Tetsuji EMURA College of Human Sciences Kinjo Gakuin University A Spatiotemporal Coupled Lorenz Model drives Emergent Creative Process.

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 − − − − − →

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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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0.720.73

0.74

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0.030.04

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2

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8

10

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14

16

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).