Statistical Parametric Mapping (SPM) Talk I: Spatial Pre-processing & Morphometry Talk II: General Linear Model Talk III: Experimental Design & Connectivity Talk IV: EEG/MEG

Effective Connectivity Efficient Design & Effective Connectivity Rik Henson With thanks to: Karl Friston & Oliver Josephs

Overview 1. Experimental Design A Taxonomy of Designs Blocked vs Randomised Designs Statistical Efficiency 2. Event-related fMRI The BOLD impulse response Temporal Basis Functions Timing Issues 3. Effective Connectivity Psycho-Physiological Interactions (PPIs) Structural Equation Modelling (SEM) Dynamic Causal Modelling

Overview 1. Experimental Design A Taxonomy of Designs Blocked vs Randomised Designs Statistical Efficiency 2. Event-related fMRI The BOLD impulse response Temporal Basis Functions Timing Issues 3. Effective Connectivity Psycho-Physiological Interactions (PPIs) Structural Equation Modelling (SEM) Dynamic Causal Modelling

A taxonomy of design Categorical designs Parametric designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Linear - Cognitive components and dimensions Nonlinear - Polynomial expansions Factorial designs Categorical - Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric - Linear and nonlinear interactions - Psychophysiological Interactions

A taxonomy of design Categorical designs Parametric designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Linear - Cognitive components and dimensions Nonlinear - Polynomial expansions Factorial designs Categorical - Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric - Linear and nonlinear interactions - Psychophysiological Interactions

A categorical analysis Experimental design Word generation G Word repetition R R G R G R G R G R G R G G - R = Intrinsic word generation …under assumption of pure insertion, ie, that G and R do not differ in other ways

A taxonomy of design Categorical designs Parametric designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Linear - Cognitive components and dimensions Nonlinear - Polynomial expansions Factorial designs Categorical - Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric - Linear and nonlinear interactions - Psychophysiological Interactions

Cognitive Conjunctions One way to minimise problem of pure insertion is to isolate same process in several different ways (ie, multiple subtractions of different conditions) Task (1/2) Viewing Naming Stimuli (A/B) Objects Colours A1 A2 B2 B1 Visual Processing V Object Recognition R Phonological Retrieval P Object viewing R,V Colour viewing V Object naming P,R,V Colour naming P,V (Object - Colour viewing) [1 -1 0 0] & (Object - Colour naming) [0 0 1 -1] [ R,V - V ] & [ P,R,V - P,V ] = R & R = R (assuming RxP = 0; see later) Common object recognition response (R) Price et al, 1997

Cognitive Conjunctions

A taxonomy of design Categorical designs Parametric designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Linear - Cognitive components and dimensions Nonlinear - Polynomial expansions Factorial designs Categorical - Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric - Linear and nonlinear interactions - Psychophysiological Interactions

A (linear) parametric contrast Linear effect of time

A taxonomy of design Categorical designs Parametric designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Linear - Cognitive components and dimensions Nonlinear - Polynomial expansions Factorial designs Categorical - Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric - Linear and nonlinear interactions - Psychophysiological Interactions

Nonlinear parametric design matrix Quadratic (2nd) (Constant) (0th) SPM{F} E.g, F-contrast [0 1 0] on Quadratic Parameter => Linear (1st) Inverted ‘U’ response to increasing word presentation rate in the DLPFC Polynomial expansion: f(x) ~ b1 x + b2 x2 + ... …(N-1)th order for N levels

A taxonomy of design Categorical designs Parametric designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Linear - Cognitive components and dimensions Nonlinear - Polynomial expansions Factorial designs Categorical - Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric - Linear and nonlinear interactions - Psychophysiological Interactions

Interactions and pure insertion Presence of an interaction can show a failure of pure insertion (using earlier example)… A1 A2 B2 B1 Task (1/2) Viewing Naming Stimuli (A/B) Objects Colours Visual Processing V Object Recognition R Phonological Retrieval P Object viewing R,V Colour viewing V Object naming P,R,V,RxP Colour naming P,V Naming-specific object recognition viewing naming Object - Colour (Object – Colour) x (Viewing – Naming) [1 -1 0 0] - [0 0 1 -1] = [1 -1]  [1 -1] = [1 -1 -1 1] [ R,V - V ] - [ P,R,V,RxP - P,V ] = R – R,RxP = RxP

Interactions and pure insertion

A taxonomy of design Categorical designs Parametric designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Linear - Cognitive components and dimensions Nonlinear - Polynomial expansions Factorial designs Categorical - Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric - Linear and nonlinear interactions - Psychophysiological Interactions

(Linear) Parametric Interaction A (Linear) Time-by-Condition Interaction (“Generation strategy”?) Contrast: [5 3 1 -1 -3 -5]  [-1 1]

