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DCM for Phase Coupling. Will Penny. Wellcome Trust Centre for Neuroimaging, University College London, UK. Brain Modes, Dec 12, 2008. Overall Aim. To study long-range synchronization processes Develop connectivity model for bandlimited data - PowerPoint PPT Presentation
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DCM for Phase CouplingDCM for Phase Coupling
Will PennyWill Penny
Wellcome Trust Centre for Neuroimaging,Wellcome Trust Centre for Neuroimaging,University College London, UKUniversity College London, UK
Brain Modes, Dec 12, 2008Brain Modes, Dec 12, 2008
Overall Aim
To study long-range synchronization processesDevelop connectivity model for bandlimited dataRegions phase couple via changes in instantaneous frequency
Region 1
Region 3
Region 2
??
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
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
Phase Reduction0 0
0 0
0
( )
( ) ( )
( )
X F X
X t T X t
X
( ) ( )
( ) _
X F X P X
X Asymptotic Phase
Stable Limit Cycle
Perturbation
n
Isochrons of a Morris-Lecar Neuron
From Erm
Isochron=SameAsymptotic Phase
Phase Reduction0 0
0 0
0
( )
( ) ( )
( )
X F X
X t T X t
X
0 00 0
0
( ) ( )
( ) ( ) ( )( ) ( ) ( )
( ) ( )( ) ( ) ( )
( ) ( )
X F X P X
d X d X d XX X F X P X
dX dX dXd X d X
X F X P XdX dX
z p
Stable Limit Cycle
Perturbation
ISOCHRON
Assume 1st orderTaylor expansion
Phase Reduction
( ) ( )z p
( ) ( )X F X P X From a high-dimensionaldifferential eq.
To a one dimensionaldiff eq.
Phase Response Curve
( )z
Perturbation function
( )p
Example: Theta rhythm
Denham et al. 2000: Hippocampus
Septum
11 1 1 13 3 3
22 2 2 21 1
13 3 3 34 4 3
44 4 4 42 2
( ) ( )
( ) ( )
( ) ( )
( ) ( )
e e CA
i i
i e CA
i i S
dxx k x z w x P
dtdx
x k x z w xdtdx
x k x z w x Pdtdx
x k x z w x Pdt
1x
2x 3x
4xWilson-Cowan style model
Four-dimensional state space
( ) ( )z p
1 1 1 2( ) ( , )z p
2 2 2 1( ) ( , )z p
Now assume thatchanges sufficiently slowly that 2nd term can be replaced by a time averageover a single cycle
1( ) ,g z t p t t dt
T
2 1
This is the ‘Phase Interaction Function’
1 1 1 2( ) ( , )z p
2 2 2 1( ) ( , )z p
Now assume thatchanges sufficiently slowly that 2nd term can be replaced by a time averageover a single cycle
1( ) ,g z t p t t dt
T
2 1
This is the ‘Phase Interaction Function’
1 2 1( )g
2 1 2( )g
Now 2nd term is only a function of phase difference
( )i i jj
g
Multiple Oscillators
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
Choice of g
1
1( ) ,
( ) sin cosfN
n nn
g z t p t t dtT
g a n b n
We use a Fourier series approximation for the PIF
This choice is justified on the following grounds …
Phase Response Curves,
• Experimentally – using perturbation method
0 1
0
T T
T
)(z
Leaky Integrate and Fire Neuron
0 5 10 15 20 25 30
-60-55-50
t (ms)
V (m
V)0 5 10 15 20 25 30
01020
t(ms)dt
/dV
0 0.2 0.4 0.6 0.8 1024
z( )
VVVdV
dt
V
VVt
tVVV
Ra
a
R
aR
1log
))/exp(1(
Z is strictly positive: Type I response
Type II(pos and neg)
Hopf Bifurcation
Stable Equilibrium Point Stable Limit Cycle
cossin)( baz
For a Hopf bifurcation (Erm & Kopell…)
Septo-Hippocampal theta rhythm
Hippocampus
Septum
A
A
B
B
Septo-Hippocampal Theta rhythm
Theta fromHopf bifurcation
PIFs
1( ) ,g z t p t t dt
T
Even if you have a type I PRC, if the perturbation is non-instantaneous, then you’ll end up with a type II first order Fourier PIF (Van Vreeswijk, alpha function synapses)
… so that’s our justification.
