Dynamic Causal Modelling forSteady State Responses
Dimitris Pinotsis
The Wellcome Trust Centre forNeuroimaging
A1 A2
NeuroimagingUCL Institute of Neurology, London, UK
SPM Course London May 2012
Dynamic Causal Modelling for SSR
A framework which uses Bayesian techniques to fitdifferential equations to steady – state data. It allowsfor comparison between competing models of brainarchitecture and furnishes estimates for parametersthat are not measured directly by exploiting
1. What is DCM for SSR?
that are not measured directly by exploitingelectrophysiological data.
Although it is based on sophisticated models fromcomputational neuroscience, its
application is straightforward and
does not require mathematical
training.
-Brain activity retains similar statistical features(e.g.variance) and frequency contentacross measurement period-Cannot describe nonlinear couplingbetween frequenciesnext talk
2. When should I use DCM for SSR?
Advantages:
Summarize activity in a compact way -no need to fit long time series (computationally expensive)Describe brain function in terms of acharacteristic frequency associated with the task understudy
Extract DataFeatures
Specifymodel
Maximise themodel evidence
(~-F)
Reveal hiddenbrain
architectures
Experimentaldata
DCM Chain
3. Which steps do we take when usingDCM for SSR?
architecturesdata
Get estimates ofhidden
parameters
DCM for SSR(Moran et al., Neuroimage, 2009)
Anaesthesia:Anaesthetic Depth in Rodents(Moran et al., Plos One, 2011) Dopamine in working memory(Moran et al., Current Biol., 2011)
4.Where has DCM for SSR been applied ?
(Moran et al., Current Biol., 2011) Beta oscillations in PD(Moran et al., Plos CB, 2011)(Marreiros – yesterday’s talk) Sleep and Coma(Boly et al., Science, 2011,J Neuro, 2012)Extension:DCM for Neural Fields(Pinotsis et al., Neuroimage, 2011,2012)
Overview
1. Data Features
2. Generative Model
3. Bayesian Inversion: Parameter Estimates and ModelComparisonComparison
4. Example: Glutamate and GABA in Rodent AuditoryCortex
5. DCM for Current Source Density
6. DCM for Neural Fields
Extract DataFeatures
Specifymodel
Maximise themodel evidence
(~-F)
Reveal hiddenbrain
architectures
Experimentaldata
DCM Chain
architecturesdata
Get estimates ofhidden
parameters
Cross Spectral DensityM
EG
–L
FP
Tim
eS
eri
es
Frequency (Hz)
Po
we
r(m
V2)
EE
G-
ME
G
1
2
3
4
Frequency (Hz)
Po
we
r(m
V
Summarizes brain responsein terms of power at eachfrequency
From Time Series to Cross Spectral DensitiesM
EG
–L
FP
Tim
eS
eri
es
Cro
ss
Sp
ec
tralD
en
sity
1
EE
G-
ME
G
Cro
ss
Sp
ec
tralD
en
sity
1
2
2 3
3
4
4
1
2
3
4
A few LFP channels or EEG/MEG spatial modes
Vector Auto-regression p-order model:
Linear prediction formulas that attempt to predict an output y[n] of a system based on
the previous outputs
npnpnnn eyyyy ....2211
}{:)( ccpA Resulting in a matrices for c Channels
From Time Series to Cross Spectral Densities
))(()( pAfg ij
ijijijij HHg )()()(
iwpijp
iwijiwijijeee
H
......
