epsilon_i(t))$$

providing adaptive triggering of communication based on real-time error and system state, avoiding both over-communication and Zeno behavior (George et al., 2018, Wang et al., 2023, Xu et al., 7 Jan 2024).

• Representation in retrieval frameworks: DEUs are linked in event graphs with temporal and entity-based relationship edges, supporting coherent multi-hop temporal and causal reasoning. Edges are weighted via

$$w_{ij} = \text{sim}(v_i, v_j) \cdot \exp(-\alpha |t_i - t_j|), \quad \text{sim}(v_i, v_j) = \frac{|E_i \cap E_j|}{|E_i \cup E_j|}$$

ensuring that traversal captures both semantic and temporal dependencies (Sun et al., 16 Jul 2025).

This adaptivity ensures that applications of DEUs are robust to structural and contextual changes in the underlying system or data.

4. DEUs in Dynamic Event-Triggered Control and Communication

In cyber-physical and multi-agent systems, DEUs provide a rigorous and resource-efficient framework for triggering updates, measurements, or communications:

• Dynamic thresholds: Internal state variables (e.g., $\eta_i(t)$, $\theta_i(t)$, $f(t)$) are DEUs that accumulate contextual information and determine event timing. A typical dynamic event-triggering function satisfies:

$$\frac{df}{dt} = \min\{-z(t)^\top \Phi z(t),\, 0\} - f(t), \qquad f(t_k^+) = \bar{f} > 0$$

and triggers an event when $f(t) \leq 0$ (Xu et al., 7 Jan 2024). This structure ensures a strictly positive minimum inter-event time, critical for system stability and practical deployment.

• Model-based estimation: Timers may be modulated based on moving average windows of Lyapunov functions,

$$VMA(t_k^i, \ell) = \frac{1}{\ell} \sum_{n=k-\ell+1}^k V_i(t_n^i)$$

leading to adjustable inter-event intervals and reduced communication frequencies, with formal guarantees of convergence (Wang et al., 2023).

• Zeno behavior avoidance: The dynamic evolution of triggering variables is mathematically constructed to ensure no infinite event accumulation in finite time (George et al., 2018, Xu et al., 7 Jan 2024).

These applications demonstrate the utility of DEUs in designing robust, low-overhead event-driven control and communication strategies.

5. Dynamic Event Units in Event Detection, Retrieval, and Reasoning

DEUs underpin advanced event-centric retrieval, reasoning, and detection in dynamic and complex data modalities:

• Event-centric retrieval and RAG: In DyG-RAG, DEUs form the event graph nodes supporting temporally grounded retrieval and robust multi-hop reasoning, enabling accurate and interpretable answers to time-sensitive queries. Time Chain-of-Thought strategies leverage DEU timelines for LLM prompting, verified by improvements in accuracy and recall on temporal QA benchmarks (Sun et al., 16 Jul 2025).
• Dynamic graph analysis: In dynamic event detection models, such as DyGED, DEUs (as graph-level embeddings over time) are learned and attended to by both structural and temporal self-attention modules. This enables detection of macro-level graph events reflecting sudden global changes (Kosan et al., 2021).
• Temporal event detection in streams: In contextual domains (e.g., emotion video analysis), the fusion of frame-level or unit-level embeddings into DEUs—detected as temporally bounded action units—outperforms approaches that do not exploit full temporal extent and context (Chen et al., 2022).
• Causal inference with event heterogeneity: In econometrics, event paper methods can be reformulated as analyses of DEUs (e.g., cohort-period specific effects), enabling estimators that are less prone to contamination and that produce interpretable, convex averages of heterogeneous dynamic effects (Sun et al., 2018).

This broad applicability reflects both the data- and model-centric roles that DEUs can assume.

6. Hardware Implementations: Spatiotemporal Processing and Scalability

DEUs are physically realized in hardware to support real-time, scalable, and low-latency event-driven computation:

• CMOS event-based circuits: Novel TDE (Time-Difference Encoding) circuits facilitate the physical instantiation of DEUs by encoding and accumulating input events with linear integration

$$V_{\text{int}}(t) = V_0 + \kappa \sum_{i=1}^n u(t - t_i)$$

and converting inter-event timing ($\Delta t$) to voltage differentials, enabling parallel processing of event streams (Greatorex et al., 17 Jan 2025).

• Scalability and robustness: By leveraging standard CMOS technology, systems can implement large heterogeneous arrays of DEUs. Calibration and design techniques ensure that device mismatch does not impair global performance, supporting deployment in edge devices and robotics.

A plausible implication is that DEUs at the hardware level enable event-driven software systems to scale without bottlenecks, thereby bridging the interface between theory and high-throughput, low-power signal processing.

7. Comparative Analysis and Limitations

Dynamic Event Units differentiate themselves from traditional event models by their:

• Temporal specificity and adaptivity: DEUs encode and reason about events in a time-sensitive, context-aware manner, surpassing static windows, frames, or tuples which lack mutability.
• Causal plasticity: The ability to dynamically rewire dependencies (adding/dropping preconditions) grants DEUs substantial expressive power for modeling exceptions, contingencies, and evolving workflows (Arbach et al., 2018).
• Resource efficiency and robustness: Their use in event-triggered control and event-based neuromorphic systems leads to significant reductions in communication, sensing, and computation, while maintaining or improving system stability (George et al., 2018, Wang et al., 2023, Greatorex et al., 17 Jan 2025).

However, the complexity of dynamic causality modeling, the challenge of precise temporal anchoring in noisy or ambiguous data, and design trade-offs between fine granularity and computational or measurement overhead may limit DEU adoption in certain domains. This suggests that future work will need to focus on automation, standardization, and scalable implementations of DEU extraction, representation, and reasoning pipelines.

Conclusion

Dynamic Event Units provide a formal, extensible abstraction for time-anchored, context-aware atomic events, with far-reaching implications for control systems, knowledge retrieval, dynamic reasoning, and event-driven hardware. Their ability to precisely encode, manipulate, and reason over the evolving structure of events has been validated across diverse real-world applications, from temporal QA in LLMs to adaptive control in cyber-physical systems and scalable neuromorphic processing architectures.