Structure of a Neuron - uni-goettingen.de€¦ · Structure of a Neuron: 2 Chemical synapse: Learning = Change of Synaptic Strength Neurotransmitter . Receptors . 3 . Machine Learning

1

At the dendrite the incoming signals arrive (incoming currents)

Molekules

Synapses

Neurons

Local Nets

Areas

Systems

CNS

At the soma current are finally integrated.

At the axon hillock action potential are generated if the potential crosses the membrane threshold

The axon transmits (transports) the action potential to distant sites

At the synapses are the outgoing signals transmitted onto the dendrites of the target neurons

Structure of a Neuron:

2

Chemical synapse: Learning = Change of Synaptic Strength

Neurotransmitter Receptors

3

M achine Learn ing C lass ica l C ondition ing Synaptic P las tic ity

D ynam ic Prog .(Be llm an Eq .)

R EIN FO R C EM EN T LEAR N IN G U N -SU PERVISED LEAR N IN Ge x a m p le b a s e d c o rre la tio n b a s e d

δ -R u le

M onte C arloC on tro l

Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner

N e u r.T D -M o d e ls(“C ritic ”)

N e u r.T D -fo rm a lism

D iffe ren tia lH ebb-R u le

(”fas t”)

STD P-M ode lsb io p h y s ic a l & n e tw o rk

EVALU ATIVE FEED BAC K (R ew ards )

N O N -EVALU ATIVE FEED BAC K (C orre la tions )

S A R S AC o rre la tio n

b a se d C o n tro l(non -eva lua t ive )

IS O -L e a rn in g

IS O -M o d e lo f S T D P

A cto r /C r iticte c h n ic a l & B a s a l G a n g l.

E lig ibility Tra ce s

H ebb-R u le


(”s low ”)

supe rv ised L .

A n tic ip a to ry C o n tro l o f A c tio n s a n d P re d ic tio n o f Va lu e s C o r re la tio n o f S ig n a ls

=

=

=

N eurona l R ew ard Sys tem s(Basa l G ang lia )

B iophys . o f Syn . P las tic ityD o p a m in e G lu ta m a te

STD P

LTP(LT D = a n ti)

IS O -C on tro l

Overview over different methods

4

Different Types/Classes of Learning

Unsupervised Learning (non-evaluative feedback) • Trial and Error Learning.

• No Error Signal.

• No influence from a Teacher, Correlation evaluation only.

Reinforcement Learning (evaluative feedback) • (Classic. & Instrumental) Conditioning, Reward-based Lng.

• “Good-Bad” Error Signals.

• Teacher defines what is good and what is bad.

Supervised Learning (evaluative error-signal feedback) • Teaching, Coaching, Imitation Learning, Lng. from examples and more.

• Rigorous Error Signals.

• Direct influence from a teacher/teaching signal.

5

Basic Hebb-Rule: = µ ui v µ << 1 dωi dt

For Learning: One input, one output.

An unsupervised learning rule:

A supervised learning rule (Delta Rule): !i! !i à ör!iE

No input, No output, one Error Function Derivative, where the error function compares input- with output- examples.

A reinforcement learning rule (TD-learning):

One input, one output, one reward.

wi! wi + ö[r(t+ 1) + í v(t+ 1)à v(t)]uà(t)

6

map

Self-organizing maps: unsupervised learning

Neighborhood relationships are usually preserved (+)

Absolute structure depends on initial condition and cannot be predicted (-)

input

7


For Learning: One input, one output





One input, one output, one reward


8

I. Pawlow

Classical Conditioning

9


For Learning: One input, one output





One input, one output, one reward


10

Supervised Learning: Example OCR

11

The influence of the type of learning on speed and autonomy of the learner

Correlation based learning: No teacher Reinforcement learning , indirect influence Reinforcement learning, direct influence Supervised Learning, Teacher Programming

Learning Speed Autonomy

12

Hebbian learning

A B

A

B

t

When an axon of cell A excites cell B and repeatedly or persistently takes part in firing it, some growth processes or metabolic change takes place in one or both cells so that A‘s efficiency ... is increased.

Donald Hebb (1949)

13




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l


14

Hebbian Learning

…Basic Hebb-Rule:

…correlates inputs with outputs by the…

= µ v u1 µ << 1 dω1

dt

v u1 ω1

Vector Notation Cell Activity: v = w . u

This is a dot product, where w is a weight vector and u the input vector. Strictly we need to assume that weight changes are slow, otherwise this turns into a differential eq.

