Chapter 1
A Coupled Oscillator Model
for Emergent Cognitive
Process
Tetsuji EMURA
College of Human Sciences
Kinjo Gakuin University
emura@kinjo-u.ac.jp
Numerous papers have been published concerning mathematical models of memory and learning of human brain activity. However, research has seldom proposed
concerning mathematical models of creative cognition process. In this paper, the author proposes a spatiotemporal coupled Lorenz model with an excitatory connection or
an inhibitory connection, which consists of temporal coupling coefficients c and spatial
coupling coefficients d. This study finds that self-organized various phase transition
phenomena appear in this model in changing the values of c and d in the case of using
the inhibitory connection. The proposed model is a device that has synchronized three
one-dimensional information codes and can be used as a device for emergent systems
and also concerns the neural populations model for autonomous agent systems.
1.1
Introduction
A great deal of research concerning clarification of characteristics of the act of
“creation”, in which things that had hitherto not existed are created, or a description of that process had been conducted, although research was conducted
as creativity research in particular primarily with a focus on the field of psychology [1][2]. Psychologists have clarified the existence of a process controlled by
imagination that precedes what is called the design act, which are deductive logic
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A Coupled Oscillator Model for Emergent Cognitive Process
operations, for the act of creation, in which things that had hitherto not existed
are created; a creative act absent this process in mental space is not possible.
In other words, a process that could be perceived as what is called conception
or idea generation before deductive logic operations substantially controls the
creative process. Naturally, completely sporadic idea generation or whatever is
perceived as original creativity is not possible; the fact that the ideas are induced
by the creator’s accumulation of experience through looking or listening things
or events that already exist preceding this creation. However, there is a great
deal of difficulty in describing the process of the conception or idea generation,
which lacks objectivity in an empirical sense, so research proposing mathematical models for the creative cognition process for direct linkage to creativity has
seldom been conducted.
1.2
1.2.1
Creation and Cognition
Conception and Imagination
Among the varied forms of creation, creation of the arts is characterized by
an extremely close relationship between the creator and creation. Taking the
creation of musical works, for example, a work that is a work on account of
the composer is a characteristic of music as an expressive art. The creation
process of musical works in topological space [15] is indicated in Figure 1.1. In
the figure, the process from a sound image X1’ in the mental space, which is in
the brain of the composer, to a requirement X1 of expression, is as shown below
when indicated as a convergent process through the combination of abduction
and deduction.
X1
X1→X1
X1
↓
X1
:abduction
X1→X1
X1
:deduction
The schema induces the requirement X1 using continuous mapping
ζ −1 :X1→X1’ through addition of the creator’s individual thought in the stage of
abduction where the requirement X1 is postulated from the sound image X1’ in
the mental space of the creator himself or herself. This is indicated as a process
that determines requirement X1 through verification of X1’ deduced from requirement X1 with X1’ that the creator previously held. However, characteristic
of the creative process for musical works is that the sound image X1’ previously
held might change due to X1’ that appears after deduction. This aspect may be
thought to be unusual in a system theory sense, although Sloboda [3] clarified the
existence of the process through verification of handwritten scores of Beethoven
and Stravinsky, who left numerous drafts during composition. If mentioned in a
design theory sense, the design environment itself is the dynamical systems that
change in the design process as well rather than something designed based on
determined specifications that converge to target values within extremely lim-
A Coupled Oscillator Model for Emergent Cognitive Process
3
Figure 1.1: Creation process of musical works in topological space.
ited tolerances and this cannot be processed with an algorithm that sequentially
executes previously programmed statements like the symbolic systems.
1.2.2
Symbolic Computation vs. Dynamical Systems
Figure 1.2: The nine bars from the sixteenth bar for the prelude to the first act from
the opera “Tristan und Isolde”, a work by Richard Wagner (1813∼1883). This piano
score is condenced from the original orchestral score [4] by the author.