1)(
221
f 2
}{:)(....1 ccpAp Resulting in a matrices for c Channels
Cross Spectral Density for channels
i,j at frequencies
..)(
..)()(
12
1211
g
gg
Overview
1. Data Features
2. Generative Model
3. Bayesian Inversion: Parameter Estimates and Model3. Bayesian Inversion: Parameter Estimates and ModelComparison
4. Example:Glutamate and GABA in Rodent AuditoryCortex
5. DCM for Cross Spectral Density
6. DCM for Neural Fields
Extract DataFeatures
Specifymodel
Maximise themodel evidence
(~-F)
Reveal hiddenbrain
architectures
Experimentaldata
DCM Chain
architecturesdata
Get estimates ofhidden
parameters
Model=equations
andparameters
A Brain Region as an Input - Output System
equations: determine the dynamics(eg. limit cycles, transients, steady –state)
1 2 3 4 5, ,{ , , , , , , , , , , , , }e i e i a F B LH H g A A A
(eg. limit cycles, transients, steady –state)parameters: fine tune the dynamics(e.g. faster, shorter)
Maximum PSP, time constants, intrinsic and extrinsic connectivity etc
Pink line =
Steady state responses
Pink line =Container (bowl)If there is no external perturbation, the ball will stay at the centreIf there is, the container will be tilted and the ball will oscillate around the centre asshownNow, imagine that the ball has a bell inside. If there is an external perturbation thebell will start ringing.CAUSE of CONTAINER TILTING NEURAL NOISEBALL BRAIN REGIONRINGINGS RESPONSES (cross spectral densities)CONTAINER MODEL (equations, parameters cf. shape/friction)STEADY STATE PERTURBATIONS means that“ the ball always stays very close to the centre” (while the bowl is tilted)
A Brain Region as an Input - Output System
ERP
(Model)Brain region
Neuronalinnovations
Predictedresponses
( , )Yg ( , )Ug k
Frequency (Hz)P
ow
er
(mV
2)
SSRSSR
Frequency Domain Generative Model(Perturbations about a fixed point)
Time DifferentialEquations
)( Buxfx
State SpaceCharacterisation
BuAxx
Transfer FunctionFrequency Domain
)(
)(
xly
Buxfx
Cxy
BuAxx
BAsICsH )()(
Linearise
mV
Stationary Time Series
Frequency Domain Generative Model(Perturbations about a fixed point)
Po
we
r(m
V2)
Po
we
r(m
V2)
Spectrum channel/mode 1 Cross-spectrum modes 1& 2Transfer FunctionFrequency Domain
..),:()(1 ,/ ieieHfH
Peaks Parameters
Frequency (Hz)
Frequency (Hz)
Frequency (Hz)
Po
we
r(m
VP
ow
er
(mV
2)
Po
we
r(m
V
Spectrum mode 2Transfer FunctionFrequency Domain
..),:()(2 ,/ ieieHfH
Cross-Transfer FunctionFrequency Domain
..),:()(12 ,/ ieieHfH
..),:()(1 ,/ ieieHfH
ERP vs Steady State Responses
Outputs ThroughLead fieldc3c1
outputneuronal
Freq DomainOutput
/)( 21fH
c2
+
Time Domain
ERPOutput
)(ty
Time Domain
Freq Domainoutputs1(t)
outputs2(t) output
s3(t)
neuronalstates
drivinginput u(t)
Freq DomainCortical Input
bf
aU
1)(
Time Domain
Pulse Input
inhibitoryinterneurons
IntrinsicConnections
EEG/MEG/LFPsignal
Neural Mass Model
Tens of thousands of neuronsapproximated by their averageresponse. Neural mass modelsdescribe the interaction of theseaverages between populations andsources
neuronal (source) model
State equations
Extrinsic Connections
,,uxFx
spiny stellatecells
interneurons
PyramidalCellsInternal
Parameters
4g
3g
=&
Excitatory spiny cells in granular layers
4g
3g
Intrinsicconnections
5g
Excitatory spiny cells in granular layers
Inhibitory cells in agranular layers
11812
102
1112511
1110
72
8938
87
2)(
2)()(
xxx
xxxSHx
xx
xxxSAAHx
xx
iiii
eeLB
ee
xx
Neural Mass Model: Equations for Voltage and Current
1g
2g
1
2
4914
41
2))(( xxuaxsHx
xx
eeeekkgk --+-=
=
&
&
1g
2g
Excitatory pyramidal cells in agranular layers
),( uxfx
12
4914
41
2))()(( xxCuxSAAHx