… and then there are delays ….
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
1
1 1
1
1
2
sin( ) cos( )
r
s c
q
Nki
i ij ki kjj
N Ns c
ij ijn ijnn n
Nq
ijn ijn c ijnq
df g
dt
g a n a n
a a u b
DCM for Phase Coupling Model
Where k denotes the kth trial. uq denotes qth modulatory input, a between trial effect
if is the frequency in the ith region (prior mean f0, dev = 3fb)
ijna
ijnb
has prior mean zero, dev=3fb
has prior mean zero, dev=3fb
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
Finger movement
Haken et al. 95
Low Freq High Freq
0 1 2 3 4 5 6-0.5
0
0.5
V(
)
0 1 2 3 4 5 6-0.5
0
0.5
G(
)
tt t
dVg
dt
(b) High Freq
Ns=2, Nc=0
Ns=1, Nc=0
Anti-Phase Unstable
0 1 2 3 4 5 6-1
-0.5
0
0.5
V(
)
0 1 2 3 4 5 6-1
-0.5
0
0.5
1G
()
(a)
PIF
Low Freq
Anti-Phase Stable
Estimating coupling coefficient
2ˆ( )E a a
LeftFinger
RightFinger
a=0.5
DCM error
EMA error
sin( )
left
right right left
f
f a
Additive noise level
Inferring the order of the PIF
LeftFinger
RightFinger
Number of trials
p(est=2|true=2)
Multiple trials requiredto adequately sample state space
High noise=0.2
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
MEG data from Visual Working Memory
+
+
+
1 sec 3 sec 5 sec 5 sec
1) No retention (control condition): Discrimination task
2) Retention I (Easy condition): Non-configural task
3) Retention II (Hard condition): Configural task
ENCODING MAINTENANCE PROBE
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] mm2. Right Occ [10,-100,0] mm3. 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 ?
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
Hipp
OccIFG
Hipp
OccIFG
Hipp
OccIFG
Hipp
OccIFG
Hipp
OccIFG
Hipp
OccIFG1
Hipp
OccIFG2
3
4
5
6
7
Master-Slave
PartialMutualEntrainment
TotalMutualEntrainment
Hippocampalsource
Occipitalsource
Frontalsource
1 2 3 4 5 6 70
50
100
150
200
250
LogEv
Model
Model Comparison
Hipp
OccIFG
0.17
0.03
0.99
0.65
0.030.03
0.00
0.13
• Intrinsic connectivity established for control task (no memory requirement)• Modulatory connections required for ‘hard’ memory task• Fronto-occipital connections increased most strongly esp. Occ->IFG
f=6.0Hz
f=5.7Hzf=5.7Hz
5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6-1
0
1Region Right-Hipp
Data
Fitted
5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6-1
0
1Region Right-Occ
Data
Fitted
5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6-1
0
1
Trial 20
Region Right-IFG
Data
Fitted
Seconds
Model Fit
Estimated Phase Interaction Functions, g
From
To
Hipp Occ IFG
Hipp
Occ
IFG
Hard
Control
0 5-0.1
0
0.1
0 5-1
0
1
0 5-0.2
0
0.2
0 5-1
0
1
0 5-1
0
1
0 5-2
0
2
Conclusions
• 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)
• 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 (N>2) oscillator networks
Neural Mass model
Neural Mass model
Input
Output
Grimbert &Faugeras
Alpha RhythmFrom HopfBifurcation
Eg. Leaky Integrate and Fire Neuron
0 5 10 15 20 25 30
-60-55-50
t (ms)V
(mV)
0 5 10 15 20 25 300
1020
t(ms)
dt/d
V
0 0.2 0.4 0.6 0.8 1024
z( )
Z is strictly positive: Type I response
Type II(pos and neg)