Dynamic Causal Models of Neurophysiology

Organization for Human Brain Mapping Educational Course on Computational Neuroscience and Modeling of Neurodynamics

June 14th 2015, Honolulu, Hawaii

Dynamic Causal Models of Neurophysiology

Rosalyn Moran

Virginia Tech Carilion Research Institute

Department of Electrical & Computer Engineering, Virginia Tech

The DCM Approach

Models of Neural Population Activity

Quantifying the Goodness of Models and Model Parameters

How it works in practice

Outline

What does a sensor say about a synapse?

time

sens

ors

sens

ors

standard

deviant

time (ms)

amplitude (μV)

Principles of organisation: some fundamentals

Functional Specialisation Functional Integration

Principles of measurement

Synchronized

Activity

Fluctuating Electrical currents at the synapse

(E/IPSPs) and fast fluctuations at cell

bodies (APs)

Slower currents at synapses are spatially coherent in pyramidal

cell dendrites

About 50000 simultaneous pyramidal

cells give rise to measureable M/EEG

signal (10nAm)

But cells around the pyramidal neurons contribute to the

fluctuations

Origin of Current Source/Sinks

Neurotransmitters

Glutamate (excitatory) : Na+ Active Synaptic Sink, distributed sources

γ-aminobutryic acid (inhibitory) : Cl- Active Synaptic Source, distributed Sinks

Neuromodulators

Dopamine

Acetylcholine

Noradrenaline

Serotonin

Neuropeptides

Hormones

Neurotransmitters

Origin of Current Source/Sinks

Doya, 2002

Glutamate (excitatory) : Na+ Active Synaptic Sink, distributed sources

γ-aminobutryic acid (inhibitory) : Cl- Active Synaptic Source, distributed Sinks

sens

ors

sens

ors

standard

deviant

time

Dynamic Causal Modelling- Motivation

Conventional Approach: Study latencies, peaks, i.e. data features

DCM Approach: How did these features arise from connected brain sources?

What connectivity exactly?

What connectivity exactly?

Synchronized

Activity Result in dynamic Sinks and Sources

In the afferent population

In connected cell

assemblies

What does a sensor say about a synapse?

time

sens

ors

sens

ors

standard

deviant

time (ms)

amplitude (μV)

Synchronized

Activity Result in dynamic Sinks and Sources

In the afferent population

In connected cell

assemblies

Model the underlying dynamics:

Dynamic Causal Modelling- Overview

),,( θuxfx =

)|(),|(

mypmyp θ

??? Build a generative model for spatiotemporal data and fit to evoked responses.

Assume that both ERs are generated by temporal dynamics of a network of a few sources

Describe temporal dynamics by differential equations

Each source projects to the sensors, following physical laws

Solve for the model parameters using Bayesian model inversion

A1 A2

Principles of functional segregation & functional

integration

Then invert

Dynamics of cell ensembles

Dynamic Causal Modelling: Generic Framework

simple neuronal model (slow time scale)

fMRI

detailed neuronal model (synaptic time scales)

EEG/MEG

),,( θuxFdtdx

=Neural state equation:

Hemodynamic forward model: neural activity BOLD

Time Domain Data

Resting State Data

Electromagnetic forward model:

neural activity EEG MEG LFP

Time Domain ERP Data

Phase Coupling Data Cross Frequency Coupling

DCM: Neurophysiological Data Features

),,( θuxFdtdx

=

Frequency (Hz)

Pow

er (m

V2 )

time (ms)

ampl

itude

(μV)

DCM for Event Related Potentials (ERPs)

DCM for Cross Spectral Densities (CSDs)

DCM for Time Frequency Modulations (TFMs)

Freq

uenc

y (H

z)

time (ms)

The DCM Approach




Outline

DCM: Neurophysiological Data Features

),,( θuxFdtdx

=

Frequency (Hz)

Pow

er (m

V2 ) “theta”

time (ms)

ampl

itude

(μV)

Neural State Equation Can Generate Different Dynamics

Identical Form of Equations F:

DCM for ERP: Input u is a brief temporal pulse

DCM for CSD: No exogenous input, endogenous noise assumed to drive dynamics

DCM for TFMs: Both plus time modulated parameters: θ(t)

How are neural dynamics modeled in DCM?

