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									Weakly Coupled Oscillators

      Will Penny, Vladimir Litvak, Lluis
    Fuentemilla, Emrah Duzel, Karl Friston



      Wellcome Trust Centre for Neuroimaging,
           University College London, UK

Symposium on Multimodal Brain Imaging, University of
           Birmingham, May 15th, 2009
        Transient Synchronization
•   Neuronal oscillations are the key to linking computational to imaging
    neuroscience and different imaging modalities to each other


•   Short range (zero-lag) sync (within single cortical area). Hebbian learning
    (STDP) results in gamma bursts signalling recognition events (Hopfield).
    Local sync necessary for long range signal transmission (Fries).


•   Long range (zero-lag) sync (between cortical areas) mediated by
    oscillations (Singer, Varela,..). Triplets of regions (Vicente, Lumer …). Long
    range sync at lower freqs ? Information relayed via pattern and/or phase
    coding (O’Keefe, Panzeri et al.) or sync necessary for chunking
    (Jensen/Lisman).

•   Sync is Transient
                 Overall Aim
To study long-range synchronization processes
Develop connectivity model for bandlimited data
Regions phase couple via changes in instantaneous frequency

 Region 1                                Region 2




                                           ?
       ?




                       Region 3
                 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 Reduction                            Stable Limit Cycle

                                            X0  F(X0)
                                            X 0 (t  T )  X 0 (t )
                                           (X0)  




 Perturbation




                X  F ( X )  P( X )
             ( X )  Asymptotic _ Phase
Isochrons of a Morris-Lecar Neuron




                                     Isochron=
                                     Same
                                     Asymptotic
  n
                                     Phase




                                     From Erm
Phase Reduction                                             Stable Limit Cycle

                                                             X0  F(X0)
                                                             X 0 (t  T )  X 0 (t )
                                                            (X0)  




 Perturbation

 X  F ( X )  P( X )                                               ISOCHRON
         d ( X )      d ( X )           d ( X )
 (X )            X           F(X )             P( X )
           dX           dX                  dX
         d ( X 0 )          d ( X 0 )
 (X )             F(X0)              P( X 0 )               Assume 1st order
            dX                  dX 0                           Taylor expansion
     z ( ) p( )
Phase Reduction
  From a high-dimensional
  differential eq.          X  F ( X )  P( X )

  To a one dimensional          z ( ) p( )
  diff eq.



  Phase Response Curve
          z ( )

   Perturbation function
          p( )
                Example: Theta rhythm

     Denham et al. 2000:                                Hippocampus

   dx1
1       x1  (ke  x1 ) ze ( w13 x3  PCA3 )
   dt                                              x1
   dx
 2 2   x2  (ki  x2 ) zi ( w21 x1 )                             x3
    dt                                                       x2
   dx
 3 1   x3  (ki  x3 ) ze ( w34 x4  PCA3 )
    dt
   dx
 4 4   x4  (ki  x4 ) zi ( w42 x2  PS )
    dt
                                                        x4        Septum
Wilson-Cowan style model
Four-dimensional state space




                    z ( ) p( )
                           1    z (1 ) p(1 , 2 )

                                                  2    z (2 ) p (2 , 1 )




Now assume that 2  1  
changes sufficiently slowly that 2nd term
can be replaced by a time average
over a single cycle

             1
     g ( )   z  t  p  t , t    dt
             T

This is the ‘Phase Interaction Function’
                           1    z (1 ) p(1 , 2 )

                                                  2    z (2 ) p (2 , 1 )




Now assume that 2  1  
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
             1
     g ( )   z  t  p  t , t    dt
             T                                        1    g (2  1 )

This is the ‘Phase Interaction Function’
                                                      2    g (1  2 )
Multiple Oscillators




  i     g (i   j )
             j
                 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
We use a Fourier series approximation for the PIF



                     1
             g ( )   z  t  p  t , t    dt
                     T

                        Nf

             g ( )   an sin n  bn cos n
                        n 1



 This choice is justified on the following grounds …
Phase Response Curves, z ( )

• Experimentally – using perturbation method
                           T0  T1
                       
                             T0
           Leaky Integrate and Fire Neuron

                                         -50




                             V (mV)
                                         -55
                                         -60
                                            0      5         10    15 20          25         30
V  VR  Va (1  exp(t /  ))                                      t (ms)
                                         20
            V  VR 

                                 dt/dV
t   log 1 
                                       10
               Va                      0
                                           0       5         10    15 20          25         30
 dt                                                                t(ms)
     
dV Va  VR  V                            4
                                  z()


                                          2
                                          0
                                           0           0.2        0.4       0.6        0.8        1
                                                                        