15

= µ v u1 µ << 1 dω1

dt Single Input

= µ v u µ << 1 dw

dt Many Inputs As v is a single output, it is scalar.

Averaging Inputs = µ <v u> µ << 1

dw

dt We can just average over all input patterns and approximate the weight change by this. Remember, this assumes that weight changes are slow.

If we replace v with w . u we can write:

= µ Q . w where Q = <uu> is the input correlation matrix

dw

dt

Note: Hebb yields an instable (always growing) weight vector!

16

Synaptic plasticity evoked artificially Examples of Long term potentiation (LTP) and long term depression (LTD). LTP First demonstrated by Bliss and Lomo in 1973. Since then induced in many different ways, usually in slice. LTD, robustly shown by Dudek and Bear in 1992, in Hippocampal slice.

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18

19

20

LTP will lead to new synaptic contacts

21

Conventional LTP = Hebbian Learning

Symmetrical Weight-change curve

Pre

tPre

Post

tPost

Synaptic change %

Pre

tPre

Post

tPost

The temporal order of input and output does not play any role

22

23

Spike timing dependent plasticity - STDP

Markram et. al. 1997

24

Pre follows Post: Long-term Depression

Pre

tPre

Post

tPost

Synaptic

change %

Spike Timing Dependent Plasticity: Temporal Hebbian Learning

Weight-change curve (Bi&Poo, 2001)

Pre

tPre

Post

tPost

Pre precedes Post: Long-term

Potentiation

25

= µ v u1 µ << 1 dω1

dt Single Input

= µ v u µ << 1 dw

dt Many Inputs As v is a single output, it is scalar.

Averaging Inputs = µ <v u> µ << 1

dw

dt We can just average over all input patterns and approximate the weight change by this. Remember, this assumes that weight changes are slow.

If we replace v with w . u we can write:

= µ Q . w where Q = <uu> is the input correlation matrix

dw

dt

Note: Hebb yields an instable (always growing) weight vector!

Back to the Math. We had:

26

= µ (v - Θ) u µ << 1 dw

dt

Covariance Rule(s)

Normally firing rates are only positive and plain Hebb would yield only LTP. Hence we introduce a threshold to also get LTD

Output threshold

= µ v (u - Θ) µ << 1 dw

dt Input vector threshold

Many times one sets the threshold as the average activity of some reference time period (training period)

Θ = <v> or Θ = <u> together with v = w . u we get:

= µ C . w, where C is the covariance matrix of the input dw

dt http://en.wikipedia.org/wiki/Covariance_matrix

C = <(u-<u>)(u-<u>)> = <uu> - <u2> = <(u-<u>)u>

27

The covariance rule can produce LTP without (!) post-synaptic output. This is biologically unrealistic and the BCM rule (Bienenstock, Cooper, Munro) takes care of this.

BCM- Rule

= µ vu (v - Θ) µ << 1 dw

dt

As such this rule is again unstable, but BCM introduces a sliding threshold

= ν (v2 - Θ) ν < 1 dΘ

dt

Note the rate of threshold change ν should be faster than then weight changes (µ), but slower than the presentation of the individual input patterns. This way the weight growth will be over-dampened relative to the (weight – induced) activity increase.

28

Evidence for weight normalization: Reduced weight increase as soon as weights are already big (Bi and Poo, 1998, J. Neurosci.)

Problem: Hebbian Learning can lead to unlimited weight growth.

Solution: Weight normalization a) subtractive (subtract the mean change of all weights from each individual weight). b) multiplicative (mult. each weight by a gradually decreasing factor).

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Examples of Applications • Kohonen (1984). Speech recognition - a map of

phonemes in the Finish language • Goodhill (1993) proposed a model for the

development of retinotopy and ocular dominance, based on Kohonen Maps (SOM)

• Angeliol et al (1988) – travelling salesman problem (an optimization problem)

• Kohonen (1990) – learning vector quantization (pattern classification problem)

• Ritter & Kohonen (1989) – semantic maps

OD ORI

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Differential Hebbian Learning of Sequences Learning to act in response to sequences of sensor events

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δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l


You are here !

32

I. Pawlow

History of the Concept of Temporally Asymmetrical Learning: Classical Conditioning

33

34

I. Pawlow


Correlating two stimuli which are shifted with respect to each other in time. Pavlov’s Dog: “Bell comes earlier than Food” This requires to remember the stimuli in the system. Eligibility Trace: A synapse remains “eligible” for modification for some time after it was active (Hull 1938, then a still abstract concept).

