Analysis Toolkit

Purpose and scope

The toolkit is structured for operational readability: each monitor returns a different failure or regime signal before the scalar order parameter alone would show anything unusual. In practice, teams use this page as a first-pass selection guide, then tune thresholds against their domain trajectories.

How operators should use this layer

Treat the toolkit as a multi-signal diagnostic funnel rather than a single alarm source:

• start with fast, broad monitors (order_parameter, lyapunov);
• add structural monitors (plv, chimera, winding) when patterns localise;
• then confirm causal or thermodynamic interpretations (coupling_est, entropy_prod, itpc, pid).

This sequencing reduces false positives and gives policy teams a reproducible rationale for each escalation before any actuation change is promoted.

The sections below are ordered from global coherence to higher-order coupling relationships because that mirrors a typical diagnostic flow: global stability -> phase alignment structure -> causality and stability risk -> topological and thermodynamic drift.

SPO provides 12 dynamical monitors — most oscillator simulators have 1-2. Each monitor detects a different aspect of the dynamics that scalar R misses.

Selecting a minimal monitor set

For a first production run, teams usually start with:

• order_parameter for synchronization trend and collapse detection,
• lyapunov for local stability margin,
• plv for pairwise synchrony topology,
• one supervisory metric (evs or pid) for domain-facing interpretability.

Expanding to all monitors is recommended only after a baseline is stable; this keeps false alarm fatigue manageable while keeping observability depth.

Order Parameter & PLV

Standard Kuramoto order parameter R = |⟨exp(iθ)⟩| and Phase-Locking Value matrix PLV_ij = |⟨exp(i(θ_i - θ_j))⟩_t|.

::: scpn_phase_orchestrator.upde.order_params

Phase-Amplitude Coupling (PAC)

Modulation index (MI) via Tort et al. 2010. Bins low-frequency phase, computes mean amplitude per bin, KL divergence from uniform. N×N PAC matrix: entry [i,j] = MI(phase_i, amplitude_j).

Central to neuroscience — cross-frequency coupling between brain oscillation bands (theta-gamma, alpha-beta).

::: scpn_phase_orchestrator.upde.pac

Chimera State Detection

Detects chimera states: coexisting coherent and incoherent clusters within the same network. Uses local order parameter R_i based on neighborhood coupling.

• Coherent: R_i > 0.7
• Incoherent: R_i < 0.3
• Boundary: in-between
• Chimera index = boundary_count / N

Detects phase transitions that global R misses.

::: scpn_phase_orchestrator.monitor.chimera options: show_bases: false show_source: false members: false

Entrainment Verification Score (EVS)

Three-criterion battery for rigorous entrainment validation:

1. ITPC (inter-trial phase coherence) persistence
2. Survival during stimulus pause
3. Frequency specificity (ratio at target vs control frequency)

Distinguishes true entrainment from broadband phase-locking artifacts.

::: scpn_phase_orchestrator.monitor.evs options: show_root_heading: false members: false

Partial Information Decomposition (PID)

Decomposes mutual information into:

• Redundancy: shared information from both oscillator groups
• Synergy: information present only in the joint group

Detects when groups carry synergistic (non-redundant) information about global phase (Williams & Beer 2010).

::: scpn_phase_orchestrator.monitor.pid

Lyapunov Exponent

Real-time estimation of the maximal Lyapunov exponent. Positive = chaos, zero = edge of chaos (critical), negative = stable attractor.

::: scpn_phase_orchestrator.monitor.lyapunov

Entropy Production

Measures thermodynamic irreversibility of the phase dynamics. Higher entropy production = system further from equilibrium.

::: scpn_phase_orchestrator.monitor.entropy_prod

Winding Number

Topological charge of phase trajectories. Counts how many times the phase wraps around the circle. Integer-valued topological invariant.

::: scpn_phase_orchestrator.monitor.winding

Inter-Trial Phase Coherence (ITPC)

Phase consistency across repeated trials or time windows. Standard neuroscience measure for event-related phase locking.

::: scpn_phase_orchestrator.monitor.itpc

Coupling Estimation from Data

Two methods for inferring coupling from observed time series:

1. Basic: least-squares fit of dθ/dt - ω = Σ K_ij sin(θ_j - θ_i)
2. Harmonics: higher Fourier harmonics for non-sinusoidal coupling

The harmonics method captures real biological coupling shapes (Stankovski 2017).

::: scpn_phase_orchestrator.autotune.coupling_est

Read with operational intent

Most monitors are most useful when compared over time and context, not as single point alarms. A practical dashboard should show short-term and rolling-window views side by side, then correlate alarms with known interventions.

The intended use is:

• detect onset conditions with one or two fast indicators,
• confirm with a slower structural monitor,
• only then trigger policy or supervisory changes.

Synthetic HCP Connectome Generation

Generates neuroscience-realistic coupling matrices inspired by the Human Connectome Project:

• Intra-hemispheric exponential distance decay
• Inter-hemispheric corpus callosum pattern
• Default Mode Network hub structure

::: scpn_phase_orchestrator.coupling.connectome

Monitoring stack as a decision chain

Treat this page as a decision chain for escalation, not a list of separate tools. The intended order is:

1. start with one global stability indicator,
2. confirm structural coherence with pairwise and topology-aware indicators,
3. apply causal or energetic checks before any bounded actuation proposal.

The sequence is designed to reduce false positives and preserve audit quality.

Minimal observability profile

A practical minimum profile for a first production pilot is:

• order_parameter for baseline synchrony,
• lyapunov for local stability trend,
• one causal or directional metric (coupling_est or itpc),
• one action governance metric (evs or pid).

This gives enough signal to decide whether a policy should stay static, reduce its scope, or escalate to broader review.