Probabilistic Inference and the Brain
SECTION 4: CRITICAL GAPS IN MODELS
JOINING THE DOTS IN THEORETICAL NEUROBIOLOGY
Karl Friston, University College London
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.
.
Does the brain use continuous or discrete
state space models?
Does the brain encode beliefs with ensemble
densities or sufficient statistics?
Does the brain use continuous or discrete
state-space models?
x 1
x

States and their
Sufficient statistics
x  {1, , N }
  [0,1]
x2
x3
x
q ( x |  )  p ( x | s , m)
Approximate Bayesian inference for
continuous states: Bayesian filtering
x

States and their
Sufficient statistics
   arg min F ( s,  )
F ( s,  )   ln p( s | m)
Free energy
q ( x |  )  p ( x | s , m)
Model evidence
“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
Hermann von Helmholtz
Richard Gregory
Impressions on the Markov blanket…
sS
Bayesian filtering and predictive coding
  (Q  )F ( s ,  )
 D      
prediction
update

  s  g ( )
prediction error
Making our own sensations
sensations – predictions
Prediction error
Action
Perception
Changing
sensations
Changing
predictions
Hierarchical generative models
what
A simple hierarchy
(3)
 v(3)

v
Ascending
prediction errors
the 
(2)
v
(2)
 x(2)

x
 x(2)
(2)
 v(2)

v
 v(1)
(1)
 x(1)

x
 x(1)
(1)
 v(1)

v
v(0)
where
Descending
predictions
  D      
Sensory
fluctuations
What does Bayesian filtering explain?
What does Bayesian filtering explain?
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
What does Bayesian filtering explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Specify a deep (hierarchical) generative model
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Simulate approximate (active) Bayesian inference by solving:
Biased competition
Visual illusions
Dissociative symptoms
Sensory attenuation
Sensorimotor integration
Optimal motor control
prediction
update
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
  D  F ( s ,  )
An example: Visual searches and saccades
Deep (hierarchical) generative model
 x, p
Parietal (where)
Frontal eye fields
v
xp
x
v
Visual cortex
v ,q
sq
x
Pulvinar salience map
 x ,q
Fusiform (what)
 (u )
xp
a
oculomotor reflex arc
v, p
sp
Superior colliculus
Saccadic eye movements
Saccadic fixation and salience maps
Action (EOG)
2
Hidden (oculomotor) states
0
-2
200
400
600
800
time (ms)
1000
1200
1400
Visual samples
Posterior belief
5
Conditional expectations
about hidden (visual) states
vs.
0
-5
200
And corresponding percept
400
600
800
time (ms)
1000
1200
1400
vs.
Is Bayesian filtering the only process theory for
approximate Bayesian inference in the brain?
x 1
What about state space models? x  {1, , N }
  [0,1]
x2
x3
x
q ( x |  )  p ( x | s , m)
A (Markovian) generative model
, 
P  o, s , u ,  | a , m   P  o | s  P  s | a  P  u |   P   | m 
P  o | s   P (o0 | s0 ) P (o1 | s1 )

P (ot | st )
P  ot | st   A
P  s | a   P ( st | st 1 , at )
P  st 1 | st , ut   B(ut )
P  u   |     (   F )
Likelihood
Control states
P ( s1 | s0 , a1 ) P ( s0 | m)
ut ,
C
, uT
Empirical priors – hidden states
B
– control states
F   EQ ( o , s | ) [ln P (o , s |  )  ln Q( s |  )]
Hidden states
st 1
st
st 1
P  o | m   C
P  s0 | m   D
P   | m   ( ,  )
A
Full priors
ot 1
ot
Generative models
Discrete states
Continuous states
v(3)
, 
 v(2)

 x(2)
 x(2)
Control states


(2)
v
ut ,
(1)
v
C
, uT
B
 x(1)
 x(1)
 v(1)
 v(0)
Hidden states

(i )
ot 1
ot
st 1
Variational Bayes
(belief updating)
  D     
(i )
st
A
Bayesian filtering
(predictive coding)
(i )
st 1
(i )
(i )
(i )
 ( i ) 0

st
st   (A  ot  Bt 1st 1  Bt  st 1 )
Variational updates
Perception
Action selection
Incentive salience
Functional anatomy
st   ( A  ot  Bt 1st 1  Bt  st 1 )
at
π   ( γ  F)
motor Cortex
β    π  ln π
   arg min F (o1 ,
st
, ot ,  )
occipital Cortex
striatum
prefrontal Cortex
F
ot
π
s
β
Performance
FE
KL
RL
success rate (%)
80
DA
60
40
20
0
0
0.2
0.4
0.6
0.8
Prior preference
Simulated behaviour
1
VTA/SN
hippocampus
Variational updating
Perception
Action selection
Incentive salience
Functional anatomy
st   ( A  ot  Bt 1st 1  Bt  st 1  ln st )
at
π   ( γ  F)
motor Cortex
β    π  ln π  β
  F (o1 ,
st
, ot ,  )
occipital Cortex
striatum
prefrontal Cortex
F
ot
π
s
β
30
25
20
15
10
VTA/SN
5
0.2
0.4
0.6
0.8
1
1.2
1.4
0.08
0.06
0.04
0.02
0
-0.02
0.25
0.5
0.75
1
1.25
1.5
0.2
0.15
0.1
0.05
0
-0.05
10
20
30
40
50
60
70
80
90
Simulated neuronal behaviour
hippocampus
What does believe updating explain?
Cross frequency coupling
Perceptual categorisation
Violation (P300) responses
Specify a deep (hierarchical) generative model
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Simulate approximate (active) Bayesian inference by solving:
Biased competition
Visual illusions
Dissociative symptoms
1
t
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
  F (o ,
, o , )
What does Bayesian filtering explain?
Local field potentials
Unit responses
0.14
0.12
0.1
8
Response
Cross frequency coupling (phase precession)
Perceptual categorisation
Violation (P300) responses
Oddball (MMN) responses
Omission responses
Attentional cueing (Posner paradigm)
Biased competition
 st 1
Visual illusions

