DCM for Phase Coupling

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DCM for Phase Coupling
Will Penny
Wellcome Trust Centre for Neuroimaging, University
College London, UK
Sir Peter Mansfield MR Centre, Nottingham
University, Wed Jan 28, 2009
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Overall Aim
To study long-range synchronization
processes Develop connectivity model for
bandlimited data Regions phase couple via changes
in instantaneous frequency
Region 2
Region 1
?
?
Region 3
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Overview

• Phase Reduction
• Choice of Phase Interaction Function (PIF)
• DCM for Phase Coupling
• Ex 1 Finger movement
• Ex 2 MEG Theta visual working memory
• Conclusions

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Overview

• Phase Reduction
• Choice of Phase Interaction Function (PIF)
• DCM for Phase Coupling
• Ex 1 Finger movement
• Ex 2 MEG Theta visual working memory
• Conclusions

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Phase Reduction
Stable Limit Cycle
Perturbation
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Isochrons of a Morris-Lecar Neuron
Isochron Same Asymptotic Phase
From Erm
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Phase Reduction
Stable Limit Cycle
Perturbation
ISOCHRON
Assume 1st order Taylor expansion
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Phase Reduction
From a high-dimensional differential eq.
To a one dimensional diff eq.
Phase Response Curve
Perturbation function
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Example Theta rhythm
Denham et al. 2000
Wilson-Cowan style model
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Four-dimensional state space
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Now assume that changes sufficiently slowly that
2nd term can be replaced by a time average over
a single cycle
This is the Phase Interaction Function
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Now assume that changes sufficiently slowly that
2nd term can be replaced by a time average over
a single cycle
Now 2nd term is only a function of phase
difference
This is the Phase Interaction Function
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Multiple Oscillators
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Overview

• Phase Reduction
• Choice of Phase Interaction Function (PIF)
• DCM for Phase Coupling
• Ex 1 Finger movement
• Ex 2 MEG Theta visual working memory
• Conclusions

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Choice of g
We use a Fourier series approximation for the PIF
This choice is justified on the following grounds

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Phase Response Curves,

• Experimentally using perturbation method

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Leaky Integrate and Fire Neuron
Type II (pos and neg)
Z is strictly positive Type I response
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Hopf Bifurcation
Stable Limit Cycle
Stable Equilibrium Point
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For a Hopf bifurcation (Erm Kopell)
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Septo-Hippocampal theta rhythm
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Septo-Hippocampal Theta rhythm
Theta from Hopf bifurcation
A
B
A
B
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Neural mass model of cortex
Jansen Ritt
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Oscillations from neural mass model
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Bifurcation analysis of neural mass model
Output
Alpha Rhythm From Hopf Bifurcation
Input
Grimbert Faugeras
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PIFs
Even if you have a type I PRC, if the
perturbation is non-instantaneous, then youll
end up with a type II first order Fourier PIF
(Van Vreeswijk, alpha function synapses)
so thats our justification. and then there
are delays .
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Overview

• Phase Reduction
• Choice of Phase Interaction Function (PIF)
• DCM for Phase Coupling
• Ex 1 Finger movement
• Ex 2 MEG Theta visual working memory
• Conclusions

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DCM for Phase Coupling Model
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Sinusoidal coupling
-0.3
-0.6
-0.3
-0.3
is a stable fixed point
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Overview

• Phase Reduction
• Choice of Phase Interaction Function (PIF)
• DCM for Phase Coupling
• Ex 1 Finger movement
• Ex 2 MEG Theta visual working memory
• Conclusions

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MEG data from Visual Working Memory
1) No retention (control condition)
Discrimination task


2) Retention I (Easy condition) Non-configural
task


3) Retention II (Hard condition) Configural task


5 sec
3 sec
5 sec
1 sec
MAINTENANCE
PROBE
ENCODING
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Questions for DCM

• Duzel et al. find different patterns of
  theta-coupling in the delay period
• dependent on task.
• Pick 3 regions based on previous source
  reconstruction
• 1. Right Hipp 27,-18,-27 mm
• 2. Right Occ 10,-100,0 mm
• 3. Right IFG 39,28,-12 mm
• Fit models to control data (10 trials) and hard
  data (10 trials). Each trial
• comprises first 1sec of delay period.
• Find out if structure of network dynamics is
  Master-Slave (MS) or
• (Partial/Total) Mutual Entrainment (ME)
• Which connections are modulated by (hard) memory
  task ?

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Data Preprocessing

• Source reconstruct activity in areas of interest
  (with fewer sources than
• sensors and known location, then pinv will do
  Baillet 01)
• Bandpass data into frequency range of interest
• Hilbert transform data to obtain instantaneous
  phase
• Use multiple trials per experimental condition
• Use first order Fourier PIFs

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MTL Master
VIS Master
IFG Master
1
IFG
3
5
VIS
IFG
VIS
IFG
VIS
Master- Slave
MTL
MTL
MTL
IFG
VIS
6
2
VIS
IFG
VIS
IFG
4
Partial Mutual Entrainment
MTL
MTL
MTL
VIS
IFG
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Total Mutual Entrainment
MTL
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Model Comparison
LogEv
Model
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Optimal model Strength of connections

• Numbers are norm of Fourier coeffs
• Intrinsic connectivity (arrow end) established
  for control task (no memory)
• Modulatory connections (dotted end) required for
  memory task

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CONTROL
Relative Phase
Background Gray-scale is So black areas
are Fixed Points
Blue arrowsFlow field
Red dots show fitted trajectories of
individual trials
Global Zero-Lag Sync
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MEMORY
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Model Fit
MTL
VIS
IFG
Seconds
One memory trial
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Overview

• Phase Reduction
• Choice of Phase Interaction Function (PIF)
• DCM for Phase Coupling
• Ex 1 Finger movement
• Ex 2 MEG Theta visual working memory
• Conclusions

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Finger movement
Haken et al. 95
Low Freq
High Freq
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Ns2, Nc0
Anti-Phase Unstable
Ns1, Nc0
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Estimating coupling coefficient
EMA error
DCM error
Additive noise level
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Inferring the order of the PIF
Multiple trials required to adequately sample
state space
Distribution of Initial Phase Difference Narrow
Wide
times true model selected
High noise s0.2
Number of trials
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Conclusions

• Model is multivariate extension of bivariate
  work by Rosenblum Pikovsky
• (EMA approach)
• On bivariate data DCM-P is more accurate than
  EMA
• Additionally, DCM-P allows for inferences about
  master-slave versus
• mutual entrainment mechanisms in multivariate
  (Ngt2) oscillator networks

• Delay estimates from DTI
• Use of phase response curves derived from
  specific neuronal models
• using XPP or MATCONT
• Stochastic dynamics (natural decoupling) see
  Kuramoto 84, Brown 04
• For within-trial inputs causing phase-sync and
  desync (Tass model)