5th March 2008 Andreina Mendez Stephanie Burnett DCM Theory 5th March 2008 Andreina Mendez Stephanie Burnett

Recap Last session: PPIs (psycho-physiological interactions) Functional vs. effective connectivity Functional connectivity: temporal correlation between spatially remote neurophysiological events Effective connectivity: the influence that the elements of a neuronal system exert over each other Standard fMRI analysis PPIs, SEM, DCM

Introduction: DCM and its place in the methods family tree Standard fMRI analysis The BOLD signal (related to brain activity in some implicit way) in some set of brain is correlated, and is also correlated with your task Task BOLD signal Of course you don’t do that, you cite converging evidence and blah blah, but still… “This is a frontoparietal network collection of brain regions involved in activated while processing coffee”

Introduction: DCM and its place in the methods family tree PPIs Represent how the (experimental) context modulates connectivity between a brain region of interest, and anywhere else E.g. (Whatever gives rise to the) signal in one brain region (V1) will lead to a signal in V5, and the strength of this signal in V5 depends on attention V1 V5 attention DCM models how neuronal activity causes the BOLD signal (forward model) i.e. your conclusions are about neural processes DCM models bidirectional and modulatory interactions, between multiple brain regions

Introduction: DCM and its place in the methods family tree Your experimental task causes neuronal activity in an input brain region, and this generates a BOLD signal. The neuronal activity in this input region, due to your task, then causes/modulates neuronal activity in other brain regions (with resultant patterns of BOLD signals across the brain) “This sounds more like something I’d enjoy writing up!”

DCM basics DCM models interactions between neuronal populations (fMRI/MEG/EEG) The aim is to estimate, and make inferences about, 1. The coupling among brain areas 2. How that coupling is influenced by changes in experimental context

DCM basics DCM starts with a realistic model of how brain regions interact Adds a forward model of how neuronal activity causes the signals you observe (e.g. BOLD) …and estimates the parameters in your model (effective connectivity), given your observed data Neural and hemodynamic models (more later on)

DCM basics DCM treats the brain as a nonlinear dynamic system The system has inputs, state variables, and outputs Your experiment is a designed perturbation of the system A primary visual area (V1) receives ‘photic’ input and has reciprocal connections with the motionsensitive area V5. ‘Motion’ input modulates the forward connection from V1 to V5. Area V5 has reciprocal connections with superior parietal cortex (SPC). ‘Attention’ input modulates the top-down connection from SPC to V5.

DCM basics Inputs In functional connectivity models (e.g. standard fMRI analysis), conceptually your input can enter anywhere In effective connectivity models (e.g. DCM), input only enters at certain places

DCM basics Inputs can exert their influence in two ways 1. Direct influence e.g. visual input to V1 2. Vicarious influence e.g. attentional modulation of the coupling between SPC and V5

DCM basics State variables neuronal activities, and other neuro/biophysical variables needed to form the outputs (later on) Neuronal priors Hemodynamic priors

DCM basics Output the BOLD signal you’ve measured in the brain regions specified in your model

Models of effective connectivity = system models. System = set of elements which interact in a spatially and temporally specific fashion. System dynamics = change of state vector in time Causal effects in the system: interactions between elements external inputs u System parameters  : specify the nature of the interactions general state equation for non- autonomous systems overall system state represented by state variables change of state vector in time Mechanistic or system models

Example: linear dynamic system FG left FG right z3 z4 LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF) visual field. LG left LG right z1 z2 RVF LVF u2 u1 state changes effective connectivity system state input parameters external inputs

Bilinear state equation in DCM state changes intrinsic connectivity modulation of connectivity system state direct inputs m external inputs

Bilinear state equation in DCM state changes intrinsic connectivity modulation of connectivity system state direct inputs m external inputs

Extension: bilinear dynamic system LG left right RVF LVF FG z1 z2 z4 z3 u2 u1 CONTEXT u3

DCM for fMRI: the basic idea Using a bilinear state equation, a cognitive system is modelled at its underlying neuronal level (which is not directly accessible for fMRI). The modelled neuronal dynamics (z) is transformed into area-specific BOLD signals (y) by a hemodynamic forward model (λ). λ z y The aim of DCM is to estimate parameters at the neuronal level such that the modelled BOLD signals are maximally similar to the experimentally measured BOLD signals.

The hemodynamic “Balloon” model 5 hemodynamic parameters: Empirically determined a priori distributions. Computed separately for each area (like the neural parameters). Buxton et al. 1998 Hemodynamic parameters, e.g. rho is the resting oxygen extraction fraction.