Type II
(pos and
                                                Z is strictly positive: Type I response
neg)
            Hopf Bifurcation




Stable Equilibrium Point   Stable Limit Cycle
For a Hopf bifurcation (Erm & Kopell…)




                 z ( )  a sin   b cos
Septo-Hippocampal theta rhythm
     Septo-Hippocampal Theta rhythm

    Hippocampus            Theta from
                           Hopf bifurcation



A


             B

                   A
          Septum




                                 B
                                 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
DCM for Phase Coupling Model

            1 dki        Nr
                    fi   gij ki  kj 
           2 dt          j 1
                         Ns                    Nc
           gij     a ijn sin(n )   a ijn cos(n )
                           s                  c

                         n 1                 n 1
                           Nq

           aijn  aijn   ucbijn q
                          q 1


Where k denotes the kth trial. uq denotes qth modulatory input, a
between trial effect
 fi   is the frequency in the ith region (prior mean f0, dev = 3fb)

 aijn has prior mean zero, dev=3fb
 bijn has prior mean zero, dev=3fb
       -0.3




       -0.6




-0.3    -0.3
                 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
                                  Anti-Phase Stable

                                                                        dV t 
                       0.5
(a) Low Freq

                                                               t                 g t 
                         0




                V()
                       -0.5


                        -1
                              0   1   2    3       4   5   6
                                                                          dt
                                               

                         1

                       0.5

         PIF                                                        Ns=2, Nc=0
                G()



                         0

                       -0.5

                        -1
                              0   1   2    3       4   5   6
                                               




                                  Anti-Phase Unstable
                       0.5

(b) High Freq
                V()




                         0




                       -0.5
                              0   1   2    3       4   5   6
                                               

                       0.5



                                                                        Ns=1, Nc=0
                G()




                         0




                       -0.5
                              0   1   2    3       4   5   6
                                               
         Estimating coupling coefficient

      Left
                 a=0.5
                         Right
                                            left  f
      Finger             Finger
                                           right  f  a sin(right  left )




                                                              EMA error
E  (a  a ) 2
         ˆ



                                                              DCM error




                         Additive noise level
                 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) No retention (control condition): Discrimination task


                                             +


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


                                             +


3) Retention II (Hard condition): Configural task


                                             +

1 sec          3 sec                       5 sec            5 sec


         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 MTL [27,-18,-27] mm
         2. Right VIS [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 ?
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
              MTL Master                    VIS Master               IFG Master


          1   IFG         VIS       3 IFG               VIS    5   IFG         VIS
Master-
Slave
                    MTL                           MTL                    MTL



          2   IFG          VIS          4   IFG          VIS   6   IFG         VIS
Partial
Mutual
                    MTL                           MTL                    MTL
Entrainment


                                        7 IFG            VIS
                          Total
                          Mutual
                          Entrainment             MTL
        450

        400

        350

        300

LogEv   250

        200

        150

        100

        50

         0
              1   2   3     4     5   6   7

                          Model
             0.77




      2.46
IFG                 VIS


                           0.89
                    2.89

              MTL
      1

MTL   0


      -1
           5   5.1   5.2   5.3   5.4   5.5   5.6   5.7   5.8   5.9   6
      1

VIS   0


      -1
           5   5.1   5.2   5.3   5.4   5.5   5.6   5.7   5.8   5.9   6
      1

IFG   0


      -1
           5   5.1   5.2   5.3   5.4   5.5   5.6   5.7   5.8   5.9   6



                                 Seconds
Control



          IFG-VIS




                      MTL-VIS
Memory


         IFG-VIS




                     MTL-VIS
                    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 (N>2) oscillator networks


• Delay estimates from DTI

• Use of phase response curves derived from specific neuronal models
  using XPP or MATCONT. Brunel and Wang, 03 !

• Stochastic dynamics (natural decoupling) … see Kuramoto 84, Brown 04
  For within-trial inputs causing phase-sync and desync (Tass model)

• What would we see in fMRI ?
Neural Mass model
         Neural Mass model

Output
                               Alpha Rhythm
                               From Hopf
                               Bifurcation




         Input
                             Grimbert &
                             Faugeras
                                                                      Type II
                                                                      (pos and
            -50                                                       neg)



  V (mV)
            -55
            -60
               0   5         10    15 20          25         30
                                    t (ms)
Z is 20
     strictly positive: Type I response
    dt/dV


            10
             0
              0    5         10    15 20          25         30
                                    t(ms)
             4
      z()




             2
             0
              0        0.2        0.4       0.6        0.8        1
                                        



       Eg. Leaky Integrate and Fire Neuron

								
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