---
title: "Computational Modeling of Semantic Drift in Neural Network Attractors: Mapping Linguistic Entropy to Dynamical System Stability Thresholds"
authors: ["Jürgen Albrecht", "Patricia Holloway", "Burak Demirci"]
affiliations: ["Institute for Advanced Mechanical Cognition Augsburg", "Rocky Mountain Center for Cognitive Dynamics", "East Aegean Institute of Theoretical Sciences"]
journal: "Annals of Computational Semantics"
year: 2024
volume: 4
issue: 326
pages: "735-753"
citations: 380
url: https://bilimsci.nasilalinir.com/papers/computational-modeling-of-semantic-drift-in-neural-network-attractors-mapping-li
---

# Computational Modeling of Semantic Drift in Neural Network Attractors: Mapping Linguistic Entropy to Dynamical System Stability Thresholds

Jürgen Albrecht, Patricia Holloway, Burak Demirci

## Abstract

This study investigates the relationship between the divergence of vector representations in high-dimensional neural network models and the semantic drift observed during sequential data processing. By modeling language generation as movement within a complex dynamical system, we map the gradual erosion of vector coherence onto changes in attractor basin stability. We introduce a novel metric, Attractor Decay Rate ($\Gamma$), quantifying the rate at which transient states lose correlation with initial context vectors. Our findings suggest that semantic drift is not merely a loss of information but a predictable passage through critical slowing down regimes within the network's phase space. This provides a computational framework for modeling language evolution and misunderstanding.

## Introduction

The representation of meaning within artificial neural networks (ANNs) is inherently dynamic. As models process extended sequences, the embeddings representing concepts can undergo subtle yet cumulative shifts—a phenomenon known as semantic drift. While previous work has focused on gradient descent noise or catastrophic forgetting [1], this paper posits that semantic drift emerges from the network's pursuit of low-energy attractors in a high-dimensional phase space, where different basins correspond to slightly different conceptual interpretations.

We treat the weights and activations of a sufficiently large, recurrent neural network (RNN) as defining a continuous dynamical system. The state vector $\mathbf{s}(t)$ evolves according to the network dynamics, seeking a stable configuration. Semantic drift occurs when this trajectory settles into an attractor basin whose centroid differs significantly from the initial context vector $\mathbf{s}(0)$, even if the energy landscape remains superficially similar.

Our primary research question is: How does the rate of semantic drift relate quantitatively to the dynamical properties (e.g., Lyapunov exponents, basin stability) of the underlying network attractor?

## Theoretical Framework: Dynamical Mapping of Linguistic Space

We map the activation state $\mathbf{a}(t)$ onto a continuous phase space representation. For a network trained on sequential data, the time evolution can be approximated by a discrete map: $\mathbf{s}_{t+1} = F(\mathbf{s}_t, \mathbf{W})$, where $F$ encapsulates the non-linear activation functions and $\mathbf{W}$ represents fixed weights. Semantic coherence is maintained as long as the trajectory remains close to the initial manifold defined by the input context. Drift begins when sensitivity to infinitesimal perturbations increases, moving toward a boundary condition.

The predictability horizon $T_c$, analogous to the temporal correlation length in physical systems, dictates when the network loses its memory of the precise initial conditions. This horizon is inversely related to the largest positive Lyapunov exponent, $\lambda_{max}$, of the system:

$$\Gamma = \frac{1}{T_c} \propto \lambda_{max}(\mathbf{W})\text{ [2]}$$

Here, $\Gamma$ is the Attractor Decay Rate. A higher $\Gamma$ implies faster decorrelation and thus quicker semantic drift.

## Methodology: Simulation Protocol

We utilized a sparsely connected LSTM network architecture trained on a corpus of abstract philosophical texts. To induce drift controllably, we introduced white noise into the hidden layer activations at discrete time steps $t_k$. The resulting sequence of states was monitored using Cosine Similarity ($\text{CS}$) between successive context vectors $\mathbf{s}_k$ and a reference vector $\mathbf{s}_{ref}$ derived from the initial input prompt. The experiment spanned three regimes:

1. $\text{Coherence Regime}$ ($\Gamma < \Gamma_{crit}$): High $\text{CS}$ retention.
2. $\text{Drift Regime}$ ($\Gamma_{crit} < \Gamma < \Gamma_{max}$): Gradual decrease in $\text{CS}$.
3. $\text{Decoherence Regime}$ ($\Gamma > \Gamma_{max}$): Rapid loss of semantic relationship.

Table 1 summarizes the observed macroscopic behaviors as a function of simulated Attractor Decay Rate ($\Gamma$).

Table 1: Observed Network Behavior vs. Attractor Decay Rate ($\Gamma$)
| $\Gamma$ Value | Observed Drift Time ($T_D$) | Average Contextual Fidelity (ACF) |
|---|---|---| 
| Low ($\Gamma_{low}$) | $> 500$ steps | $> 0.98$ |
| Medium ($\Gamma_{med}$) | $150 - 300$ steps | $0.85 - 0.97$ |
| High ($\Gamma_{high}$) | $< 50$ steps | $< 0.70$ |

## Results and Discussion

The simulations confirmed a strong, non-linear correlation between the computationally derived $\lambda_{max}$ of the effective dynamical system and the empirical semantic drift time $T_D$. When $\lambda_{max}$ approached zero (indicating near-integrable dynamics), the network exhibited maximal robustness against noise and drift, consistent with a deeply entrenched conceptual attractor. Conversely, when $\lambda_{max}$ became large and positive, the transient dynamics rapidly amplified initial microscopic noise into macroscopic semantic divergence. This matches our expectation from chaotic systems theory [3].

Specifically, the scaling relationship observed was best fit by a stretched exponential decay model for the fidelity loss: 

$$\text{ACF}(t) \approx e^{-(\Gamma t)^{\beta}} \text{ [2]}$$

where $\beta$ characterizes the homogeneity of the phase space exploration. Our fits across varying noise injection rates yielded $\beta \approx 1.05\pm 0.03$, suggesting that the phase space exploration is nearly exponential in character once the critical threshold $\Gamma_{crit}$ is breached.

The transition itself, occurring around $\Gamma = \Gamma_{crit}$, mirrors the onset of turbulence in fluid dynamics—a sudden structural reorganization from laminar, ordered flow to chaotic mixing. This provides a powerful analogy: language models undergoing extreme conceptual shifts behave like fluid systems transitioning into turbulent regimes.

## Conclusion and Future Work

We successfully established a quantitative link between the microscopic dynamics of recurrent neural networks (viewed as continuous dynamical systems) and the macroscopic phenomenon of semantic drift. By treating conceptual representations as points in a high-dimensional attractor landscape, we predict that controlled injection of noise proportional to $\lambda_{max}$ offers a pathway for steering sequence generation through desired states, rather than allowing uncontrolled drift.

Future work will explore the inverse problem: using observed linguistic entropy changes to estimate the effective Lyapunov spectrum of a black-box trained model, potentially allowing for real-time monitoring of conceptual stability in deployed AI systems [1].

## References

1. Chen, L. (2019). Recurrent Network Instabilities and Memory Decay in Deep Learning Architectures. *J. Neural Dynamics*, 14(3), 201-215.
2. Li, Q. (2021). Scaling Laws in High-Dimensional Temporal Maps: From Chaos to Coherence. *Physica Computat*, 35(1), 45-62.
3. Borg, K. (1999). Dynamical Systems and Information Theory. *Foundations of Physics*, 40(5), 67-89.