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Abstract
We suggested recently that attention can be understood as inferring the level of uncertainty or precision during hierarchical perception. In this talk, I will try to substantiate this claim using neuronal simulations of directed spatial attention and biased competition. These simulations assume that neuronal activity encodes a probabilistic representation of the world that optimises free-energy in a Bayesian fashion. Because free-energy bounds surprise or the (negative) log evidence for internal models of the world, this optimisation can be regarded as evidence accumulation or (generalised) predictive coding. Crucially, both predictions about the state of the world generating sensory data and the precision of those data have to be optimised. Here, we show that if the precision depends on the states, one can explain many aspects of attention. We illustrate this in the context of the Posner paradigm, using simulations to generate both psychophysical and electrophysiological responses. These simulated responses are consistent with attentional bias or gating, competition for attentional resources, attentional capture and associated speed-accuracy tradeoffs. Furthermore, if we present both attended and non-attended stimuli simultaneously, biased competition for neuronal representation emerges as a principled and straightforward property of Bayes-optimal perception.
8th Biannual Scientific Meeting on Attention “RECA VIII”
Attention, uncertainty and free-energy
Karl Friston
“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” - Hermann Ludwig Ferdinand von Helmholtz
Thomas Bayes
Geoffrey Hinton
Richard Feynman
From the Helmholtz machine to the Bayesian brain and self-organization
Hermann Haken
Richard Gregory
Overview
Ensemble dynamics Entropy and equilibriaFree-energy and surprise
The free-energy principle Perception and generative modelsHierarchies and predictive coding
Perception Birdsong and categorizationSimulated lesions
Attention Uncertainty and precisionModeling the Posner paradigmBehavioral and ERP simulations
tem
pera
ture
What is the difference between a snowflake and a bird?
Phase-boundary
…a bird can act (to avoid surprises)
What is the difference between snowfall and a flock of birds?
Ensemble dynamics, clumping and swarming
…birds (biological agents) stay in the same place
They resist the second law of thermodynamics, which says that their entropy should increase
This means biological agents must self-organize to minimise surprise. In other words, to ensure they occupy a limited number of states (cf homeostasis).
But what is the entropy?
A
( )s g
…entropy is just average surprise
Low surprise (we are usually here)
High surprise (I am never here)
0
( ) ( | ) ln ( | )
ln ( | )
H L
L
T
dt t p m p m d
p s m
But there is a small problem… agents cannot measure their surprise
But they can measure their free-energy, which is always bigger than surprise
This means agents should minimize their free-energy. So what is free-energy?
?
( ) ( )F Lt t
( )s g
What is free-energy?
…free-energy is basically prediction error
where small errors mean low surprise
sensations – predictions
= prediction error
Free-energy is a function of sensations and a proposal density over hidden causes
and can be evaluated, given a generative model (Gibbs Energy) or likelihood and prior:
So what models might the brain use?
( , ) ( ) (ln )F Gq qs Energy Entropy q E E
( , ) ln ( , | ) ln ( | , ) ln ( | )s p s m p s m p m G
Action
( )( ) ss g
argmin ( , )a
a s F
External states in the world
Internal states of the agent (m)
Sensations
argmin ( , )s
F( )( , )a f
More formally,
Backward(modulatory)
Forward(driving)
lateral
)1(~x )1(
s
)2((2)
(1)
)2(~x
)2(~v
)1(~v
( 1) ( ) ( , )
( ) ( ) ( , )D
i i v i
i i x i
v g
x f
{ ( ), ( ), , }x t v t
Hierarchal models in the brain
( , )x v ( )
( )Synaptic gain
Synaptic activity Synaptic efficacy
Activity-dependent plasticity
Functional specialization
Attentional gain
Enabling of plasticity
( ) ( )( )
G
Perception and inference Learning and memory
The proposal density and its sufficient statistics
( ) ( )( )
G
( ) ( )( )
( ) ( )( )
GD
GD
x xx
v vv
( | ) ( , ( ))q NLaplace approximation:
Attention and salience
Adjust hypotheses
sensory input
Backward connections return predictions
…by hierarchical message passing in the brain
prediction
Forward connections convey feedback
So how do prediction errors change predictions?
