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Modeling Biological Multifunctionality with Echo State Networks

Anastasia-Maria Leventi-Peetz, Jörg-Volker Peetz, Kai Weber, Nikolaos Zacharis

TL;DR

The work develops a 3D reaction-diffusion-based electrophysiology model with morphogen coupling and nonlinear sine interactions to capture rich spatiotemporal dynamics. A data-driven Echo State Network, implemented via ReservoirPy, learns a one-step-ahead map from the full 81{,}000-field state, enabling fast, parallel forecasting of the coupled $C$, $V$, and $H$ fields. Across three model versions, the ESN achieves close agreement with numerical solutions and extended prediction horizons (up to ~500 steps in favorable settings) while revealing the influence of weak chaos through Lyapunov diagnostics. This demonstrates the feasibility and efficiency of high-dimensional, multifunctional ESN surrogates for complex biological PDE systems, with implications for morphogenesis modeling and control.

Abstract

In this work, a three-dimensional multicomponent reaction-diffusion model has been developed, combining excitable-system dynamics with diffusion processes and sharing conceptual features with the FitzHugh-Nagumo model. Designed to capture the spatiotemporal behavior of biological systems, particularly electrophysiological processes, the model was solved numerically to generate time-series data. These data were subsequently used to train and evaluate an Echo State Network (ESN), which successfully reproduced the system's dynamic behavior. The results demonstrate that simulating biological dynamics using data-driven, multifunctional ESN models is both feasible and effective.

Modeling Biological Multifunctionality with Echo State Networks

TL;DR

The work develops a 3D reaction-diffusion-based electrophysiology model with morphogen coupling and nonlinear sine interactions to capture rich spatiotemporal dynamics. A data-driven Echo State Network, implemented via ReservoirPy, learns a one-step-ahead map from the full 81{,}000-field state, enabling fast, parallel forecasting of the coupled , , and fields. Across three model versions, the ESN achieves close agreement with numerical solutions and extended prediction horizons (up to ~500 steps in favorable settings) while revealing the influence of weak chaos through Lyapunov diagnostics. This demonstrates the feasibility and efficiency of high-dimensional, multifunctional ESN surrogates for complex biological PDE systems, with implications for morphogenesis modeling and control.

Abstract

In this work, a three-dimensional multicomponent reaction-diffusion model has been developed, combining excitable-system dynamics with diffusion processes and sharing conceptual features with the FitzHugh-Nagumo model. Designed to capture the spatiotemporal behavior of biological systems, particularly electrophysiological processes, the model was solved numerically to generate time-series data. These data were subsequently used to train and evaluate an Echo State Network (ESN), which successfully reproduced the system's dynamic behavior. The results demonstrate that simulating biological dynamics using data-driven, multifunctional ESN models is both feasible and effective.

Paper Structure

This paper contains 12 sections, 6 equations, 17 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Reaction-Diffusion models and solutions
  3. The electrophysiology model
  4. Model variables and equations
  5. The model with ReservoirPy
  6. Discrete time updates:
  7. Echo State Network mapping:
  8. ESN setup:
  9. ESN calculations and results
  10. Conclusions
  11. Future Work:
  12. System Configuration and Runtimes:

Figures (17)

  • Figure 1: ESN architecture featuring shared input and output vectors and a single reservoir. The interdependence of the variables is captured by this reservoir, which is larger (contains more nodes) than each of the three reservoirs in the separate-input-channel architecture, depicted in Figure \ref{['f:esn-2']}.
  • Figure 2: Architecture proposed in Figure 1 of b06.
  • Figure 3: ESN architecture with separate input channels, three reservoirs, and three readouts. Each reservoir is connected to all readouts to model the interdependence among the inputs.
  • Figure 4: Comparison of numerical simulation results and ESN predictions for time step 100, model version 1.
  • Figure 5: Comparison of numerical simulation results and ESN predictions for time step 150, model version 1.
  • ...and 12 more figures