A suite of neuronal population models including neural masses, fields and

conductance-based models…expressed in terms of sets of differential equations

Macro- and meso-scale

internal granular layer: spiny stellate cells internal pyramidal layer: pyramidal cells

external pyramidal Layer: pyramidal cells

external granular Layer: inhibitory interneurons

macro-scale meso-scale micro-scale

The state of a neuron comprises a number of attributes, membrane potentials, conductances etc.

Modelling these states can become intractable. Mean field approximations summarise the states in

terms of their ensemble density. Neural mass models consider only point densities and describe

the interaction of the means in the ensemble

Meso-scale dynamics

AP generation zone

⊗synapses

AP generation zone

eH

eτ

1ρ

x Connectivity parameter eH

internal granular layer: spiny stellate cells internal pyramidal layer: pyramidal cells

external pyramidal Layer: pyramidal cells


1

0

0.5 fir

ing

rate

(Hz)

1

0

0.5 fir

ing

rate

(Hz)

Meso-scale dynamics: the Sigmoid

Wilson & Cowan, 1972

From first principles, the sigmoidal firing rate curve arrives by assuming an average potential excitation at each cell and a distribution of individual neuronal

threshold potentials

Or

A distribution of number of synapses per cell and the same threshold

then integrate over excitation levels or above-threshold cells for:

Unimodal: ρ1 the precision/gain

Bimodal

AP generation zone

1ρ

1

0

0.5 fir

ing

rate

(Hz)

Meso-scale dynamics: & connectivity

Post-synaptic effects on pyramidal cells

<

≥−=

00

0)exp()( //

/

t

tttHth ie

ie

ie ττ

internal granular layer: spiny stellate cells

internal pyramidal layer: pyramidal cells


⊗ ),( ρvS


⊗iH


<

≥−=

00

0)exp()( //

/

t

tttHth ie

ie

ie ττ




Synapses activated by GABA from inhibitory interneurons

iτiiρ

1

0

0.5 fir

ing

rate

(Hz)

AP generation ii

⊗

1γ

),( ρvS

⊗

Synapses activated by glutamate from spiny stellate cells

iH

eτ


<

≥−=

00

0)exp()( //

/

t

tttHth ie

ie

ie ττ




Synapses activated by GABA from inhibitory interneurons

eH

iτ

⊗

AP generation ss

iiρ

1

0

0.5 fir

ing

rate

(Hz)

AP generation ii

ssρ

⊗

1

0

0.5 fir

ing

rate

(Hz)

