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A statistical framework to assess cross-frequency coupling while accounting for confounding analysis effects

  1. Jessica K Nadalin
  2. Louis-Emmanuel Martinet
  3. Ethan B Blackwood
  4. Meng-Chen Lo
  5. Alik S Widge
  6. Sydney S Cash
  7. Uri T Eden
  8. Mark A Kramer  Is a corresponding author
  1. Boston University, United States
  2. Massachusetts General Hospital, United States
  3. University of Minnesota, United States
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Cite this article as: eLife 2019;8:e44287 doi: 10.7554/eLife.44287
13 figures and 1 additional file

Figures

The gamma distribution provides a good fit to example human data.

Three examples of 20 s duration recorded from a single electrode during a human seizure. In each case, the gamma fit (red curve) provides an acceptable fit to the empirical distributions of the high frequency amplitude.

https://doi.org/10.7554/eLife.44287.002
Distribution of the number of control points (n) that minimize the AIC.

Values of n between 7 and 12 minimize the AIC in a simulation with phase-amplitude coupling and amplitude-amplitude coupling.

https://doi.org/10.7554/eLife.44287.003
Example model surfaces used to determine RPAC and RAAC.

(A,B,C) Three example surfaces (ASAlow, (BSϕlow, and (CSAlow,ϕlow in the three-dimensional space (Alow, ϕlow, Ahigh). (D) The maximal distance between the surfaces SAlow (red) and SAlow,ϕlow (yellow) is used to compute RPAC. (E) The maximal distance between the surfaces Sϕlow (blue) and SAlow,ϕlow (yellow) is used to compute RAAC.

https://doi.org/10.7554/eLife.44287.004
Illustration of synthetic time series with PAC and AAC.

(A) Example simulation of Vlow (blue) and modulation signal M (red). When the phase of Vlow is near 0 radians, M increases. (B) Example simulation of PAC. When the phase of Vlow is approximately 0 radians, the high frequency amplitude (yellow) increases. (C) Example simulations of AAC. When the amplitude of Vlow is large, so is the amplitude of the high frequency signal (purple).

https://doi.org/10.7554/eLife.44287.005
The statistical modeling framework successfully detects different types of cross-frequency coupling.

(A–C) Simulations with no CFC. (A) When no CFC occurs, the low frequency signal (blue) and high frequency signal (orange) evolve independently. (B) The surfaces SAlow, Sϕlow, and SAlow,ϕlow suggest no dependence of Ahigh on ϕlow or Alow. (C) Significant (p<0.05) values of 𝐑PAC and 𝐑AAC from 1000 simulations. Very few significant values for the statistics R are detected. (D–G) Simulations with PAC only. (D) When the phase of the low frequency signal is near 0 radians (red tick marks), the amplitude of the high frequency signal increases. (E) The surfaces SAlow, Sϕlow, and SAlow,ϕlow suggest dependence of Ahigh on ϕlow. (F) In 1000 simulations, significant values of RPAC frequently appear, while significant values of 𝐑AAC rarely appear. (G) As the intensity of PAC increases, so do the significant values of 𝐑PAC (black), while any significant values of 𝐑AAC remain small. (H–K) Simulations with AAC only. (H) The amplitudes of the high frequency signal and low frequency signal are positively correlated. (I) The surfaces SAlow, Sϕlow, and SAlow,ϕlow suggest dependence of Ahigh on Alow. (J) In 1000 simulations, significant values of 𝐑AAC frequently appear. (K) As the intensity of AAC increases, so do the significant values of 𝐑AAC (blue), while any significant values of 𝐑PAC remain small. (L–O) Simulations with PAC and AAC. (L) The amplitude of the high frequency signal increases when the phase of the low frequency signal is near 0 radians and the amplitude of the low frequency signal is large. (M) The surfaces SAlow, Sϕlow, and SAlow,ϕlow suggest dependence of Ahigh on ϕlow and Alow. (N) In 1000 simulations, significant values of 𝐑PAC and 𝐑AAC frequently appear. (O) As the intensity of PAC and AAC increase, so do the significant values of 𝐑PAC and 𝐑AAC. In (G,K,O), circles indicate the median, and x’s the 5th and 95th quantiles.

https://doi.org/10.7554/eLife.44287.006
The two measures of PAC increase with intensities near zero.

