non-hermitian-gnw-consciousness

name	non-hermitian-gnw-consciousness
description	Non-Hermitian Potential Well Formalism for modeling the Global Neuronal Workspace (GNW) consciousness framework. Uses nonlinear Schrödinger-type equation in imaginary time with non-Hermitian, non-normal Hamiltonian and Lotka-Volterra-type term to reproduce subliminal-preconscious-conscious hierarchy. Maps conscious access to bound-state emergence in complex-valued landscape. Use for consciousness modeling, neural field theory, GNW theory, and quantum-inspired cognitive dynamics.
activation	non-hermitian consciousness, GNW model, global neuronal workspace, consciousness formalism, subliminal processing, preconscious buffer, cloud function, neural field theory, Lotka-Volterra neural dynamics, bound state consciousness, potential well neural model
tags	["neuroscience","consciousness","neural-field-theory","gnw","non-hermitian","mathematical-modeling","cognitive-science"]
version	1.0.0
author	Hermes Agent
category	neuroscience
created	"2026-07-13T00:00:00.000Z"
source_arxiv	2607.08302v1

Non-Hermitian Potential Well Formalism for GNW Consciousness

Source Paper

Title: A Non-Hermitian Potential Well Formalism for Conscious–Preconscious–Subliminal Processing
Authors: Vasily Lubashevskiy (Tokyo International University), Ihor Lubashevsky (HSE University, Moscow)
arXiv: 2607.08302v1 [q-bio.NC, nlin.AO]
Published: 2026-07-09
License: CC BY 4.0

Core Innovation

Proposes a phenomenological model of the Global Neuronal Workspace (GNW) using a non-Hermitian Schrödinger-type equation in imaginary time with a Lotka–Volterra-type nonlinear term. This provides the first tractable dynamical framework that unifies sensory encoding, attention, and conscious access within a single mathematical formalism.

Key Concepts

1. Cloud Function Ψ(x,t)

• High-level stimulus representations encoded as cloud functions in Hilbert space ℍ = L²(ℝᴺ)
• |Ψ(x,t)|² interpreted as normalized density over perceptual configurations
• Nonlocality in ℝᴺ represents perceptual uncertainty from neural sensory processing
• Combines holistic structure of mental images with neural implementation

2. Complex-Valued GNW Landscape Ω(x)

• Early sensory processing generates an effective complex-valued potential landscape
• Ω(x) = V(x) + iW(x), where:
  • V(x) (real/Hermitian part): Recognition via dissipative localization at landscape minima
  • W(x) (imaginary/anti-Hermitian part): Information broadcasting via spatial spreading
• Landscape depth proportional to bottom-up sensory activation strength

3. Governing Equation

∂Ψ/∂t = -ĤΨ + nonlinear Lotka-Volterra term

where Ĥ is a non-Hermitian, non-normal Hamiltonian:

• Hermitian component → dissipative dynamics driving Ψ toward landscape minima (recognition)
• Anti-Hermitian component → spatial spreading / broadcasting across state space
• Nonlinear term → preserves norm (∫|Ψ|² = 1) while enabling spatially nonlocal interactions

4. Three Processing Regimes

Regime	Bottom-Up Strength	Top-Down Attention	Outcome
Subliminal	Below threshold	Any	No conscious access
Preconscious (supraliminal unattended)	Above threshold	Unavailable	Nonconscious buffer maintenance
Conscious (supraliminal attended)	Above threshold	Available (A > A_c)	Bound state emergence = global ignition

5. Conscious Access as Bound State Emergence

• Conscious access corresponds to emergence of a bound state in the GNW landscape
• Requires both conditions:
  1. GNW landscape depth exceeds threshold (strong enough bottom-up activation)
  2. Degree of top-down attention A exceeds critical value A_c
• Stepwise emergence: stimulus representations appear gradually as A crosses A_c

Mathematical Framework

Hilbert Space Structure

• State space: ℝᴺ (N-dimensional perceptual state space)
• Cloud functions: Ψ(x,t) ∈ L²(ℝᴺ)
• Inner product: ⟨Ψ₁|Ψ₂⟩ = ∫ Ψ₁*(x)Ψ₂(x) dx
• Normalization: ∫ |Ψ(x,t)|² dx = 1

Landscape Construction

• V(x) = -U · exp(-|x-x₀|²/(2σ²)) — Gaussian potential well centered at stimulus x₀
• Depth U ∝ bottom-up sensory activation strength
• Width σ ∝ perceptual uncertainty