35

Σ ω0 = 1

ω1

Unconditioned Stimulus (Food)

Conditioned Stimulus (Bell)

Response

Σ

X

∆ω1 + Stimulus Trace E

The first stimulus needs to be “remembered” in the system

Classical Conditioning: Eligibility Traces

36

I. Pawlow


Eligibility Traces

Note: There are vastly different time-scales for (Pavlov’s) hehavioural experiments:

Typically up to 4 seconds

as compared to STDP at neurons:

Typically 40-60 milliseconds (max.)

37

Defining the Trace In general there are many ways to do this, but usually one chooses a trace that looks biologically realistic and allows for some analytical calculations, too.

EPSP-like functions: α-function:

Double exp.:

This one is most easy to handle analytically and, thus, often used.

Dampened Sine wave:

Shows an oscillation.

h(t) =n

0 t<0hk(t) tõ0

h(t) = teàatk

h(t) = b1 sin(bt) eàat

k

h(t) = î1 (eàatà eàbt)

k

38




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l


Mathematical formulation of learning rules is

similar but time-scales are much different.

39

Σ

ω Early: “Bell”

Late: “Food”

x

)( )( )( tytutdtd

ii ′µ=ω

Differential Hebb Learning Rule

Xi

X0

Simpler Notation x = Input u = Traced Input

V

V’(t)

ui

u0

40

Convolution used to define the traced input, Correlation used to calculate weight growth.

)()()()()()()( xfxgxgxfduuxgufxh ==−= ∫∞

∞−

u

)()()()()()()( xgxfxfxgduxugufxh ∗=/∗=−= ∫∞

∞−

w

41

Produces asymmetric weight change curve (if the filters h produce unimodal „humps“)

)(' )( )( tvtutdtd

ii µω =

Derivative of the Output

Filtered Input

∑= )( )()( tuttv iiω

Output

∆ω

T

Differential Hebbian Learning

42

Conventional LTP

Symmetrical Weight-change curve

Pre

tPre

Post

tPost

Synaptic change %

Pre

tPre

Post

tPost

The temporal order of input and output does not play any role

43

Produces asymmetric weight change curve (if the filters h produce unimodal „humps“)

)(' )( )( tvtutdtd

ii µω =

Derivative of the Output

Filtered Input

∑= )( )()( tuttv iiω

Output

∆ω

T

Differential Hebbian Learning

44 Weight-change curve

(Bi&Poo, 2001)

T=tPost - tPre ms

Pre follows Post: Long-term Depression

Pre

tPre

Post

tPost

Synaptic change % Pre

tPre

Post

tPost

Pre precedes Post: Long-term

Potentiation

Spike-timing-dependent plasticity (STDP): Some vague shape similarity

45




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l


You are here !

46

Plastic Synapse

NMDA/AMPA

Postsynaptic: Source of Depolarization

The biophysical equivalent of Hebb’s postulate

Presynaptic Signal (Glu)

Pre-Post Correlation, but why is this needed?

47

inout

inout

Plasticity is mainly mediated by so called N-methyl-D-Aspartate (NMDA) channels. These channels respond to Glutamate as their transmitter and they are voltage depended:

48

Biophysical Model: Structure

x NMDA synapse

v

Hence NMDA-synapses (channels) do require a (hebbian) correlation between pre and post-synaptic activity!

Source of depolarization:

1) Any other drive (AMPA or NMDA)

2) Back-propagating spike

49

Local Events at the Synapse

ΣLocal

Current sources “under” the synapse: • Synaptic current

Isynaptic

ΣGlobal IBP

• Influence of a Back-propagating spike • Currents from all parts of the dendritic tree

IDendritic

u1

x1

v

51

Σ

ω

Pre-syn. Spike

BP- or D-Spike

* 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 2 4 6 8 10

V*h

gNMDA

0 40 80 t [ms]

g [nS]NMDA

0.1

On „Eligibility Traces“

Membrane potential:

Weight Synaptic input

Depolarization source

deprest

iii

ii IR

tVVVEttVdtdC +

−+−∆+= ∑ )())((g )()( ωω

ω1

ω0

X

v

v’

ISO-Learning

h

hΣ

x

x0

1

52

• Dendritic compartment

• Plastic synapse with NMDA channels Source of Ca2+ influx and coincidence detector

