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Will Penny DCM for Time-Frequency DCM Course, Paris, DCM for Induced Responses 2. DCM for Phase Coupling

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Dynamic Causal Models NeurophysiologicalPhenomenological DCM for ERP DCM for SSR DCM for Induced Responses DCM for Phase Coupling spiny stellate cells inhibitory interneurons Pyramidal Cells Time Frequency Phase Source locations not optimizedElectromagnetic forward model included

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Region 1 Region 2 ? ? Changes in power caused by external input and/or coupling with other regions Model comparisons: Which regions are connected? E.g. Forward/backward connections (Cross-)frequency coupling: Does slow activity in one region affect fast activity in another ? 1. DCM for Induced Responses Time Frequency Time

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Connection to Neural Mass Models First and Second order Volterra kernels From Neural Mass model. Strong (saturating) input leads to cross-frequency coupling

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Single region u2u2 u1u1 z1z1 z2z2 z1z1 u1u1 a 11 c cf. Neural state equations in DCM for fMRI

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Multiple regions u2u2 u1u1 z1z1 z2z2 z1z1 z2z2 u1u1 a 11 a 22 c a 21 cf. DCM for fMRI

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Modulatory inputs u2u2 u1u1 z1z1 z2z2 u2u2 z1z1 z2z2 u1u1 a 11 a 22 c a 21 b 21 cf. DCM for fMRI

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u1u1 u2u2 z1z1 z2z2 a 11 a 22 c a 12 a 21 b 21 Reciprocal connections u2u2 u1u1 z1z1 z2z2 cf. DCM for fMRI

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dg(t)/dt=A∙g(t)+C∙u(t) DCM for induced responses Where g(t) is a K x 1 vector of spectral responses A is a K x K matrix of frequency coupling parameters Also allow A to be changed by experimental condition Time Frequency

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G=USV’ Use of Frequency Modes Where G is a K x T spectrogram U is K x K’ matrix with K frequency modes V is K x T and contains spectral mode responses over time Hence A is only K’ x K’, not K x K Time Frequency

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Connection to Neurobiology From Neural Mass model. Strong (saturating) input leads to cross-frequency coupling

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Connection to Neurobiology Strong (saturating) input leads to cross-frequency coupling

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Connection to Neurobiology Weak input does not

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Differential equation model for spectral energy Nonlinear (between-frequency) coupling Linear (within-frequency) coupling Extrinsic (between-source) coupling Intrinsic (within-source) coupling How frequency K in region j affects frequency 1 in region i

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Modulatory connections Extrinsic (between-source) coupling Intrinsic (within-source) coupling

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Example: MEG Data

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OFA FFA input The “core” system

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nonlinear (and linear) linear Forward Backward linearnonlinear linear nonlinear FLBLFLBL FNBLFNBL FLNBFLNB FNBNFNBN OFA Input FFA FLBLFLBL Input FNBLFNBL OFA FFA FLBNFLBN OFA Input FFA FNBNFNBN OFA Input FFA Face selective effects modulate within hemisphere forward and backward cxs

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FLBLFLBL FNBLFNBL FLBNFLBN *F N B N backward linearbackward nonlinear forward linear forward nonlinear Model Inference Winning model: F N B N Both forward and backward connections are nonlinear

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Parameter Inference: gamma affects alpha Right backward - inhibitory - suppressive effect of gamma-alpha coupling in backward connections Left forward - excitatory - activating effect of gamma-alpha coupling in the forward connections From 32 Hz (gamma) to 10 Hz (alpha) t = 4.72 ; p = SPM t df 72; FWHM 7.8 x 6.5 Hz Frequency (Hz) From 30Hz To 10Hz

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For studying synchronization among brain regions Relate change of phase in one region to phase in others Region 1 Region 3 Region 2 ? ? 2. DCM for Phase Coupling Phase Interaction Function

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Synchronization achieved by phase coupling between regions Model comparisons: Which regions are connected? E.g. ‘master-slave’/mutual connections Parameter inference: (frequency-dependent) coupling values Region 1 Region 2 ? ?

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Synchronization Gamma sync synaptic plasticity, forming ensembles Theta sync system-wide distributed control (phase coding) Pathological (epilepsy, Parkinsons) Phase Locking Indices, Phase Lag etc are useful characterising systems in their steady state Sync – Steve Strogatz

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One Oscillator

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Two Oscillators

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Two Coupled Oscillators 0.3 Here we assume the Phase Interaction Function (PIF) is a sinewave

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Different initial phases 0.3

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Stronger coupling 0.6

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Bidirectional coupling 0.3

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DCM for Phase Coupling Phase interaction function is an arbitrary order Fourier series Allow connections to depend on experimental condition

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Example: MEG data Fuentemilla et al, Current Biology, 2010

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Delay activity (4-8Hz) Visual Cortex (VIS) Medial Temporal Lobe (MTL) Inferior Frontal Gyrus (IFG)

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Questions 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 Find out if structure of network dynamics is Master-Slave (MS) or (Partial/Total) Mutual Entrainment (ME) Which connections are modulated by memory task ?

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MTL VISIFG MTL VISIFG MTL VISIFG MTL VIS IFG MTL VISIFG MTL VIS IFG 1 MTL VISIFG Master- Slave Partial Mutual Entrainment Total Mutual Entrainment MTL MasterVIS MasterIFG Master

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Analysis 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. Data that we model are unwrapped phase time series in multiple regions. Use multiple trials per experimental condition Model inversion

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LogEv Model MTL VISIFG 3

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MTL VISIFG Connection Strengths

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Jones and Wilson, PLoS B, 2005 Recordings from rats doing spatial memory task:

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MTL - VIS IFG - VIS Control

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MTL - VIS IFG - VIS Memory

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Connection to Neurobiology: Septo-Hippocampal theta rhythm Denham et al. 2000: Hippocampus Septum Wilson-Cowan style model

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Four-dimensional state space

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Hippocampus Septum A A B B Hopf Bifurcation

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For a generic Hopf bifurcation (Erm & Kopell…) See Brown et al. 04, for PRCs corresponding to other bifurcations

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