Emily P. Stephen
Assistant Professor
Boston University
June 26, 2023

Single Unit
50 μm

Local Field Potential Electrocorticography(LFP)(ECoG)
Multiunit Activity
3mm300 μm

Electroencephalography
(EEG)
>2 cm
2 of 22

Obien et al (2015).

3 of 22

Buzsáki (2006) Rhythms of the Brain

4 of 22

(Here, they haveremoved the log slope)Buzsáki (2006) Rhythms of the Brain

5 of 22

Purdon et al (2013) PNAS

6 of 22

Human
EEG Mu
Human
ECoG
Beta
Rodent
LFP ThetaRodent
LFP SlowCole and Voytek (2017) Trends Cog SciSimulated
PING

Simulated
Slow

Rodent
LFP alpha
7 of 22

SleepSpindles in
two humanparticipants

Stokes et al. (2023) Sleep

8 of 22

Troughmax canprecede LOC

Mukamel et al (2014) J Neurosci

Peakmax only
occurs when thesubject isunarousable

9 of 22

Amount of the total power that is
captured by the first eigenvalue ofthe cross-spectral matrix

What is the best way toquantify all of these effects?
What is the best model
for a rhythm?
Purdon et al (2013) PNAS

10 of 22

 Defining rhythms
 Sinusoids
 Analytic Signal
 State Space Oscillators
 Rhythm separation
 Decomposition of a univariate signal into multiple time-domain oscillatory processes Phase amplitude coupling Modeling interactions between rhythms
 Functional connectivity
 Three different kinds of rhythmic networks
11 of 22

Fourier Transform:
Signal = projection onto
Real axis

ImaginaryPhasor
Amplitude = vector length
Phase = starting angle
InstantaneousPhase

RealBut neural rhythms aren’t sinusoidal!
https://upload.wikimedia.org/wikipedia/commons/8/89/Unfasor.gif 12 of 22

(3a) Instantaneous Phase(1) Bandpass Filter(2) Compute Analytic
Signal(3b) Instantaneous Amplitude

If Xt is narrowband
But bandpass filtering imposes arbitrary cutoffs that
may leave out signal and include noise

14 of 22

A rotating 2-D vector autoregressive process can beinterpreted as a noisy phasor:

Decay(forstability)

Rotate byat each timestep

GaussianNoise

This is the foundation of the StateSpace Oscillator (SSO) Model

15 of 22

Hidden state: K oscillators

Observations: sum of real parts of the oscillators
Inst. AmplitudeInst. Phase
Matsuda and Komaki (2017). Neural Computation

16 of 22

Beck, Stephen, Purdon (2018) IEEE EMBC

17 of 22

 Defining rhythms
 Sinusoids
 Analytic Signal
 State Space Oscillators
 Rhythm separation
 Decomposition of a univariate signal into multiple time-domain oscillatory processes Phase amplitude coupling Modeling interactions between rhythms
 Functional connectivity
 Three different kinds of rhythmic networks
18 of 22

Better at capturing transient andnon-sinusoidal rhythms
Beck, Stephen, Purdon (2018) IEEE EMBC

19 of 22

 Defining rhythms
 Sinusoids
 Analytic Signal
 State Space Oscillators
 Rhythm separation
 Decomposition of a univariate signal into multiple time-domain oscillatory processes Phase amplitude coupling Modeling interactions between rhythms
 Functional connectivity
 Three different kinds of rhythmic networks
20 of 22

Magnitude ofCoupling
Soulat, Stephen, Beck, Purdon (2022) Scientific Reports

Phase ofCoupling

FittedAmplitude
21 of 22

Soulat, Stephen, Beck, Purdon (2022) Scientific Reports

22 of 22

Soulat, Stephen, Beck, Purdon (2022) Scientific Reports

23 of 22 Defining rhythms Sinusoids
 Analytic Signal
 State Space Oscillators
 Rhythm separation
 Decomposition of a univariate signal into multiple time-domain oscillatory processes Phase amplitude coupling Modeling interactions between rhythms
 Functional connectivity
 Three different kinds of rhythmic networks
24 of 22

Common Oscillator Model

Multivariate observations

Hsin, Eden, Stephen (2022) IEEE SSC

Common Oscillator Model
Correlated Noise Model
Directed Influence Model

25 of 22

Common Oscillator
Alpha rhythm B matrices

Global Coherence
Magnitude at alpha

26 of 2227 of 22

 State space oscillators are a powerful, flexible framework for

capturing neural oscillations
Decomposing signals in the time domain Phase amplitude coupling Functional networks Future directions:

Other network models and applications
 Other observation models


28 of 22

29 of 22

Beck, He, Gutierrez, Purdon (2022) Neuroscience

30 of 22