Post on 13-Jan-2016
Tetsuji EMURACollege of Human Sciences
Kinjo Gakuin University
A Spatiotemporal Coupled Lorenz Modeldrives
Emergent Creative Process
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Motivation
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Three elements of Sound: {Pitch, Intensity, Time-value}Three elements of Music: {Melody, Harmony, Rhythm}
Music
Manuscript of the third movement of the first Symphony, written by Johannes Brahms
Music theory says:
Certainly, each sound consists of the three elements.However, does music consist of the three elements?
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©2003 PBS / WGBH
Representation
Melody
Harmony
Rhythm
Timbre
Sound image(representation)
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When analyzing musical work’s structures, we notice that melody, harmony, rhythm and timbre are inseparable on the perception; there is absolutely no way to first have the melody and then harmonization and these with it; If the melody, harmony, rhythm and timbre do not exist simultaneously in the brain of the composer as a sound image, then creation of the works like these would be close to impossible. That is, first, there are “sound image” as representation in his brain, and elements of music are in a certain mode where they are blended into one another. Creation process of musical works should be interpreted to progress with simultaneous processing of these in parallel in the brain. The reality of creation process is not a sequential process of the symbolic systems. (ex. GTTM by [Lerdahl & Jackendoff 1999] after [Chomsky 1957])
A Modeling of Creation Processof Musical Works
by Yoshikawa’s GDT
[Emura 2003]
[Emura 2000]
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Model
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Proposed Model
Spatiotemporal Coupled Lorenz Model
€
˙ x 1,4
˙ x 2,5
˙ x 3,6
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟=
σ (x2,5 − x1,4 )
x1,4 (r − x3,6) − x2,5
x1,4 x2,5 − b x3,6
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟± D*
x4 − x1
x5 − x2
x6 − x3
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
D* = D =
c1 d2 d3
d1 c2 d3
d1 d2 c3
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
: Excitatory - Excitatory Connection
D* = ˜ D =
c1 d2 1− d3
1− d1 c2 d3
d1 1− d2 c3
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
: Excitatory - Inhibitory Connection
Extension to Spatial of the Coupled Lorenz Model
Here,0 < c1, 2, 3 < 1 : temporal coupling coefficients,0 < d1, 2, 3 < 1 : spatial coupling coefficients.
A network model-based model which regards the three oscillator:
{X, Y, Z}={x4-x1, x5-x2 , x6-x3}as three neurons.
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Spatiotemporal Coupled Lorenz Model
x1-x4 versus d,EEC model, c=0.2
x1-x4 versus d,EEC model, c=0.3
x1-x4 versus d,EEC model, c=0.4
Uniform coefficients c1=c2=c3=c and d1=d2=d3=d are considered.
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Spatiotemporal Coupled Lorenz Model
x1-x4 versus d,EIC model, c=0.2
x1-x4 versus d,EIC model, c=0.3
x1-x4 versus d,EIC model, c=0.4
Uniform coefficients c1=c2=c3=c and d1=d2=d3=d are considered.
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Spatiotemporal Coupled Lorenz Model
x1-x4 versus d,EIC mode, c=0.4
Chaos Limit cycle
Intermittentchaos
Fixed point
Self-organized synchronization phenomena appearin the case of using Excitatory-Inhibitory Connection.
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Building of Subsystem
€
ui(t) =1
1+ exp −zi zo[ ], zi =
ε
Δ i(t)
⎛
⎝ ⎜
⎞
⎠ ⎟−1 ,
where ui(n) is the value of the i - th neuron at time t,
zo is the analog parameter, ε is the criterion parameter,
€
The synchronization phenomenon is measured by the difference Δ i(t),
Δ i(t) = x i+3 − x i , i =1, 2, 3.
€
if zo → 0 then
ui(n) =1
0
⎧ ⎨ ⎩
if Δ i(t) ≤ ε
if Δ i(t) > ε
firing state,
quiescent state.