))(()( tShtv ⊗=

1γ2γ

),( ρvS



<

≥−=

00

0)exp()( //

/

t

tttHth ie

ie

ie ττ




⊗

21, ,,,,,, γγττρρθ ieiessii HH⊆

2///

/ )()(2)(ieieie

ie tvtvSHtvτττ

−−=

))(()( tShtv ⊗=

1γ2γ

),( ρvS

PiPePyr

PeePeesseePe

PePe

PiiPiiiiiiPi

PiPi

iivvivSHi

ivvivSHi

iv

−=−−=

=−−=

=

22

21

2)(

2)(

κκγκ

κκγκ

Changes in PSP : inhibitory

Changes in PSP : excitatory

Net change in PSP

Model Parameters so far


Meso-scale dynamics: the convolution model

54321 ,,,,,,,,, γγγγγττρθ ieie HH⊆

1γ




2γ

PiPePyr

PeePeesseePe

PePe

PiiPiiiiiiPi

PiPi

iivvivSHi

ivvivSHi

iv

−=−−=

=−−=

=

22

21

2)(

2)(

κκγκ

κκγκ Net change in Pyramidal PSP

Model Parameters for one source with intrinsic

connections

IiIeII

IeeIeePyreeIe

IeIe

IiiIiiIIiiIi

IiIi

iivvivSHi

ivvivSHi

iv

−=

−−=

=−−=

=

24

23

2)(

2)(

κκγκ

κκγκ

3γ

4γ

Net change in Interneuronal PSP

SSeSSePyreeSS

SSSS

viuvSHiiv

25 2))(( κκγκ −−+=

=

Net change in Spiny Stellate PSP

5γ

Extrinsic Brain Connectivity

Extrinsic forward

connections spiny

stellate cells

inhibitory interneurons

pyramidal cells

4γ 3γ

214

014

41

2))()((ee

LF

e

e xxCuxSIAAHx

xx

ττγ

τ−−+++=

=

1γ 2γ)( 0xSAF

)( 0xSAL

)( 0xSABExtrinsic backward connections

Intrinsic connections

Extrinsic lateral connections ( )θ,,uxfx =

0x

278

038

87

2))()((ee

LB

e

e xxxSIAAHx

xx

ττγ

τ−−++=

=

236

746

63

225

1205

52

650

2)(

2))()()((

iii

i

ee

LB

e

e

xxxSHx

xx

xxxSxSAAHx

xxxxx

ττγ

τ

ττγ

τ

−−=

=

−−++=

=−=

LBFieie AAAHH ,,,,,,,,,,,, 54321 γγγγγττρθ ⊆

Model Parameters source with intrinsic and extrinsic

connections

Spiny stellate

Pyramidal cells

Inhibitory interneuron

Extrinsic Output

Extrinsic Forward Input

Extrinsic Backward Input


GABA Receptors AMPA Receptors NMDA Receptors

Γ+−∑−=Γ+−=

))(()(

, gVgVVgVC

affthresholdaffaff

rev

µσγκ

Conductance-Based Neural Mass Models in DCM

Spiny stellate

Superficial pyramidal

Inhibitory interneuron

Deep pyramidal

4-subpopulation Canonical Microcircuit

Backward Extrinsic Output

Forward Extrinsic Output

Extrinsic Forward Input



Moran et al. 2011, Neuroimage

The Full Generative Model

),,( θuxfx =

Source dynamics f

states x parameters θ

Input u

data y

),( θxLy =

Spatial forward model g

White Noise (+ parameterized color noise)

The DCM Approach




Outline

Dynamic Causal Modelling: Inversion Framework

Generative M

odel

Baye

sian

Inve

rsio

n

Empirical Data

Model Structure/ Model Parameters

),,( θuxFdtdx

=

Frequency (Hz)

Pow

er (m

V2 )

time (ms)

ampl

itude

(μV)

Neural State Equation Can Generate Different Dynamics

)()|()|( θθθ pypyp ∝posterior ∝ likelihood ∙ prior

)|( θyp )(θp

Bayes theorem allows one to formally incorporate prior knowledge into computing statistical probabilities.

In DCM for Neurphysiology priors include time constants, Extrinsic connectivity strengths, PSP, delays etc.

The “posterior” probability of the parameters given the data is an optimal combination of prior knowledge and new data, weighted by their relative precision.

new data prior knowledge

Bayesian Statistics

)|(),(),|(),|(

)(),|()|(

mypmpmypmyp

dpmypmyp

θθθ

θθθ

=

= ∫

),(),( θθµθ Σ= Nmp

Invert model Using variational method

Make inferences

Define likelihood model

Specify priors

Neural Parameters: Dynamic Model

Observer function: Forward Spatial Model

Inference on models

Inference on parameters

0)( xLy ϑ=

LBFaieie AAAgHH ,,,,,,,,,,,, ,54321 γγγγγκκκθ =

Bayesian Inversion

VB in a nutshell (mean-field approximation)