Plastic Synapse NMDA/AMPA

depi

ii IVEt~dtdV

+−∑ ))((g

NMDA/AMPA g BP spike

Source of Depolarization

Dendritic spike

• Source of depolarization: 1. Back-propagating spike 2. Local dendritic spike

Model structure

53

Plasticity Rule (Differential Hebb)

NMDA synapse -Plastic synapse

depi

ii IVEtdtdV

+−∑ ))((g ~

NMDA/AMPA g

NMDA/AMPA

Source of depolarization

Instantenous weight change:

)(' )( )( tFtctdtd

Nµ=ω

Presynaptic influence Glutamate effect on NMDA channels

Postsynaptic influence

54

0 40 80 t [ms]

g [nS]NMDA

0.1

Normalized NMDA conductance:

NMDA channels are instrumental for LTP and LTD induction (Malenka and Nicoll, 1999; Dudek and Bear ,1992)

V

tt

N eMgeec γ−+

τ−τ−

η+−

=][1 2

// 21

Pre-synaptic influence


depi

ii IVEtdtdV

+−∑ ))((g ~

NMDA/AMPA g

NMDA/AMPA


)(' )( )( tFtctdtd

Nµ=ω

55

0 10

0

-40-60

-20

20V [mV]

20 t [ms]

0 10

0

-40-60

-20

20V [mV]

20 t [ms]

0 10

0

-40-60

-20

20V [mV]

20 t [ms]

0 10

0

-40-60

-20

20V [mV]

20 t [ms]

Dendritic spikes

Back-propagating spikes

(Larkum et al., 2001

Golding et al, 2002

Häusser and Mel, 2003)

(Stuart et al., 1997)

Depolarizing potentials in the dendritic tree

56


depi

ii IVEtdtdV

+−∑ ))((g ~

NMDA/AMPA g

NMDA/AMPA


Postsyn. Influence

)(' )( )( tFtctdtd

Nµ=ω

For F we use a low-pass filtered („slow“) version of a back-propagating or a dendritic spike.

57

0 10

0

-40-60

-20

20V [mV]

20 t [ms]

0 10

0

-40-60

-20

20V [mV]

20 t [ms]

0 50 150 t [ms]100

0

-40

-60

-20

V [mV]

0 50 150 t [ms]100

0

-40

-60

-20

V [mV]

0 20 80 t [ms]40 60

0

-40

-60

-20

V [mV]

0 20 80 t [ms]40 60

0

-40

-60

-20

V [mV]

0 10

0

-40-60

-20

20V [mV]

20 t [ms]

0 10

0

-40-60

-20

20V [mV]

20 t [ms]

BP and D-Spikes

58

0 10

0

-40-60

-20

20V [mV]

20 t [ms]

0 10

0

-40-60

-20

20V [mV]

20 t [ms]

0-20 40 T [ms]-40 20

-0.01

-0.03

-0.01

0.01∆ω

0-20 40 T [ms]-40 20

-0.01

-0.03

-0.01

0.01∆ω

Back-propagating spike

Weight change curve

T

NMDAr activation

Back-propagating spike

T=tPost – tPre

Weight Change Curves Source of Depolarization: Back-Propagating Spikes

59

Plastic Synapse

NMDA/AMPA

Postsynaptic: Source of Depolarization

The biophysical equivalent of Hebb’s PRE-POST CORRELATION postulate:

THINGS TO REMEMBER

Presynaptic Signal (Glu)

Possible sources are: BP-Spike Dendritic Spike Local Depolarization

60

One word about

Supervised Learning

61




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l

Overview over different methods – Supervised Learning

And many more

62

Supervised learning methods are mostly non-neuronal and will therefore not

be discussed here.

63

So Far:

• Open Loop Learning

All slides so far !

64

CLOSED LOOP LEARNING

• Learning to Act (to produce appropriate behavior)

• Instrumental (Operant) Conditioning

All slides to come now !

65

Sensor 2

conditioned Input

Bell

Food

Salivation

Pavlov, 1927

Temporal Sequence

66

Adaptable Neuron

Env.

Closed loop

Sensing Behaving

67

Instrumental/Operant Conditioning

68




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l

Overview over different methods – Closed Loop Learning

69

Behaviorism “All we need to know in order

to describe and explain behavior is this: actions

followed by good outcomes are likely to recur, and

actions followed by bad outcomes are less likely to

recur.” (Skinner, 1953)

Skinner had invented the type of experiments called operant conditioning.