Dynamics Properties

1. Ground state stability: Uniform Ψ₀ converges to localized ground state when U > U_c
2. Supraliminal attended processing: Cloud function localizes at landscape minimum
3. Spatial spreading: Anti-Hermitian component enables broadcasting across state space
4. Norm preservation: Nonlinear term maintains ∫|Ψ|² = 1

Numerical Results (from paper)

• Simulations use Crank–Nicolson (2nd order implicit) + Adams–Bashforth (2nd order explicit)
• Domain: [0, L] with L=60, periodic boundary conditions
• Parameters: dx=0.005, dt=0.001, c=2
• Key findings:
  • Figure 3: Cloud function dynamics for supraliminal attended processing
  • Figure 4: Stepwise emergence of representations when A exceeds A_c
  • Figure 5: Dynamics near landscape maximum (potential sign reversed)

Applications

When to Use This Skill

• Modeling consciousness and conscious access mechanisms
• Implementing GNW-inspired neural field models
• Studying subliminal vs. preconscious vs. conscious processing
• Designing attention-gated neural architectures
• Quantum-inspired cognitive modeling (Schrödinger-type equations in cognition)
• Non-Hermitian dynamics in neuroscience

Research Connections

• Global Neuronal Workspace theory (Dehaene et al.)
• Neural field theory (continuous population-level dynamics)
• Non-Hermitian quantum mechanics (open systems, PT symmetry)
• Lotka–Volterra dynamics (competitive/cooperative interactions)
• Perceptual consciousness (subliminal/preconscious/conscious taxonomy)

Implementation Patterns

Pattern 1: Basic GNW Cloud Function Simulation

import numpy as np

def gnw_landscape(x, x0, U, sigma):
    """GNW potential well from sensory activation."""
    return -U * np.exp(-(x - x0)**2 / (2 * sigma**2))

def gnw_hamiltonian(V, W):
    """Non-Hermitian Hamiltonian: H = V + iW."""
    return V + 1j * W

def gnw_evolution(Psi, H, alpha, dt, nonlinear=True):
    """One step of GNW cloud function evolution."""
    # Linear part: -H * Psi
    dPsi_linear = -H @ Psi
    # Nonlinear Lotka-Volterra term (norm-preserving)
    if nonlinear:
        norm = np.sum(np.abs(Psi)**2)
        dPsi_nonlinear = alpha * Psi * (1 - norm)
    else:
        dPsi_nonlinear = 0
    return Psi + dt * (dPsi_linear + dPsi_nonlinear)

Pattern 2: Conscious Access Threshold Check

def conscious_access(bottom_up_strength, top_down_attention, U_threshold, A_threshold):
    """Determine processing regime."""
    if bottom_up_strength < U_threshold:
        return "subliminal"
    elif top_down_attention < A_threshold:
        return "preconscious"
    else:
        return "conscious"  # Bound state emerges

Pattern 3: GNW Landscape Construction from Sensory Input

def build_gnw_landscape(sensory_input, N=100):
    """Construct complex GNW landscape from sensory features."""
    x = np.linspace(-10, 10, N)
    # Real part: recognition potential from feature similarity
    V = -np.sum([s * np.exp(-(x - mu)**2 / (2*sig**2))
                 for s, mu, sig in sensory_input], axis=0)
    # Imaginary part: broadcasting strength from attentional gain
    W = np.ones(N) * attention_level
    return V, W, x

Open Questions & Extensions

1. N-dimensional extension: Current paper focuses on 1D; generalization to high-D perceptual spaces needed
2. Learning mechanism: How does the landscape Ω(x) adapt through experience?
3. Multi-stimulus competition: How do multiple bound states compete for workspace access?
4. Neural implementation: How to map cloud functions to actual neural population activity?
5. Connection to IIT: Relationship with Integrated Information Theory's Φ measure

Related Skills

• consciousness-usk-framework — Uncommon Self-Knowledge consciousness theory
• canonical-functionalism-consciousness — Mathematical refinement of computational functionalism
• neuromorphic-artificial-consciousness — Neuromorphic correlates of consciousness
• abstraction-fallacy-ai-consciousness — Physicalism framework on simulation vs instantiation
• ctm-ai-consciousness-blueprint — CTM-AI blueprint for general AI

Keywords

GNW, Global Neuronal Workspace, consciousness, non-Hermitian, Schrödinger equation, imaginary time, cloud function, Hilbert space, Lotka-Volterra, bound state, subliminal, preconscious, perceptual state space, neural field theory, complex-valued landscape, dissipative localization, spatial broadcasting, top-down attention, bottom-up activation, global ignition