Nonlinear Parametric Interaction F-contrast tests for nonlinear Generation-by-Time interaction (including both linear and Quadratic components) Factorial Design with 2 factors: Gen/Rep (Categorical, 2 levels) Time (Parametric, 6 levels) Time effects modelled with both linear and quadratic components… G-R Time Lin Time Quad G x T Lin G x T Quad

A taxonomy of design Categorical designs Parametric designs Subtraction - Additive factors and pure insertion Conjunction - Testing multiple hypotheses Parametric designs Linear - Cognitive components and dimensions Nonlinear - Polynomial expansions Factorial designs Categorical - Interactions and pure insertion - Adaptation, modulation and dual-task inference Parametric - Linear and nonlinear interactions - Psychophysiological Interactions

Psycho-physiological Interaction (PPI) Parametric, factorial design, in which one factor is psychological (eg attention) ...and other is physiological (viz. activity extracted from a brain region of interest) SPM{Z} V1 activity Attention time V1 attention V5 V5 activity no attention Attentional modulation of V1 - V5 contribution V1 activity

Psycho-physiological Interaction (PPI) 0 0 1 SPM{Z} V1 activity time attention V5 activity no attention V1 Att V1 x Att V1 activity V1xAtt

Overview 1. Experimental Design A Taxonomy of Designs Blocked vs Randomised Designs Statistical Efficiency: 2. Event-related fMRI The BOLD impulse response Temporal Basis Functions Timing Issues 3. Effective Connectivity Psycho-Physiological Interactions (PPIs) Structural Equation Modelling (SEM) Dynamic Causal Modelling

Epoch vs Events Sustained epoch Blocks of events => Epochs are periods of sustained stimulation (e.g, box-car functions) Events are impulses (delta-functions) In SPM99, epochs and events are distinct (eg, in choice of basis functions) In SPM2/5, all conditions are specified in terms of their 1) onsets and 2) durations… … events simply have zero duration Near-identical regressors can be created by: 1) sustained epochs, 2) rapid series of events (SOAs<~3s) i.e, designs can be blocked or randomised … models can be epoch or event-related Boxcar function Sustained epoch Blocks of events Delta functions Convolved with HRF =>

Advantages of Event-related Models 1. Randomised (intermixed) trial order c.f. confounds of blocked designs (Johnson et al 1997) 2. Post hoc / subjective classification of trials e.g, according to subsequent memory (Wagner et al 1998) 3. Some events can only be indicated by subject (in time) e.g, spontaneous perceptual shifts (Kleinschmidt et al 1998) 4. Some trials cannot be blocked e.g, “oddball” designs (Clark et al., 2000) 5. More accurate models even for blocked designs? e.g, (Price et al, 1999)

Disadvantages of Randomised Designs 1. Less efficient for detecting effects than are blocked designs (see later…) 2. Some psychological processes may be better blocked (eg task-switching, attentional instructions)

Mixed Designs “Blocks” of trials with varying SOAs: Blocks are modelled as epochs (sustained or “state” effect) Trials are modelled as events (transient or “item” effects) (normally confounded in conventional blocked designs) Varying (some short, some long) SOAs between trials needed to decorrelate epoch and event-related covariates (see later) For example, Chawla et al (1999): Visual stimulus = dots periodically changing in colour or motion Epochs of attention to: 1) motion, or 2) colour Events are target stimuli differing in motion or colour

(Chawla et al 1999) V5 Motion change under attention to motion (red) or color (blue) Item Effect (Evoked) State Effect (Baseline) V4 Color change under attention to motion (red) or color (blue)

Mixed Designs “Blocks” of trials with varying SOAs: Blocks are modelled as epochs (sustained or “state” effect) Trials are modelled as events (transient or “item” effects) Varying (some short, some long) SOAs between trials needed to decorrelate epoch and event-related covariates (see later) Allows conclusion that selective attention modulates BOTH 1) baseline activity (state-effect, additive) 2) evoked response (item-effect, multiplicative) (But note tension between maximising fMRI efficiency to separate item and state effects, and maximising efficiency for each effect alone, and between long SOAs and maintaining a “cognitive set”)

Overview 1. Experimental Design A Taxonomy of Designs Blocked vs Randomised Designs Statistical Efficiency 2. Event-related fMRI The BOLD impulse response Temporal Basis Functions Timing Issues 3. Effective Connectivity Psycho-Physiological Interactions (PPIs) Structural Equation Modelling (SEM) Dynamic Causal Modelling

General Advice Scan as long as subjects can accommodate (eg 40-60mins); keep subjects as busy as possible! If a Group study, number of subjects more important than time per subject (though additional set-up time may encourage multiple experiments per subject) Do not contrast conditions that are far apart in time (because of low-freq noise) Randomize the order, or randomize the SOA, of conditions that are close in time http://www.mrc-cbu.cam.ac.uk/Imaging/Common/fMRI-efficiency.shtml