xx
eeLF
ee
659
32
61246
63
22
51295
52
2)(
2))()()((
xxx
xxxSHx
x
xxxSxSAAHx
xx
iiii
eeLB
ee
Extrinsic Connections:
Forward
Backward
Lateral
4g
3g
=&
Excitatory spiny cells in granular layers
4g
3g
Intrinsicconnections
5g
Excitatory spiny cells in granular layers
Inhibitory cells in agranular layers
11812
102
1112511
1110
72
8938
87
2)(
2)()(
xxx
xxxSHx
xx
xxxSAAHx
xx
iiii
eeLB
ee
xx
Synaptic ‘alpha’ kernel
Neural Mass Model: Two Transformations
1g
2g
1
2
4914
41
2))(( xxuaxsHx
xx
eeeekkgk --+-=
=
&
&
1g
2g
Excitatory pyramidal cells in agranular layers
),( uxfx
12
4914
41
2))()(( xxCuxSAAHx
xx
eeLF
ee
Sigmoid function
659
32
61246
63
22
51295
52
2)(
2))()()((
xxx
xxxSHx
x
xxxSxSAAHx
xx
iiii
eeLB
ee
ExtrinsicConnections:
Forward
Backward
Lateral
Neural Mass Model: 1st Transformation
4g
3g
=&
Excitatory spiny cells in granular layers
4g
3g
Intrinsicconnections
5g
Excitatory spiny cells in granular layers
Inhibitory cells in agranular layers
11812
102
1112511
1110
72
8938
87
2)(
2)()(
xxx
xxxSHx
xx
xxxSAAHx
xx
iiii
eeLB
ee
xx
Synaptic ‘alpha’ kernel
ieH /
ie/
hrv
: Receptor Density
1g
2g
1
2
4914
41
2))(( xxuaxsHx
xx
eeeekkgk --+-=
=
&
&
1g
2g
Excitatory pyramidal cells in agranular layers
),( uxfx
12
4914
41
2))()(( xxCuxSAAHx
xx
eeLF
ee
659
32
61246
63
22
51295
52
2)(
2))()()((
xxx
xxxSHx
x
xxxSxSAAHx
xx
iiii
eeLB
ee
ExtrinsicConnections:
Forward
Backward
Lateral
hrv
Input: presynaptic rateOutput: Average synapticdepolarization
Neural Mass Model: 2nd Transformation
4g
3g
Excitatory spiny cells in granular layers
4g
3g
Intrinsicconnections
5g
Excitatory spiny cells in granular layers
Inhibitory cells in agranular layers
11812
102
1112511
1110
72
8938
87
2)(
2)()(
xxx
xxxSHx
xx
xxxSAAHx
xx
iiii
eeLB
ee
Input: Avg synapticdepolirizationOutput: firing rate
1g
2g
1
2
4914
41
2))(( xxuaxsHx
xx
eeeekkgk --+-=
=
&
&
1g
2g
Excitatory pyramidal cells in agranular layers
),( uxfx
12
4914
41
2))()(( xxCuxSAAHx
xx
eeLF
ee
Sigmoid function
659
32
61246
63
22
51295
52
2)(
2))()()((
xxx
xxxSHx
x
xxxSxSAAHx
xx
iiii
eeLB
ee
ExtrinsicConnections:
Forward
Backward
Lateral
)(vSr
: Firing Rate
Overview
1. Data Features
2. Generative Model
3. Bayesian Inversion: Parameter Estimates and ModelComparisonComparison
4. Example: Glutamate and GABA in Rodent AuditoryCortex
5. DCM for Cross Spectral Density
6. DCM for Neural Fields
Extract DataFeatures
Specifymodel
Maximise themodel evidence
(~-F)
Reveal hiddenbrain
architectures
Experimentaldata
Bayesian Inversion
DCM Chain
architecturesdata
Get estimates ofhidden
parameters
Frequency (Hz)
Po
wer
Time Domain
Freq Domain
c3c1
NMM
NMM
NMM
Freq DomainOutput
Freq DomainCortical Input
)( fH
bf
aU
1)(
c2
+
• predictedresponses
• parameters• Hyper
parameters
Generativemodel
• variance
Priors
Roadmap
•Modelevidence
•Posteriorestimates ofparameters
BayesianInference
Baye
sia
nIn
vers
ion
Inference on models Inference on parameters
DATA PREDICTIONS
Variational Bayes Algorithm
• Iterative procedure• Minimize free energy and optimize parameters•Maximum accuracy over complexity constraints
Model Fit and Comparison
Baye
sia
nIn
vers
ion
Model 1
Model 2
Model 1
),()( mypq
Model comparison via Bayes factor:
)|(
)|(
2
1
myp
mypBF
Overview
1. Data Features
2. Generative Model
3. Bayesian Inversion: Parameter Estimates and ModelComparisonComparison
4. Example: Glutamate and GABA in Rodent AuditoryCortex
5. DCM for Cross Spectral Density
6. DCM for Neural Fields
Pharmacological Manipulation of Glutamate andGABA
Questions of Study:
Are our estimates of excitation and inhibition veridical,
e.g. ?