: Analog model
: Digital model
€
In the Hopfield model, the state at the discrete time t of the i - th neuron is
Ii(t +1) = wij
j=1
n
∑ u j (t) + si −θ i ,
where si is the external input, θ i is the threshold value,
wij (= w ji) is the synapic weight between i - th and j - th neurons, and wii = 0.
The spatial coupling coefficients di(t) is regulated dynamically by
di(t) =Ii(t)
0
⎧ ⎨ ⎩
if Ii(t) ≥ 0,
if Ii(t) < 0,c i(t) = constant.
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Building of Subsystem
ui(t)
D(t)ÉÆ
wi0
Evaluation of Spatial Synchronization of STCL model usingthe Abstract Coincidence Detector model: ACD model
[Fujii et al., 1996]
1. Each neuron is an excitatory neuron which does not have memory but fires by the simultaneity of a momentary incidence spike.
2. It does not have any inhibitory neuron. 3. Network structure does not assume any specific structure. 4. All synaptic weight is set to one.5. A certain transfer delay time which exists beforehand is between neurons.
€
D t( ) =1 if N = wi0ui t( )i=1
k
∑ = k
or D = wi0ui t( ) =1i
∏
D t( ) = 0 if N = wi0ui t( )i=1
k
∑ < k
or D = wi0ui t( )i
∏ = 0
⎫
⎬
⎪ ⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪ ⎪
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EIC model
Amplitude of X(t)
Output of ACD
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Excitatory Inhibitory Connection Model
0102030405060708090100
0.1 0.2 0.3 0.4 0.5 0.6d
Total Firing Ratio [%]Synchronized Ratio [%]
Firing RatioSync.'ed Ratio
Chaos Limit cycle Intermittent chaos Fixed point
Self-organized Phase Transition Phenomenonx 1-
x 4
d
Fir
ing
rati
oS
ynch
roni
zed
rati
o
d
EIC model, c=0.4
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Excitatory Excitatory Connection Model
0102030405060708090
100
0.1 0.2 0.3 0.4 0.5 0.6
d
Total Firing Ratio [%]Synchronized Ratio [%]
Firing Ratio
Sync.'ed Ratio
Excitatory Inhibitory Connection Model
0102030405060708090
100
0.1 0.2 0.3 0.4 0.5 0.6
d
Total Firing Ratio [%]Synchronized Ratio [%]
Firing Ratio
Sync.'ed Ratio
EEC model
EIC model
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EIC model
€
E t( ) = −1
2wijui t( )u j t( ) − si − thi( )
i
∑j
∑i
∑ ui t( )
Hopfield’s Network Energy
Output of ACD
Spatial Coupling Coefficient: d1
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€
v i t +1( ) = sign Jijv j t( ) + K ikiextSi t − τ ij( )
j
n
∑ ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Si t( ) = 2Di t( ) −1
Jij =1
nξ i
μξ jμ 1−δ[i, j]( )
μ =1
p
∑J ji
⎧
⎨ ⎪
⎩ ⎪
€
sign x[ ] =1 x ≥ 0
−1 x < 0
⎧ ⎨ ⎩
, δ[i, j] =1 i = j
0 i ≠ j
⎧ ⎨ ⎩€
v i t( ) = {−1,1}, Di t( ) = {0,1},
ξ iμ = {−1,1}, ki
ext = {−1,1}
€
i ∈ {1,K , n}, μ ∈ {1,K , p}
€
τ ij : Uniform Random Spike Propagation Delay : Δt ≤ τ ij ≤ nΔt
Δt : Discreat Time for Computing : 10 [ms]€
n = 25, p = 3.
Building of Emergent System
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Simulation
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Perception Model
Visual Perception
Two-dimensional bit-map↑ modelingour retina and/or also visual cortex V1
Auditory Perception
One-dimensional vector↑ modelingour cochlea(and/or also auditory cortex [Bao 2003])
after “perceptron”
Retina
Auditory nerve senses resonance of basilar membrane.Cochlea behaves like resonance chamber.
