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Phantom Limb Phenomena. Hand movement observation by individuals born without hands: phantom limb experience constrains visual limb perception. Funk M , Shiffrar M , Brugger P .

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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.

summary
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?
connectionist models basics

Connectionist Models: Basics

Jerome Feldman

CS182/CogSci110/Ling109

Spring 2007

neural networks abstract from the details of real neurons
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)
the mccullough pitts neuron

yj

wij

yi

xi

f

ti : target

xi = ∑j wij yj

yi = f(xi – qi)

The McCullough-Pitts Neuron

yj: output from unit j

Wij: weight on connection from j to i

xi: weighted sum of input to unit i

Threshold

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

A Simple Example

.

different activation functions
Different Activation Functions
  • Threshold Activation Function (step)
  • Piecewise Linear Activation Function
  • Sigmoid Activation Funtion
  • Gaussian Activation Function
    • Radial Basis Function

BIAS UNIT

With X0 = 1

the sigmoid function20
The Sigmoid Function

Output=1

y=a

Output=0

x=neti

the sigmoid function21
The Sigmoid Function

Output=1

Sensitivity to input

y=a

Output=0

x=neti

stochastic units
Stochastic units
  • Replace the binary threshold units by binary stochastic units that make biased random decisions.
    • The “temperature” controls the amount of noise

temperature

types of neuron parameters
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.
computing various functions
Computing various functions
  • McCollough-Pitts Neurons can compute logical functions.
    • AND, NOT, OR
computing other functions the or function

i1

w01

w02

i2

y0

b=1

w0b

x0

f

Computing other functions: the OR function
  • 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

i2

i1

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)

decision hyperplane
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.
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.

Linearly Separable Patterns

the perceptron32
The Perceptron

Input Pattern

the perceptron33
The Perceptron

Input Pattern

Output Classification

pattern space
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.
conjunctive or sigma pi nodes
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(x1*x2) where w12>0 (for a inhibiting influence of input 2 we would, of course, have w12<0 ).
  • w1*x1 + w2*x2 +w3*x3 + w12*(x1*x2) + w23(x2*x3) +w13*(x1*x3)
biological evidence for sigma pi units
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(x1*x2)
  • [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.
biological evidence for sigma pi units45
Biological Evidence for Sigma-Pi units

[presynaptic inhibition]

[axo-dendritic synapse]

[synapse cluster]

computing other relations
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.
representing concepts using triangle nodes
Representing concepts using triangle nodes

triangle nodes:

when two of the neurons fire, the third also fires

they all rose
“They all rose”

triangle nodes:

when two of the neurons fire, the third also fires

model of spreading activation

basic ideas behind the model
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
5 levels of neural theory of language
5levels 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

Quiz

Midterm

Finals

can we formalize model these intuitions
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?