Can we obtain hierarchical relationships between brainregions?
,e iH H
regions?
AIM:
NOT to explain mechanisms of isoflurane BUT
to exploit isoflurane to induce known changes in synaptictransmission and THEN
Use LFP recordings and DCM for SSR to infer changes
Moran, Tetsuya, Jung, Endepols, Graf, Dolan, Friston, Stephan, Tittgemeyer,PLoSONE, 2011
Use animal LFP recordings from primary auditory cortex (A1) & posteriorauditory field (PAF)
Manipulate neurotransmitter processing via anaesthetic agent Isoflurane
4 levels of anaesthesia: each successively decreasing glutamate andincreasing GABA (Larsen et al Brain Research 1994; Lingamaneni et alAnesthesiology 2001; Caraiscos et al J Neurosci 2004 ; de Sousa et alAnesthesiology 2000
White noise stimulus & Silence
1.4 % 1.8 % 2.4 % 2.8 %
Pharmacological Manipulation of Glutamate and GABA
-0.06
0
0.06
0.12
mV
A2
LFP
-0.06
0
0.06
0.12
mV
-0.06
0
0.06
0.12
mV
-0.06
0
0.06
0.12
mV
A1
1.4 %Isoflurane
1.8 %Isoflurane
2.4 %Isoflurane
2.8 %Isoflurane
Pharmacological Manipulation of Glutamate and GABA
Data
Cross-spectra white noise
Po
wer
A1
Po
wer
A1
-P
AF
0.04
0.06
0.04
0.06
1.4 %1.8 %2.4 %2.8 %
Frequency (Hz) Frequency (Hz)
Frequency (Hz)
Po
wer
A1
Po
wer
PA
FP
ow
er
A1
0 5 10 15 20 25 300
0.02
0.04
0 5 10 15 20 25 300
0.02
0.04
0 5 10 15 20 25 30
0
0.02
0.04
0.06
Predicted Observed
Pharmacological Manipulation of Glutamate and GABA
Model
PAF
backward
Or?
5gInhibitory cells in supra(infra) granular
layers4 4v g
ExtrinsicConnections:Forward
M2
AI
PAFforward
backward
M1
AI
PAF
forward
4g 3g
2g
12
4914
41
2))(( xxuaxsHxxx
eeee kkgk --+-==
&&
Excitatory spiny cells in granular layers
3g
1g 2g
Intrinsic connections
Excitatory spiny cells
in granular layers
Pyramidal cells in infra(supra) granularlayers
layers4 4
2 24 6 3 6 4 4
5 5
25 5 7 5 5
7 4 5
( ) ( ) 2
( ) 2
B regione e e e e e
i i i i
v g
g H A S v H S v g v
v g
g H S v g v
v g g
1 1
2 21 6 1 6 1 1( ) ( ) 2F region
e e e e e e
v g
g H A S v H S v g v
6
2 2
2 22 2 1 2 2
3 3
23 4 7 3 3
6 2 3
( ) ( ) 2
( ) 2
B regione e e e e e
i i i i
v g
i H A S v H S v g v
v g
g H S v g v
v g g
ForwardBackwardLateral
AI
PAF
lateral
lateral
Or?
