phase dynamics of neural populations under impact of ... · phase dynamics of neural populations...

Phase dynamics of neural populations under Phase dynamics of neural populations under impact of statistical feedbackimpact of statistical feedback

Andreas DaffertshoferAndreas Daffertshofer

Research Institute MOVEResearch Institute MOVEFaculty of Human Movement SciencesFaculty of Human Movement Sciences

VU University Amsterdam, The NetherlandsVU University Amsterdam, The Netherlands

Leiden, May 24, 2009Leiden, May 24, 2009

Experimental approaches Experimental approaches ……

Manipulate M/EEG Manipulate M/EEG dynamics via cognitive factorsdynamics via cognitive factors

Changes in Changes in corticocortico--cortical and cortical and corticocortico--spinal synchronizationspinal synchronization

Conceptual framework: movementConceptual framework: movement (in(in--)stabilities)stabilitiesCognitive factors, e.g., attentionCognitive factors, e.g., attention(Motor) learning(Motor) learning

…… Mathematical modeling of information exchange between Mathematical modeling of information exchange between neural ensemblesneural ensembles

• Experimental approacheso Overviewo Design

• Synchronizationo Corticalo Spinal

• Phase dynamicso Jirsa & Hakeno Sakaguchi & Kuramoto

• Statistical feedbacko Mean fieldo Simulated annealing

MEG CentreAmsterdam

MEG CentreAmsterdam

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

-100

-50

0

50

100

150

[0.10,-117.27]

[0.10,140.01]

time [sec]

LC15LP11

motor eventmotor event

Experimental framework Experimental framework ………… methods, procedures, analysesmethods, procedures, analyses





Learning rhythmic bimanual tappingLearning rhythmic bimanual tapping

Houweling, Daffertshofer et al., Houweling, Daffertshofer et al., NeuroImageNeuroImage 20082008

1:1

2:3

ββ--amplitude amplitude modulation modulation

corticocortico--spinal spinal

entrainmententrainmentin in ββ--bandband

learninglearningtim

etim

e





Which part of activity is transferred to the spinal cord?Which part of activity is transferred to the spinal cord?

Van Wijk, Daffertshofer, et al., Van Wijk, Daffertshofer, et al., Cerebral Cortex Cerebral Cortex (2009)(2009)

--4s 4s --2s 2s --0s0s





““Example descending (red) and ascending (blue) pathways that coulExample descending (red) and ascending (blue) pathways that could mediate d mediate corticomuscularcorticomuscular coherence.coherence.””(after Baker, (after Baker, Current Opinion in BiologyCurrent Opinion in Biology 2007)2007)





Experimental approaches Experimental approaches …… conclusionsconclusions

Neural Neural dynamics can be altered via cognitive factors dynamics can be altered via cognitive factors

Changes in Changes in corticocortico--cortical and cortical and corticocortico--spinal synchronization spinal synchronization are functionally relevantare functionally relevant

MovementMovement (in(in--)stabilities provide excellent window into dynamical )stabilities provide excellent window into dynamical changes changes

(Motor) learning alters (Motor) learning alters corticocortico--spinal entrainmentspinal entrainment

…… Mathematical modeling of information exchange between Mathematical modeling of information exchange between neural ensemblesneural ensembles

KNAW-MEG CentreAmsterdam

KNAW-MEG CentreAmsterdam





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Jirsa & Haken, Jirsa & Haken, Physical Review LettersPhysical Review Letters 19961996





•• EE((xx,,tt)) / / II((xx,,tt)) are pulse (action potential) densities of excitatory / inhibitoare pulse (action potential) densities of excitatory / inhibitory ry ((sub)populationssub)populations at time at time tt and position and position xx

•• ff(e/i)(e/i)(x,X(x,X)) are the densities of excitatory and inhibitory connections are the densities of excitatory and inhibitory connections between sites between sites xx and and XX

•• ppdd is the pulse densityis the pulse density

•• tt’’=|=|xx--X|/cX|/c with with cc as transmission velocity along the axon (6as transmission velocity along the axon (6--9 9 m/sm/s) )

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Freeman, Freeman, Societies of BrainsSocieties of Brains 1975, 1975, ErmentroutErmentrout & Cowan, & Cowan, Biological CyberneticsBiological Cybernetics 19791979Robinson et al., Robinson et al., Physical Review EPhysical Review E 19981998

