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The free-energy principle: a
rough guide to the brain?
Karl Friston
Presented by : Gokrna Poudel
Guiding question.
Q1: Explain the following terms: KL divergence, entropy, ergodic, free
energy, Bayesian surprise, generative model, recognition density,
sufficient statistics.
Q2: Explain the free-energy principle of the brain, i.e. the fact that selforganizing biological agents resist a tendency to disorder and therefore
minimize the entropy of their sensory states. Give various forms of free
energy.
Q3: How can action reduce free energy? How can perception reduce free
energy? How can active sampling of the sensorium contribute to the free
energy reduction?
Q4: Explain the neurobiological architecture for implementing the freeenergy principle in Figure 1 in Box 1. Describe each of the modules in
the figure and their functions as well as the quantities that define the free
energy.
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Guiding question.
Q5: Describe the sufficient statistics representing a hierarchical dynamic
model of the world in the brain in Figure 1 in Box 2. How are they related
with each other? How are the changes in synaptic activity, connectivity, and
gain involved with perceptual inference, learning and attention?
Q6: Formulate and describe the neuronal architecture for the hierarchical
dynamic model in Figure 1 in Box 3. How are the forward prediction errors
computed? How are the backward predictions made? What are the sources
of the forward and backward connections in terms of brain anatomy?
Q7: A key implementational issue is how the brain encodes the recognition
density. There are two forms of probabilistic neuronal codes: free forms and
fixed forms. Give examples of each form and explain them.
Q8: What kinds of optimization schemes does the brain use? Does it use
deterministic search on free energy to optimize action and perception? Or,
does it use stochastic search? What is your opinion?
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KL[Kullback-Leibler] divergence:


Information divergence, information gain, cross
or relative entropy is a non-commutative
measure of the difference between two
probability distributions.
In other words KL Divergence is a nonsymmetric measure of the difference between
two probability distributions .

p( x)
DKL ( P || Q)   p( x) log
dx

q( x)
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Ergodic

:
The process is ergodic if its long term time-average converges to
its ensemble average. Ergodic processes that evolve for a long
time forget their initial states.
Entropy:
 The average surprise of outcomes sampled from a probability
density. A density with low entropy means, on average, the
outcome is relatively predictable.
 The second law of thermo dynamics states that the entropy of
closed systems increases with time. Entropy is a measure of
disorder or, more simply, the number of ways the elements of a
system can be rearranged.
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Generative model :

It is a forward model and is a probabilistic mapping from causes to
observed consequences (data). It is usually specified in terms of the
likelihood of getting some data given their causes (parameters of a
model) and priors on the parameters
Recognition density :

It is an approximating conditional density is an approximate
probability distribution of the causes of data. It is the product
inference or inverting a generative model.
sufficient statistics :

quantities which are sufficient to parameterize a probability density
(e.g., mean and covariance of a Gaussian density).
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Bayesian theory

Bayesian probability theory, one of these “events” is the
hypothesis, H, and the other is data, D, and we wish to
judge the relative truth of the hypothesis given the
data. According to Bayes’ rule, we do this via the
relation
Bayesian surprise :
 A measure of salience based on the divergence
between the recognition and prior densities. It
measures the information in the data that can be
recognized.
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Free Energy .
 It is an attempt to explain the structure
and function of the brain, starting from
the fact that exist.
 Free-energy is an information theory
quantity that bounds the evidence for a
model of data.
 free-energy is greater than the negative
log-evidence or ‘surprise’ in sensory data,
given a model of how they were
generated.
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Action , perception and sensorium
contribution on free energy reduction

We are open systems in exchange with the environment; the
environment acts on us to produce sensory impressions, and
we act on the environment to change its states.

On changing the environment or our relationship to it, then
sensory input changes. Therefore, action can reduce freeenergy by changing the sensory input predicted.

perception reduces free-energy by changing predictions.

we sample the world to ensure our predictions become a selffulfilling prophecy and surprises are avoided. In this view,
perception is enslaved by action to provide veridical predictions
that guides active sampling of the sensorium.
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No. 4

Explain the neurobiological architecture for
implementing the free-energy principle in Figure
1 in Box 1. Describe each of the modules in the
figure and their functions as well as the
quantities that define the free energy.
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Neurobiological architecture for
implementing the free-energy principle
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Neurobiological architecture for
implementing the free-energy principle

Upper panel: schematic detailing the quantities
that define free-energy.

