Computational Neuroanatomy John Ashburner johnfil'ion'ucl'ac'uk

1
Statistical Parametric Mapping (SPM) Talk
I Spatial Pre-processing Morphometry Talk
II General Linear Model Talk III
Experimental Design Connectivity Talk IV
EEG/MEG
2
Efficient Design Effective Connectivity Rik
Henson With thanks to Karl Friston,
3
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-Physio
logical Interactions (PPIs) Structural
Equation Modelling (SEM) Dynamic Causal
Modelling
4
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-Physio
logical Interactions (PPIs) Structural
Equation Modelling (SEM) Dynamic Causal
Modelling
5
A taxonomy of design

• Categorical 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

6
A taxonomy of design

• Categorical 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

7
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
8
A taxonomy of design

• Categorical 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

9
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)

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)
10
Cognitive Conjunctions
11
A taxonomy of design

• Categorical 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

12
A (linear) parametric contrast
Linear effect of time
13
A taxonomy of design

• Categorical 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

14
Nonlinear parametric design matrix
E.g, F-contrast 0 1 0 on Quadratic Parameter gt
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
15
A taxonomy of design

• Categorical 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

16
Interactions and pure insertion

• Presence of an interaction can show a failure of
  pure insertion (using earlier example)

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
(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
17
Interactions and pure insertion
18
A taxonomy of design

• Categorical 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

19
(Linear) Parametric Interaction
A (Linear) Time-by-Condition Interaction (Genera
tion strategy?)
Contrast 5 3 1 -1 -3 -5 ? -1 1
20
Nonlinear Parametric Interaction

• Factorial Design with 2 factors
• Gen/Rep (Categorical, 2 levels)
• Time (Parametric, 6 levels)
• Time effects modelled with both linear and
  quadratic components

21
A taxonomy of design

• Categorical 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

22
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)
V1 activity
time
attention
V5 activity
no attention
Attentional modulation of V1 - V5 contribution
V1 activity
23
Psycho-physiological Interaction (PPI)
0 0 1
V1 activity
time
attention
V5 activity
no attention
V1 activity
V1xAtt
24
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-Physio
logical Interactions (PPIs) Structural
Equation Modelling (SEM) Dynamic Causal
Modelling
25
Epoch vs 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
  (SOAslt3s)
• i.e, designs can be blocked or randomised
  models can be epoch or event-related

26
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)
27
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)
28

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

29

(Chawla et al 1999)
30

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)

31
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-Physio
logical Interactions (PPIs) Structural
Equation Modelling (SEM) Dynamic Causal
Modelling
32
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-e
  fficiency.shtml

33
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
34
Fixed SOA 16s
Stimulus (Neural)
HRF
Predicted Data
?

Not particularly efficient
35
Fixed SOA 4s
Stimulus (Neural)
HRF
Predicted Data
Very Inefficient
36
Randomised, SOAmin 4s
Stimulus (Neural)
HRF
Predicted Data
More Efficient
37
Blocked, SOAmin 4s
Stimulus (Neural)
HRF
Predicted Data
Even more Efficient
38
Blocked, epoch 20s
Stimulus (Neural)
HRF
Predicted Data
?
Blocked-epoch (with small SOA) and Time-Freq
equivalences
39
Sinusoidal modulation, f 1/33s
Stimulus (Neural)
HRF
Predicted Data
The most efficient design of all!
40
High-pass Filtering

• fMRI contains low frequency noise
• Physical (scanner drifts)
• Physiological (aliased)
• cardiac (1 Hz)
• respiratory (0.25 Hz)

41
Blocked (80s), SOAmin4s, highpass filter
1/120s
Stimulus (Neural)
HRF
Predicted Data
Dont have long (gt60s) blocks!
42
Randomised, SOAmin4s, highpass filter 1/120s
Stimulus (Neural)
HRF
Predicted Data
(Randomised design spreads power over frequencies)
43
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
44
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)?

45
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

46
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
• gt ABBBABAABABAAA...