B.F. Skinner (1904-1990)

70

Operant behavior: occurs without an observable external stimulus. Operates on the organism’s environment. The behavior is instrumental in securing a stimulus more representative of everyday learning.

Skinner Box

71

OPERANT CONDITIONING TECHNIQUES

• POSITIVE REINFORCEMENT = increasing a behavior by administering a reward

• NEGATIVE REINFORCEMENT = increasing a behavior by removing an aversive stimulus when a behavior occurs

• PUNISHMENT = decreasing a behavior by administering an aversive stimulus following a behavior OR by removing a positive stimulus

• EXTINCTION = decreasing a behavior by not rewarding it

72




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l


You are here !

73

How to assure behavioral & learning convergence ??

This is achieved by starting with a stable reflex-like action and learning to supercede it by an anticipatory action.

Remove before being hit !

74

Controller ControlledSystem

ControlSignals

Feedback

DisturbancesSet-Point

X0

Reflex Only

(Compare to an electronic closed loop controller!)

This structure assures initial (behavioral) stability (“homeostasis”)

Think of a Thermostat !

75

Robot Application

Σ ω

x Early: “Vision”

Late: “Bump”

76

Robot Application

Initially built-in behavior: Retraction reaction whenever an obstacle is touched.

Learning Goal: Correlate the vision signals with the touch signals and navigate without collisions.

77

Robot Example

78

Controller ControlledSystem

ControlSignals

Feedback

DisturbancesSet-Point

X0X1early late

What has happened during learning to the system ?

The primary reflex re-action has effectively been eliminated and replaced by an anticipatory action

Reinforcement Learning (RL) Learning from rewards (and punishments) Learning to assess the value of states.

Learning goal directed behavior.

RL has been developed rather independently from two different fields:

1) Dynamic Programming and Machine Learning (Bellman Equation).

2) Psychology (Classical Conditioning) and later Neuroscience (Dopamine System in the brain)

I. Pawlow

Back to Classical Conditioning

U(C)S = Unconditioned Stimulus U(C)R = Unconditioned Response CS = Conditioned Stimulus CR = Conditioned Response

Less “classical” but also Conditioning ! (Example from a car advertisement)

Learning the association CS → U(C)R

Porsche → Good Feeling




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l

Overview over different methods – Reinforcement Learning

You are here !




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l


And later also here !

US = r,R = “Reward” CS = s,u = Stimulus = “State1” CR = v,V = (Strength of the) Expected Reward = “Value” UR = --- (not required in mathematical formalisms of RL) Weight = ω = weight used for calculating the value; e.g. v=ωu Action = a = “Action” Policy = π = “Policy”

1 Note: The notion of a “state” really only makes sense as soon as there is more than one state.

Notation

A note on “Value” and “Reward Expectation” If you are at a certain state then you would value this state according to how much reward you can expect when moving on from this state to the end-point of your trial. Hence: Value = Expected Reward ! More accurately: Value = Expected cumulative future discounted reward. (for this, see later!)

1) Rescorla-Wagner Rule: Allows for explaining several types of conditioning experiments.

2) TD-rule (TD-algorithm) allows measuring the value of states and allows accumulating rewards. Thereby it generalizes the Resc.-Wagner rule.

3) TD-algorithm can be extended to allow measuring the value of actions and thereby control behavior either by ways of a) Q or SARSA learning or with b) Actor-Critic Architectures

Types of Rules




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l


You are here !

Rescorla-Wagner Rule

Pavlovian: Extinction: Partial:

Train Result

u→r

u→r u→●

Pre-Train

u→r u→●

u→v=max

u→v=0

u→v<max

We define: v = ωu, with u=1 or u=0, binary and ω → ω + µδu with δ = r - v

This learning rule minimizes the avg. squared error between actual reward r and the prediction v, hence min<(r-v)2>

We realize that δ is the prediction error.

The associability between stimulus u and reward r is represented by the learning rate µ.

Extinction 10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

reward expected reward

prediction error

Pawlovian

Pawlovian

Extinction Partial

Stimulus u is paired with r=1 in 100% of the discrete “epochs” for Pawlovian and in 50% of the cases for Partial.

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Partial (50% reward)

Rescorla-Wagner Rule, Vector Form for Multiple Stimuli

We define: v = w.u, and w → w + µδu with δ = r – v Where we minimize δ.