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Dynamics of the rotating phase of each oscillatorDynamics of the rotating phase of each oscillator

SakaguchiSakaguchi, , ProgProg. Theo. Phys.. Theo. Phys. 1988; 1988; SakaguchiSakaguchi, , ShinomotoShinomoto & Kuramoto, & Kuramoto, ProgProg. Theo. Phys.. Theo. Phys. 1988 1988


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1

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Φ = + − Φ −ΦΦ +−Ω − Γ∑

,

1 for 1, ,,

1 otherwisen m n m l

l MK Kv v v

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heterogeneityheterogeneity

nonnon--rotating phaserotating phase(unimodal freq. dist.)(unimodal freq. dist.)

external forceexternal forceattracts phaseattracts phase





Dynamics of the rotating phase of each oscillatorDynamics of the rotating phase of each oscillator


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Mean field approximation yields a FokkerMean field approximation yields a Fokker--Planck equationPlanck equation……

……that is nonthat is non--linear in the probability densitylinear in the probability density






x1 x2

Haken et al., Haken et al., BiologBiolog. . CybernCybern.. 19851985SchSchöönerner et al., et al., BiologBiolog. . CybernCybern.. 19861986Kay et al., Kay et al., J. Exp Psychol.J. Exp Psychol. 1987, 19911987, 1991Fuchs et al., Fuchs et al., BiologBiolog. . CybernCybern.. 19961996Daffertshofer et al., Daffertshofer et al., PhysicaPhysica DD 19991999Beek et al., Beek et al., Brain & Brain & CognCogn.. 20022002……

sin sin 2d A Bdtφ φ φ= − −






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φ





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20


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φ φ μ φ χ χ χ φ φφ φ⎡ ⎤∂ ∂ ∂

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Nonlinear FokkerNonlinear Fokker--Planck EquationsPlanck Equations•• can be related to noncan be related to non--extensive entropies via extensive entropies via MaxEntMaxEnt principles (Frank & principles (Frank &

Daffertshofer, Daffertshofer, PhysicaPhysica A, 1999); A, 1999);

e.g., for e.g., for TsallisTsallis generalized entropy we find generalized entropy we find

( )

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2

1( ) 1 ( ) 1

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q

S p p x dxq

P x t h x P x t P x tt x xε

= −−

∂ ∂ ∂= − + ⎡ ⎤⎣ ⎦∂ ∂ ∂

∫





Frank, Frank, NonNon--linear Fokkerlinear Fokker--Planck equationsPlanck equations Springer 2004Springer 2004

•• often obey poweroften obey power--laws, i.e. they mimic longlaws, i.e. they mimic long--range correlationsrange correlations

Frank, Daffertshofer & Beek, Frank, Daffertshofer & Beek, J. Biological PhysicsJ. Biological Physics 20002000

coupling via coupling via PP((xx,,tt)) = statistical feedback: example = statistical feedback: example ‘‘simulated annealingsimulated annealing’’

( ) ( )

2d 1d 1 ,

n

x h xt P x t

βα

⎡ ⎤= + Γ⎢ ⎥

+⎢ ⎥⎣ ⎦





Frank, Daffertshofer & Beek, Frank, Daffertshofer & Beek, J. Biological PhysicsJ. Biological Physics 20002000

coupling via coupling via PP((xx,,tt)) = statistical feedback: example simulated annealing= statistical feedback: example simulated annealing

1, 1, 10n α β= = = 1, 100, 10n α β= = =





( ) ( )

2d 1d 1 ,

n

x h xt P x t

βα

⎡ ⎤= + Γ⎢ ⎥

+⎢ ⎥⎣ ⎦

Thanks for your attentionThanks for your attention

Peter BeekPeter BeekLieke PeperLieke PeperTill FrankTill FrankTjeerd BoonstraTjeerd BoonstraSanne HouwelingSanne HouwelingAlistair VardyAlistair VardyBernadette van WijkBernadette van Wijk

Kees StamKees StamBob van DijkBob van DijkPeter PraamstraPeter PraamstraNick RoachNick RoachAlan WingAlan WingGuido NolteGuido NolteMichael BreakspearMichael BreakspearHermann HakenHermann Haken

phase dynamics of neural populations under impact of ... · phase dynamics of neural populations...

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