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A Spatiotemporal Coupled Lorenz Model
drives
Emergent Creative Process
Tetsuji EMURA
College of Human Sciences
Kinjo Gakuin University
Motivation
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Music
Music theory says:
Three elements of Sound: {Pitch, Intensity, Time-value}
Three elements of Music: {Melody, Harmony, Rhythm}
Manuscript of the third movement of the first Symphony, written by Johannes Brahms
Certainly, each sound consists of the three elements.
However, does music consist of the three elements?
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Representation
Rhythm
Harmony
Timbre
Melody
Sound image
(representation)
©2003 PBS / WGBH
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A Modeling of Creation Process
of Musical Works
by Yoshikawa’s GDT
[Emura 2000]
[Emura 2003]
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])
Model
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Proposed Model

Spatiotemporal Coupled Lorenz Model
Extension to Spatial of the Coupled Lorenz Model
xÝ1, 4    (x  x ) 
x 4  x1 
2, 5
1, 4
  


* 
xÝ2, 5  x1, 4 (r  x 3, 6 )  x 2, 5  D x 5  x 2 



Ý   x x  b x
x

x


x
1,
4
2,
5
3,
6
6
3


 3, 6 
c1 d2 d3 


D*  D  d1 c 2 d3  : Excitatory - Excitatory Connection


d1 d2 c 3 
 c1
d2
1 d3 

˜  1 d1
D*  D
c
d
2
3

 : Excitatory - Inhibitory Connection


1 d2
c 3 
 d1
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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Here,
0 < c1, 2, 3 < 1 : temporal coupling coefficients,
0 < d1, 2, 3 < 1 : spatial coupling coefficients.

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
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Uniform coefficients c1=c2=c3=c and d1=d2=d3=d are considered.

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
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Uniform coefficients c1=c2=c3=c and d1=d2=d3=d are considered.

Spatiotemporal Coupled Lorenz Model
Self-organized synchronization phenomena appear
in the case of using Excitatory-Inhibitory Connection.
x1-x4 versus d,
EIC mode, c=0.4
Chaos
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Limit
cycle
Intermittent
chaos
Fixed point
Building of Subsystem
The synchronization phenomenon is measured by the difference
 i (t),
 i (t)  x i3  x i , i  1, 2, 3.
  
1
ui (t) 
, z i  
1 ,
1 exp  zi z o 

(t)
 i 

: Analog model
where ui (n) is the value of the i - th neuron at time t,
zo is the analog parameter,  is the criterion parameter,
if zo  0 then

1
ui (n)  
0
if  i (t)  
firing state,
if  i (t)   quiescent state.
: Digital model
In the Hopfield model, the state at the discrete time t of the i - th neuron is
n

Ii (t  1)   w ij u j (t)  si   i ,
j1
where si is the external input,
i is the threshold value,
w ij ( w ji ) is the synapic weight between
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i - th and j - th neurons, and w ii  0.
The spatial coupling coefficients
di (t) is regulated dynamically by
Ii (t)
di (t)  
 0
c i (t)  constant.
if Ii (t)  0,
if Ii (t)  0,
Building of Subsystem
Evaluation of Spatial Synchronization of STCL model using
the Abstract Coincidence Detector model: ACD model
1.
2.
3.
4.
5.
Each neuron is an excitatory neuron which does not have memory but
fires by the simultaneity of a momentary incidence spike.
It does not have any inhibitory neuron.
Network structure does not assume any specific structure.
All synaptic weight is set to one.
A certain transfer delay time which exists beforehand is between neurons.
[Fujii et al., 1996]

Dt   1 if N   w i0 ui t   k 
i1

or D   w i0 ui t   1 

i

k
Dt   0 if N   w i0 ui t   k 

i1

or D   w i0 ui t   0

i
k
ui(t)
wi0
Π
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D(t)