Numerous methods actively using computers have been researched in order
to aid human creative activity. These have been used in the creation of musical works and Computer-aided Composition, a support environment for musical
work creation, currently exists, although this is mostly providing a fixed melody
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A Coupled Oscillator Model for Emergent Cognitive Process
and harmonization with it. However, when analyzing a musical work structure
as in Figure 1.2, for example, like this, a melody like the motive is present here,
although the melody and harmony are inseparable; there is absolutely no way
to first have the melody and then harmonization with it. The melody does not
first exist as the melody, the melody is a harmonic progression that changes
with time. If there is a melody here, whatever stands out from the harmonic
progression must be interpreted as the melody. That is, if melody and harmony
do not exist simultaneously in the brain of the composer as a sound image,
then creation of a work like this would be close to impossible. Moreover, individual motives are allocated to individual instruments such as woods, brasses
and strings for respective sounds and with harmonic progression are changed
to be extremely effective as motives; if changes in both harmonic progression
and timbre in the process of creation do not exist simultaneously in the brain
of the composer as a sound image, creation of a musical work like this would
be impossible. That is, harmony, melody, and timbre are in a mode where they
are blended into one another and creation must be interpreted to progress with
simultaneous processing of these in parallel in the brain. The reality of creation
process is not a sequence process of the symbolic systems.
Recently, Inoue et al. [5] proposed a model of the process in cognitive interpretation via on-off intermittency [6][7] as appears in a coupled chaos oscillator, and
the research perceiving the conduct of brain neuron activity as a synchronization
that appears in coupled oscillator has attracted attention [8][9]. The following
section proposes a new device for emergent systems using a coupled nonlinear
oscillator model.
1.3
1.3.1
Self-Organizing Device for Emergence
The Coupled Lorenz Model
Two continuous-time autonomous dynamical systems Xa and Xb are considered
in n-dimensional Euclidean space Rn .
Ẋa = F(Xa ),
Ẋb = F(Xb )
· · · (1)
Here, F is considered to be the Lorenz system [10] for both with n=3, where
individual vector components are
Xa = [x1 , x2 , x3 ] ,
Xb = [x4 , x5 , x6 ]
· · · (2)
These are bi-directionally coupled and indicated as below. Here, 0< c <1 is
a coupling coefficient.
ẋi = F(xi ) + c (xi±3 − xi )
· · · (3)
When the temporal coupling coefficient c is sufficiently small for Xa and Xb
in bi-directionally coupled Equation (3), Xa and Xb depict independent trajectories, although when c is greater than certain value even if |Xa (0)−Xb (0)|>0,
A Coupled Oscillator Model for Emergent Cognitive Process
5
the two trajectories are entrained bi-directionally and synchronized after a moment; the two then depict exactly the same trajectory. This means that the
coupled system for Equation (3) is six-dimensional space, although the two trajectories are constrained to three-dimensional invariant manifold; near this synchronous/desynchronous boundary, on-off intermittent chaos where the laminar
phase and burst phase appear intermittently is observed at Xa (t)−Xb (t). Where
the two attractors are completely synchronized, the attractor is constrained to
one plane of (x1 , x3 ); identically, x1 −x4 becomes zero. These hyperplanes where
attractors are constrained at the laminar phase are the (x1 , x2 ) plane and (x2 ,
x3 ) plane in addition to the (x1 , x3 ) plane. The attractor when the two are
completely synchronized converge to one point of (x1 − x4 , x2 − x5 , x3 − x6 )=(0,
0, 0), although when on-off intermittent chaos occurs, they repeatedly have irregular and unpredictable intermittency with wandering into three-dimensional
space (x1 − x4 , x2 − x5 , x3 − x6 ) from one point of (0, 0, 0). This wandering
in a three-dimensional space on the burst phase with seeking and gathering of
valuable information from this, synchronized stabilization in the laminar phase
for (0, 0, 0) can be interpreted as a process that intermittently and irregularly
repeats.
1.3.2
A Proposed Model
The coupled Lorenz model in the previous equations is one device, although the
Lorenz model itself is a model of the smooth manifold in three-dimensional space.
