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ABSTRACT: My treatment of critical gaps in models of probabilistic inference will focus on the potential of unified
theories to “close the gaps” between probabilistic models of perception and evidence accumulation - and how these
models can be understood in terms of embodied inference and action. Formally speaking, models of motor control
and choice behaviour can be cast in terms of (active) probabilistic inference; however, there are two key outstanding
issues. Both pertain to the implementation of active inference in the brain. The first speaks to the distinction between
discrete and continuous state-space models – and the requisite message passing schemes (e.g., variational Bayes
versus predictive coding). The second key distinction is between mean field descriptions in terms of population
dynamics (i.e., sufficient statistics) and their microscopic implementation in terms of spiking neurons (i.e., sampling
approaches to probabilistic inference). These are potentially important issues that constrain the interpretation of
empirical data – and how these data can be used to adjudicate among different models of the Bayesian brain.
.
Probabilistic Inference and the Brain
SECTION 4: CRITICAL GAPS IN MODELS
JOINING THE DOTS IN THEORETICAL NEUROBIOLOGY
Karl Friston, University College London
Page 2
Does the brain use continuous or discrete
state space models?
Does the brain encode beliefs with ensemble
densities or sufficient statistics?
Page 3
Does the brain use continuous or discrete
state-space models?
( | ) ( | , )q x p x s m
{1, , }
[ 0 , 1 ]
x N
x
States and their
Sufficient statistics
1x
2x
3x
x
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Approximate Bayesian inference for
continuous states: Bayesian filtering
a r g m i n ( , )
( , ) l n ( | )
F s
F s p s m
x
States and their
Sufficient statistics
( | ) ( | , )q x p x s m
Free energy Model evidence
Page 5
“Objects are always imagined as being present in the field of
vision as would have to be there in order to produce the same
impression on the nervous mechanism” - von Helmholtz
Richard Gregory Hermann von Helmholtz
Impressions on the Markov blanket…
s S
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Bayesian filtering and predictive coding
( ) ( , )Q F s
D
prediction update
( )s g
prediction error
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Making our own sensations
Changing
sensations
sensations – predictions
Prediction error
Changing
predictions
Action Perception
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the
(1 )
x
(1 )
v
( 2 )
x
( 2 )
v
( 2 )
x
( 3 )
v
( 2 )
v
(1 )
x
(1 )
v
( 0 )
v
Descending
predictions
Ascending
prediction errors
A simple hierarchy what where
Sensory
fluctuations
Hierarchical generative models
(1 )
x
(1 )
v
( 2 )
x
( 2 )
v
( 2 )
x
( 3 )
v
( 2 )
v
( 1 )
x
( 1 )
v
( 0 )
v
D
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What does Bayesian filtering explain?
Page 10
What does Bayesian filtering explain?
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What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 12
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 13
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 14
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 15
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 16
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 17
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 18
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 19
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 20
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 21
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 22
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 23
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 24
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 25
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 26
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 27
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 28
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 29
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 30
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 31
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Specify a deep (hierarchical) generative model
Simulate approximate (active) Bayesian inference by solving:
( , )D F s
prediction update
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An example: Visual searches and saccades
Page 33
,x p Frontal eye fields
ps
,v p
qs
x
v
v
,x q
x
,v q
px
Pulvinar salience map Fusiform (what)
Superior colliculus
Visual cortex
oculomotor reflex arc
( )u
px
Parietal (where)
a
Deep (hierarchical) generative model
Page 34
200 400 600 800 1000 1200 1400-2
0
2
Action (EOG)
time (ms)
200 400 600 800 1000 1200 1400
-5
0
5
Posterior belief
time (ms)
Saccadic fixation and salience maps
Visual samples
Conditional expectations
about hidden (visual) states
And corresponding percept
Saccadic eye movements
Hidden (oculomotor) states
vs. vs.
Page 35
Is Bayesian filtering the only process theory for
approximate Bayesian inference in the brain?
What about state space models?