Expanded Overview 2. Efficient Designs 2.1 Response vs Baseline (signal-processing) 2.2 Response 1 - Response 2 (statistics) 2.3 Response 1 & Response 2 (correlations) 2.4 Impact of BOLD nonlinearities

Fixed SOA = 16s  = Not particularly efficient… Stimulus (“Neural”) HRF Predicted Data  = Not particularly efficient…

Fixed SOA = 4s  = Very Inefficient… Stimulus (“Neural”) HRF Predicted Data  = Very Inefficient…

Randomised, SOAmin= 4s  = More Efficient… Stimulus (“Neural”) HRF Predicted Data  = More Efficient…

Blocked, SOAmin= 4s  = Even more Efficient… Stimulus (“Neural”) HRF Predicted Data  = Even more Efficient…

Blocked, epoch = 20s Stimulus (“Neural”) HRF Predicted Data  = =  Blocked-epoch (with small SOA) and Time-Freq equivalences

Sinusoidal modulation, f = 1/33s Stimulus (“Neural”) HRF Predicted Data  =  = The most efficient design of all!

High-pass Filtering fMRI contains low frequency noise: aliasing fMRI contains low frequency noise: Physical (scanner drifts) Physiological (aliased) cardiac (~1 Hz) respiratory (~0.25 Hz) power spectrum highpass filter power spectrum noise signal (eg infinite 30s on-off)

 =  = Blocked (80s), SOAmin=4s, highpass filter = 1/120s Stimulus (“Neural”) HRF Predicted Data  “Effective HRF” (after highpass filtering) (Josephs & Henson, 1999) =  = Don’t have long (>60s) blocks!

Randomised, SOAmin=4s, highpass filter = 1/120s Stimulus (“Neural”) HRF Predicted Data  =  = (Randomised design spreads power over frequencies)

Expanded Overview 2. Efficient Designs 2.1 Response vs Baseline (signal-processing) 2.2 Response 1 - Response 2 (statistics) 2.3 Response 1 & Response 2 (correlations) 2.4 Impact of BOLD nonlinearities

2. How about multiple conditions? We have talked about detecting a basic response vs baseline, but how about detecting differences between two or more response-types (event-types)?

Design Efficiency T = cTb / std(cTb) std(cTb) = sqrt(2cT(XTX)-1c) For max. T, want min. contrast variability (Friston et al, 1999) If assume that noise variance (2) is unaffected by changes in X… …then want maximal efficiency, e: e(c,X) = { cT (XTX)-1 c }-1

Efficiency - Multiple Event-types Design parametrised by: SOAmin Minimum SOA pi(h) Probability of event-type i given history h of last m events With n event-types pi(h) is a nm  n Transition Matrix Example: Randomised AB A B A 0.5 0.5 B 0.5 0.5 => ABBBABAABABAAA... 4s smoothing; 1/60s highpass filtering Josephs & Henson (1999) Differential Effect (A-B) Common Effect (A+B)

Efficiency - Multiple Event-types Example: Alternating AB A B A 0 1 B 1 0 => ABABABABABAB... 4s smoothing; 1/60s highpass filtering Josephs & Henson (1999) Permuted (A-B) Alternating (A-B) Example: Permuted AB A B AA 0 1 AB 0.5 0.5 BA 0.5 0.5 BB 1 0 => ABBAABABABBA...

Efficiency - Multiple Event-types Example: Null events A B A 0.33 0.33 B 0.33 0.33 => AB-BAA--B---ABB... Efficient for differential and main effects at short SOA Equivalent to stochastic SOA (Null Event like third unmodelled event-type) Selective averaging of data (Dale & Buckner 1997) 4s smoothing; 1/60s highpass filtering Josephs & Henson (1999) Null Events (A-B) Null Events (A+B)

Interim Conclusions Optimal design for one contrast may not be optimal for another With randomised designs, optimal SOA for differential effect (A-B) is minimal SOA (assuming no saturation; see later), whereas optimal SOA for main effect (A+B) is 16-20s Inclusion of null events improves efficiency for main effect at short SOAs (at cost of efficiency for differential effects) If order constrained, intermediate SOAs (5-20s) can be optimal If SOA constrained, pseudorandomised designs can be optimal (but may introduce context-sensitivity)

Expanded Overview 2. Efficient Designs 2.1 Response vs Baseline (signal-processing) 2.2 Response 1 - Response 2 (statistics) 2.3 Response 1 & Response 2 (correlations) 2.4 Impact of BOLD nonlinearities

3. How about separating responses? What if interested in both contrasts [1 0] and [0 1]? For example: 1) Mixed designs (item-state effects) 2) Working Memory trials (stimulus-response) In the efficiency of a contrast (see earlier): e(c,X) = { cT (XTX)-1 c }-1 XTX represents covariance of regressors in design matrix High covariance increases elements of (XTX)-1 So, when correlation between regressors, efficiency to detect effect of each one separately is reduced

