Golden Ratio and

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Title
Golden Ratio and Optimal Neural Information
Reprint/Preprint Download at http//www.math.unl.
edu/bdeng
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intro
Golden Ratio f

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f2
f
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intro

• 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.
• Paciolis 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)

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Nature
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Neurons models
Neurons Models
Rinzel Wang (1997)
Bechtereva Abdullaev (2000)
time
(1994)
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seedtuning
SEED Implementation
Spike Excitation Encoding Decoding(SEED)
3 2 4 3 3 2 2 1 1 3
Mistuned
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Bit rate
Entropy
Information System Alphabet A 0,1
Message s 11100101 Information System
Ensemble of messages, characterized by
symbol probabilities
P(0) p0 , P( 1) p1
In general, if A 0, , n-1, P(0) p0 ,,
P(n 1) pn 1, then each average symbol
contains E(p) ( p0 ln p0
pn 1 ln pn 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
Probability for a particular message s0 sn 1
is ps0 psn p0 of 0s p1 of 1s , where
of 0s of 1s n The average symbol
probability for a typical message is (ps0 psn
) 1/n p0( of 0s) / n p1( of 1s) / n p0p0
p1p1
Entropy Let p0 (1/2) log ½ p0 (1/2)-ln p0/
ln 2 , p1 (1/2) log ½ p1 (1/2)-ln p1/ ln 2
Then the average symbol probability for a
typical message is (ps0 psn ) 1/n p0p0
p1p1 (1/2)( p0ln p0 p1ln p1) / ln 2
(1/2)E( p0) By definition, the entropy of the
system is E(p) ( p0ln p0
p1ln p1) / ln 2 in bits per symbol
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Bit rate
Golden Ratio Distribution
SEED Encoding Sensory Input Alphabet Sn A1
, A2 , , An with probabilities p1, ,
pn.
Isospike Encoding En 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
Tave (p) Tp1 2Tp2 nTpn
And, The bit per unit time is
rn (p) E (p) / Tave (p)
time
Theorem (Golden Ratio Distribution) For each n
r 2 rn maxrn (p) p1 p2 pn 1,
pk r 0 _ ln p1 / (T ln 2) for which pk
p1k and p1 p12 p1n 1. In particular,
for n 2, p1 f, p2 f 2 . In addition,
p1(n) ? ½ as n ? .
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Bit rate
Golden Ratio Distribution
Generalized Golden Ratio Distribution Special
Case Tk m k, Tk / T1 k
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GoldenSequence
Golden Sequence of 1s
of 0s Total (Rule 1?10, 0?1)
(Fn) (Fn-1) (Fn Fn 1
Fn 1) 1
1 0 10
1
1 101
2 1 10110
3 2 10110101
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3 1011010110110 8
5 101101011011010110101 13
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( of 1s)/(
of 0s) Fn /Fn-1 ?1/f, Fn1 Fn Fn -1,
gt Distribution 1 Fn /Fn1 Fn -1 /Fn1 ,
gt p1 ? f , p0 ? f 2
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Title