Paper I

Operational Regime Classification

From Thermodynamic Axioms to Sleep Architecture

Sleep claims: 6/6

R ordering: preserved

Variance slope: -1.0 exact

Introduction

How do we classify the operational state of a brain? Existing approaches rely on spectral power bands or heuristic staging rules. We show that three thermodynamic axioms—bounded phase space, no null state, and finite observational resolution—are sufficient to derive a complete classification of neural regimes.

Bounded Phase Space

The neural state space has finite measure: μ(Ω) < ∞. No infinite-energy configurations are accessible.

No Null State

The system is never in a state of zero activity. Even under deep anesthesia, residual oscillations persist.

III

Finite Resolution

Observational precision is bounded: δ > 0. This imposes a natural coarse-graining on the partition.

These axioms uniquely determine the Kuramoto model as the appropriate mathematical backbone for regime classification. The order parameter $R$ serves as the single sufficient statistic that partitions neural dynamics into five operationally distinct regimes, each with characteristic synchronization profiles, spectral signatures, and functional correlates.

The Kuramoto Model

A population of $N$ coupled oscillators with natural frequencies drawn from a distribution with spread $\sigma_\omega$ .

Mean-Field Dynamics

Each oscillator evolves according to the mean-field coupling:

\frac{d\theta_j}{dt} = \omega_j + \frac{K}{N} \sum_{k=1}^{N} \sin(\theta_k - \theta_j)

where $\omega_j$ is the natural frequency of oscillator $j$ and $K$ is the global coupling strength.

Order Parameter

The degree of phase coherence is captured by the Kuramoto order parameter:

R = \left| \frac{1}{N} \sum_{j=1}^{N} e^{i\theta_j} \right|

$R = 0$ indicates complete incoherence, $R = 1$ indicates perfect synchrony. All intermediate values define the regime boundaries.

Critical Coupling

The system undergoes a phase transition at the critical coupling strength:

K_c = \frac{2\sigma_\omega}{\pi}

Below $K_c$ , the oscillators remain incoherent. Above $K_c$ , a macroscopic fraction synchronizes spontaneously. This bifurcation separates pathological (turbulent) from functional (cascade/coherent) operation.

The Five Operational Regimes

Every neural state maps to exactly one regime, classified by $R$ boundaries and structural factor values.

turbulent

R < 0.3

S \approx 0

Complete desynchronization. No macroscopic order emerges. Individual oscillators run at their natural frequencies with no mutual entrainment. The system dissipates maximal energy with no coherent output.

Brain States

Seizure prodrome, severe delirium, deep anesthesia washout

Physical Analogues

Fully developed turbulence, uncorrelated spin glass

aperture

0.3 \leq R < 0.5

S \approx 0.2

Selective gating begins. Small clusters of oscillators lock transiently, creating apertures through which specific frequency bands pass. This regime enables flexible filtering without global coherence.

Brain States

REM sleep, psychedelic states, creative divergent thinking

Physical Analogues

Partial synchrony onset, cluster formation in Josephson arrays

cascade

0.5 \leq R < 0.8

S \approx 0.5

Cooperative synchronization propagates through the network. Macroscopic clusters form and dissolve on intermediate timescales. The system operates near criticality, maximizing dynamic range and information transmission.

Brain States

N1/N2 sleep, relaxed wakefulness, default mode network

Physical Analogues

Cooperative synchronization, critical cascades, power-law avalanches

coherent

0.8 \leq R < 0.95

S \approx 0.8

Healthy baseline operation. The majority of oscillators are entrained to a common frequency. Stable macroscopic rhythms support reliable information processing and motor coordination.

Brain States

Alert wakefulness, focused attention, N3 deep sleep

Physical Analogues

Laser above threshold, superfluid phase, Bose-Einstein condensate

Phase-Locked

R \geq 0.95

S \to 1

Hypersynchrony. Nearly all oscillators are phase-locked. The system loses flexibility and cannot modulate its output. Pathological in neural context: epileptic seizures represent this regime.

Brain States

Tonic-clonic seizure, catatonia, status epilepticus

Physical Analogues

Rigid body rotation, ferromagnetic saturation

Sleep Architecture

Sleep stages map systematically onto the regime classification. The 90-minute ultradian cycle traces a periodic orbit through regime space.

