Transcript Connectionist Models: Basics

Motor cortex
Somatosensory cortex
Sensory associative
cortex
Pars
opercularis
Visual associative
cortex
Broca’s
area
Visual
cortex
Primary
Auditory cortex
Wernicke’s
area
Connectionist Models
[Adapted from Neural Basis of Thought and Language
Jerome Feldman, Spring 2007, [email protected]
Neural networks abstract from
the details of real neurons
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Conductivity delays are neglected
An output signal is either discrete (e.g., 0 or
1) or it is a real-valued number (e.g., between
0 and 1)
Net input is calculated as the weighted sum of
the input signals
Net input is transformed into an output signal
via a simple function (e.g., a threshold
function)
The McCullough-Pitts Neuron
yj
wij
xi
f
yi
ti : target
xi = ∑j wij yj
yi = f(xi – qi)
Threshold
yj: output from unit j
Wij: weight on connection from j to i
xi: weighted sum of input to unit i
Mapping from neuron
Nervous System
Computational Abstraction
Neuron
Node
Dendrites
Input link and propagation
Cell Body
Axon
Combination function,
threshold, activation function
Output link
Spike rate
Output
Synaptic strength
Connection strength/weight
Simple Threshold Linear Unit
Simple Neuron Model
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A Simple Example
a = x1w1+x2w2+x3w3... +xnwn
.
a= 1*x1 + 0.5*x2 +0.1*x3
x1 =0, x2 = 1, x3 =0
Net(input) = f = 0.5
Threshold bias = 1
Net(input) – threshold bias< 0
Output = 0
Simple Neuron Model
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Simple Neuron Model
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Simple Neuron Model
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Simple Neuron Model
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Different Activation Functions
BIAS UNIT
With X0 = 1
Threshold Activation Function (step)
 Piecewise Linear Activation Function
 Sigmoid Activation Funtion
 Gaussian Activation Function
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 Radial
Basis Function
Types of Activation functions
The Sigmoid Function
y=a
x=neti
The Sigmoid Function
Output=1
y=a
Output=0
x=neti
The Sigmoid Function
Output=1
Sensitivity to input
y=a
Output=0
x=neti
Changing the exponent k
K >1
K<1
Radial Basis Function
f ( x)  e
 ax 2
Stochastic units
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Replace the binary threshold units by binary
stochastic units that make biased random
decisions.
 The
“temperature” controls the amount of
noise
p( si 1)

1 e
1
  s j wij
j
T
temperature
Types of Neuron parameters
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The form of the input function - e.g. linear,
sigma-pi (multiplicative), cubic.
The activation-output relation - linear, hardlimiter, or sigmoidal.
The nature of the signals used to communicate
between nodes - analog or boolean.
The dynamics of the node - deterministic or
stochastic.
Computing various functions
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McCollough-Pitts Neurons can compute
logical functions.
 AND,
NOT, OR
Computing other functions: the OR function
i1
i2
b=1
w01
w02
w0b
x0
f
y0
i1
i2
y0
0
0
0
0
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0
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• Assume a binary threshold activation function.
• What should you set w01, w02 and w0b to be so that
you can get the right answers for y0?
Many answers would work
y = f (w01i1 + w02i2 + w0bb)
i2
recall the threshold function
the separation happens when
w01i1 + w02i2 + w0bb = 0
i1
move things around and you get
i2 = - (w01/w02)i1 - (w0bb/w02)
Decision Hyperplane
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The two classes are therefore separated by the
`decision' line which is defined by putting the
activation equal to the threshold.
It turns out that it is possible to generalise this
result to TLUs with n inputs.
In 3-D the two classes are separated by a
decision-plane.
In n-D this becomes a decision-hyperplane.
Linearly separable patterns
Linearly Separable Patterns
PERCEPTRON is an architecture which can
solve this type of decision boundary problem.
An "on" response in the output node
represents one class, and an "off" response
represents the other.
The Perceptron
The Perceptron
Input Pattern
The Perceptron
Input Pattern
Output Classification
A Pattern Classification
Pattern Space
 The space in which the inputs reside is
referred to as the pattern space.
 Each pattern determines a point in the
space by using its component values as
space-coordinates.
 In general, for n-inputs, the pattern space
will be n-dimensional.
