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Phantom Limb Phenomena

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Phantom Limb Phenomena Summary Both genetic factors and activity dependent factors play a role in developing the brain architecture and circuitry. – PowerPoint PPT presentation

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Title: Phantom Limb Phenomena


1
Phantom Limb Phenomena
2
Hand movement observation by individuals born
without hands phantom limb experience constrains
visual limb perception.Funk M, Shiffrar M,
Brugger P.
We investigated the visual experiences of two
persons born without arms, one with and the other
without phantom sensations. Normally-limbed
observers perceived rate-dependent paths of
apparent human movement . The individual with
phantom experiences showed the same perceptual
pattern as control participants, the other did
not. Neural systems matching action observation,
action execution and motor imagery are likely
contribute to the definition of body schema in
profound ways.
3
Summary
  • Both genetic factors and activity dependent
    factors play a role in developing the brain
    architecture and circuitry.
  • There are critical developmental periods where
    nurture is essential, but there is also a great
    ability for the adult brain to regenerate.
  • Next lecture What computational models satisfy
    some of the biological constraints.
  • Question What is the relevance of neural
    development and learning in language and thought?

4
Connectionist Models Basics
  • Jerome Feldman
  • CS182/CogSci110/Ling109
  • Spring 2007

5
(No Transcript)
6
Realistic Biophysical Neuron SimulationsNot
covered in any UCB class?Genesis and Neuron
systems
7
Neural networks abstract from the details of real
neurons
  • 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)

8
The McCullough-Pitts Neuron
Threshold
  • yj output from unit j
  • Wij weight on connection from j to i
  • xi weighted sum of input to unit i

9
Mapping from neuron
Nervous System Computational Abstraction
Neuron Node
Dendrites Input link and propagation
Cell Body Combination function, threshold, activation function
Axon Output link
Spike rate Output
Synaptic strength Connection strength/weight
10
Simple Threshold Linear Unit
11
Simple Neuron Model
1
12
A Simple Example
  • a x1w1x2w2x3w3... xnwn
  • a 1x1 0.5x2 0.1x3
  • x1 0, x2 1, x3 0
  • Net(input) f 0.5
  • Threshold bias 1
  • Net(input) threshold biaslt 0
  • Output 0

.
13
Simple Neuron Model
1
1
1
1
14
Simple Neuron Model
1
1
1
1
1
15
Simple Neuron Model
0
1
1
1
16
Simple Neuron Model
0
1
0
1
1
17
Different Activation Functions
BIAS UNIT With X0 1
  • Threshold Activation Function (step)
  • Piecewise Linear Activation Function
  • Sigmoid Activation Funtion
  • Gaussian Activation Function
  • Radial Basis Function

18
Types of Activation functions
19
The Sigmoid Function
ya
xneti
20
The Sigmoid Function
Output1
ya
Output0
xneti
21
The Sigmoid Function
Output1
Sensitivity to input
ya
Output0
xneti
22
Changing the exponent k(neti)
K gt1
K lt 1
23
Nice Property of Sigmoids
24
Radial Basis Function
25
Stochastic units
  • Replace the binary threshold units by binary
    stochastic units that make biased random
    decisions.
  • The temperature controls the amount of noise

temperature
26
Types of Neuron parameters
  • The form of the input function - e.g. linear,
    sigma-pi (multiplicative), cubic.
  • The activation-output relation - linear,
    hard-limiter, or sigmoidal.
  • The nature of the signals used to communicate
    between nodes - analog or boolean.
  • The dynamics of the node - deterministic or
    stochastic.

27
Computing various functions
  • McCollough-Pitts Neurons can compute logical
    functions.
  • AND, NOT, OR

28
Computing other functions the OR function
i1 i2 y0
0 0 0
0 1 1
1 0 1
1 1 1
  • 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?

