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

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. 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

9. Mapping from neuron

10. Simple Threshold Linear Unit

11. Simple Neuron Model 1

12. 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 .

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 • Threshold Activation Function (step) • Piecewise Linear Activation Function • Sigmoid Activation Funtion • Gaussian Activation Function • Radial Basis Function BIAS UNIT With X0 = 1

18. Types of Activation functions

19. The Sigmoid Function y=a x=neti

20. The Sigmoid Function Output=1 y=a Output=0 x=neti

21. The Sigmoid Function Output=1 Sensitivity to input y=a Output=0 x=neti

22. Changing the exponent k(neti) K >1 K < 1

23. Radial Basis Function

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

25. 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.

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

27. 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?

28. 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)

29. 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.

30. 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

31. The Perceptron

32. The Perceptron Input Pattern

33. The Perceptron Input Pattern Output Classification

34. A Pattern Classification

35. 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.

36. The XOR Function

37. The Input Pattern Space

38. The Decision planes

39. Multi-layer Feed-forward Network

40. Pattern Separation and NN architecture

41. 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)

42. Sigma-Pi units

43. Sigma-Pi Unit

44. 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.

45. Biological Evidence for Sigma-Pi units [presynaptic inhibition] [axo-dendritic synapse] [synapse cluster]

46. Link to Vision: The Necker Cube

48. Constrained Best Fit in Nature inanimate animate

49. 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.

50. A B C Triangle nodes and McCullough-Pitts Neurons? A B C