EIC model
Amplitude of X(t)
Output of ACD
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Self-organized Phase Transition Phenomenon
Chaos
Limit cycle
Intermittent chaos
Fixed point
x1-x4
EIC model, c=0.4
d
Firing ratio
Total Firing Ratio [%]
ratio
Synchronized
Synchronized Ratio [%]
Excitatory Inhibitory Connection Model
Firing Ratio
Sync.'ed Ratio
100
90
80
70
60
50
40
30
20
10
0
0.1
0.2
0.3
0.4
d
d
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0.5
0.6
Total Firing Ratio [%]
Synchronized Ratio [%]
Excitatory Excitatory Connection Model
Firing Ratio
Sync.'ed Ratio
100
90
80
70
60
50
40
30
20
10
0
0.1
0.2
0.3
0.4
0.5
0.6
d
EEC model
Total Firing Ratio [%]
Synchronized Ratio [%]
Excitatory Inhibitory Connection Model
Firing Ratio
Sync.'ed Ratio
100
90
80
70
60
50
40
30
20
10
0
0.1
0.2
0.3
0.4
d
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0.5
0.6
EIC model
EIC model
Spatial Coupling
Coefficient: d1
Output of ACD
Hopfield’s Network Energy
E t   
1
  wij ui tu j t    si  thi ui t 
2 i j
i
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Building of Emergent System
 n

ext
v i t  1  sign  J ijv j t   K i k i Si t   ij 


 j

Si t   2Di t  1
1 p
  i j 1 [i, j]
J ij  n  1

J ji

v i t   {1,1}, Di t   {0,1},
i  {1,1}, kiext  {1,1}

 1 x  0
1 i  j
sign x   
, [i, j]  
1 x  0
0 i  j

i  {1,

n  25,

 ij : Uniform Random Spike Propagation Delay
t : Discreat Time for Computing
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, n},   {1,
: 10 
[ms]
, p}
p  3.
: t   ij  nt
Simulation
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Perception Model
Retina
Visual Perception
Two-dimensional bit-map
↑ modeling
our retina and/or also visual cortex V1
after “perceptron”
Auditory Perception
One-dimensional vector
↑ modeling
our cochlea
(and/or also auditory cortex [Bao 2003])
Auditory nerve senses resonance of basilar membrane.
Cochlea behaves like resonance chamber.
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Numerical Simulations
Natural Harmonics
f n  n  f 0,
n{1,
,25}

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
1

 1
 1

 1
 1

 1
1

1

1
1

 1
1

i   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
k iext
 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.
Subsystems: digital EIC models
→DDN model
Subsystems: analog EIC models
→ADN model
ADN model, Ki = 0.2
Retieval dynamics of
ordinary associative memory,
retrieved vector:μ=1.
ADN model, Ki = 0.9
Only external vector is retrieved,
and all embedded vectors are
destroyed by external stimuli.
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ADN model, Ki = 0.72
Autonomous Retrieval Dynamics
Attractor :μ= inv. 1 →
Attractor :μ= inv. 2 →
Attractor :μ= inv. 3 − − − − − →
Attractor :μ=3 →
Attractor :μ=2 →
Attractor :μ=1 →
n
Evaluated by Ininerancy   i  v i (t)
i1
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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]
winner-take-all competition
Auditory perception
← NOT winner-take-all competition
Contextual modulation
↑
Functional connectivity
↑
addressed from Chaotic itinerancy [Tsuda 1992]
Brain
as
Dynamical Systems
Representation
as
Long-term Memory
by
Hebbian Rule
Activation
by
External Stimuli
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Contextual Modulation
by
Chaotic Ininerancy
in
Multi-moduled
Mutually Connected
Neural Networks
Triggering
Subsystems
consist of
Coupled Oscillators
and Coincidence
Detectors

Future
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.
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Retrieval dynamics of proposed model
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Retrieval dynamics of proposed model
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Retrieval dynamics of proposed model
16
14
10
8
6
4
2
0
0.04
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0.03
0.02
Zo
0.01
0.70
0.71
0.72
0.73
Ki
0.74
Event Number
12
Emergent Parameters
ICP: Internal Control Parameter *
n
p
    i  1
i1  1
m
 t
 Di (t) if Sz (t)   v k (t)  m
z0 (t)   t 0
k1

otherwise
 0.02

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* [Keijzer 2001]
Sz (t)

z0 (t)

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Retrieval dynamics of each layer with ICP
without ICP
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with ICP
Application
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A
Musical Work
dedié à Edward N. Lorenz
Tetsuji EMURA
Les Papillons de Lorenz
le paysage non périodique déterminé du printemps
pour orchestre
Gérard Billaudot Editeur, Paris (1999)
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Thank you
Emura, T., Physics Letters A, 349, 306-313 (2006).
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