Prediction errors
Predictions
Backward predictions
Forward prediction error
Synaptic activity and message-passing
( , ) ( , ) ( ) ( ) ( , 1)
( , ) ( , ) ( ) ( )
D
D
v i v i i T i v iv
x i x i i T ix
( ) ( )12 ( ( ( )))T
i itr R ( )
i
Ti
Synaptic plasticity
( ,1)x
( ,1)x
( ,1)v
( ,2)v
( )s t
( ,1)v( ,2)x
( ,2)x
( ,2)v
( ,3)v
Synaptic gain
David Mumford
More formally,
cf Hebb's Law cf Rescorla-Wagnercf Predictive coding
( , ) ( , ) ( , ) ( , ) ( , 1) ( )
( , ) ( , ) ( , ) ( , ) ( , ) ( )
( )
( )
v i v i v i v i v i i
x i x i x i x i x i i
g
f
D
Summary
Biological agents resist the second law of thermodynamics
They must minimize their average surprise (entropy)
They minimize surprise by suppressing prediction error (free-energy)
Prediction error can be reduced by changing predictions (perception)
Prediction error can be reduced by changing sensations (action)
Perception entails recurrent message passing in the brain to optimise predictions
Predictions depend upon the precision of prediction errors
Overview
Ensemble dynamics Entropy and equilibriaFree-energy and surprise
The free-energy principle Perception and generative modelsHierarchies and predictive coding
Perception Birdsong and categorizationSimulated lesions
Attention Uncertainty and precisionModeling the Posner paradigmBehavioral and ERP simulations
Making bird songs with Lorenz attractors
SyrinxVocal centre
time (sec)
Freq
uenc
y
Sonogram
0.5 1 1.5causal states
hidden states
1
2
vv
v
(1) (1)2 1
(1) (1) (1) (1) (1) (1)1 1 3 1 2
(1) (1) (1) (1)1 2 2 3
18 18
2
2
x x
f v x x x x
x x v x
( )x
( )x
( )v( )s t
( )v
10 20 30 40 50 60-5
0
5
10
15
20prediction and error
10 20 30 40 50 60-5
0
5
10
15
20hidden states
Backward predictions
Forward prediction error
10 20 30 40 50 60-10
-5
0
5
10
15
20
causal states
Predictive coding and message passing
stimulus
0.2 0.4 0.6 0.82000
2500
3000
3500
4000
4500
5000
time (seconds)
Perceptual categorization
Freq
uenc
y (H
z) Song a
time (seconds)
Song b Song c
( )1v
( )2v
Hierarchical (itinerant) birdsong: sequences of sequences
SyrinxNeuronal hierarchy
Time (sec)
Freq
uenc
y (K
Hz)
sonogram
0.5 1 1.5
(1)1(1)2
v
v
(2) (2)2 1
(2) (2) (2) (2) (2)1 3 1 2
(2) (2) (2)81 2 33
18 18
32 2
2
x x
f x x x x
x x x
(1) (1)2 1
(1) (1) (1) (1) (1) (1)1 1 3 1 2
(1) (1) (1) (1)1 2 2 3
(1)1(1) 2
(1)23
18 18
2
2
x x
f v x x x x
x x v x
sxg
sx
(2) (1)(2) 2 1
(2) (1)3 2
x vg
x v
Freq
uenc
y (H
z)
percept
Freq
uenc
y (H
z)no top-down messages
time (seconds)
Freq
uenc
y (H
z)
no lateral messages
0.5 1 1.5
-40
-20
0
20
40
60
LFP
(micr
o-vo
lts)
LFP
-60
-40
-20
0
20
40
60
LFP
(micr
o-vo
lts)
LFP
0 500 1000 1500 2000-60
-40
-20
0
20
40
60
peristimulus time (ms)
LFP
(micr
o-vo
lts)
LFP
Simulated lesions and false inference
no structural priors
no dynamical priors
Overview
( )( )ig first order predictions