M3
150
200
250
Lo
gG
rou
pB
ayes
Facto
r
Pharmacological Manipulation of Glutamate and GABA
Model
AI
PAF
forward
backward
PAbackward
M1
M1
M2
M3
0
50
100
150
White Noise
Silence
Lo
gG
rou
p
AI
PAF
forward
AI
PAF
lateral
lateral
M2
M3
DCM recovers known neuronatomy
Decreased Glutamate Release
Increased gabaergic transmission
1.4 %Isoflurane
1.8 %Isoflurane
2.4 %Isoflurane
2.8 %Isoflurane
Pharmacological Manipulation of Glutamate and GABA
Physiological Parameters
AI
PAF
Excitatory spiny cellsin granular layers
Excitatory pyramidal cellsin agranular layers
Inhibitory cells inagranular layers
Synaptic ‘alpha’ kerneleH
eH
iH
Pharmacological Manipulation of Glutamate and GABA
Parametric Effectof Isoflurane
-1.5
-1
-0.5
0.5
1
**
**
**
Do
se
Sp
ec
ific
Ga
in
0
Excitatory spiny cellsin granular layers
Excitatory pyramidal cellsin agranular layers
Inhibitory cells inagranular layers
eH
eH
iH
1.4 1.8 2.4 2.8-1.5
Isoflurane Dose
DCM recovers known drug-induced changes
Overview
1. Data Features
2. Generative Model
3. Bayesian Inversion: Parameter Estimates and Model3. Bayesian Inversion: Parameter Estimates and ModelComparison
4. Example: Glutamate and GABA in Rodent AuditoryCortex
5. DCM for Cross Spectral Density
6. DCM for Neural Fields
Specifymodel
Extract DataFeatures
Maximise themodel evidence
(~-F)
Test models orMAP parameters
Find yourexperimental
data
DCM Chain
1. Interface Additions
2. New CSD routines, similar to SSR
3. SPM_NLSI_GN accommodates imaginary numbers, slopes,curvatures
4. A host of new results features, in channel and source space!
dataAlso report lags,
coherence,covariance &phase delays
Friston, Bastos, Litvak, Stephan, Fries, Moran, Neuroimage, 2012
Interface Additions
Frequency Domain Generative Model(Perturbations about a fixed point)
Time DifferentialEquations
)( Buxfx
State SpaceCharacterisation
BuAxx
Transfer FunctionFrequency Domain
Complex NumberComplex Number
)(
)(
xly
Buxfx
Cxy
BuAxx
BAsICsH )()(
Linearise
mV
Im
r
φb
What is a complex number ?
The Fourier transform of a signal is a continuouscomplex function
Re
z=a + ib
i2 = -1
= r(cosφ + isinφ)
H(ω) = F(f)= ∫∞
f(t)e-2πωit dt-∞
= reiφ
a
1.5
2
2.5
3
3.5
real
prediction and response: E-Step: 32
-0.2
0
0.2
0.4
0.6
0.8
1
imag
inar
y
prediction and response: E-Step: 32
DCM for CSD: data fits have two parts:real and imaginary
0 10 20 30 40 500
0.5
1
1.5
Frequency (Hz)0 10 20 30 40 50
-1
-0.8
-0.6
-0.4
-0.2
Frequency (Hz)im
agin
ary
| H2(ω) . H2*(ω) |
DCM for SSR
Predictionsand Data
| H1(ω) . H1*(ω) |
| H1(ω) . H2*(ω) |
4g 3g2g
12491441
2))(( xxuaxsHxxx eeee kkgk --+-==&&
3g
1g 2g
5g
Excitatory spiny cells
Pyramidal cells
Inhibitory cells4g 3g
2g124914
412))(( xxuaxsHxxx eeee kkgk --+-==&
&3g
1g 2g
5g
Excitatory spiny cells
Pyramidal cells
Inhibitory cells
H2(ω) . H2*(ω)
Predictions
DCM for Cross Spectral Density
H1(ω) . H1*(ω)
H1(ω) . H2*(ω)
4g 3g2g
12491441
2))(( xxuaxsHxxx eeee kkgk --+-==&&
3g
1g 2g
5g
Excitatory spiny cells
Pyramidal cells
Inhibitory cells4g 3g
2g124914
412))(( xxuaxsHxxx eeee kkgk --+-==&
&3g
1g 2g
5g
Excitatory spiny cells
Pyramidal cells
Inhibitory cells
H2(ω) . H2*(ω)
DCM for Cross Spectral Density
Predictionsand Data
H1(ω) . H1*(ω)
H1(ω) . H2*(ω)
4g 3g2g
12491441
2))(( xxuaxsHxxx eeee kkgk --+-==&&
3g
1g 2g
5g
Excitatory spiny cells
Pyramidal cells
Inhibitory cells4g 3g
2g124914
412))(( xxuaxsHxxx eeee kkgk --+-==&
&3g
1g 2g
5g
Excitatory spiny cells
Pyramidal cells
Inhibitory cells
Spectra Abs(H (ω) . H *(ω)) , Abs(H (ω) . H *(ω)) …
4g 3g2g
12491441
2))(( xxuaxsHxxx eeee kkgk --+-==&&
3g
1g 2g
5g
Excitatory spiny cells
Pyramidal cells
