non-hermitian-gnw-consciousness

name	non-hermitian-gnw-consciousness
description	Non-Hermitian potential well formalism for the subliminal-preconscious-conscious processing hierarchy in the Global Neuronal Workspace. Uses nonlinear Schrödinger-type equation in imaginary time with non-Hermitian, non-normal Hamiltonian to model conscious access as bound state emergence. Activation: GNW, consciousness, non-Hermitian, neural field theory, bound states, sensory processing hierarchy, cloud functions, global neuronal workspace.
tags	["neuroscience","consciousness","GNW","neural-field-theory","non-Hermitian","quantum-analogue"]
version	1.0.0
author	agent
date	2026-07-12T00:00:00.000Z

Non-Hermitian Potential Well Formalism for Conscious–Preconscious–Subliminal Processing

arXiv: 2607.08302 | q-bio.NC, nlin.AO | Lubashevskiy & Lubashevsky

Overview

A phenomenological model of the Global Neuronal Workspace (GNW) that reproduces the tripartite taxonomy of sensory processing (subliminal, preconscious, conscious) using a non-Hermitian, non-normal Hamiltonian framework. Conscious access emerges as a bound state when both landscape depth and top-down attention exceed thresholds.

Core Framework

Cloud Functions and GNW Hilbert Space

• GNW modeled as Hilbert space H = L²(Rᴺ) over N-dimensional perceptual state space
• High-level representations encoded as cloud functions Ψ(x,t)
• |Ψ(x,t)|² interpreted as normalized density over perceptual configurations
• Nonlocality represents perceptual uncertainty from neural processing mechanisms
• Normalization: ∫|Ψ(x,t)|² dx = 1

Governing Equation

The cloud function evolves according to a nonlinear Schrödinger-type equation in imaginary time:

τ ∂Ψ/∂t = -Ĥ Ψ + ⟨Ψ|Ĥ|Ψ⟩ Ψ

where τ ~ 200ms (characteristic time of high-level visual processing), and Ĥ is a non-Hermitian, non-normal Hamiltonian.

Priority Hamiltonian Decomposition

Ĥ = Ĥ' + iĤ'' decomposes into complementary processes:

Hermitian component (Recognition):

Ĥ' = A(x,t) [-ℓ²∇² + Ω(x)]

• Drives Ψ toward minima of the GNW landscape Ω(x)
• Proportional to attention degree A (0 ≤ A ≤ 1)
• Acts as dissipative localization → stimulus recognition

Anti-Hermitian component (Broadcasting):

Ĥ'' = [-c_η·ℓ²∇² - c_ω·Ω(x)]

• Promotes delocalization of Ψ → information broadcasting
• Operates without selective attention
• Minima of Ω act as potential barriers in this component

GNW Landscape

• Effective potential Ω(x) shaped by early sensory processing
• Bridges early (feedforward) and late (recurrent) processing stages
• Minima correspond to established stimulus representations

Tripartite Processing Taxonomy

The model naturally reproduces three regimes based on landscape depth U and attention A:

I. Subliminal Processing

• U < U_c²(c): Stimulus too weak → no bound state
• Neural activity cannot trigger global ignition regardless of attention

II. Preconscious (Supraliminal Unattended)

• U > U_c²(c) but A < A_c: Stimulus strong enough, but insufficient attention
• Representation exists but is unstable (preconscious buffer)
• Bound state exists but Re E₀ < 0 → unstable

III. Conscious (Supraliminal Attended)

• U > U_c²(c) and A > A_c: Both conditions met
• Stable bound state emerges → conscious access
• Information broadcast throughout GNW

Phase Transition

• Emergence of bound state at A = A_c is a first-order phase transition
• Bound state appears with finite spatial extent (not diverging)
• Contrast with Hermitian wells (second-order, diverging localization length)

Pöschl–Teller Potential Well Model

For a single potential well in 1D:

Ω(η) = -U / cosh²(η)

Ground state eigenfunction: Ψ₀(η) = Z₀ / cosh^μ(η)

where μ(μ+1) = ((A - ic)/(A + ic))·U

Stability criteria:

1. Existence: Re μ > 0 ⟺ U > U_c¹(g) = ¼(g² - 1)/g², where g = c/A
2. Stability: Re E₀ > 0 ⟺ U > U_c²(g) (computed numerically)

Key Insights

1. Dual role of GNW landscape:

   • Minima → attractors (recognition via Hermitian part)
   • Minima → barriers (broadcasting via anti-Hermitian part)

2. Conscious access as bound state emergence:

   • Requires BOTH sufficient stimulus strength (U > threshold) AND attention (A > A_c)
   • Mathematically formalizes GNW's two-condition theory

3. Maxima don't support recognition:

   • Bound states at landscape maxima are all unstable
   • Only minima contribute to stimulus recognition

4. First-order vs second-order transition:

   • Non-Hermitian: bound state appears with finite extent (first-order)
   • Hermitian: bound state energy approaches continuum edge (second-order)

Mathematical Properties

• Non-normal operator: eigenfunctions non-orthogonal → winner-takes-all competition
• Nonlinear norm-preserving term: ⟨Ψ|Ĥ|Ψ⟩Ψ enables transitions between eigenstates
• Complex-valued landscape: combines recognition (real) and broadcasting (imaginary)
• Spatially nonlocal interactions: via convolution structure of Ĥ

Applications

• Modeling conscious access dynamics
• Explaining attention-dependent perception
• Bridge between first-person phenomenology and neural implementation
• Previously applied to: power law of working memory, change-of-mind in decision-making

Pitfalls

• Phenomenological model: Not derived from first-principles neural dynamics
• Short-range approximation: Only two leading terms of convolution kept
• Constant attention assumption: A(x,t) = constant; spatially varying attention needs separate analysis
• 1D simplification: Full N-dimensional case may have richer dynamics

Related Skills

• consciousness-usk-framework - USK consciousness theory
• canonical-functionalism-consciousness - Canonical functionalism
• neural-dynamics-analysis-methodology - Neural dynamics analysis
• quantum-cognition - Quantum probability for cognitive modeling