Computational and physiological modelsPart 2

Daniel RenzComputational Psychiatry Seminar: Computational Neuropharmacology14 March, 2014

Overview

• Dynamic Causal Modeling for fMRI• Example: visuomotor modulation of putamen• Dynamic Causal Modeling for EEG/MEG• Example: Reduction of synaptic plasticity using ketamine• Example: Dopaminergic modulation of neurotransmission

What is Dynamic Causal Modeling (DCM)?• Procedure for inferring the architecture of

coupled dynamic systems using Bayesian model comparison

• Need to specify competing models• neural state equation models neural dynamics• Observer function links neural dynamics to

measurements

• Infer model from measurements (model inversion), maximizing similarity between measurements and model predictions

𝑓

𝑔

inference

Neural dynamics - effective connectivity• Effective connectivity: causation• How to formalize? Deterministic differential equations

• : hidden states, : sensory input, : form and strength of influence

• We approximate to model neural population dynamics (differently for fMRI and EEG/MEG)

Sporns 2007, Scholarpedia

DCM for fMRI – approximation to neural population dynamics• All processes are assumed to occur instantaneously• Neural state variable represent a summary index of neural population

dynamics in the respective brain region• Approximation of using Taylor expansion around x=0, u=0•

�⃑� 𝑓= {𝐴 ,𝐵( 𝑗 ) ,𝐶 }

DCM for fMRI – neural population dynamics example

• : endogenous activity: modulation of connection j: direct inputs

𝑓 ( �⃑� , �⃑� , �⃑� 𝑓 )=(− 1 𝑎12

𝑎21 −1)(𝑥1

𝑥2)+( 0 0

𝑏21 0)𝑢2(𝑥1

𝑥2)+(𝑐0)𝑢1

x1

x2

u2

u1

a21 a12

-

-

Non-linear DCM for fMRI

• Extension of linear DCM to account for modulation of connectivity by brain regions

• Additional term corresponding to the second-order Taylor expansion in the states:

Non-linear DCM for fMRI

• : endogenous activity: modulation of connections by input j: direct inputs: modulation of connections by region i

x1

x2

u2

u1

a21

d12

a12

𝑓 ( �⃑� , �⃑� , �⃑� 𝑓 )=(− 1 𝑎12

𝑎21 −1)(𝑥1

𝑥2)+( 0 0

𝑏21 0)𝑢2(𝑥1

𝑥2)+(0 𝑑12

0 0 )𝑥1(𝑥1

𝑥2)+(𝑐0)𝑢1

-

-

DCM parameters are rate constants

• exp()x1

-

The coupling parameter thus describes the speed ofthe exponential change in x(t)

00.5x

a

2ln

Coupling parameter a is inverselyproportional to the half life of x(t):

0

0

( ) 0.5

exp( )

x x

x a

2ln

a❑⇒

DCM for fMRI – observer function

• Set of differential equations describing how neural activity causes the BOLD signal

• The forward model is important for model fitting, but of no interest for statistical inference.

• Computed separately for each region x1

x2

u2

u1

a21

a12

y1

y2

-

-

HRF

sf

tionflow induc

(rCBF)

s

v

stimulus functions

v

q q/vvEf,EEfqτ /α

dHbchanges in

100 )( /αvfvτ

volumechanges in

1

f

q

)1(

fγsxs

signalryvasodilato

u

s

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dx m

j

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)( neural state equation f

1

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3

002

001

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k

TEErk

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vkv

qkqkV

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f

Balloon model

BOLD signal change equation

6 hemodynamic parameters:

Friston et al. 2000, NeuroImageStephan et al. 2007, NeuroImage

},,,,,{ h },,,,,{ h

},,,,,{ h⃑𝜗𝑔

v: blood volumeq: deoxyhemoglobin content

Hemodynamic model

Bayesian Inversion

)|(

),(),|(),|(

)(),|()|(

myp

mpmypmyp

dpmypmyp

Make inferences

Define likelihood model

Specify priors

Neural state function

Observer function

Inference on models

Inference on parameters

𝑓 ( �⃑� , �⃑� , �⃑� 𝑓 )

𝑔 ( �⃑� , �⃑�𝑔)

Iterative optimization using Expectation-Maximization:1.Compute model response2.Compare with data3.Improve parameters, if possible

𝑝 (�⃑� 𝑓 ,𝑚)𝑝 (�⃑�𝑔 ,𝑚)Empirical prior

Conservative shrinkage prior

Overview

• Dynamic Causal Modeling for fMRI• Example: visuomotor modulation of putamen• Dynamic Causal Modeling for EEG/MEG• Example: Reduction of synaptic plasticity using ketamine• Example: Dopaminergic modulation of neurotransmission

Modulation of visuomotor integration by putamen• How do (failures of) learned predictions

about visual stimuli influence subsequent motor responses?