Inhibitory cells
4g 3g2g
12491441
2))(( xxuaxsHxxx eeee kkgk --+-==&&
3g
1g 2g
5g
Excitatory spiny cells
Pyramidal cells
Inhibitory cells
H1(ω)H2(ω)
Spectra
Coherence
Delay at particular frequencies
Covariance (lags over time, collapsed across frequencies )
Abs(H1(ω) . H1*(ω)) , Abs(H1(ω) . H2*(ω)) …
|(H1(ω).H2*(ω))|2 / { (H1(ω).H1*(ω)) + (H2(ω).H2*(ω))}
arg(H1(ω).H2*(ω))/2πf
Real( F-1(H1(ω).H2*(ω)))
Can also optimize complex-valued quantities Understand how biophysical parameters affect conventionallinear systems measures
Overview
1. Data Features
2. Generative Model
3. Bayesian Inversion: Parameter Estimates and ModelComparisonComparison
4. Example: Glutamate and GABA in Rodent AuditoryCortex
5. DCM for Cross Spectral Density
6. DCM for Neural Fields
Specifymodel
Extract DataFeatures
Maximise themodel evidence
(~-F)
Test models orMAP parameters
Find yourexperimental
data
DCM Chain
New NFM routines
data
And also reportspatial extent of
intrinsic connectionsand conduction speed
Pinotsis, Moran, Friston, Neuroimage 2012
Main difference with previous models:Brain activity is deployed on a cortical patch as opposed tobeing centred around a point
P
2
1
2
( , ) expx
L x
Lead field modified to enable a mapping of spatially distributedactivity (coloured waves) to a time series at P
Lateral interactions in the visual cortex
Novelty of DCM for NFs: Can get estimates of parametersrelating to spatial properties of sources when there is NOSPATIAL INFO in the data
intrinsic connection strength
spatial decay rate connection extent
c xK x = ae
a
c
Inhibitory cells in supragranular layers
23 23,a c32 32,a c
Exogenous input
Hidden states
22 2 2 2
2 2 22 23 2 2 23 2 23 23 3 3
2
2 ( ) ( ( ) ( ))
i i i i
xx
v v v m
sc s c s c v s v
Jansen and Rit Neural Field Models
( )U t
( , ) ( , )V x t f V U
Augment old equations with wave equations describingpropagation of afferent spike rate between points on the cortex
Exogenousinput
13 13,a c
( )U t Excitatory spiny cells in granular layers
Excitatory pyramidal cells in supra- and infragranular layers
Endogenousoutput 3( )v t
31 31,a c
23 23,a c32 32,a cObserved signals
23 3 3 3
2 2 23 31 3 3 31 3 31 31 1 2 1 2
2
2 ( ) ( ( ) ( )) ( ) ( )
e e e e
xx
v v v m
sc s c s c v v s v v
21 1 1 1
2 2 21 13 1 1 13 1 13 13 3 3
2 ( )
2 ( ) ( ( ) ( ))
e e e e
xx
v v v m U
sc s c s c v s v
( ) ( , ) ( , )y t L x V x t dx
conduction velocitys
Inference on Model Parameters
Summary
• DCM is a generic framework for asking mechanistic questions based onneuroimaging data (e.g. drug-induced changes in balance of synaptictransmission)
• Neural mass models parameterise intrinsic and extrinsic ensemble connectionsand synaptic measures (time constants, effective connectivity,…)
• DCM for SSR and CSD provide a compact characterisation of multi- channel LFPor EEG data in the frequency domainor EEG data in the frequency domain
• Bayesian inversion provides parameter estimates and allows model comparisonfor competing hypothesised architectures
• Neural field models incorporate propagation of activity on a cortical patch, so onecan distinguish between spatial effects and other factors such as cortico-thalamicinteractions or intrinsic cell properties
• Neural field models yield estimates of parameters related to topographicproperties of the sources such as spatial decay rate of synaptic connections andintrinsic conduction speed, even when using spatially unresolved data
Thanks to FIL Methods GroupKarl Friston
Rick Adams
Harriet Brown
Jean Daunizeau
Marta Garrido
Guillaume Flandin
Stefan Kiebel
VladimirVladimir LitvakLitvak
…and thank you !