29
Many answers would work
  • y f (w01i1 w02i2 w0bb)
  • recall the threshold function
  • the separation happens when w01i1 w02i2 w0bb
    0
  • move things around and you get
  • i2 - (w01/w02)i1 - (w0bb/w02)

30
Decision Hyperplane
  • 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.

31
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.
Linearly Separable Patterns
32
The Perceptron
33
The Perceptron
Input Pattern
34
The Perceptron
Input Pattern
Output Classification
35
A Pattern Classification
36
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.
  • Clearly, for nD, the pattern space cannot be
    drawn or represented in physical space. This is
    not a problem we shall return to the idea of
    using higher dimensional spaces later. However,
    the geometric insight obtained in 2-D will carry
    over (when expressed algebraically) into n-D.

37
The XOR Function
X1/X2 X2 0 X2 1
X1 0 0 1
X1 1 1 0
38
The Input Pattern Space
 
39
The Decision planes
 
40
Multi-layer Feed-forward Network
41
Pattern Separation and NN architecture
42
Conjunctive or Sigma-Pi nodes
  • The previous spatial summation function supposes
    that each input contributes to the activation
    independently of the others. The contribution to
    the activation from input 1 say, is always a
    constant multiplier ( w1) times x1.
  • Suppose however, that the contribution from input
    1 depends also on input 2 and that, the larger
    input 2, the larger is input 1's contribution.
  • The simplest way of modeling this is to include a
    term in the activation like w12(x1x2) where
    w12gt0 (for a inhibiting influence of input 2 we
    would, of course, have w12lt0 ).
  • w1x1 w2x2 w3x3 w12(x1x2) w23(x2x3)
    w13(x1x3)

43
Sigma-Pi units
44
Sigma-Pi Unit
45
Biological Evidence for Sigma-Pi Units
  • axo-dendritic synapse The stereotypical synapse
    consists of an electro-chemical connection
    between an axon and a dendrite - hence it is an
    axo-dendritic synapse
  • presynaptic inhibition However there is a large
    variety of synaptic types and connection
    grouping. Of special importance are cases where
    the efficacy of the axo-dendritic synapse between
    axon 1 and the dendrite is modulated (inhibited)
    by the activity in axon 2 via the axo-axonic
    synapse between the two axons. This might
    therefore be modelled by a quadratic like
    w12(x1x2)
  • synapse cluster Here the effect of the
    individual synapses will surely not be
    independent and we should look to model this with
    a multilinear term in all the inputs.

46
Biological Evidence for Sigma-Pi units
presynaptic inhibition
axo-dendritic synapse
synapse cluster
47
Link to Vision The Necker Cube
48
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49
Constrained Best Fit in Nature
inanimate animate
physics lowest energy state
chemistry molecular minima
biology fitness, MEU Neuroeconomics
vision threats, friends
language errors, NTL
50
Computing other relations
  • The 2/3 node is a useful function that activates
    its outputs (3) if any (2) of its 3 inputs are
    active
  • Such a node is also called a triangle node and
    will be useful for lots of representations.

51
Triangle nodes and McCullough-Pitts Neurons?
A
B
C
52
Representing concepts using triangle nodes
triangle nodes when two of the neurons fire, the
third also fires
53
They all rose
  • triangle nodes
  • when two of the neurons fire, the third also
    fires
  • model of spreading activation

54
Basic Ideas behind the model
  • 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

55
5 levels of Neural Theory of Language
Spatial Relation
Motor Control
Metaphor
Grammar
Cognition and Language
Computation
Structured Connectionism
abstraction
Neural Net
SHRUTI
Computational Neurobiology
Triangle Nodes
Biology
Neural Development
Midterm
Quiz
Finals
56
Can we formalize/model these intuitions
  • What is a neurally plausible computational model
    of spreading activation that captures these
    features.
  • What does semantics mean in neurally embodied
    terms
  • What are the neural substrates of concepts that
    underlie verbs, nouns, spatial predicates?
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