Blocking:

Train Result

u1+u2→r

Pre-Train

u1→v=max, u2→v=0 u1→r

For Blocking: The association formed during pre-training leads to δ=0. As ω2 starts with zero the expected reward v=ω1u1+ω2u2 remains at r. This keeps δ=0 and the new association with u2 cannot be learned.


Inhibitory: Train Result Pre-Train

u1+u2→●, u1→r u1→v=max, u2→v<0

Inhibitory Conditioning: Presentation of one stimulus together with the reward and alternating presenting a pair of stimuli where the reward is missing. In this case the second stimulus actually predicts the ABSENCE of the reward (negative v). Trials in which the first stimulus is presented together with the reward lead to ω1>0. In trials where both stimuli are present the net prediction will be v=ω1u1+ω2u2 = 0. As u1,2=1 (or zero) and ω1>0, we get ω2<0 and, consequentially, v(u2)<0.


Overshadow: Train Result Pre-Train

u1+u2→r u1→v<max, u2→v<max

Overshadowing: Presenting always two stimuli together with the reward will lead to a “sharing” of the reward prediction between them. We get v= ω1u1+ω2u2 = r. Using different learning rates µ will lead to differently strong growth of ω1,2 and represents the often observed different saliency of the two stimuli.


Secondary:

Train Result Pre-Train

u1→r u2→u1 u2→v=max

Secondary Conditioning reflect the “replacement” of one stimulus by a new one for the prediction of a reward. As we have seen the Rescorla-Wagner Rule is very simple but still able to represent many of the basic findings of diverse conditioning experiments. Secondary conditioning, however, CANNOT be captured.

Predicting Future Reward

Animals can predict to some degree such sequences and form the correct associations. For this we need algorithms that keep track of time. Here we do this by ways of states that are subsequently visited and evaluated.

The Rescorla-Wagner Rule cannot deal with the sequentiallity of stimuli (required to deal with Secondary Conditioning). As a consequence it treats this case similar to Inhibitory Conditioning lead to negative ω2.

Prediction and Control

The goal of RL is two-fold: 1) To predict the value of states (exploring the state space

following a policy) – Prediction Problem. 2) Change the policy towards finding the optimal policy –

Control Problem.

• State, • Action, • Reward, • Value, • Policy

Terminology (again):

Markov Decision Problems (MDPs)

1 2 3 4 5 6 7 8

9 10 11 12

13 14

15 16

r1 r2a2 a15a14a1

s

te rm ina l sta tes

states

actions rewards

If the future of the system depends always only on the current state and action then the system is said to be “Markovian”.

What does an RL-agent do ? An RL-agent explores the state space trying to accumulate as much reward as possible. It follows a behavioral policy performing actions (which usually will lead the agent from one state to the next). For the Prediction Problem: It updates the value of each given state by assessing how much future (!) reward can be obtained when moving onwards from this state (State Space). It does not change the policy, rather it evaluates it. (Policy Evaluation).

For the Control Problem: It updates the value of each given action at a given state and of by assessing how much future reward can be obtained when performing this action at that state (State-Action Space, which is larger than the State Space). and all following actions at the following state moving onwards. Guess: Will we have to evaluate ALL states and actions onwards?

p(N) = 0.5p(S) = 0.125p(W) = 0.25p(E) = 0.125

Policy:

x x x x x

R R

0.0

value = 0.0everywherereward R=1

possible startlocations

0.9

0.9

0.8

0.1 0.1 0.1 0.1 0.1

etc

Policy Evaluationgive values of states

Exploration – Exploitation Dilemma: The agent wants to get as much cumulative reward (also often called return) as possible. For this it should always perform the most rewarding action “exploiting” its (learned) knowledge of the state space. This way it might however miss an action which leads (a bit further on) to a much more rewarding path. Hence the agent must also “explore” into unknown parts of the state space. The agent must, thus, balance its policy to include exploitation and exploration.

What does an RL-agent do ?

Policies 1) Greedy Policy: The agent always exploits and selects the

most rewarding action. This is sub-optimal as the agent never finds better new paths.

Policies 2) ε-Greedy Policy: With a small probability ε the agent

will choose a non-optimal action. *All non-optimal actions are chosen with equal probability.* This can take very long as it is not known how big ε should be. One can also “anneal” the system by gradually lowering ε to become more and more greedy.