Priors on biophysical parameters

Conceptual overview Input u(t) neuronal z states λ y y y y BOLD Neural state equation The bilinear model effective connectivity modulation of connectivity Input u(t) direct inputs c1 integration neuronal states λ z y b23 Schematic summary of the conceptual basis of DCM. The dynamics in a system of interacting neuronal populations (blue boxes), which are not directly observable by fMRI, is modeled using a bilinear state equation (green box). Integrating the state equation gives predicted neural dynamics (z) that enter a model of the hemodynamic response (λ) to give predicted BOLD responses (y) (green boxes). The parameters at both neural and hemodynamic levels are adjusted such that the diferences between predicted and measured BOLD series are minimized. Critically, the neural dynamics are determined by experimental manipulations. These enter the model in the form of external or driving inputs. Driving inputs (u1; e.g. sensory stimuli) elicit local responses directly that are propagated through the system according to the intrinsic connections. The strengths of these connections can be changed by modulatory inputs (u2; e.g. changes in cognitive set, attention, or learning). a12 activity z2(t) activity z1(t) activity z3(t) hemodynamic model y y y BOLD Friston et al. 2003, NeuroImage

Estimating model parameters Bayes Theorem posterior  likelihood ∙ prior ) ( | q p y × µ DCMs are biologically plausible (i.e. complicated) - they have lots of free parameters A Bayesian framework is a good way to embody the constraints on these parameters

Use Bayes’ theorem to estimate model parameters posterior  likelihood ∙ prior ) ( | q p y × µ Priors – empirical (haemodynamic parameters) and non-empirical (eg. shrinkage priors, temporal scaling) Likelihood derived from error and confounds (eg. drift) Calculate the Posterior probability for each effect, and the probability that it exceeds a set threshold You need shrinkage priors to estimate the posterior – they make the system stable. That’s why the prior distribution is so narrow. Inferences about the strength (= speed) of connections between the brain regions in your model

Interpretation of parameters EM algorithm – works out the parameters in a model Bayesian model selection to test between alternative models Single subject analysis Use the cumulative normal distribution to test the probability with which a certain parameter is above a chosen threshold γ:  ηθ|y

Model comparison and selection A good model of your data will balance model fit with complexity (overfitting models noise) You find this by taking evidence ratios (the “Bayes factor”) The “Bayes factor” is a summary of the evidence in favour of one model as opposed to another

Bayesian Model Selection Bayes’ theorem: Model evidence: The log model evidence can be represented as: Bayes factor: Basically this is how you maximise model evidence. The best model has the highest free energy. Penny et al. 2004, NeuroImage

Interpretation of parameters - Group analysis: Like “random effects” analysis in SPM, 2nd level analysis can be applied to DCM parameters: Separate fitting of identical models for each subject Selection of bilinear parameters of interest One sample t-test: Parameter > 0? Paired t-test: Parameter 1 > parameter 2? rm ANOVA: For multiple sessions per subject

New stuff in DCM 1. DCM now accounts for the slice timing problem

Extension I: Slice timing model Potential timing problem in DCM: temporal shift between regional time series because of multi-slice acquisition 2 slice acquisition 1 visual input Solution: Modelling of (known) slice timing of each area. Slice timing extension now allows for any slice timing differences. Long TRs (> 2 sec) no longer a limitation. (Kiebel et al., 2007)

New stuff in DCM 1. DCM now accounts for the slice timing problem (SPM5) 2. Biological plausibility: each brain area can have two states (SPM8) (exc./inh.)

Extension II: Two-state model input Single-state DCM Intrinsic (within-region) coupling Extrinsic (between-region) coupling Two-state DCM

New stuff in DCM 1. DCM now accounts for the slice timing problem (SPM5) 2. Biological plausibility: each brain area can have two states (SPM8) (exc./inh.) 3. Biological plausibility: more complex balloon model (SPM5) 4. Non-linear version of DCM as well as bilinear (SPM8)

Expectation-maximization Posterior distribution fMRI DCM Diagram Dynamic Causal Modelling of fMRI Network dynamics Haemodynamic response Priors Model comparison State space Model In the same way like SPM for fMRI Use SPM2‘s methods! However, model needs to be adapted to make proper inference Comparison with fMRI analysis to aid illustration Also, conventional model (to be shown) in the same framework Model inversion using Expectation-maximization Posterior distribution of parameters fMRI data y

SUMMARY DCM good because Next week: practical issues in DCM Causal – models effective connectivity, not functional connectivity Neuronally plausible Forward model of how neuronal activity causes BOLD signal Inputs only enter at certain places; can model vicarious (modulatory) input; can have reciprocal connections and loops Next week: practical issues in DCM

REFERENCES Karl J. Friston. Dynamic Causal Modelling. Human brain function. Chapter 22. Second Edition. http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/ K.J Friston, L. Harrison and W. Penny. Dynamic Causal Modelling. Neuroimage 2003; 19:1273-1302. SPM Manual Last year’s presentation

THANK YOU Special thanks to: - Andre Marreiros - Maria Joao