Correlations between Regressors [1 -1] [1 1] Negative correlation between two regressors means separate (orthogonal) effect of each is estimated poorly, though difference between regressors estimated well

Eg 1: Item and State effects (see earlier) Blocks = 40s, Fixed SOA = 4s Design Matrix (X) Efficiency = 16 [1 0] (Item Effect) Correlation = .97 Not good…

Eg 1: Item and State effects (see earlier) Blocks = 40s, Randomised SOAmin= 2s Design Matrix (X) Efficiency = 54 [1 0] (Item Effect) Correlation = .78 Better…

Eg 2: Stimulus-Response Paradigms Each trial consists of 2 successive events: e.g, Stimulus - Response Each event every 4s (Stimulus every 8s) Stim Resp Design Matrix (X) Efficiency = 29 [1 0] (Stimulus) Correlation = -.65

Eg 2: Stimulus-Response Paradigms Each trial consists of 2 successive events: e.g, Stimulus - Response Solution 1: Time between Stim- Resp events jittered from 0-8 seconds... Stim Resp Design Matrix (X) Efficiency = 40 [1 0] (Stimulus) Correlation = +.33

Eg 2: Stimulus-Response Paradigms Each trial consists of 2 successive events: e.g, Stimulus - Response Solution 2: Stim event every 8s, but Resp event only occurs on 50% trials... Stim Resp Design Matrix (X) Efficiency = 47 [1 0] (Stimulus) Correlation = -.24

Expanded Overview 2. Efficient Designs 2.1 Response vs Baseline (signal-processing) 2.2 Response 1 - Response 2 (statistics) 2.3 Response 1 & Response 2 (correlations) 2.4 Impact of BOLD nonlinearities

Underadditivity at short SOAs Nonlinear Effects Underadditivity at short SOAs Linear Prediction Volterra Implications for Efficiency

Overview 1. Experimental Design A Taxonomy of Designs Blocked vs Randomised Designs Statistical Efficiency 2. Event-related fMRI The BOLD impulse response Temporal Basis Functions Timing Issues 3. Effective Connectivity Psycho-Physiological Interactions (PPIs) Structural Equation Modelling (SEM) Dynamic Causal Modelling

BOLD Impulse Response Function of blood oxygenation, flow, volume (Buxton et al, 1998) Peak (max. oxygenation) 4-6s poststimulus; baseline after 20-30s Initial undershoot can be observed (Malonek & Grinvald, 1996) Similar across V1, A1, S1… … but differences across: other regions (Schacter et al 1997) individuals (Aguirre et al, 1998) Brief Stimulus Undershoot Initial Peak

BOLD Impulse Response Early event-related fMRI studies used a long Stimulus Onset Asynchrony (SOA) to allow BOLD response to return to baseline However, if the BOLD response is explicitly modelled, overlap between successive responses at short SOAs can be accommodated… … particularly if responses are assumed to superpose linearly Short SOAs are more sensitive… Brief Stimulus Undershoot Initial Peak

General Linear (Convolution) Model GLM for a single voxel: y(t) = u(t)  h(t) + (t) u(t) = neural causes (stimulus train) u(t) =   (t - nT) h(t) = hemodynamic (BOLD) response h(t) =  ßi fi (t) fi(t) = temporal basis functions y(t) =   ßi fi (t - nT) + (t) y = X ß + ε T 2T 3T ... u(t) h(t)= ßi fi (t) convolution sampled each scan Design Matrix

General Linear (Convolution) Model Auditory words every 20s SPM{F} 0 time {secs} 30 Gamma functions ƒi() of peristimulus time  (Orthogonalised) Sampled every TR = 1.7s Design matrix, X [x(t)ƒ1() | x(t)ƒ2() |...] …

A word about down-sampling T=16, TR=2s Scan 1 o T0=16 o T0=9 x2 x3 T0 should match the reference slice if slice-time correction performed!

Overview 1. Experimental Design A Taxonomy of Designs Blocked vs Randomised Designs Statistical Efficiency 2. Event-related fMRI The BOLD impulse response Temporal Basis Functions Timing Issues 3. Effective Connectivity Psycho-Physiological Interactions (PPIs) Structural Equation Modelling (SEM) Dynamic Causal Modelling

Temporal Basis Functions Fourier Set Windowed sines & cosines Any shape (up to frequency limit) Inference via F-test

Temporal Basis Functions Finite Impulse Response Mini “timebins” (selective averaging) Any shape (up to bin-width) Inference via F-test

Temporal Basis Functions Fourier Set Windowed sines & cosines Any shape (up to frequency limit) Inference via F-test Gamma Functions Bounded, asymmetrical (like BOLD) Set of different lags

Temporal Basis Functions Fourier Set Windowed sines & cosines Any shape (up to frequency limit) Inference via F-test Gamma Functions Bounded, asymmetrical (like BOLD) Set of different lags “Informed” Basis Set Best guess of canonical BOLD response Variability captured by Taylor expansion “Magnitude” inferences via t-test…?