Josephs Henson (1999)
47
Efficiency - Multiple Event-types

• Example Alternating AB
• A B A 0 1
• B 1 0
• gt ABABABABABAB...

Josephs Henson (1999)

• Example Permuted AB
• A B
• AA 0 1
• AB 0.5 0.5
• BA 0.5 0.5
• BB 1 0
• gt ABBAABABABBA...

48
Efficiency - Multiple Event-types

• Example Null events
• A B
• A 0.33 0.33
• B 0.33 0.33
• gt 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)

Josephs Henson (1999)
49
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 (AB) 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)

50
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
51
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

52
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
53
Eg 1 Item and State effects (see earlier)
Blocks 40s, Fixed SOA 4s
Efficiency 16 1 0 (Item Effect)
Correlation .97
Not good
54
Eg 1 Item and State effects (see earlier)
Blocks 40s, Randomised SOAmin 2s
Efficiency 54 1 0 (Item Effect)
Correlation .78
Better
55
Eg 2 Stimulus-Response Paradigms
Each trial consists of 2 successive events e.g,
Stimulus - Response Each event every
4s (Stimulus every 8s)
Efficiency 29 1 0 (Stimulus)
Correlation -.65
56
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...
Efficiency 40 1 0 (Stimulus)
Correlation .33
57
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...
Efficiency 47 1 0 (Stimulus)
Correlation -.24
58
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
59
Nonlinear Effects

60
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-Physio
logical Interactions (PPIs) Structural
Equation Modelling (SEM) Dynamic Causal
Modelling
61
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)

62
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

63
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 e
sampled each scan
Design Matrix
64
General Linear (Convolution) Model

65
A word about down-sampling
x2
x3
T0 should match the reference slice if slice-time
correction performed!
66
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-Physio
logical Interactions (PPIs) Structural
Equation Modelling (SEM) Dynamic Causal
Modelling
67
Temporal Basis Functions

• Fourier Set
• Windowed sines cosines
• Any shape (up to frequency limit)
• Inference via F-test

68
Temporal Basis Functions

• Finite Impulse Response
• Mini timebins (selective averaging)
• Any shape (up to bin-width)
• Inference via F-test

69
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
• Inference via F-test

70
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
• Inference via F-test
• Informed Basis Set
• Best guess of canonical BOLD response Variabilit
  y captured by Taylor expansion Magnitude
  inferences via t-test?

71
Temporal Basis Functions
72
Temporal Basis Functions

• Informed Basis Set
• (Friston et al. 1998)
• Canonical HRF (2 gamma functions)

Canonical
73
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
74
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
75
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
76
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
77
(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

78
Temporal Basis Sets Which One?
In this example (rapid motor response to faces,
Henson et al, 2001)
FIR
Dispersion
Temporal
Canonical
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)
79
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-Physio
logical Interactions (PPIs) Structural
Equation Modelling (SEM) Dynamic Causal
Modelling
80
Timing Issues Practical
TR4s
Scans

• Typical TR for 48 slice EPI at 3mm spacing is 4s

81
Timing Issues Practical
TR4s
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)
SOA8s
Sampling rate4s
82
Timing Issues Practical
TR4s
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
  SOA1.5TR

Stimulus (asynchronous)
SOA6s
Sampling rate2s
83
Timing Issues Practical
TR4s
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
  SOA1.5TR 2. Random Jitter eg
  SOA(20.5)TR

Stimulus (random jitter)
Sampling rate2s
84
Timing Issues Practical
TR4s
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
  SOA1.5TR 2. Random Jitter eg
  SOA(20.5)TR
• Better response characterisation (Miezin et al,
  2000)

Stimulus (random jitter)
Sampling rate2s
85
Timing Issues Practical
Bottom Slice
Top Slice

• but Slice-timing Problem
• (Henson et al, 1999)
• Slices acquired at different times, yet
  model is the same for all slices
• gt 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

TR3s
SPMt
SPMt
Interpolated
SPMt
Derivative
SPMF
86
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(tdt) ? f(t) ? f (t) dt
1st-order Taylor

• If the fitted response, R(t), is modelled by
  canonicaltemporal 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)

• (similar logic applicable to other partial
  derivatives)

87
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(tdt) ? 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)

88
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-Physio
logical Interactions (PPIs) Structural
Equation Modelling (SEM) Dynamic Causal
Modelling (DCM)
89
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...