• Hypothesis: Visuomotor connections are modulated by activity in the putamen

• This activity increases with the size of the prediction error (surprise; free energy)

Den Ouden 2009

Bayesian learning model

• Speed and accuracy of motor responses increased significantly with predictability of stimulus

• Response speed is a good linear predictor of choice error, but unrealistic• Hierarchical Bayesian model estimates posterior PDF of both the probabilistic

associations and their volatility

Den Ouden 2009

Imaging analysis

• Responses in putamen (and PMd) were negatively correlated with probability of any visual stimulus

• Responses in the fusiform face area (FFA) were negatively correlated with probability of faces

• Responses in the parahippocampal place area (PPA) were negatively correlated with probability of houses (and positively correlated with probability of faces)

Den Ouden 2009

DCM analysis

DCM analysis

• Does putamen really gate visuomotor connections, or does PMd gate connections between visual areas and putamen?

DCM analysis

• Does putamen really gate visuomotor connections, or does PMd gate connections between visual areas and putamen?

Den Ouden 2009

Overview

• Dynamic Causal Modeling for fMRI• Example: visuomotor modulation of putamen• Dynamic Causal Modeling for EEG/MEG• Example: Reduction of synaptic plasticity using ketamine• Example: Dopaminergic modulation of neurotransmission

DCM for EEG/MEG – neural mass model

David et al., NeuroImage, 2006

DCM for EEG/MEG – forward model

measurementslead field matrix, coupling electrical sources to EEG channels membrane potential of pyramidal cellsobservation error

Kiebel et al., NeuroImage, 2006Daunizeau et al., NeuroImage, 2009

Overview

• Dynamic Causal Modeling for fMRI• Example: visuomotor modulation of putamen• Dynamic Causal Modeling for EEG/MEG• Example: Reduction of synaptic plasticity using ketamine• Example: Dopaminergic modulation of neurotransmission

How does Ketamine modulate synaptic plasticity underlying mismatch negativity?

Schmidt, Diaconescu, et al 2012

Roving paradigm

Mismatch negativity (MMN): ERP component elicited when the brain detects that an established pattern in sensory input has been violated.

Mismatch negativity

• Functionally, MMN is thought to serve two roles:• Current prediction error signal caused by previous neuronal spike-frequency

adaptation (memory trace formation) in auditory cortex• Model adjustment to minimize future prediction error, reflected by

glutamatergic long-range connections (between temporal and frontal regions)

• Free-energy principle: Suggests an overarching physiological and computational process of minimizing prediction error that requires both adaptation and model adjustment

Mismatch negativity and pathopsychology• MMN is a potential index for pathopsychology• Patients with schizophrenia show deficits in auditory sensory memory.

They show impaired ability to match tones, accompanied by deficient generation of MMN

• MMN can also be reduced using NMDA-antagonists, e.g. ketamine

Neurophysiology

• In animal studies it was shown that NMDAR plays a key role in MMN generation

• Both spike-frequency adaptation and glutamatergic plasticity are regulated by NMDARs

• Spike-frequency adaptation results from potassium channel-dependent hyperpolarization which relies on intracellular calcium influx modulated by NMDAR status

• NMDARs can lead to rapid changes in the strength of glutamatergic synapses, for example, via phosphorylation of AMPA receptors

ResultsInter-regional Synaptic Coupling

Adaptation and Inter-regional Synaptic Coupling

Schmidt, Diaconescu, et al 2012

ResultsInter-regional Synaptic Coupling

Adaptation and Inter-regional Synaptic Coupling

The model with plasticity in forward and backward connections as well as adaptation (expressed via post-synaptic gain modulation in A1) had the largest model evidence in both conditions

Schmidt, Diaconescu, et al 2012

Results

A significant reduction of synaptic plasticity, following ketamine administration, of the forward connection from left A1 to left STG was found.

There was a significant linear relation between drug effects on “control and cognition” ratings and drug effects on plasticity of the left A1-STG connection.

Schmidt, Diaconescu, et al 2012

Overview

• Dynamic Causal Modeling for fMRI• Example: visuomotor modulation of putamen• Dynamic Causal Modeling for EEG/MEG• Example: Reduction of synaptic plasticity using ketamine• Example: Dopaminergic modulation of neurotransmission

Delayed match-to-sample

• How does dopamine influence WM?• L-Dopa vs placebo in delayed match-to sample task• MEG recordings

Moran 2011

MEG results

• MEG measured greater activity during memory condition in delta, theta and alpha bands at predicted locations

• Prominent theta-activity in right superior frontal gyrus during memory maintenance. Spectra were boosted by L-Dopa

Moran 2011

Dopaminergic Synaptic Effects - Hypotheses• An enhancement in the conductance of GABAA and NMDA channels• A decreased conductance at AMPA receptor-associated channels (at

synapses between exogenous glutamatergic inputs and layer III pyramidal cells)

Moran 2011

Dopaminergic Synaptic Effects - Results• The only parameter with a differential contribution to theta band at

peak of interaction under L-DOPA was the NMDA non-linearity parameter

• This was further enhanced during the memory condition

NMDA nonlinear function. As alpha increases, the voltage-dependent magnesium switch becomes highly nonlinear.

Dopaminergic Synaptic Effects - Results• Can we link WM performance to synaptic parameters?• Greater L-Dopa mediated improvement in memory performance was

correlated with increased NMDA signaling and decreased AMPA signaling