Andre Marreiros
Jeremie Mattout
Rosalyn MoranRosalyn Moran
Ashwini Oswal
Will Penny
Ged Ridway
Klaas Stephan
Yen Yu
Field Power Spectra
2
1
2
( , ) expx
L x
( ) ( , ) ( , ) ( )Y Y Ng g g
Exogenous input
Hidden states
Observed signals
( ) ( , ) ( , )y t L x V x t dx
( , ) ( , )V x t f V U
( )U t
'( , ) , , ,
( , ) ( , )
Re( ) ~ (0, ( , )) Im( ) ~ (
Y m U mj
N UN N U U
j j j j jg L T g T L
g g k
0, ( , ))
Mass Power Spectra
( , ) ( )L x L
( ) ( , ) ( , ) ( )
( , ) ( ) , ,
Y Y N
k
g g
g L T g T L
g
Exogenous input
Hidden states
Observed signals
( )U t
( )y t L V
( ) ( , )V t f V U
'
1
( , ) ( ) , ,
( , ) ( , )
, ( , )
( , )
Re( ) ~ (0, ( , )) Im
kY m U mk
N UN N U U
k k j tm m
km
k
g L T g T L
g g k
T t e dt
g ft e
x u
( ) ~ (0, ( , ))
Maximum postsynaptic depolarization8, 32 (mV)
Postsynaptic time constants1/4, 1/28 (ms-1)
Amplitude of intrinsic connectivity kernels2000, 8000, 2000, 1000
Intrinsic connectivity decay constant0.32 (mm-1)0.32 (mm-1)
Sigmoid parameters(post synaptic firing rate function)0.54, 0,0.135
Conduction velocity3 m/s
Radius of cortical source50 (mm)
Exogenous input
Neural Mass Neural Field
( )U t
( )U t
Neural Mass vs Neural Field
( , )Yg Difference in predicted spectra because of difference in underlying model:
Hidden states
Observed signals
, ( , )D Q D x x t t Q x t dx dt
2 2V B V BV ABF V GU
( )y t L V
2 2V B V BV ABD F V GU
( ) ( , ) ( , )y t L x V x t dx
Inverse conduction speedS
pect
rald
ensi
ty
0 20 400
10
0 20 400
5
10
0 20 400
5
10
0 20 400
10
0 20 400
5
0 20 400
2
4
10
Connectivity decayconstant
Changing model parameters
New peaks appear:- as intrinsic speed decreases- as connectivity extent increases
Spe
ctra
lden
sity
Frequency (Hz)
0 20 400
0 20 400
10
20
0 20 400
50
0 20 400
0 20 400
20
0 20 400
10
Pinotsis, Moran, Friston, Neuroimage 2012
Roadmap
•predictedresponses
•parameters•hyperparameters
Generativemodel
•variancePriors
•Model Evidence•Posteriors
BayesianInference
( ) ( , ) ( , ) ( )
( , )
Y Y N
N
g g
g
g
( , ) ( , )p m N
( | , ) ( ( ), ( , ))
( | ) ( | , ) ( )
( | , ) ( , )( | , )
( | )
Yp G m N
p G m p G m p d
p G m p mp G m
p G m
g
( , )
Re( ) ~ (0, ( , )) Im( ) ~ (0, ( , ))
NN Ng
( , ) ( , )p m N
Measured data Specify generative forward model(with prior distributions of parameters)
Variational Laplace Algorithm
Iterative procedure:
1. Compute model response usingcurrent set of parameters andhyperparameters
log ( ) ( ( ) ( , ))
log ( , ) ( ( ) ( ))q
F p y m D q p y m
p y m D q p m
Bayesian Model Inversion
Maximize a free energy bound to model evidence :
Iterative procedure: hyperparameters2. Compare model response with data3. Improve parameters and
hyperparameters
)|(
)|(
2
1
myp
mypBF
Model comparison via Bayes factor:
Maximum accuracy over complexity constraints
),()( mypq