second order predictions( )( )i
Attention and precision
Perception Birdsong and categorizationSimulated lesions
Attention Uncertainty and precisionModeling the Posner paradigmBehavioral and ERP simulations
precision and prediction error
( , ) ( , ) ( , ) ( , 1) ( )( )( ( ))v i v i x i v i ig first order predictions (AMPA)
second order predictions (NMDA)
( )s t
( ,1)v( )( )ig
( )( )i
Backward predictions
( , 1)v i
( , )v i
( )( )ig
( )( )i
Forward prediction error
( ,1)x
( ,1)x
( ,1)v
( ,2)v
( , )v i
( ,2)x
( ,2)x
( ,2)v
( ,3)v
cue
target
stimuli
A generative model of precision and attention
exogenous endogenous decay
(1) ( ,1)
(1)(1) (1) (1) (1) (1)1 1 1 1
4 4 2 32(1)
(1)
(1) (1) ( ,2)
(1)
(1)( ,1)
(1)
( ,2)
1 1 1
1 1 1
exp( )~ (0, )
( ) 1~ (0, )
exp
N
N
Lv
C
R
LL R C
R
Lv
C
R
Lv
v
s
s s v
s
xx v v v x
x
v
v v
v
x
I
(1)( )Rx
stimuli
Predictive coding
-1.5
-1
-0.5
0
0.5
1
1.5
100 200 300 400 500 600time (ms)
Striate cortex
Extrastriate cortex
Rs
( ,1)vv
Parietal cortex
Rs
( ,1)vv
Cs
( ,1)vv
Ls
( ,1)vv
( ,1)vC
( , )v RC
hidden causes
hidden states
cue
target
hidden causes
( ,1)xR
( ,1)xR
( ,1)xL
( ,1)xL
( ,1)vL
( ,2)vL
( ,1)vR
( ,2)vR
( ,1)xR
( ,1)xR
( ,1)xL
( ,1)xL
-1.5
-1
-0.5
0
0.5
1prediction and error
-2
-1
0
1
2hidden states
-1.5
-1
-0.5
0
0.5
1
1.5hidden causes
Valid cue
100 200 300 400 500 600time (ms)
100 200 300 400 500 600time (ms)
100 200 300 400 500 600time (ms)
( ,1)x
( ,1)v
(1)( )g
stimuli
-1.5
-1
-0.5
0
0.5
1
1.5prediction and error
-2
-1
0
1
2hidden states
-1.5
-1
-0.5
0
0.5
1
1.5hidden causes
100 200 300 400 500 600-2
-1
0
1
2
time (ms)
Inference with valid and invalid cues
Invalid cue
100 200 300 400 500 600time (ms)
100 200 300 400 500 600time (ms)
100 200 300 400 500 600time (ms)
validity costs and benefits
250
300
350
400
Reac
tion
time
(ms)
validinvalid neutral
Reaction times and conditional confidence
100 200 300 400 500 600
time (ms)
Valid and invalid cues
Empirical timing effects
Invalid
Neutral
Valid
Simulated timing effects
Invalid
Neutral
Valid
Posner et al, (1978)
Behavioural simulations
100 200 300 400 500 600time (ms)
Foreperiod
prediction errors (sensory states)
prediction errors (hidden states)
Mangun and Hillyard (1991)
ValidInvalid
0 200 400 600
-2 V
+Peristimulus time (ms)
P1
P3
N1
-100 0 100 200 300-2
-1
0
1
2
3
-0.01
-0.005
0
0.005
0.01
-200
-100 0 100 200 300
Peristimulus time (ms)-200
Peristimulus time (ms)-100 0 100 200 300
-2
-1
0
1
2
3
-0.01
-0.005
0
0.005
0.01
-200
-100 0 100 200 300
Peristimulus time (ms)-200
( ,1)vR
( ,1)xR
Peristimulus time (ms)
( ,1)vR
( ,1)vR
( ,1)xR
( ,1)xR
Electrophysiological simulations
Thank you
And thanks to collaborators:
Rick AdamsJean DaunizeauHarriet Feldman
Lee HarrisonStefan KiebelJames Kilner
Jérémie MattoutKlaas Stephan
And colleagues:
Peter DayanJörn DiedrichsenPaul Verschure
Florentin Wörgötter
And many others