Lower panel: alternative expressions for the freeenergy that show what its minimization entails.
For action, free-energy can only be suppressed
by increasing the accuracy of sensory data (i.e.
selectively sampling data that are predicted by
the representation).
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No. 5

Describe the sufficient statistics representing a
hierarchical dynamic model of the world in the
brain in Figure 1 in Box 2. How are they related
with each other? How are the changes in
synaptic activity, connectivity, and gain involved
with perceptual inference, learning and
attention?
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Hierarchical dynamic of the brain
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Hierarchical dynamic of the brain
Key architecture is the hierarchy.
 The recognition density is encoded in terms of
its sufficient statistics.
 On the fig. three sorts of representations
pertaining to the states: {x,v}, parameters: θ and
precisions : λ of a hierarchical dynamic model,
these are encoded by neural activity, synaptic
connectivity and gain respectively. Crucially, the
optimization of any one representation depends
on the others.

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Hierarchical dynamic of the brain

The equations associated with this
partition
represent a gradient descent on free- energy and
correspond to
(i) Perceptual inference on states of the world
(i.e. optimizing synaptic activity);
(ii) Perceptual learning of the parameters
underlying causal regularities (i.e. optimizing
synaptic efficacy) and
(iii) Attention or optimizing the expected
precision of states in the face of random
fluctuations and uncertainty (i.e. optimizing
synaptic gain).
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No. 6

Formulate and describe the neuronal architecture
for the hierarchical dynamic model in Figure 1 in
Box 3. How are the forward prediction errors
computed? How are the backward predictions
made? What are the sources of the forward and
backward connections in terms of brain anatomy?
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Neuronal architecture for the hierarchical
dynamic model
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Neuronal architecture for the hierarchical
dynamic model
Schematic detailing the neuronal architectures
that might encode a density on the states of
hierarchical dynamic model.
 This shows the speculative cells of origin of
forward driving connections that convey
prediction error from a lower area to a higher
area and backward connections that construct
predictions .
 These predictions try to explain away prediction
error in lower levels. In this scheme, the sources
of forward and backward connections are
superficial and deep pyramidal cells, respectively.

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No.7

A key implementational issue is how the
brain encodes the recognition density.
There are two forms of probabilistic
neuronal codes: free forms and fixed
forms. Give examples of each form and
explain them.
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Brain encoding recognition density

The free-energy principle induces the
recognition density, which has to be represented
by its sufficient statistics. It is therefore a given
that the brain represents probability
distributions over sensory causes .
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Probabilistic neuronal codes
Free-form
and fixed-form:
o Free form :

particle filtering :
the recognition density is represented by the
sample density of neuronal ensembles, whose
activity encodes the location of particles in
state-space.
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Probabilistic neuronal codes
o probabilistic population code:
Method to represent stimuli by using the joint
activities of a number of neurons, each neuron has
a distribution of responses over some set of inputs,
the responses of many neurons may be combined
to determine some value about the inputs.
Probabilistic neuronal codes
o Fix
form :
 multinomial or Gaussian
Multinomial forms assume the world is
in one of several discrete states and are
usually associated with hidden Markov
models.
The Gaussian or Laplace assumption
allows for continuous and correlated states.
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No 8

What kinds of optimization schemes does
the brain use? Does it use deterministic
search on free energy to optimize action
and perception? Or, does it use stochastic
search? What is your opinion?
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Optimization schemes by the brain
According to the free-energy principle, the
sufficient statistics representing will change to
minimize free-energy,
 It provides principled explanation for perception,
memory and attention,
 it accounts for perceptual inference (optimization
of synaptic activity to encode the states of the
environment),
 perceptual learning and memory (optimization of
synaptic connections that encode contingencies
and causal regularities) and
 Attention (neuromodulatory optimization of
synaptic gain that encodes the precision of states)

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Optimization schemes by the brain
Assumption is that the brain uses a deterministic
gradient descent on free-energy to optimize
action and perception.
 It might also use stochastic searches; sampling the
sensorium randomly for a percept with low freeenergy.
 Evidence is our eye movements implement an
optimal stochastic strategy. This raises interesting
questions about the role of stochastic searches
from visual search to foraging, in both perception
and action

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Summary
It provides a comprehensive measure of how
individual represent and come to sample it
adaptively.
 It is the goal to minimize the prediction error
(suppress Free Energy)
 Changes in synaptic activity, connectivity and
gain can be understood as perceptual
inference, learning and attention.

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References
Friston. The free-energy principle: a unified
brain theory?. Nat Rev Neurosci (2010) vol. 11
(2) pp. 127-38
 Friston. The free-energy principle: a rough
guide to the brain?. Trends Cogn Sci (Regul
Ed) (2009) vol. 13 (7) pp. 293-301
 Friston etal. A free energy principle for the
brain, Journal of Physiology - Paris 100 (2006)
70–87

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Thank You
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