Download presentation

Presentation is loading. Please wait.

1
Title Reprint/Preprint Download at: http://www.math.unl.edu/~bdeng

2
intr o Golden Ratio : 1 22 + f 2 = 1 f =

3
intr o 1 f f3f3 f2f2 Pythagoreans (570 – 500 B.C.) were the first to know that the Golden Ratio is an irrational number. Euclid (300 B.C.) gave it a first clear definition as ‘ the extreme and mean ratio’. Pacioli’s book ‘The Divine Proportion’ popularized the Golden Ratio outside the math community (1445 – 1517). Kepler (1571 – 1630) discovered the fact that Jacques Bernoulli (1654 – 1705) made the connection between the logarithmic spiral and the golden rectangle. Binet Formula (1786 – 1856) Ohm (1835) was the first to use the term ‘Golden Section’.

4
Nature

5
Neurons Models Neurons models Rinzel & Wang (1997) Bechtereva & Abdullaev (2000) (1994) time 1T1T3T3T 1 f

6
seedtuning SEED Implementation Signal Encode Decode Channel Mistuned Spike Excitation Encoding & Decoding(SEED) 3 2 4 3 3 2 2 1 1 3 …

7
Bit rate Entropy Information System Alphabet: A = {0,1} Message: s = 11100101… Information System: Ensemble of messages, characterized by symbol probabilities: P({0})= p 0, P( {1})= p 1 Probability for a particular message s 0 … s n –1 is p s 0 … p s n = p 0 # of 0s p 1 # of 1s, where # of 0s + # of 1s = n The average symbol probability for a typical message is (p s 0 … p s n ) 1/n = p 0 (# of 0s) / n p 1 (# of 1s) / n ~ p 0 p 0 p 1 p 1 Entropy Let p 0 = (1/2) log ½ p 0 = (1/2) -ln p 0 / ln 2, p 1 = (1/2) log ½ p 1 = (1/2) -ln p 1 / ln 2 Then the average symbol probability for a typical message is (p s 0 … p s n ) 1/n ~ p 0 p 0 p 1 p 1 = (1/2) (– p 0 ln p 0 – p 1 ln p 1 ) / ln 2 : = (1/2) E( p 0 ) By definition, the entropy of the system is E(p) = (– p 0 ln p 0 – p 1 ln p 1 ) / ln 2 in bits per symbol In general, if A = {0, …, n-1}, P({0}) = p 0,…, P({n –1}) = p n –1, then each average symbol contains E(p) = (– p 0 ln p 0 – … – p n –1 ln p n –1 ) / ln 2 bits of information, call it the entropy. In general, if A = {0, …, n-1}, P({0}) = p 0,…, P({n –1}) = p n –1, then each average symbol contains E(p) = (– p 0 ln p 0 – … – p n –1 ln p n –1 ) / ln 2 bits of information, call it the entropy. Example: Alphabet: A = {0, 1}, w/ equal probability P({0})=P({1})=0.5. Message: …011100101… Then each alphabet contains E = ln 2 / ln 2 = 1 bit of information Example: Alphabet: A = {0, 1}, w/ equal probability P({0})=P({1})=0.5. Message: …011100101… Then each alphabet contains E = ln 2 / ln 2 = 1 bit of information

8
Bit rate Golden Ratio Distribution SEED Encoding: Sensory Input Alphabet: S n = {A 1, A 2, …, A n } with probabilities {p 1, …, p n }. SEED Encoding: Sensory Input Alphabet: S n = {A 1, A 2, …, A n } with probabilities {p 1, …, p n }. Isospike Encoding: E n = {burst of 1 isospike, …, burst of n isospikes} Message: SEED isospike trains… Idea Situation: 1) Each spike takes up the same amount of time, T, 2) Zero inter-spike transition Then, the average time per symbol is T ave (p) = Tp 1 + 2Tp 2 +… +nTp n And, The bit per unit time is r n (p) = E (p) / T ave (p) Theorem: (Golden Ratio Distribution) For each n r 2 r n * = max{r n (p) | p 1 + p 2 +… +p n = 1, p k r 0} = _ ln p 1 / (T ln 2) for which p k = p 1 k and p 1 + p 1 2 +… + p 1 n = 1. In particular, for n = 2, p 1 = f, p 2 = f 2. In addition, p 1 (n) ½ as n . Theorem: (Golden Ratio Distribution) For each n r 2 r n * = max{r n (p) | p 1 + p 2 +… +p n = 1, p k r 0} = _ ln p 1 / (T ln 2) for which p k = p 1 k and p 1 + p 1 2 +… + p 1 n = 1. In particular, for n = 2, p 1 = f, p 2 = f 2. In addition, p 1 (n) ½ as n . time 1T1T3T3T 8

9
Bit rate Golden Ratio Distribution Generalized Golden Ratio Distribution = Special Case: T k = m k, T k / T 1 = k

10
Go lde nS equ enc e Golden Sequence # of 1s # of 0s Total (Rule: 1 10, 0 1) (F n ) (F n-1 ) (F n + F n –1 = F n +1 ) 1 1 0 10 1 1 101 2 1 10110 3 2 10110101 5 3 1011010110110 8 5 101101011011010110101 13 8 (# of 1s)/(# of 0s) = F n /F n-1 1/ f, F n+1 = F n + F n -1, => Distribution: 1 = F n /F n+1 + F n -1 /F n+1, => p 1 f, p 0 f 2 P{fat tile} f P{thin tile} f 2

11
Title

Similar presentations

OK

Noise, Information Theory, and Entropy (cont.) CS414 – Spring 2007 By Karrie Karahalios, Roger Cheng, Brian Bailey.

Noise, Information Theory, and Entropy (cont.) CS414 – Spring 2007 By Karrie Karahalios, Roger Cheng, Brian Bailey.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on new zealand culture and tradition Ppt on any one mathematician jobs Ppt on power grid failure usa Ppt on next generation 2-stroke engine fuel Eye movements in reading ppt on ipad Ppt on class 9 motion powerpoint Ppt on leverages in financial management Ppt on diode circuits application Ppt on rutherford's gold foil experiment Ppt on self development