# Fractal Neural Dynamics and Memory Encoding Through Scale Relativity

## TL;DR

This paper proposes a new computational model for memory encoding using fractal neural dynamics and wave-like activity in non-differentiable space-time.

## Contribution

It introduces a novel framework based on Scale Relativity Theory to explain distributed memory formation through nonlinear wave dynamics.

## Key findings

- The model reproduces biological features like place fields, grid cells, and orientation maps.
- Interference-driven plasticity and cross-frequency coupling generate complex memory structures.
- The system shows robustness to noise and learning dynamics similar to empirical observations.

## Abstract

Background/Objectives: Synaptic plasticity is fundamental to learning and memory, yet classical models such as Hebbian learning and spike-timing-dependent plasticity often overlook the distributed and wave-like nature of neural activity. We present a computational framework grounded in Scale Relativity Theory (SRT), which describes neural propagation along fractal geodesics in a non-differentiable space-time. The objective is to link nonlinear wave dynamics with the emergence of structured memory representations in a biologically plausible manner. Methods: Neural activity was modeled using nonlinear Schrödinger-type equations derived from SRT, yielding complex wave solutions. Synaptic plasticity was coupled through a reaction–diffusion rule driven by local activity intensity. Simulations were performed in one- and two-dimensional domains using finite difference schemes. Analyses included spectral entropy, cross-correlation, and Fourier methods to evaluate the organization and complexity of the resulting synaptic fields. Results: The model reproduced core neurobiological features: localized potentiation resembling CA1 place fields, periodic plasticity akin to entorhinal grid cells, and modular tiling patterns consistent with V1 orientation maps. Interacting waveforms generated interference-dependent plasticity, modeling memory competition and contextual modulation. The system displayed robustness to noise, gradual potentiation with saturation, and hysteresis under reversal, reflecting empirical learning and reconsolidation dynamics. Cross-frequency coupling of theta and gamma inputs further enriched trace complexity, yielding multi-scale memory structures. Conclusions: Wave-driven dynamics in fractal space-time provide a hypothesis-generating framework for distributed memory formation. The current approach is theoretical and simulation-based, relying on a simplified plasticity rule that omits neuromodulatory and glial influences. While encouraging in its ability to reproduce biological motifs, the framework remains preliminary; future work must benchmark against established models such as STDP and attractor networks and propose empirical tests to validate or falsify its predictions.