3) Softmax Policy: ε-greedy can be problematic because of (*). Softmax ranks the actions according to their values and chooses roughly following the ranking using for example:

P

b=1

n

exp(T

Qb)

exp( TQa) where Qa is value of the currently

to be evaluated action a and T is a temperature parameter. For large T all actions have approx. equal probability to get selected.




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l


You are here !

Back to the question: To get the value of a given state, will we have to evaluate ALL states and actions onwards?

There is no unique answer to this! Different methods exist which assign the value of a state by using differently many (weighted) values of subsequent states. We will discuss a few but concentrate on the most commonly used TD-algorithm(s).

Temporal Difference (TD) Learning

Towards TD-learning – Pictorial View In the following slides we will treat “Policy evaluation”: We define some given policy and want to evaluate the state space. We are at the moment still not interested in evaluating actions or in improving policies.

Formalising RL: Policy Evaluation with goal to find the optimal value function of the state space We consider a sequence st, rt+1, st+1, rt+2, . . . , rT , sT . Note, rewards occur downstream (in the future) from a visited state. Thus, rt+1 is the next future reward which can be reached starting from state st. The complete return Rt to be expected in the future from state st is, thus, given by:

where γ≤1 is a discount factor. This accounts for the fact that rewards in the far future should be valued less. Reinforcement learning assumes that the value of a state V(s) is directly equivalent to the expected return Eπ at this state, where π denotes the (here unspecified) action policy to be followed.

Thus, the value of state st can be iteratively updated with:

We use α as a step-size parameter, which is not of great importance here, though, and can be held constant. Note, if V(st) correctly predicts the expected complete return Rt, the update will be zero and we have found the final value. This method is called constant-α Monte Carlo update. It requires to wait until a sequence has reached its terminal state (see some slides before!) before the update can commence. For long sequences this may be problematic. Thus, one should try to use an incremental procedure instead. We define a different update rule with:

The elegant trick is to assume that, if the process converges, the value of the next state V(st+1) should be an accurate estimate of the expected return downstream to this state (i.e., downstream to st+1). Thus, we would hope that the following holds:

Indeed, proofs exist that under certain boundary conditions this procedure, known as TD(0), converges to the optimal value function for all states.

This is why it is called TD (temp. diff.) Learning

| {z }




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l

Reinforcement Learning – Relations to Brain Function I

You are here !

Trace

ω

δ

1

X

x1

r

vv ’Σ

E

Σu1

How to implement TD in a Neuronal Way

Now we have:


We had defined: (first lecture!)

X 0

X 1

X n

v( t)

x

x

v ’

re w a rd

(n - i)τ

δ

How to implement TD in a Neuronal Way

v(t+1)-v(t)

Note: v(t+1)-v(t) is acausal (future!). Make it “causal” by using delays.

x

w = 10X 0

X 1

re w a rd

τ τδ

v (t)v (t- )τ

r

Serial-Compound representations X1,…Xn for defining an eligibility trace.




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l

Reinforcement Learning – Relations to Brain Function II

You are here !

TD-learning & Brain Function N o v e lty R e s p o n s e :n o p re d ic tio n ,re w a rd o c c u rs

n o C S r

A fte r le a rn in g :p re d ic te d re w a rd o c c u rs

C S r

DA-responses in the basal ganglia pars compacta of the substantia nigra and the medially adjoining ventral tegmental area (VTA).

This neuron is supposed to represent the δ-error of TD-learning, which has moved forward as expected.

A fte r le a rn in g :p re d ic te d re w a rd d o e s n o to c c u r

C S 1 .0 s

Omission of reward leads to inhibition as also predicted by the TD-rule.

TD-learning & Brain Function

1 .5 srTr

R e w a rdE x p e c ta tio n

This neuron is supposed to represent the reward expectation signal v. It has extended forward (almost) to the CS (here called Tr) as expected from the TD-rule. Such neurons are found in the striatum, orbitofrontal cortex and amygdala.

1 .0 s

R e w a rd E x p e c ta tio n(P o p u la tio n R e s p o n s e )

T r r

This is even better visible from the population response of 68 striatal neurons

Reinforcement Learning – The Control Problem So far we have concentrated on evaluating and unchanging policy. Now comes the question of how to actually improve a policy π trying to find the optimal policy.

We will discuss: 1) Actor-Critic Architectures But not: 2) SARSA Learning 3) Q-Learning

Abbreviation for policy: π




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l

Reinforcement Learning – Control Problem I

You are here !