( | ) ( | , )q x p x s m
{1, , }
[ 0 , 1 ]
x N
1x
2x
3x
x
Page 36
Full priors
– control states
Empirical priors – hidden states
Likelihood
A (Markovian) generative model
Hidden states ts
to
, ,t T
u u
Control states
C
,
B
A
1ts
1ts
1to
0 0 1 1
1 1 0 1 0
1
( , | )
0
, , , | , | | | |
| ( | ) ( | ) ( | )
|
| ( | , ) ( | , ) ( | )
| , ( )
| ( )
[ l n ( , | ) l n ( | ) ]
|
t t
t t
t t t
t t t t
Q o s
P o s u a m P o s P s a P u P m
P o s P o s P o s P o s
P o s
P s a P s s a P s s a P s m
P s s u u
P u
E P o s Q s
P o m
P s
A
B
F
F
C
|
| ( , )
m
P m
D
Page 37
Generative models
Hidden states ts
to
, ,t T
u u
Control states
C
,
B
A
1ts
1ts
1to
(1 )
x
(1 )
v
( 2 )
x
( 2 )
v
( 2 )
x
( 3 )
v
( 2 )
v
(1 )
x
(1 )
v
( 0 )
v
( ) ( ) ( ) ( ) ( )i i i i i
D ( )i
1 1 1( )
t t t t t to
s A B s B s
ts
Continuous states Discrete states
Bayesian filtering
(predictive coding)
Variational Bayes
(belief updating)
( )0
i
Page 38
Variational updates
Perception
Action selection
Incentive salience
Functional anatomy
0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
Performance
Prior preference
succ
ess
rate
(%
)
FE
KL
RL
DA
Simulated behaviour
1 1 1( )
( )
l n
t t t t t to
s A B s B s
π γ F
β π π
1a r g m in ( , , , )
tF o o
β
ta
VTA/SN
motor Cortex
occipital Cortex striatum
to
π
ts
s
prefrontal Cortex
F
hippocampus
Page 39
Variational updating
Perception
Action selection
Incentive salience
Functional anatomy
Simulated neuronal behaviour
1 1 1( l n )
( )
l n
t t t t t t to
s A B s B s s
π γ F
β π π β
1( , , , )
tF o o
0.2 0.4 0.6 0.8 1 1.2 1.4
5
10
15
20
25
30
0.25 0.5 0.75 1 1.25 1.5
-0.02
0
0.02
0.04
0.06
0.08
10 20 30 40 50 60 70 80 90
-0.05
0
0.05
0.1
0.15
0.2
β
ta
VTA/SN
motor Cortex
occipital Cortex striatum
to
π
ts
s
prefrontal Cortex
F
hippocampus
Page 40
What does believe updating explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Specify a deep (hierarchical) generative model
Simulate approximate (active) Bayesian inference by solving:
1( , , , )
tF o o
Page 41
What does Bayesian filtering explain?
Unit responses
time (seconds) 0.25 0.5 0.75
8
16
24
0.25 0.5 0.75
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Local field potentials
time (seconds)
Res
pons
e
Cross frequency coupling (phase precession)
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
1
2
3
t
t
t
s
s
s
Page 42
What does Bayesian filtering explain?
Time-frequency (and phase) response
time (seconds)
freq
uenc
y
1 2 3 4 5 6
5
10
15
20
25
30
0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5 5.75 6
-0.2
0
0.2
0.4
Local field potentials
time (seconds)
Res
pons
e
50 100 150 200 250 300 350
0
0.05
0.1
0.15
Phasic dopamine responses
time (updates)
chan
ge in
pre
cisi
on
Cross frequency coupling (phase synchronisation)
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 43
What does Bayesian filtering explain?
Time-frequency (and phase) response
time (seconds)
freq
uenc
y
0.2 0.4 0.6 0.8 1 1.2 1.4
5
10
15
20
25
30
0.25 0.5 0.75 1 1.25 1.5
-0.02
0
0.02
0.04
0.06
0.08
Local field potentials
time (seconds)
Res
pons
e
10 20 30 40 50 60 70 80 90
-0.05
0
0.05
0.1
0.15
0.2
Phasic dopamine responses
time (updates)
chan
ge in
pre
cisi
on
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 44
What does Bayesian filtering explain?
0.25 0.5 0.75 1 1.25 1.5
-0.1
0
0.1
0.2
0.3
0.4
Local field potentials
time (seconds)
Res
pons
e
10 20 30 40 50 60 70 80 90
0
0.05
0.1
0.15
0.2
Phasic dopamine responses
time (updates)
chan
ge in
pre
cisi
on
50 100 150 200 250 300 350 -0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Time (ms)
LFP
Difference waveform (MMN)
oddball
standard
MMN
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 45
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance (transfer of responses)
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 46
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (reversal learning)
Interoceptive inference
Communication and hermeneutics
Page 47
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
Motor gain control
Oculomotor control and smooth pursuit
Visual searches and saccades
Action observation and mirror neuron responses
Dopamine and affordance
Action sequences (devaluation)
Interoceptive inference
Communication and hermeneutics
Page 48
Does the brain use continuous or discrete
state space models or both?