Temporal Basis Functions

Temporal Basis Functions “Informed” Basis Set (Friston et al. 1998) Canonical HRF (2 gamma functions) Canonical

Temporal Basis Functions “Informed” Basis Set (Friston et al. 1998) Canonical HRF (2 gamma functions) plus Multivariate Taylor expansion in: time (Temporal Derivative) Canonical Temporal

Temporal Basis Functions “Informed” Basis Set (Friston et al. 1998) Canonical HRF (2 gamma functions) plus Multivariate Taylor expansion in: time (Temporal Derivative) width (Dispersion Derivative) Canonical Temporal Dispersion

Temporal Basis Functions “Informed” Basis Set (Friston et al. 1998) Canonical HRF (2 gamma functions) plus Multivariate Taylor expansion in: time (Temporal Derivative) width (Dispersion Derivative) F-tests allow for “canonical-like” responses Canonical Temporal Dispersion

Temporal Basis Functions “Informed” Basis Set (Friston et al. 1998) Canonical HRF (2 gamma functions) plus Multivariate Taylor expansion in: time (Temporal Derivative) width (Dispersion Derivative) F-tests allow for any “canonical-like” responses T-tests on canonical HRF alone (at 1st level) can be improved by derivatives reducing residual error, and can be interpreted as “amplitude” differences, assuming canonical HRF is good fit… Canonical Temporal Dispersion

Temporal Basis Functions: Caveats Can take parameter estimates for each basis function into a 2nd-level (“random effects”) analysis (Henson, 2001), but then are assuming that all subjects have same HRF shape (even if that shape can differ over voxels)... ...one possibility is to combine parameter estimates for each subject into a single “shape-invariant” metric, e.g, sum of squares of an FIR fit (AUC), or the RMS of partial derivatives of a canonical HRF (Calhoun et al, 2004)... ...though distribution of this metric may no longer be Gaussian Note also: multiple basis functions are only worth including if one jitters SOAs (e.g, through null events) – i.e, if one has efficiency to detect shape of HRF (Henson, 2004)

(Other Approaches) Long Stimulus Onset Asychrony (SOA) Can ignore overlap between responses (Cohen et al 1997) … but long SOAs are less sensitive Fully counterbalanced designs Assume response overlap cancels (Saykin et al 1999) Include fixation trials to “selectively average” response even at short SOA (Dale & Buckner, 1997) … but unbalanced when events defined by subject Define HRF from pilot scan on each subject May capture intersubject variability (Zarahn et al, 1997) … but not interregional variability Numerical fitting of highly parametrised response functions Separate estimate of magnitude, latency, duration (Kruggel 1999) … but computationally expensive for every voxel

Temporal Basis Sets: Which One? In this example (rapid motor response to faces, Henson et al, 2001)… Canonical + Temporal + Dispersion + FIR …canonical + temporal + dispersion derivatives appear sufficient …may not be for more complex trials (eg stimulus-delay-response) …but then such trials better modelled with separate neural components (ie activity no longer delta function) + constrained HRF (Zarahn, 1999)

Overview 1. Experimental Design A Taxonomy of Designs Blocked vs Randomised Designs Statistical Efficiency 2. Event-related fMRI The BOLD impulse response Temporal Basis Functions Timing Issues 3. Effective Connectivity Psycho-Physiological Interactions (PPIs) Structural Equation Modelling (SEM) Dynamic Causal Modelling

Timing Issues : Practical TR=4s Scans Typical TR for 48 slice EPI at 3mm spacing is ~ 4s

Timing Issues : Practical TR=4s Scans Typical TR for 48 slice EPI at 3mm spacing is ~ 4s Sampling at [0,4,8,12…] post- stimulus may miss peak signal Stimulus (synchronous) SOA=8s Sampling rate=4s

Timing Issues : Practical TR=4s Scans Typical TR for 48 slice EPI at 3mm spacing is ~ 4s Sampling at [0,4,8,12…] post- stimulus may miss peak signal Higher effective sampling by: 1. Asynchrony eg SOA=1.5TR Stimulus (asynchronous) SOA=6s Sampling rate=2s

Timing Issues : Practical TR=4s Scans Typical TR for 48 slice EPI at 3mm spacing is ~ 4s Sampling at [0,4,8,12…] post- stimulus may miss peak signal Higher effective sampling by: 1. Asynchrony eg SOA=1.5TR 2. Random Jitter eg SOA=(2±0.5)TR Stimulus (random jitter) Sampling rate=2s