Control Loops

C o n tro lle r C o n tro lle dS yste m

C o n tro lS ig n a ls

Fe e d b a ck

D istu rb a n ce sS e t-P o in t

X 0

A basic feedback–loop controller (Reflex) as in the slide before.

A cto r(C o n tro lle r)

E n viro n m e n t(C o n tro lle d S y s te m )

Fe e d b a ck

D istu rb a n ce s

C o n te xtC ritic

A ctio n s(C o n tro l S ig n a ls )

R e in fo rce m e n tS ig n a l

X 0

Control Loops

An Actor-Critic Architecture: The Critic produces evaluative, reinforcement feedback for the Actor by observing the consequences of its actions. The Critic takes the form of a TD-error which gives an indication if things have gone better or worse than expected with the preceding action. Thus, this TD-error can be used to evaluate the preceding action: If the error is positive the tendency to select this action should be strengthened or else, lessened.

ù(s; a) = Pb e

p(s;b)ep(s;a)

Example of an Actor-Critic Procedure

Action selection here follows the Gibb’s Softmax method:

where p(s,a) are the values of the modifiable (by the Critic!) policy parameters of the actor, indicating the tendency to select action a when being in state s.

p(st; at) p(st; at) + ìît

We can now modify p for a given state action pair at time t with:

where δt is the δ-error of the TD-Critic.




δ -R u le


Q -Learn ing

TD ( )o ften = 0

λλ

TD (1) TD (0 )

R escorla /W agner




(”fas t”)






IS O -L e a rn in g




H ebb-R u le


(”s low ”)

supe rv ised L .


=

=

=



STD P

LTP(LT D = a n ti)

IS O -C on tro l

Reinforcement Learning – Control I & Brain Function III

You are here !

C ortex (C ) Fron ta lC ortex

V P S N r G P i

D A -S ys tem(S N c ,V TA ,R R A )

Tha lam us

S tria tum (S )G P e

S TN

Actor-Critics and the Basal Ganglia

VP=ventral pallidum, SNr=substantia nigra pars reticulata, SNc=substantia nigra pars compacta, GPi=globus pallidus pars interna, GPe=globus pallidus pars externa, VTA=ventral tegmental area, RRA=retrorubral area, STN=subthalamic nucleus.

The basal ganglia are a brain structure involved in motor control. It has been suggested that they learn by ways of an Actor-Critic mechanism.

So called striosomal modules of the Striatum S fulfill the functions of the adaptive Critic. The prediction-error (δ) characteristics of the DA-neurons of the Critic are generated by: 1) Equating the reward r with excitatory input from the lateral hypothalamus. 2) Equating the term v(t) with indirect excitation at the DA-neurons which is initiated from striatal striosomes and channelled through the subthalamic nucleus onto the DA neurons. 3) Equating the term v(t−1) with direct, long-lasting inhibition from striatal striosomes onto the DA-neurons. There are many problems with this simplistic view though: timing, mismatch to anatomy, etc.

C

S

STN

D A r+

-Cortex=C, striatum=S, STN=subthalamic Nucleus, DA=dopamine system, r=reward.

Actor-Critics and the Basal Ganglia: The Critic

D AG lu

C o r tico -s tr ia ta l( ”p re ” )

N ig ro -s tr ia ta l( ”D A ”)

M e d iu m -s iz e d S p in y P ro je c tio nN e u ro n in th e S tria tu m (”p o s t”)

C DA

δ

v(t-1)

v(t)

LH

121

Literature (all of this is very mathematical!)

General Theoretical Neuroscience:

„Theoretical Neuroscience“, P.Dayan and L. Abbott, MIT Press (there used to be a version of this on the internet)

„Spiking Neuron Models“, W. Gerstner & W.M. Kistler, Cambridge University Press. (there is a version on the internet)

Neural Coding Issues: „Spikes“ F. Rieke, D. Warland, R. de Ruyter v. Steveninck, W. Bialek, MIT Press

Artificial Neural Networks: „Konnektionismus“, G. Dorffner, B.G. Teubner Verlg. Stuttgart

„Fundamentals of Artificial Neural Networks“, M.H. Hassoun, MIT Press

Hodgkin Huxley Model: See above „Spiking Neuron Models“, W. Gerstner & W.M. Kistler, Cambridge University Press.

Learning and Plasticity: See above „Spiking Neuron Models“, W. Gerstner & W.M. Kistler, Cambridge University Press.

Calculating with Neurons: Has been compiled from many different sources.

Maps: Has been compiled from many different sources.

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