---
id: wiki-2026-0508-phase-amplitude-coupling
title: Phase Amplitude Coupling
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [PAC, Cross-Frequency Coupling, Theta-Gamma PAC]
duplicate_of: none
source_trust_level: A
confidence_score: 0.85
verification_status: applied
tags: [neuroscience, signal-processing, eeg, oscillations]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: mne/numpy
---

# Phase Amplitude Coupling

## 매 한 줄
> **"매 slow rhythm 의 phase 가 매 fast rhythm 의 amplitude 를 모듈레이션."**. 2006 Canolty et al. theta-gamma PAC in human ECoG → 2010s memory/working-memory 의 neural mechanism 으로 자리잡음. 2026 BCI, sleep staging, anesthesia depth monitoring 에 활용.

## 매 핵심

### 매 What is PAC
- Two oscillations: phase φ_low(t) ∈ [−π, π], amplitude A_high(t) ≥ 0.
- PAC: A_high 가 φ_low 의 특정 phase에 preferentially elevated.
- Canonical pair: theta (4–8 Hz) phase × gamma (30–80 Hz) amplitude.

### 매 PAC Metrics
- **Modulation Index (MI, Tort)**: KL divergence of phase-binned amplitude from uniform.
- **Mean Vector Length (MVL, Canolty)**: |⟨A·e^(iφ)⟩|.
- **Phase-Locking Value (PLV)**: |⟨e^(i(φ_low − φ_amp_envelope))⟩|.
- **GLM-based**: regress A on sin/cos(φ).

### 매 응용
1. Working memory (theta-gamma in hippocampus, PFC).
2. Sleep stage classification (slow oscillation × spindle).
3. Anesthesia depth (alpha-gamma, alpha-beta PAC).
4. Parkinson's DBS biomarker (beta-gamma PAC).

## 💻 패턴

### Filter + Hilbert (extract phase & amplitude)
```python
import numpy as np
from scipy.signal import butter, sosfiltfilt, hilbert

def phase_amp(x, fs, f_low=(4,8), f_high=(30,80)):
    sos_lo = butter(4, f_low, btype='band', fs=fs, output='sos')
    sos_hi = butter(4, f_high, btype='band', fs=fs, output='sos')
    x_lo = sosfiltfilt(sos_lo, x)
    x_hi = sosfiltfilt(sos_hi, x)
    phi = np.angle(hilbert(x_lo))
    amp = np.abs(hilbert(x_hi))
    return phi, amp
```

### Tort Modulation Index
```python
def tort_mi(phi, amp, n_bins=18):
    edges = np.linspace(-np.pi, np.pi, n_bins+1)
    bin_idx = np.digitize(phi, edges) - 1
    bin_idx = np.clip(bin_idx, 0, n_bins-1)
    P = np.array([amp[bin_idx == k].mean() for k in range(n_bins)])
    P = P / P.sum()
    H = -np.sum(P * np.log(P + 1e-12))
    H_max = np.log(n_bins)
    return (H_max - H) / H_max  # MI ∈ [0, 1]
```

### Canolty MVL with surrogate test
```python
def mvl(phi, amp):
    return np.abs(np.mean(amp * np.exp(1j * phi)))

def mvl_significance(phi, amp, n_surr=200):
    obs = mvl(phi, amp)
    surr = []
    for _ in range(n_surr):
        shift = np.random.randint(len(amp))
        surr.append(mvl(phi, np.roll(amp, shift)))
    p = np.mean(np.array(surr) >= obs)
    z = (obs - np.mean(surr)) / np.std(surr)
    return obs, z, p
```

### Comodulogram (PAC across freq pairs)
```python
def comodulogram(x, fs, lo_freqs, hi_freqs):
    MI = np.zeros((len(lo_freqs)-1, len(hi_freqs)-1))
    for i in range(len(lo_freqs)-1):
        for j in range(len(hi_freqs)-1):
            phi, amp = phase_amp(x, fs,
                f_low=(lo_freqs[i], lo_freqs[i+1]),
                f_high=(hi_freqs[j], hi_freqs[j+1]))
            MI[i, j] = tort_mi(phi, amp)
    return MI
```

### MNE-Python ready pipeline
```python
import mne
from mne_connectivity import phase_slope_index

raw = mne.io.read_raw_edf("eeg.edf", preload=True)
raw.filter(1, 100).notch_filter(60)
epochs = mne.make_fixed_length_epochs(raw, duration=2.0)
# pactools for PAC
from pactools import Comodulogram
estimator = Comodulogram(fs=raw.info['sfreq'],
                          low_fq_range=np.linspace(2, 12, 11),
                          high_fq_range=np.linspace(20, 100, 17),
                          method='tort')
estimator.fit(epochs.get_data()[0, 0])
estimator.plot()
```

## 매 결정 기준
| 상황 | Method |
|---|---|
| Quick screening | Tort MI (robust to amp distribution) |
| Phase preference angle | Canolty MVL (gives complex vector) |
| Phase-phase coupling | n:m PLV |
| Need significance | Surrogate w/ time shifts (≥200) |
| Continuous data | sliding-window comodulogram |

**기본값**: Tort MI + 200 time-shift surrogates + FDR correction across frequency pairs.

## 🔗 Graph
- 부모: [[Cross-Frequency Coupling (CFC)]] · [[Signal-Processing-Foundations]]
- 변형: [[Theta-Gamma Coupling]]
- Adjacent: [[Neural Ignition]]

## 🤖 LLM 활용
**언제**: PAC method 선택 explain, comodulogram 결과 interpretation.
**언제 X**: 매 raw EEG 매 LLM에 stream (use MNE/pactools locally).

## ❌ 안티패턴
- **Spurious PAC from sharp transients**: 매 epileptic spikes / artifacts → broadband amp. Always inspect raw + reject artifacts.
- **No surrogate test**: MI > 0 always — significance ≠ value.
- **Filter bandwidth too narrow**: ringing → spurious phase locking.
- **Edge effects**: 매 hilbert 에서 매 처음/끝 cycle 버려야.

## 🧪 검증 / 중복
- Verified (Tort et al. 2010 *J Neurophysiol*; Canolty & Knight 2010 *Trends Cogn Sci*).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — Tort MI / Canolty MVL / comodulogram 패턴, surrogate test |