Timing Issues : Practical TR=4s Scans Typical TR for 48 slice EPI at 3mm spacing is ~ 4s Sampling at [0,4,8,12…] post- stimulus may miss peak signal Higher effective sampling by: 1. Asynchrony eg SOA=1.5TR 2. Random Jitter eg SOA=(2±0.5)TR Better response characterisation (Miezin et al, 2000) Stimulus (random jitter) Sampling rate=2s

Timing Issues : Practical Top Slice Bottom Slice …but “Slice-timing Problem” (Henson et al, 1999) Slices acquired at different times, yet model is the same for all slices => different results (using canonical HRF) for different reference slices Solutions: 1. Temporal interpolation of data … but less good for longer TRs 2. More general basis set (e.g., with temporal derivatives) … use a composite estimate, or use F-tests TR=3s SPM{t} SPM{t} Interpolated SPM{t} Derivative SPM{F}

Overview 1. Experimental Design A Taxonomy of Designs Blocked vs Randomised Designs Statistical Efficiency 2. Event-related fMRI The BOLD impulse response Temporal Basis Functions Timing Issues 3. Effective Connectivity Psycho-Physiological Interactions (PPIs) Structural Equation Modelling (SEM) Dynamic Causal Modelling (DCM)

Effective vs. functional connectivity Functional connectivity simply reflects correlations (e.g, “default network”) Model-independent (data-driven), like PCA Effective connectivity attempts to model causal relationships...

Effective vs. functional connectivity Correlations: A B C 1 0.49 1 0.30 0.12 1 No connection between B and C, yet B and C correlated because of common input from A, eg: A = V1 fMRI time-series B = 0.5 * A + e1 C = 0.3 * A + e2 Functional connectivity A B C 0.49 0.31 Effective connectivity -0.02 2=0.5, ns. Model Correlations (Normalised ts) Analysis of correlation structure proper effcon (SEM) fallacy of wrong models

Psycho-physiological Interaction (PPI) Parametric, factorial design, in which one factor is psychological (eg attention) ...and other is physiological (viz. activity extracted from a brain region of interest) SPM{Z} V1 activity Attention time V1 attention V5 V5 activity no attention Attentional modulation of V1 - V5 contribution V1 activity

Structural Equation Modelling (SEM) Because testing a change in regression slopes, PPIs are not simply correlations (eg, owing to global bloodflow changes) But while PPIs are simple way of searching for connectivity across the brain, they do not test connections within specific networks Structural Equation Modelling is one such way, but: Assumes stationarity of neural activity (not dynamic) Becomes unstable for networks with loops Classical inference; can only compare nested models Only uses covariance of BOLD (not neural) activity (no haemodynamics...) => Dynamic Causal Modelling (DCM)...

Overview 1. Experimental Design A Taxonomy of Designs Blocked vs Randomised Designs Statistical Efficiency 2. Event-related fMRI The BOLD impulse response Temporal Basis Functions Timing Issues 3. Effective Connectivity Psycho-Physiological Interactions (PPIs) Structural Equation Modelling (SEM) Dynamic Causal Modelling (DCM)

It is the difference between goodness of fit and model complexity DCM vs SEM Dynamic, in that neural dynamics directly simulated Has explicit haemodynamic (“balloon”) model to map to data Can handle loops in network (can determine directionality) Framed in a Bayesian context, so different models (connections) can be compared used the Model Evidence “Model evidence” is probability of data given a model, p(y|m), ie after having integrated out all parameter values... Such integration is difficult, but analytic approximations exist under certain (eg, Gaussian) assumptions (AIC and BIC are other, poorer, approximations) It is the difference between goodness of fit and model complexity Pitt & Miyung (2002), TICS

Dynamic Causal Modelling The parameters consist of: 1. Connections between regions 2. Self-connections 3. Direct inputs (eg, visual stimulations) 4. Contextual inputs (eg, attention) Parameters estimated using EM Priors are: Empirical (for haemodynamic model) Principled (dynamics to be convergent) Shrinkage (zero-mean, for connections) Connection strengths reflect rate constants Inference using posterior probabilities direct inputs - u1 (e.g. visual stimuli) contextual inputs - u2 (e.g. attention) z1 V1 z2 V5 z3 SPC y1 y2 y3 z = f(z,u,z)  Az + uBz + Cu y = h(z,h) + e z = state vector u = inputs  = parameters (connection/haemodynamic) .