Dissociative symptoms
  st 2
Sensory attenuation
s
Sensorimotor integration
 t 3
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
16
0.08
0.06
0.04
0.02
0
-0.02
24
0.25
0.5
time (seconds)
0.75
0.25
0.5
time (seconds)
0.75
What does Bayesian filtering explain?
Time-frequency (and phase) response
30
frequency
25
20
15
10
5
1
2
3
4
time (seconds)
5
6
Local field potentials
Response
0.4
0.2
0
-0.2
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
time (seconds)
Phasic dopamine responses
change in precision
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
0.15
0.1
0.05
0
50
100
150
200
time (updates)
250
300
350
What does Bayesian filtering explain?
Time-frequency (and phase) response
30
frequency
25
20
15
10
5
0.2
0.4
0.6
0.8
1
time (seconds)
1.2
1.4
Local field potentials
0.08
Response
0.06
0.04
0.02
0
-0.02
0.25
0.5
0.75
1
time (seconds)
1.25
1.5
Phasic dopamine responses
0.2
change in precision
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
0.15
0.1
0.05
0
-0.05
10
20
30
40
50
time (updates)
60
70
80
90
What does Bayesian filtering explain?
0.14
Difference waveform (MMN)
0.12
0.1
oddball
0.08
LFP
0.06
0.04
standard
0.02
0
-0.02
Local field potentials
-0.04
-0.06
50
Response
0.4
MMN
100
150
200
250
Time (ms)
0.3
0.2
0.1
0
-0.1
0.25
0.5
0.75
1
time (seconds)
1.25
1.5
Phasic dopamine responses
0.2
change in precision
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
0.15
0.1
0.05
0
10
20
30
40
50
time (updates)
60
70
80
90
300
350
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
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
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
Does the brain use continuous or discrete
state space models or both?
Does the brain encode beliefs with ensemble
densities or sufficient statistics?
Does the brain use continuous or discrete
state space models or both?
, 

Control states
ut ,
st 1
st
 v(2)
 x(2)
 x(2)
v(2)
 v(1)
 x(1)
 x(1)
 v(1)
 v(0)
C
, uT
B
Hidden states
v(3)
st 1
A
ot 1
ot
Does the brain use continuous or discrete
state space models or both?
Does the brain encode beliefs with ensemble
densities or sufficient statistics?
Ensemble code
Laplace code
or
u
u
 

u 

q( x | u )  N   1n i ui ,  
Ensemble density
1
n
 (u   )(u   ) 
T
i
i
i
 
q( x |  )  N   , ( ) 
Parametric density
Variational filtering with
ensembles
hidden states
Variational filtering
1.5
v (t )
F GH
G  ln p ( s , u )
H  ln q (u )
q
 U (u )
1
0.5
q
0
q
cause
-0.5
u  D  U ( s , u )  
u  (v, v, v, x, x, x, )
-1
5
10
15
20
25
30
cause
1.2
1
0.8
time
0.6
u
0.4
u
0.2
u 
0
x(t )
-0.2
-0.4
5
10
15
20
time {bins}
25
30
From ensemble coding to predictive coding (Bayesian filtering)
Taking the expectation of the ensemble dynamics, under the Laplace
assumption,we get:
u  D  U   
  D  F    D  F
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.
  D   0  F  0   u F  0
c.f., Hamilton's principle of stationary action.
predicted response and error
0.3
hidden states
1
predicted response and error
hidden states
1
0.2
0.5
0.5
states (a.u.)
states (a.u.)
0.2
0.1
0
0
-0.5
-0.1
-0.2
5
10
15
20
time {bins}
25
30
-1
0.1
0
0
-0.5
-0.1
5
10
15
20
time {bins}
25
-0.2
30
5
Deconvolution with variational filtering
(SDE) – free-form
0.8
0.6
0.6
0.2
states (a.u.)
states (a.u.)
0.8
u  D  U  
-0.2
-0.2
10
15
20
time {bins}
25
30
-1
5
10
15
20
time {bins}
25
  D  F
0.2
0
5
30
0.4
0
-0.4
25
causal states - level 2
1
0.4
15
20
time {bins}
Deconvolution with Bayesian filtering
(ODE) – fixed-form
causal states - level 2
1
10
-0.4
5
10
15
20
time {bins}
25
30
30
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?
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
u
Laplace code
 
u
u
u 

u

q( x | u )  N  
u 
1
n
 u , ( ) 
i
i
Parametric description of ensemble density
 
In other words, do we have:
An approximate description of
(nearly) exact Bayesian inference
A (nearly) exact description of
approximate Bayesian inference
or
Ensemble code
u
Laplace code
 
u
u
u 

u

q( x | u )  N  
u 
1
n
 u , ( ) 
i
i
Parametric description of ensemble density
 
Thank you
And thanks to collaborators:
And colleagues:
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
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