1. Dynamic Causal Modelling
Karl Friston, Lee Harrison, Will Penny
Wellcome Department of Imaging Neuroscience,
University College London, UK
Neuronal Variability and Noise: Challenges and Promises
NIMH, Washington, September 2002

2. Neuronal Variability Neurons often vary in their response to
identical stimuli
Multi-unit recordings suggest that variability
previously attributed to single neuron noise may instead reflect system-wide changes
“Noise” in linear systems analysis may be “signal” in nonlinear systems analysis

3. The brain as a nonlinear dynamical systemZ2 Z1 Z4 Z3 Z5
Stimuli u1
Set u2
Nonlinear,
systems-level
model

4. Bilinear Dynamics
a53
Set u2
Stimuli u1

5. Bilinear Dynamics: Oscillatory transients
Stimuli u1
Set u2 u 1 Z 2 - + Z1 - - + Z2 -
Seconds -

6. Bilinear Dynamics: Positive transients
Stimuli u1
Set u2 u 1 Z 2 - + Z1 - + + Z2 - -

7. DCM: A model for fMRI Causality: set of differential
Stimuli u1
Causality:
set of differential
equations relating
change in one area
to change in
another

8. The hemodynamic model
Buxton,
Mandeville,
Hoge,
Mayhew.

9. Impulse
response
Hemodynamics
BOLD is
sluggish

10. Neuronal Transients and BOLD: I
300ms
500ms
Seconds
Seconds
More enduring transients produce bigger BOLD signals

11. Neuronal Transients and BOLD: II
Seconds
Seconds
BOLD is sensitive to frequency
content of transients
Relative timings of transients are
amplified in BOLD
Seconds

12. Model estimation and inference
Unknown neural parameters, N={A,B,C}
Unknown hemodynamic parameters, H
Vague (stability) priors, p(N)
Informative priors, p(H)
Observed BOLD time series, B.
Data likelihood, p(B|H,N) = Gauss (B-Y)
Bayesian inference p(N|B) a p(B|N) p(N)
Laplace
Approximation

13. Single word processing at different rates
SPM{F}
“Dog”
“Mountain”
“Gate”
Functional localisation of primary and secondary
auditory cortex and Wernicke’s area

14. Time Series
Auditory stimulus, u1 A1 WA A2
Adaptation variable, u2

15. Dynamic Causal Model Auditory stimulus, u1 Adaptation variable, u2 u1
u1 allowed to
affect all intrinsic
self-connections A2
Model
allows for
full intrinsic
connectivity A1 .
Adaptation variable, u2 u1
u2 allowed to
affect all intrinsic
connections between
regions . WA

16. Posterior Distributions
P(A(ij)) = N (mA(i,j),SA(ij))
P(B(ij)) = N (mB(i,j),SB(ij))
P(C(ij)) = N (mC(i,j),SC(ij)) mA mB mC A1 A2 WA
Show connections for which
A(i,j) > Thresh
with probability > 90%

17. Inferred Neural NetworkWA A1
.92
(100%)
.38
(94%)
.47
(98%)
.37 (91%)
-.62 (99%)
-.51 (99%)
.37 (100%)
Intrinsic connections
are feed-forward
Neuronal saturation
with increasing
stimulus frequency
in A1 & WA
Time-dependent
change in A1-WA
connectivity

18. Summary Brain as a nonlinear dynamical system
Bilinear neural dynamics, hemodynamic model
Bayesian estimation and inference to detect changes in connectivity

19. Bilinear Dynamics: Positive transients
Stimuli u1
Set u2
a23=0.2 - + Z1 - + Z3 Z3 + +
a23 Z2 - -
a23=0.1