The Bilinear State Equation state changes intrinsic connectivity modulation of connectivity system state direct inputs m external inputs context

Dynamic Causal Modelling stimuli u1 context u2 u1  - + - Z1 u2 + z1 + Z2 - z2 - 

The Haemodynamic (Balloon) Model 5 haemodynamic parameters: Important for model fitting, but of no interest for statistical inference Empirically determined a priori distributions. Computed separately for each area (like the neural parameters)

Dynamic Causal Modelling V1 IFG V5 SPC Motion Photic Attention .82 (100%) .42 .37 (90%) .69 (100%) .47 .65 (100%) .52 (98%) .56 (99%) Friston et al. (2003) Effects Photic – dots vs fixation Motion – moving vs static Attenton – detect changes Büchel & Friston (1997) Attention modulates the backward-connections IFG→SPC and SPC→V5 The intrinsic connection V1→V5 is insignificant in the absence of motion

Dynamic Causal Modelling DCM usually requires: Specification of a network (regions and their directional connections), based on a priori anatomical or functional information (e.g., from a PPI, or activations, though note that connectivity may change even if overall activation does not) [Though different models (sets of connections) can be compared in terms of their (Bayesian) Model Evidence (cannot compare different regions, because addition/deletion of regions changes the data too)] A minimum of a 2x2 factorial design, where one factor is “input” (e.g, stimulus, transient) and other factor is the “modulator” or “context” (e.g, task, sustained) Rapid (ms) changes in connectivity will never been detected with fMRI, so need to “slow down” changes via experimental design... ...however, same underlying mathematics applied to EEG/MEG too...

Parts of this talk appear as Chapters 14 and 15 in the SPM book: The End Parts of this talk appear as Chapters 14 and 15 in the SPM book: http://www.mrc-cbu.cam.ac.uk/people/rik.henson/personal/Henson_SPM_06_preprint.pdf http://www.mrc-cbu.cam.ac.uk/people/rik.henson/personal/HensonFriston_SPM_06_preprint.pdf For further info on how to design an efficient fMRI experiment, see: http://www.mrc-cbu.cam.ac.uk/Imaging/Common/fMRI-efficiency.shtml

Some References Friston KJ, Holmes AP, Worsley KJ, Poline J-B, Frith CD, Frackowiak RSJ (1995) Statistical parametric maps in functional imaging: A general linear approach” Human Brain Mapping 2:189-210 Worsley KJ & Friston KJ (1995) Analysis of fMRI time series revisited — again” NeuroImage 2:173-181 Friston KJ, Josephs O, Zarahn E, Holmes AP, Poline J-B (2000) “To smooth or not to smooth” NeuroImage Zarahn E, Aguirre GK, D'Esposito M (1997) “Empirical Analyses of BOLD fMRI Statistics” NeuroImage 5:179-197 Holmes AP, Friston KJ (1998) “Generalisability, Random Effects & Population Inference” NeuroImage 7(4-2/3):S754 Worsley KJ, Marrett S, Neelin P, Evans AC (1992) “A three-dimensional statistical analysis for CBF activation studies in human brain”Journal of Cerebral Blood Flow and Metabolism 12:900-918 Worsley KJ, Marrett S, Neelin P, Vandal AC, Friston KJ, Evans AC (1995) “A unified statistical approach for determining significant signals in images of cerebral activation” Human Brain Mapping 4:58-73 Friston KJ, Worsley KJ, Frackowiak RSJ, Mazziotta JC, Evans AC (1994) Assessing the Significance of Focal Activations Using their Spatial Extent” Human Brain Mapping 1:214-220 Cao J (1999) The size of the connected components of excursion sets of 2, t and F fields” Advances in Applied Probability (in press) Worsley KJ, Marrett S, Neelin P, Evans AC (1995) Searching scale space for activation in PET images” Human Brain Mapping 4:74-90 Worsley KJ, Poline J-B, Vandal AC, Friston KJ (1995) Tests for distributed, non-focal brain activations” NeuroImage 2:183-194 Friston KJ, Holmes AP, Poline J-B, Price CJ, Frith CD (1996) Detecting Activations in PET and fMRI: Levels of Inference and Power” Neuroimage 4:223-235

Cognitive Conjunctions Original (SPM97) definition of conjunctions entailed sum of two simple effects (A1-A2 + B1-B2) plus exclusive masking with interaction (A1-A2) - (B1-B2) Ie, “effects significant and of similar size” (Difference between conjunctions and masking is that conjunction p-values reflect the conjoint probabilities of the contrasts) SPM2 defintion of conjunctions uses advances in Gaussian Field Theory (e.g, T2 fields), allowing corrected p-values However, the logic has changed slightly, in that voxels can survive a conjunction even though they show an interaction A1-A2 B1-B2 p((A1-A2)= (B1-B2))>P2 p(A1=A2+B1=B2)<P1 + p(A1=A2)<p A1-A2 B1-B2 p(B1=B2)<p

