dynamic causal models of neurophysiology
TRANSCRIPT
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
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
Models of Neural Population Activity
Quantifying the Goodness of Models and Model Parameters
How it works in practice
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
external granular Layer: inhibitory interneurons
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
external granular Layer: inhibitory interneurons
⊗ ),( ρvS
Meso-scale dynamics: & connectivity
⊗iH
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
external granular Layer: inhibitory interneurons
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
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
external granular Layer: inhibitory interneurons
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
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
external granular Layer: inhibitory interneurons
⊗
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: & connectivity
Meso-scale dynamics: the convolution model
54321 ,,,,,,,,, γγγγγττρθ ieie HH⊆
1γ
internal granular layer: spiny stellate cells
internal pyramidal layer: pyramidal cells
external granular Layer: inhibitory interneurons
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
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
Extrinsic Backward Input
Extrinsic Backward 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
Models of Neural Population Activity
Quantifying the Goodness of Models and Model Parameters
How it works in practice
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
Model Selection & Hypothesis Testing
data y Model
selection:
)|( 1mypModel 1
Model 2
...
Model n
)|( 2myp
)|( nmyp
),|( 1myp θ
),|( 2myp θ
),|( nmyp θ
)|( imyp
STG STG
A1 A1
STG STG
A1 A1
)()|()|( 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
Models of Neural Population Activity
Quantifying the Goodness of Models and Model Parameters
How it works in practice
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
Garrido et al., (2007), NeuroImage
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
Garrido et al., (2007), NeuroImage
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