The mean (circles) and 5th to 95th quantiles (x’s) of (A𝐑PAC and (B) MI for intensity values between 0 and 0.5. Black bars indicate pPAC or pMI is below 0.05 for ≥95% of simulations; gray bars indicate pPAC is not below 0.05 for ≥95% of simulations. While both measures increase with intensity, MI detects more instances of significant PAC than does 𝐑PAC for very small values of IPAC.

https://doi.org/10.7554/eLife.44287.007
Increases in the amplitude of the low frequency signal, and the amplitude-amplitude coupling (AAC), increase the modulation index more than RPAC.

(A,B) Distributions of (A) RPAC and (B) MI when Alow is small (blue) and when Alow is large (red). (C,D) Distributions of (C) RPAC and (D) MI when AAC is small (blue) and when AAC is large (red).

https://doi.org/10.7554/eLife.44287.008
PAC events restricted to a subset of occurrences are still detectable.

(A) The low frequency signal (blue), amplitude envelope (yellow), and threshold (black dashed). (B–C) The modulation signal increases (B) at every occurrence of ϕlow=0, or (C) only when Alow exceeds the threshold and ϕlow=0.

https://doi.org/10.7554/eLife.44287.009
PAC with AAC is accurately detected with the proposed method, but not with the modulation index.

(A) The low frequency signal (blue), amplitude envelope (yellow), and threshold (black dashed). (B) The modulation signal (red) increases when ϕlow=0 and Alow>T, and deceases when ϕlow=0 and Alow<T. (C) The modulated Ahigh signal (purple) increases and decreases with the modulation signal. (D) The proportion of significant detections (out of 1000) for MI and RPAC.

https://doi.org/10.7554/eLife.44287.010
RPAC, but not MI, detects phase-amplitude coupling in a simple stochastic spiking neuron model.

(A) The phase and amplitude of the low frequency signal (blue) modulate the probability of a high frequency spike (orange). (B) The surfaces SAlow (red) and SAlow,ϕlow (yellow). The phase of maximal Ahigh modulation depends on Alow. (C) The modulation index fails to detect this type of PAC.

https://doi.org/10.7554/eLife.44287.011
The proposed method detects cross-frequency coupling in an in vivo human recording.

(A,B) Voltage recording (A) and spectrogram (B) from one MEA electrode over the course of a seizure; PAC and AAC were computed for the time segment outlined in red. (C) The 10 s voltage trace (blue) corresponding to the outlined segment in (A), and Vlow (red), Vhigh (yellow), and Alow (purple). (D) A 2 s subinterval of the voltage trace (blue), Vlow (red), Vhigh (yellow), Alow (purple), and ϕlow (green). (EAlow (purple) and Ahigh (red) for the 10 s segment in (C), normalized and smoothed.

https://doi.org/10.7554/eLife.44287.012
The SAlow,ϕlow surface shows how PAC changes with the low frequency amplitude and phase during an interval of human seizure.

(A) The full model surface (blue) in the (ϕlow, Alow, Ahigh) space, and components of that surface when (BAlow is small (black), and Alow is large (red).

https://doi.org/10.7554/eLife.44287.013
Example simulated Vlow (blue) and Vhigh (orange) signals for which (A) PAC increases at 20 s (indicated by black dashed line), and (B) no increase in PAC occurs.
https://doi.org/10.7554/eLife.44287.014

Data availability

In vivo human data available at https://github.com/Eden-Kramer-Lab/GLM-CFC (copy archived at https://github.com/elifesciences-publications/GLM-CFC). In vivo rat data available at https://github.com/tne-lab/cl-example-data (copy archived at https://github.com/elifesciences-publications/cl-example-data).

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