90
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
91
(No Transcript)
92
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)
V1 activity
time
attention
V5 activity
no attention
Attentional modulation of V1 - V5 contribution
V1 activity
93
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...)
• gt Dynamic Causal Modelling (DCM)...

94
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-Physio
logical Interactions (PPIs) Structural
Equation Modelling (SEM) Dynamic Causal
Modelling (DCM)
95
(No Transcript)
96
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

Akaike information criterion (AIC) or Bayesian
information criterion (BIC) assess evidence in
terms of accuracy of a model given its complexity
97
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

z3 SPC
z1 V1
z2 V5
y1
y2
y3
98
The Bilinear State Equation
state changes
intrinsic connectivity
m externalinputs
systemstate
direct inputs
modulation of connectivity
context
99
Dynamic Causal Modelling
stimuli u1
context u2
u1
?
-

-
Z1
u2

z1

Z2
-
z2
-
?
100
The Haemodynamic (Balloon) Model

• 5 haemodynamic parameters

• Important for model fitting, but of no interest
  for statistical inference
• Empirically determineda priori distributions.
• Computed separately for each area (like the
  neural parameters)

101
Dynamic Causal Modelling
Büchel Friston (1997)
102
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...

103
The End
Parts of this talk appears as Chapter 15 in the
SPM bookhttp//www.mrc-cbu.cam.ac.uk/rh01/Henso
n_Design_SPMBook_2006_preprint.pdf
For further info on how to design an efficient
fMRI experiment, seehttp//www.mrc-cbu.cam.ac.uk
/Imaging/Common/fMRI-efficiency.shtml
104
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
2189-210 Worsley KJ Friston KJ (1995) Analysis
of fMRI time series revisited again NeuroImage
2173-181 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 5179-197 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
brainJournal of Cerebral Blood Flow and
Metabolism 12900-918 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 458-73 Friston
KJ, Worsley KJ, Frackowiak RSJ, Mazziotta JC,
Evans AC (1994) Assessing the Significance of
Focal Activations Using their Spatial Extent
Human Brain Mapping 1214-220 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
474-90 Worsley KJ, Poline J-B, Vandal AC,
Friston KJ (1995) Tests for distributed,
non-focal brain activations NeuroImage
2183-194 Friston KJ, Holmes AP, Poline J-B,
Price CJ, Frith CD (1996) Detecting Activations
in PET and fMRI Levels of Inference and Power
Neuroimage 4223-235
105
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

106
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 lt2s 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

107
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

108
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
• gt simply corresponds to choice of basis
  functions
• detection canonical HRF
• estimation FIR

109
Efficiency - Single Event-type

• Design parametrised by
• SOAmin Minimum SOA
• p(t) Probability of event at
  each SOAmin
• Deterministic p(t)1 iff tnT
• Stationary stochastic p(t)constant
• Dynamic stochastic
• p(t) varies (eg blocked)

Blocked designs most efficient! (with small
SOAmin)
110
PCA/SVD and Eigenimages

A time-series of 1D images 128 scans of 32
voxels Expression of 1st 3 eigenimages Ei
genvalues and spatial modes The time-series
reconstituted
111
PCA/SVD and Eigenimages
...
Y USVT s1U1V1T s2U2V2T
...
112
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 zx 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 ?

113
Attention - No attention
No attention
Attention
Changes in effective connectivity
114
Second-order Interactions

Modulatory influence of parietal cortex on V1 to
V5
115
Blocked
116
(No Transcript)
117
(No Transcript)
118
Time
119
Blocked Design
Data
Model
Epoch model