heteroclinic-neural-field-cognition

name	heteroclinic-neural-field-cognition
description	Heteroclinic dynamics with discrete neural-field equations for modeling sequential cognitive states. Uses Universal Approximation Theorem to approximate target heteroclinic dynamics by Amari-type neural-field systems. Activates: heteroclinic cycle, sequential cognitive states, neural field dynamics, Lotka-Volterra neural, focused attention meditation modeling, cyclic brain activity.

Heteroclinic Neural-Field Dynamics for Sequential Cognitive States

Models cyclic and sequential brain activity patterns by combining heteroclinic dynamics with discrete neural-field equations, using universal approximation to bridge Lotka-Volterra dynamics with biologically realistic neural-field systems.

Metadata

• Source: arXiv:2605.02365
• Authors: M Virginia Bolelli, Luca Greco, Dario Prandi
• Published: 2026-05-04
• Categories: math.DS, q-bio.NC

Core Methodology

Key Innovation

Bridges the gap between heteroclinic dynamics (which capture sequential state transitions) and biologically realistic neural-field models by using the Universal Approximation Theorem to approximate any target heteroclinic dynamics with a high-dimensional Amari-type neural-field system.

Theoretical Results

1. Impossibility: Spatial-discrete neural-field equations with biologically realistic equilibria cannot support heteroclinic cycles
2. Bridge: Lotka-Volterra systems exhibit heteroclinic dynamics but lack direct neuronal interpretation
3. Solution: Universal Approximation Theorem enables approximating any target dynamics (including heteroclinic cycles) by an interpretable Amari-type neural-field system
4. Result: The approximating vector field generates a periodic trajectory that closely follows the heteroclinic connection

Mathematical Framework

• Target dynamics: Heteroclinic cycle (sequential state transitions)
• Approximator: High-dimensional Amari-type neural-field system (neural network)
• Connection: Universal approximation ensures the neural-field system reproduces the heteroclinic trajectory

Implementation Guide

Step-by-Step

1. Define Target Dynamics: Specify the heteroclinic cycle encoding desired state sequence
2. Construct Neural-Field System: Build Amari-type discrete neural-field equations
3. Approximation: Use Universal Approximation Theorem to train neural network approximating the target vector field
4. Verification: Show the approximating system generates periodic trajectory following heteroclinic connections
5. Neural Interpretation: Provide biological interpretation of the approximating dynamics

Case Study Application

• Focused-attention meditation: Sequential transitions among cognitive states (wandering → attention → awareness → reset)
• Each cognitive state corresponds to an equilibrium point
• Transitions follow heteroclinic connections between equilibria

Applications

• Sequential cognitive process modeling (meditation, task switching, working memory)
• Neural interpretation of dynamical systems models
• Understanding state transitions in brain networks
• Designing neuromorphic systems with sequential computation

Pitfalls

• Pure neural-field equations cannot directly support heteroclinic cycles — approximation is necessary
• Approximation quality depends on network dimensionality
• Biological realism of approximating system requires careful validation
• Case study on meditation is illustrative; generalization to other cognitive tasks needs empirical support

Related Skills

• neural-population-dynamics
• attractor-metadynamics-neural
• working-memory-heterogeneous-delays
• neural-dynamics-decision-making
• neural-emulator-theory