Does the brain encode beliefs with ensemble
densities or sufficient statistics?
Page 49
Does the brain use continuous or discrete
state space models or both?
Hidden states ts
to
, ,t T
u u
Control states
C
,
B
A
1ts
1ts
1to
(1 )
x
(1 )
v
( 2 )
x
( 2 )
v
( 2 )
x
( 3 )
v
( 2 )
v
(1 )
x
(1 )
v
( 0 )
v
Page 50
Does the brain use continuous or discrete
state space models or both?
Does the brain encode beliefs with ensemble
densities or sufficient statistics?
u
u
u
Ensemble code
1 1
( | ) , ( ) ( )T
i i in ni iq x u N u u u
Laplace code
( | ) , ( )q x N
Ensemble density Parametric density
or
Page 51
Variational filtering with
ensembles
Page 52
( , )
( , , , , , , )
u D U s u
u v v v x x x
( )x t
( )v t
5 10 15 20 25 30 -1
-0.5
0
0.5
1
1.5
hidden states
5 10 15 20 25 30 -0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
cause
time {bins}
time
cause
Variational filtering
l n ( , ) ( )
ln ( )
q q
q
F G H
G p s u U u
H q u
u
u
u
Page 53
Taking the expectation of the ensemble dynamics, under the Laplace
assumption,we get:
This can be regarded as a gradient ascent in a frame of reference that
moves along the trajectory encoded in generalised coordinates. The
stationary solution, in this moving frame of reference, maximises
variational action by the Fundamental lemma.
c.f., Hamilton's principle of stationary action.
u D U
D F D F
0 0 0u
D F F
From ensemble coding to predictive coding (Bayesian filtering)
Page 54
5 10 15 20 25 30-0.2
-0.1
0
0.1
0.2
0.3predicted response and error
time {bins}
sta
tes
(a.u
.)
5 10 15 20 25 30-1
-0.5
0
0.5
1hidden states
time {bins}
5 10 15 20 25 30-0.4
-0.2
0
0.2
0.4
0.6
0.8
1causal states - level 2
time {bins}
sta
tes
(a.u
.)
Deconvolution with variational filtering
(SDE) – free-form
5 10 15 20 25 30-0.2
-0.1
0
0.1
0.2
predicted response and error
time {bins}
sta
tes
(a.u
.)
5 10 15 20 25 30-1
-0.5
0
0.5
1hidden states
time {bins}
5 10 15 20 25 30-0.4
-0.2
0
0.2
0.4
0.6
0.8
1causal states - level 2
time {bins}
sta
tes
(a.u
.)
Deconvolution with Bayesian filtering
(ODE) – fixed-form
u D U D F
Page 55
Does the brain use continuous or discrete
state space models or both?
Does the brain encode beliefs with ensemble
densities or sufficient statistics or both?
Page 56
u
u
u
Does the brain use continuous or discrete
state space models or both?
Does the brain encode beliefs with ensemble
densities or sufficient statistics or both?
Ensemble code
1
( | ) , ( )in i
q x u N u
Laplace code
Parametric description of ensemble density
u
u
u
Page 57
u
u
u
Ensemble code
1
( | ) , ( )in i
q x u N u
Laplace code
Parametric description of ensemble density
u
u
u
An approximate description of
(nearly) exact Bayesian inference
A (nearly) exact description of
approximate Bayesian inference or
In other words, do we have:
Page 58
And thanks to collaborators:
Rick Adams
Ryszard Auksztulewicz
Andre Bastos
Sven Bestmann
Harriet Brown
Jean Daunizeau
Mark Edwards
Chris Frith
Thomas FitzGerald
Xiaosi Gu
Stefan Kiebel
James Kilner
Christoph Mathys
Jérémie Mattout
Rosalyn Moran
Dimitri Ognibene
Sasha Ondobaka
Will Penny
Giovanni Pezzulo
Lisa Quattrocki Knight
Francesco Rigoli
Klaas Stephan
Philipp Schwartenbeck
And colleagues:
Micah Allen
Felix Blankenburg
Andy Clark
Peter Dayan
Ray Dolan
Allan Hobson
Paul Fletcher
Pascal Fries
Geoffrey Hinton
James Hopkins
Jakob Hohwy
Mateus Joffily
Henry Kennedy
Simon McGregor
Read Montague
Tobias Nolte
Anil Seth
Mark Solms
Paul Verschure
And many others
Thank you