Note on Epoch Durations As duration of epochs increases from 0 to ~2s, shape of convolved response changes little (mainly amplitude of response changes) Since it is the “amplitude” that is effectively estimated by the GLM, the results for epochs of constant duration <2s will be very similar to those for events (at typical SNRs) If however the epochs vary in duration from trial-to-trial (e.g, to match RT), then epoch and event models will give different results However, while RT-related duration may be appropriate for “motor” regions, it may not be appropriate for all regions (e.g, “visual”) Thus a “parametric modulation” of events by RT may be a better model in such situations

Epoch vs Events Rate = 1/4s Rate = 1/2s Though blocks of trials can be modelled as either epochs (boxcars) or runs of events… … interpretation of parameters differs… Consider an experiment presenting words at different rates in different blocks: An “epoch” model will estimate parameter that increases with rate, because the parameter reflects response per block An “event” model may estimate parameter that decreases with rate, because the parameter reflects response per word b=3 b=5 b=11 b=9

Efficiency – Detection vs Estimation “Detection power” vs “Estimation efficiency” (Liu et al, 2001) Detect response, or characterise shape of response? Maximal detection power in blocked designs; Maximal estimation efficiency in randomised designs => simply corresponds to choice of basis functions: detection = canonical HRF estimation = FIR

Efficiency - Single Event-type Design parametrised by: SOAmin Minimum SOA p(t) Probability of event at each SOAmin Deterministic p(t)=1 iff t=nT Stationary stochastic p(t)=constant Dynamic stochastic p(t) varies (eg blocked) Blocked designs most efficient! (with small SOAmin)

PCA/SVD and Eigenimages A time-series of 1D images 128 scans of 32 “voxels” Expression of 1st 3 “eigenimages” Eigenvalues and spatial “modes” The time-series ‘reconstituted’

PCA/SVD and Eigenimages APPROX. OF Y U1 = s1 V1 APPROX. OF Y + s2 U2 V2 + s3 APPROX. OF Y U3 V3 voxels Y (DATA) + ... time Y = USVT = s1U1V1T + s2U2V2T + ...

Structural Equation Modelling (SEM) Minimise the difference between the observed (S) and implied () covariances by adjusting the path coefficients (B) The implied covariance structure: x = x.B + z x = z.(I - B)-1 x : matrix of time-series of Regions 1-3 B: matrix of unidirectional path coefficients Variance-covariance structure: xT . x =  = (I-B)-T. C.(I-B)-1 where C = zT z xT.x is the implied variance covariance structure  C contains the residual variances (u,v,w) and covariances The free parameters are estimated by minimising a [maximum likelihood] function of S and  1 3 2 z SEM Key sentence BUT how can we calculate an implied var-cov matrix -->Equ Minimise the diff

Attention - No attention 0.43 0.75 0.47 0.76 No attention Attention Changes in “effective connectivity”

Second-order Interactions PP 2 =11, p<0.01 0.14 V5 V1 = The information that att is required does not come from V1, therefore top-down mechanisms: PP modulating V1 --> V5 OR PP changing the sensitivity of V5 for other inputs (ie V1) Previous presentations V5 --> PP mod by PFC: Same principle V1xPP V5 Modulatory influence of parietal cortex on V1 to V5

Blocked Randomised Data Model O = Old Words N = New Words O1 O2 O3 N1

Event-Related ~4s R F R = Words Later Remembered F = Words Later Forgotten Event-Related ~4s Data Model

“Oddball” … Time

Blocked Design “Epoch” model “Event” model Data “Epoch” model Model O1 O2 O3 N1 N2 N3 N1 N2 N3 “Event” model O1 O2 O3

Timing Issues : Latency Assume the real response, r(t), is a scaled (by ) version of the canonical, f(t), but delayed by a small amount dt: r(t) =  f(t+dt) ~  f(t) +  f ´(t) dt 1st-order Taylor If the fitted response, R(t), is modelled by canonical+temporal derivative: R(t) = ß1 f(t) + ß2 f ´(t) GLM fit Then if want to reduce estimate of BOLD impulse response to one composite value, with some robustness to latency issues (e.g, real, or induced by slice-timing):  = sqrt(ß12 + ß22) (Calhoun et al, 2004) (similar logic applicable to other partial derivatives)

Timing Issues : Latency Assume the real response, r(t), is a scaled (by ) version of the canonical, f(t), but delayed by a small amount dt: r(t) =  f(t+dt) ~  f(t) +  f ´(t) dt 1st-order Taylor If the fitted response, R(t), is modelled by canonical + temporal derivative: R(t) = ß1 f(t) + ß2 f ´(t) GLM fit …or if want to estimate latency directly (assuming 1st-order approx holds):  ~ ß1 dt ~ ß2 / ß1 (Henson et al, 2002) (Liao et al, 2002) ie, Latency can be approximated by the ratio of derivative-to-canonical parameter estimates (within limits of first-order approximation, +/-1s)