---
title: "Phase Synchronization in Spatio-Temporal Neural Oscillator Arrays: Modeling Emergent Global Attractor States via Localized Synaptic Decay Kinetics"
authors: ["Geraldine Olsen", "Klaus Schneider", "Burak Demirci"]
affiliations: ["Northwood Institute for Advanced Dynamics", "Hohenfels Academy of Applied Mechanics", "East Aegean Institute of Theoretical Sciences"]
journal: "Annals of Cybernetic Biophysics"
year: 2024
volume: 45
issue: 699
pages: "383-394"
citations: 799
url: https://bilimsci.nasilalinir.com/papers/phase-synchronization-in-spatio-temporal-neural-oscillator-arrays-modeling-emerg
---

# Phase Synchronization in Spatio-Temporal Neural Oscillator Arrays: Modeling Emergent Global Attractor States via Localized Synaptic Decay Kinetics

Geraldine Olsen, Klaus Schneider, Burak Demirci

## Abstract

This study investigates the emergence of large-scale, coherent firing patterns in coupled arrays of non-linear neural oscillators. We introduce a model where the decay kinetics of excitatory potentials, modulated by localized synaptic plasticity, dictates the system's transition between asynchronous, quasi-periodic, and fully synchronized states. By tuning the decay rate constants ($\gamma$), we observe critical slowing down preceding phase-locking events, providing a predictive framework for information throughput bottlenecks in biological neural circuits. The simulation successfully reproduces experimentally observed transitions observed in vitro neuronal cultures, linking microscopic kinetic parameters to macroscopic dynamical states.

## Introduction

The computational modeling of neural activity relies heavily on understanding how localized electrochemical events scale up to generate global brain function. In large-scale neural networks, stability and information processing often hinge on the ability of disparate oscillators to achieve phase coherence—a phenomenon known as synchronization. While mean-field models successfully predict macroscopic firing rates, they often fail to capture the transient dynamics leading to stable attractors, particularly those involving noise or heterogeneous connectivity.

We posit that the decay dynamics of post-synaptic potentials (PSPs) are not merely passive sinks for charge but active, dynamical variables influencing the phase relationship between coupled units. Specifically, we examine how variations in the intrinsic decay constant ($\gamma$) across the array influence the system's approach to a globally synchronized state.

## Theoretical Framework

We model each individual neuron $i$ as a simple FitzHugh–Nagumo oscillator coupled linearly to its neighbors $j \in N(i)$. The membrane potential $V_i$ evolves according to the equation:

$$\frac{dV_i}{dt} = f(V_i) - g(V_i) + \sum_{j \in N(i)} K_{ij} [h(V_j) - \gamma_i(t)]$$

where $f(V_i)$ and $g(V_i)$ describe the intrinsic membrane dynamics, $K_{ij}$ is the coupling strength, and $\gamma_i(t)$ represents the effective decay rate modulated by local synaptic activity. For simplicity in our primary analysis, we treat $\gamma_i$ as a quasi-constant parameter influenced by the local history of firing activity, representing synaptic decay kinetics [1].

Table: Dependence of Synchronization Time Constant ($\tau_{sync}$) on Decay Rate ($\gamma$)
| $\gamma$ Value | $\tau_{sync}$ (ms) | Observed State |
| :---: | :---: | :---: |
| 0.1 | 85.2 $\pm$ 1.1 | Quasi-periodic |
| 0.5 | 22.9 $\pm$ 0.5 | Near Critical |
| 1.2 | 4.1 $\pm$ 0.3 | Fully Synchronized |

This table illustrates the non-linear relationship between the intrinsic decay timescale of the postsynaptic current and the macroscopic time scale required for the network to enter a stable, high-coherence state [2].

## Simulation Methodology

Our simulations employed a 2D lattice structure of $N=1000$ coupled oscillators. We introduced controlled heterogeneity in the decay parameter $\gamma_i$, drawing values from a truncated normal distribution $\mathcal{N}(\mu_\gamma, \sigma_\gamma^2)$ to mimic biological variability. The critical threshold for phase transition was determined by monitoring the Kuramoto order parameter, $R(t)$, defined as:

$$R(t) = \left| \frac{1}{N} \sum_{j=1}^{N} e^{i \phi_j(t)} \right|$$

We varied the mean decay rate $\mu_\gamma$ while keeping the coupling strength $K_{ij}$ constant to isolate the effect of kinetic decay on global coherence. The simulation ran for $T = 5000$ ms, allowing for sufficient time for transients to decay and steady states to establish [3].

## Results and Discussion

The primary finding confirms a sharp, critical transition in the synchronization manifold dependent on $\mu_\gamma$. Below a critical mean decay rate, $\mu_{\gamma, crit} \approx 0.35$, the system exhibits persistent clusters of asynchronous activity, characterized by low $R(t)$ values ($R < 0.5$) and high variance in instantaneous firing frequency. As $\mu_\gamma$ crosses this threshold, the system undergoes a rapid phase transition into a high-coherence state ($R > 0.95$) within $\tau_{sync} \sim 50$ ms.

This transition mirrors the Kuramoto model's behavior but critically attributes the abruptness to the coupling between the firing dynamics and the decay timescale, suggesting that decay kinetics effectively mediate the network's susceptibility to global forcing [4].

In contrast, when the decay rate is excessively high ($\mu_\gamma > 1.5$), the oscillators effectively reset faster than the coupling can establish phase relationship, leading to high-frequency, uncorrelated spiking, indicative of information saturation or pathological firing patterns.

## Conclusion

We successfully mapped the macroscopic dynamical state of a neural oscillator array onto a microscopic parameter ($\gamma$) governing local synaptic decay kinetics. This work establishes that the temporal integration window, dictated by decay rates, plays an equally vital, if not dominant, role alongside coupling strength in determining functional brain states. Future work will incorporate directional coupling dependencies into the decay equation to model cortical microcircuit flows more realistically [1].

## References

1. Li, B. (2021). Synaptic Decay as a Dynamical Variable in Neural Networks. *Proc. Neur. Sci.*, 5(1), 10-15.
2. Chen, C. (2022). Scaling Laws for Coherence in Large-N Systems. *Phys. Rev. X*, 10(4), 041001.
3. Tanaka, D. (2019). Transient Dynamics in Coupled FitzHugh–Nagumo Models. *Comput. Biol. Phys.*, 88(6), 112-125.
4. Garcia, E. (2023). Criticality in Biological Phase Transitions. *Physica Comput. Appl.*, 45(3), 500-515.
5. Smith, A. (2020). Non-Equilibrium Dynamics in Coupled Oscillator Systems. *J. Theor. Phys.*, 1(2), 3-4.