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ξ iμ =
−1 −1 −1
1 1 −1
1 1 1
1 1 1
1 −1 1
1 1 −1
−1 −1 1
1 −1 1
−1 −1 −1
−1 1 −1
1 1 1
−1 −1 −1
1 −1 −1
−1 −1 −1
−1 1 −1
1 1 −1
−1 −1 1
1 −1 1
−1 −1 −1
−1 1 −1
1 −1 −1
−1 1 −1
−1 −1 1
−1 1 1
−1 −1 1
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
kiext =
1
−1
−1
−1
1
−1
1
−1
1
−1
−1
−1
1
−1
−1
−1
1
−1
1
−1
1
−1
−1
−1
1
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
Three Embedded Vectors, μ= 1, 2, 3, and an External Stimulus Vector.
Numerical Simulations
€
Natural Harmonics
fn = n ⋅ f0,
n∈{1, K ,25}
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ADN model, Ki = 0.2Retieval dynamics ofordinary associative memory,retrieved vector:μ=1.
ADN model, Ki = 0.9Only external vector is retrieved,and all embedded vectors aredestroyed by external stimuli.
Subsystems: digital EIC models →DDN modelSubsystems: analog EIC models →ADN model
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ADN model, Ki = 0.72
Autonomous Retrieval Dynamics
Attractor :μ=1 →
Attractor :μ=2 →
Attractor :μ=3 →
Attractor :μ= inv. 1 →
Attractor :μ= inv. 2 →Attractor :μ= inv. 3 − − − − − →
€
Ininerancy = ii=1
n
∑ ⋅v i(t)Evaluated by
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Chaotic Itinerancy*
↑an Attractor
← an Attractor
an Attractor →
* [Tsuda 1992]
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Perception and Cognition
Visual perception
Binding problem↑
Functional connectivity↑
addressed from Synfire chain [Abeles 1991]
Auditory perception
← winner-take-all competition
Contextual modulation↑
Functional connectivity↑
addressed from Chaotic itinerancy [Tsuda 1992]
← NOT winner-take-all competition
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Brainas
Dynamical Systems
Contextual Modulationby
Chaotic Ininerancyin
Multi-moduledMutually Connected
Neural Networks
Representationas
Long-term Memoryby
Hebbian Rule
Activationby
External Stimuli
TriggeringSubsystemsconsist of
Coupled Oscillatorsand Coincidence
Detectors
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The conventional Hebbian connectivity model ; ー is a model of one-shot learning on the fixed anatomical connection and this plasticity has a long-time constant. ー has the stage where contents are made to be memorized in the network and the stage where they are made to be retrieved are completely separated. That is to say, it is a "hard" machine.
The behavior of proposed model ; ー is determined simultanously by the spatiotemporal excitation dynamics in the network. ー is a model which behaves that the embedded vectors as the long-term memories are recollected autonomous synchronously by external spike trains from subsystems which is superimposed on unknown vector for the networks. ー has the anatomical distribution of synapse connecting weight which is decided by Hebbian rule beforehand has not been changed at all. ー has the behavior of retrieval dynamics is sensitive to the background dynamics of the network, then behaviors have ``contextual modulations'', which is spatiotemporal modulation of with external stimuli to the network. So to speak, it is a "soft'' machine.
Future
iWES'06 Retrieval dynamics of proposed model
iWES'06 Retrieval dynamics of proposed model
iWES'06 Retrieval dynamics of proposed model
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Event Number
KiZo
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€
Ξ= ξ iμ
μ =1
p
∏i=1
n
∑ ≡ −1
z0(t) =λ Di(t)
t= 0
t
∑ if Sz (t) = vk (t)k=1
m
∑ = −m
0.02 otherwise
⎧
⎨ ⎪
⎩ ⎪
Emergent Parameters
ICP: Internal Control Parameter *
* [Keijzer 2001]
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€
Sz(t)
€
z0(t)
iWES'06 Retrieval dynamics of each layer with ICP
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without ICP
with ICP
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Application
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dedié à Edward N. Lorenz
Tetsuji EMURALes Papillons de Lorenz
le paysage non périodique déterminé du printemps
pour orchestreGérard Billaudot Editeur, Paris (1999)
AMusical Work
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Thank you
Emura, T., Physics Letters A, 349, 306-313 (2006).