| 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
- Impossibility: Spatial-discrete neural-field equations with biologically realistic equilibria cannot support heteroclinic cycles
- Bridge: Lotka-Volterra systems exhibit heteroclinic dynamics but lack direct neuronal interpretation
- Solution: Universal Approximation Theorem enables approximating any target dynamics (including heteroclinic cycles) by an interpretable Amari-type neural-field system
- 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
- Define Target Dynamics: Specify the heteroclinic cycle encoding desired state sequence
- Construct Neural-Field System: Build Amari-type discrete neural-field equations
- Approximation: Use Universal Approximation Theorem to train neural network approximating the target vector field
- Verification: Show the approximating system generates periodic trajectory following heteroclinic connections
- 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