When the coupled Lorenz attractor leaves invariant manifold, it simultaneously
leaves the three hyperplanes. Thus, the author has considered this to be a model
of the coincidence detector [11][12] of neural populations; a device with three onedimensional information codes {X, Y , Z}={x1 − x4 , x2 − x5 , x3 − x6 } is coupled
to each of the three by coupling of the coupled Lorenz model in Equation (3)
spatially as well and a new device was considered. This is indicated in Equation
(4) and Figure 1.3, where 0< c1,2,3 <1 are temporal coupling coefficients, 0<
d1,2,3 <1 are spatial coupling coefficients. Then, each of {X, Y , Z} corresponds
to each state space which governs the creation process of an object.

 



ẋ1, 4
σ (x2, 5 − x1, 4 )
x4 − x1
 ẋ2, 5  =  x1, 4 (r − x3, 6 ) − x2, 5  ± D  x5 − x2  · · · (4)
ẋ3,6
x1, 4
x2, 5 − b x3, 6
x6 − x3
c1 d2 d3
D =  d1 c2 d3  : excitatory connection
 d1 d2 c3

c1
d2
1 − d3
 : inhibitory connection
D̃ =  1 − d1 c2
d3
d1
1 − d2 c3
In the previous paper [16], the author has presented that the c and d control
on-off intermittent chaos, although they have no direct effect on individual vectors and the c and d work as independent parameters without providing internal
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A Coupled Oscillator Model for Emergent Cognitive Process
Figure 1.3: Spatiotemporal coupled oscillator model as a network model-based subsystem for emergent systems
disturbance. In this paper, the difference in behavior of the model with the case
where the excitatory connection or the inhibitory connection is used are shown in
Figures 1.4-1.9, when the uniform spatial coupling coefficients d1 = d2 = d3 = d
and the uniform temporal coupling coefficients c1 = c2 = c3 = c is considered.
These figures show the behaviors of {x1 − x4 , x2 − x5 , x3 − x6 } to change of the
values of d at the value of certain c. (Note: Only x1 − x4 is illustrated in the
figures. Each fugure is plotted in t=0∼100000, d=0∼1. d is changing linearly
with t where d=0.00001t.) When the excitatory connection is used, as shown in
Figures 1.4-1.6, if the value of c becomes large, the value of d becomes small with
which the {x1 − x4 , x2 − x5 , x3 − x6 } synchronizes, and an on-off intermittent
domain also becomes narrow, and d works as an effect like a switch. As shown
in Figures 1.7-1.9, when the inhibitory connection is used, the domain of d separates to two places where the {x1 − x4 , x2 − x5 , x3 − x6 } does desynchronize.
Furthermore, it should mention specially as shown in Figure 1.9, in the domain
of certain c, the various phase transition phenomena such as chaos → limit cycle
→ intermittent chaos → laminar phase appear in this model in changing the
value of d and using the inhibitory connection.
Figure 1.4: x1 − x4 vs. d (indicated by 105 ), excitatory, c=0.2
A Coupled Oscillator Model for Emergent Cognitive Process
Figure 1.5: x1 − x4 vs. d (indicated by 105 ), excitatory, c=0.3
Figure 1.6: x1 − x4 vs. d (indicated by 105 ), excitatory, c=0.4
Figure 1.7: x1 − x4 vs. d (indicated by 105 ), inhibitory, c=0.2
Figure 1.8: x1 − x4 vs. d (indicated by 105 ), inhibitory, c=0.3
Figure 1.9: x1 − x4 vs. d (indicated by 105 ), inhibitory, c=0.4
7
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A Coupled Oscillator Model for Emergent Cognitive Process
1.4
Discussion
Numerous results for methods of Hopfield model [13] have been cited, although
in actuality these results are substantially controlled by how synapses that link
neurons are set; when the quantity of information increases, the handling time for
this setting increases tremendously. The reality is that neural networks manifest
certain functions even on a small-scale structure [14] like a functional device as a
proposed model as well as functions produced by a large-scale structure. A new
type of emergence, which has not been expected until now, can be realizable by
regarding this proposed model as a functional subsystem, for example, mounting
on the artificial neural networks or the autonomous agent systems. The topics
of these applications will be reported in the next papers in combination with an
actual example [17].
This work was supported by the special research grant from the Kinjo Gakuin
University.
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