( ) ( ) ( ) ( ), | ,p y q q qθ λ θ λ θ λ≈ =

( ) ( ) ( )

( ) ( ) ( )( )

( )

exp exp ln , ,

exp exp ln , ,

q

q

q I p y

q I p y

θ λ

λ θ

θ θ λ

λ θ λ

∝ = ∝ =

( ) ( ) ( )( ) ( ) ( )

ln | , , , |

ln , , , , , |q

p y m F KL q p y

F p y KL q p m

θ λ θ λ

θ λ θ λ θ λ

= + = −

Free-energy approximation to model evidence

Mean Field Partition

Maximise variational energies via gradient ascent

Using Laplace assumption: local covariance is a function of the mean

Inference on models

Dynamic Causal Modelling: Framework

Baye

sian

Inve

rsio

n

)|()|(),|(),|(

mypmpmypmyp θθθ =Bayes’ rules:

Model 1 Model 2

Free Energy: )),()(()(ln mypqKLmypF θθ−=max

)|()|(

2

1

mypmypBF =

Model comparison via Bayes factor:

B12 p(m1|y) Evidence

1 to 3 50-75% weak

3 to 20 75-95% positive

20 to 150 95-99% strong

≥ 150 ≥ 99% Very strong

Kass & Raftery 1995, J. Am. Stat. Assoc.

Inference on models

Inference on parameters

Dynamic Causal Modelling: Framework

Baye

sian

Inve

rsio

n

)|()|(

2

1

mypmypBF =

Model comparison via Bayes factor:

)|()|(),|(),|(

mypmpmypmyp θθθ =Bayes’ rules:

Model 1 Model 2 Model 1

Free Energy: )),()(()(ln mypqKLmypF θθ−=max

-2 -1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

),()( mypq θθ ≈

%1.99)|0( => yconnp

A Neural Mass Model

Bayesian Model Comparison The model goodness: Negative Free Energy

( ) ( )[ ]mypqKLmypF ,|,)|(log θθ−=

[ ]

( ) ( )θθθθθθθ µµµµ

θθ

−−+−= −y

Tyy CCC

mpqKL

|1

|| 21ln

21ln

21

)|(),(

Accuracy - Complexity

The complexity term of F is higher the more independent the prior parameters (↑ effective DFs) the more dependent the posterior parameters the more the posterior mean deviates from the prior mean

STG STG

A1 A1

STG STG

A1 A1

The DCM Approach




Outline

pseudo-random auditory sequence

80% standard tones – 500 Hz

20% deviant tones – 550 Hz

time

standards deviants

Mismatch negativity (MMN) – DCM Motivation

time (ms)

μV

Garrido et al., (2007), NeuroImage

Paradigm

Raw Data

(128 channel EEG)

Preprocessing Epoched Source Priors

Model for mismatch negativity


Models for Deviant vs Standard Response Generation

Which Data,

Which Model?

What time?

What Conditions

Design Matrix?

Sources?

Connectivity?

Condition Specific Effects?

GUI: The FB Model

Hit Invert!

The F is Rising!

Trial Specific Effects

time (ms)

μV

Rare Events Increased Forward Connectivity

Rare Events Decreased Backward Connectivity

Bayesian Model Comparison

Forward (F)

Backward (B)

Forward and Backward (FB)

subjects

log

-evi

denc

e

Group level

Group model comparison


Summary

1. DCM enables testing hypotheses about how brain sources communicate.

2. DCM is based on a neurobiologically plausible generative model of evoked responses.

3. Differences between conditions are modelled as modulation of connectivity.

4. Inference using Bayesian model selection, Bayesian parameter testing, or mix Bayes and classical stats!

Thank You

Neurphysiological DCM Developers Jean Daunizeau Karl Friston Marta Garrido Stephan Kiebel Vladimir Litvak Andre Marreiros Will Penny Dimitris Pinotsis Klaas Stephan

Dynamic Causal Models of Neurophysiology

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