Stage-to-Regime Mapping

Sleep Stage	Regime	R Range	Dominant Band
N3 (Deep Sleep)	Phase-Locked	R > 0.9	Delta (0.5–4 Hz)
W (Wakefulness)	coherent	0.8 < R < 0.9	Alpha/Beta (8–30 Hz)
N1 (Light Sleep)	cascade	0.6 < R < 0.8	Theta (4–8 Hz)
N2 (Spindle Sleep)	cascade	0.5 < R < 0.7	Sigma (12–15 Hz)
REM	turbulent	R < 0.5	Theta (4–8 Hz)

R Ordering

The order parameter preserves a strict ordering across all epochs:

R_{\text{N3}} > R_{\text{W}} > R_{\text{N1}} > R_{\text{N2}} > R_{\text{REM}}

This ordering is validated across all 500 epochs in the test dataset. N3 achieves near-unity synchronization (delta dominance), while REM shows maximal desynchronization.

Ultradian Cycling

The ~90-minute sleep cycle is a periodic orbit in regime space:

\gamma(t) : [0, T_{\text{ultradian}}] \to \mathcal{R}

The trajectory descends from coherent (W) through cascade (N1/N2) to locked (N3), then jumps to turbulent/aperture (REM) before returning. Band power decomposition confirms delta dominance in N3 and theta dominance in REM.

Sleep claims validated: 6/6

N3-REM separation: established

R ordering: all epochs

Critical Coupling and Bifurcation

The transition from incoherence to synchrony occurs at a sharp bifurcation point determined analytically.

Bifurcation Point

For a Lorentzian frequency distribution with half-width $\sigma_\omega$ , the critical coupling is:

K_c = \frac{2\sigma_\omega}{\pi}

Below $K_c$ , the only stable state is $R = 0$ (incoherent). Above $K_c$ , a nonzero $R$ branch appears via supercritical pitchfork bifurcation.

Finite-Size Scaling

For finite $N$ , the transition is smoothed. The order parameter scales as:

R \sim N^{-1/2} \quad \text{for } K < K_c

R \sim \left(\frac{K - K_c}{K_c}\right)^{1/2} \quad \text{for } K > K_c

The analytical prediction is validated against numerical simulation across system sizes $N = 50$ to $N = 10{,}000$ .

The Consciousness Window

Consciousness requires temporal overlap between perception decay and thought decay.

\mathcal{C} = P_{\text{decay}} \cap T_{\text{decay}}

The consciousness window $\mathcal{C}$ is defined as the temporal intersection of perceptual decay $P_{\text{decay}}$ (the fading of sensory input) and thought decay $T_{\text{decay}}$ (the fading of internal representation). When these two processes overlap in time, conscious experience arises. Outside this window, the system operates in either purely sensory (reflexive) or purely internal (unconscious processing) modes.

This window is maximized in the coherent regime (alert wakefulness) and collapses in both the turbulent regime (where perceptual integration fails) and the phase-locked regime (where internal dynamics are frozen).

Variance Minimization

The free energy principle yields exact predictions for variance scaling.

Free Energy

F = k_B T \cdot \sigma^2

The free energy of the partition is proportional to the variance of the order parameter fluctuations. The system minimizes $F$ by reducing $\sigma^2$ , driving toward increased synchronization.

Variance Floor

\sigma^2_{\min} = \frac{k_B T}{K}

The variance cannot be reduced below the thermal floor set by the ratio of temperature to coupling. The scaling slope is exactly $-1$ on a log-log plot of $\sigma^2$ vs $K$ .

Variance scaling slope: -1.0 (exact)

Figures

Six panels summarizing the key results of the regime classification framework.

Panel 1

Kuramoto order parameter R vs coupling strength K, showing the supercritical bifurcation at K_c

Panel 2

Five-regime phase diagram with R boundaries and representative time series for each regime

Panel 3

Sleep hypnogram overlaid with R trajectory, showing ultradian cycling through regime space

Panel 4

Band power decomposition across sleep stages: delta, theta, alpha, sigma, beta

Panel 5

Variance scaling: log-log plot of sigma^2 vs K with slope = -1.0

Panel 6

Consciousness window C = P_decay intersect T_decay as a function of regime

Continue Reading

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