1. Hypercubes and Neural Networks bill wolfe 9/21/2005

20. Modeling

21. “activation level” “Net Input”

22. 0 <= a i <= 1 Saturation

23. da j /dt = Net j (1-a j )(a j ) Dynamics

24. 3 Neuron Example

25. Brain State:

26. “Thinking”

27. Binary Model a j = 0 or 1 Neurons are either “on” or “off”

28. Binary Stability a j = 1 and Net j >=0 Or a j = 0 and Net j <=0

29. Hypercubes

32. 4-Cube

36. 5-Cube

40. http://www1.tip.nl/~t515027/hypercube.html Hypercube Computer Game

41. 2-Cube Adjacency Matrix: Hypercube Graph

42. Recursive Definition

43. Theorem 1: If v is an eigenvector of Q n-1 with eigenvalue x then the concatenated vectors [v,v] and [v,-v] are eigenvectors of Q n with eigenvalues x+1 and x-1 respectively. Eigenvectors of the Adjacency Matrix

44. Proof

45. Generating Eigenvectors and Eigenvalues

46. Walsh Functions for n=1, 2, 3

49. 1 000 001 010 011 100 101 110 111 eigenvectorbinary number

50. n=3

51. Theorem 3: Let k be the number of +1 choices in the recursive construction of the eigenvectors of the n-cube. Then for k not equal to n each Walsh state has 2 n-k-1 non adjacent subcubes of dimension k that are labeled +1 on their vertices, and 2 n-k-1 non adjacent subcubes of dimension k that are labeled -1 on their vertices. If k = n then all the vertices are labeled +1. (Note: Here, "non adjacent" means the subcubes do not share any edges or vertices and there are no edges between the subcubes).

52. n=5, k= 3n=5, k= 2

53. Inhibitory Hypercube

54. Theorem 5: Each Walsh state with positive, zero, or negative eigenvalue is an unstable, weakly stable, or strongly stable state of the inhibitory hypercube network, respectively.