The XOR Function
X1/X2
X2 = 0
X2 = 1
X1= 0
0
1
X1 = 1
1
0
The Input Pattern Space
The Decision planes
From:
S. Harris Computer Cartoons
http://www.sciencecartoonsplus.com/galcomp2.htm
Multi-layer Feed-forward Network
Pattern Separation and NN
architecture
Triangle nodes and
McCullough-Pitts Neurons?
A
B
C
A
B
C
Representing concepts using
triangle
triangle nodes
nodes:
when two
of the
neurons
fire, the
third also
fires
Basic Ideas
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Parallel activation streams.
Top down and bottom up activation combine to
determine the best matching structure.
Triangle nodes bind features of objects to values
Mutual inhibition and competition between
structures
Mental connections are active neural
connections
Bottom-up vs. Top-down Processes
Bottom-up: When processing is driven by
the stimulus
 Top-down: When knowledge and context
are used to assist and drive processing
 Interaction: The stimulus is the basis of
processing but almost immediately topdown processes are initiated
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Stroop Effect
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Interference between form and meaning
Name the words
Book Car Table Box Trash Man Bed
Corn Sit Paper Coin Glass House Jar
Key Rug Cat Doll Letter Baby Tomato
Check Phone Soda Dish Lamp Woman
Name the print color of the words
Blue Green Red Yellow Orange Black Red
Purple Green Red Blue Yellow Black Red
Green White Blue Yellow Red Black Blue
White Red Yellow Green Black Purple
Connectionist Model
McClelland & Rumelhart (1981)
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Knowledge is distributed and processing occurs
in parallel, with both bottom-up and top-down
influences
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This model can explain the Word-Superiority
Effect because it can account for context effects
Connectionist Model of
Word Recognition
Do rhymes compete?
 Cohort (Marlsen-Wilson):
 onset
similarity is primary because of the
incremental nature of speech
 Cat activates cap, cast, cattle, camera, etc.
 NAM (Neighborhood Activation Model):
 global
similarity is primary
 Cat activates bat, rat, cot, cast, etc.
 TRACE (McClelland & Elman):
 global
similarity constrained by incremental
nature of speech
TRACE predictions
Do rhymes compete?
Sequence recognition for beaker
Sequence recognition for beaker - probability scale
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1
beaker
beetle
speaker
carriage
0.9
0.8
0.6
0.4
0.6
Matching level
Matching level
0.7
0.5
0.4
0.2
0
-0.2
0.3
-0.4
0.2
-0.6
0.1
-0.8
0
beaker
beetle
speaker
carriage
0.8
1
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4
Input letter
 Temporal
 global
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Input letter
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Sequence Learning in LTM
similarity constrained by incremental nature of speech
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A 2-step Lexical Model
Semantic Features
FOG
f
r
d
Onsets
k
DOG
m
CAT
ae
RAT
o
Vowels
MAT
t
g
Codas
Linking memory and tasks
From:
S. Harris Computer Cartoons
http://www.sciencecartoonsplus.com/galcomp2.htm
Distributed vs Local Representation
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John
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0
0
John
1
0
0
0
Paul
0
1
1
0
Paul
0
1
0
0
George
0
0
1
1
George
0
0
1
0
Ringo
1
0
0
1
Ringo
0
0
0
1
What happens if you want to
represent a group?
How many persons can you
represent with n bits? 2^n
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What happens if one neuron
dies?
How many persons can you
represent with n bits? n
Visual System
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1000 x 1000 visual map
For each location,
encode:
orientation
direction
…
of motion
speed
size
color
depth
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Blows up
combinatorically!
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Computational Model of Object Recognition
(Riesenhuber and Poggio, 1999)
invariance
invariance
Eye Movements: Beyond Feedforward Processing
1) Examine scene freely
2) estimate material
circumstances of family
3) give ages of the people
4) surmise what family has
been doing before arrival
of “unexpected visitor”
5) remember clothes worn by
the people
6) remember position of people
and objects
7) estimate how long the “unexpected
visitor” has been away from family
How does activity lead to structural
change?
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The brain (pre-natal, post-natal, and adult) exhibits a
surprising degree of activity dependent tuning and
plasticity.
To understand the nature and limits of the tuning and
plasticity mechanisms we study
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How activity is converted to structural changes (say the ocular
dominance column formation)
It is centrally important to understand these mechanisms
to arrive at biological accounts of perceptual, motor,
cognitive and language learning
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Biological Learning is concerned with this topic.