Amoeba-Based Chaotic Neurocomputing: Combinatorial

Transcription

Amoeba-Based Chaotic Neurocomputing: Combinatorial Optimization by Coupled Biological Oscillators1
Amoeba-Based Chaotic Neurocomputing:
Combinatorial Optimization by Coupled
Biological Oscillators
Masashi AONO
Advanced Science Institute, RIKEN
2-1 Hirosawa, Wako, Saitama 351-0198 JAPAN
[email protected]
Yoshito HIRATA
Institute of Industrial Science, The University of Tokyo
4-6-1 Komaba, Meguro, Tokyo 153-8505 JAPAN
[email protected]
Masahiko HARA
Advanced Science Institute, RIKEN
2-1 Hirosawa, Wako, Saitama 351-0198 JAPAN
[email protected]
Kazuyuki AIHARA
Institute of Industrial Science, The University of Tokyo
and ERATO Aihara Complexity Modelling Project, JST
4-6-1 Komaba, Meguro, Tokyo 153-8505 JAPAN
[email protected]
Received 28 October 2008
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Kazuyuki AIHARA
Abstract
We demonstrate a neurocomputing system incorporating
an amoeboid unicellular organism, the true slime mold Physarum, known
to exhibit rich spatiotemporal oscillatory behavior and sophisticated computational capabilities. Introducing optical feedback applied according to
a recurrent neural network model, we induce that the amoeba’s photosensitive branches grow or degenerate in a network-patterned chamber in
search of an optimal solution to the traveling salesman problem (TSP),
where the solution corresponds to the amoeba’s stably relaxed configuration (shape), in which its body area is maximized while the risk of
being illuminated is minimized. Our system is capable of reaching the
optimal solution of the four-city TSP with a high probability. Moreover,
our system can find more than one solution, because the amoeba can
coordinate its branches’ oscillatory movements to perform transitional
behavior among multiple stable configurations by spontaneously switching between the stabilizing and destabilizing modes. We show that the
optimization capability is attributable to the amoeba’s fluctuating oscillatory movements. Applying several surrogate data analyses, we present
results suggesting that the amoeba can be characterized as a set of coupled chaotic oscillators.
Keywords Multilevel Self-Organization, Coupled Oscillators, Chaotic
Neural Network, Chaotic Itinerancy, Self-Disciplined Computing.
§1
Introduction
Can a biocomputer with hardware incorporating biological materials
implement some unique functions that are difficult for conventional digital computers to deal with? A biological organism is a hierarchically structured system
in which a number of self-organization processes run simultaneously on their
characteristic spatiotemporal scales at multiple levels, such as the molecules,
genes, proteins, cells, tissues, organs, and body parts, as well as the whole body.
Because the self-organization process at each level involves a certain kind of
benefit optimization, such as energy minimization and stability maximization, it
would be reasonable to consider the organism as a particular kind of concurrent
computing system in which a number of computing processes to solve different
benefit optimization problems are executed concurrently by sharing common
computational resources such as energies and structured substances.
For these optimization processes, there is no decision program for assigning priority orders to the multiple levels to mediate their potentially conflicting
interests. Despite this, if the multilevel optimization processes are capable of
making a self-disciplined decision, for example, a decision to accept a loss in
short-term benefits of body parts for the sake of long-term gains of the organism’s whole body, the decision capability may be exploited for performing some
unprogrammed but reasonable operations when incorporated in a biocomputer.
Additionally, if the essential dynamics of the multilevel optimization processes
could be extracted, the computing scheme may be implemented by other faster
materials capable of multilevel self-organization.
With the above expectations, we focus on an amoeboid unicellular organism, Physarum polycephalum, that is regarded as a kind of self-organizing
oscillatory medium 1, 2, 3, 4) capable of sophisticated computing 5, 6, 7, 8) . Aono
and Gunji proposed a computing scheme to implement a nonclassical computational model utilizing the fluctuating photoavoidance behavior of the amoeba’s
oscillating branches induced by optical feedback 9) . Optical feedback creates a
dynamic environmental condition provided for the amoeba and enables us to
set various types of problems simply by changing some parameters. Based on
this proposal, some of the authors created amoeba-based neurocomputing systems with optical feedback to implement recurrent neural network models and
demonstrated that the systems work as associative memory 10) , constraint satisfaction problem solvers 11, 12, 13) , combinatorial optimization problem solvers 14) ,
and autonomous meta-problem solvers 14) . Other researchers have employed the
amoeba to implement logic operations
problem solving 17, 18) .
15)
, robot control
16)
, and unconventional
In this paper, first we measure our amoeba-based neurocomputing system’s optimization capability of solving the four-city traveling salesman problem.
Second, we examine how the amoeba’s oscillatory movements contribute to enhancement of the system’s optimization capability. Third, analyzing the time
series data of the oscillatory movements by several surrogate data tests, we verify
our hypothesis that the oscillatory movements can be characterized as behavior
generated by coupled chaotic oscillators, where chaotic dynamics is capable of
exponentially amplifying tiny fluctuations at the microscopic level to extensive
instabilities at the macroscopic level. Finally, we discuss what benefits can be
gained from the existence of chaos in our computing scheme.
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§2
Materials and Methods
2.1
True Slime Mold Physarum
[1]
Body architecture
A plasmodium of the true slime mold Physarum polycephalum is a huge
unicellular multinucleated amoeboid organism (Fig. 1A). Because numerous cell
nuclei are distributed throughout the whole body of an individual amoeba (i.e.,
a single cell), a part of the amoeba divided from the individual survives as
another self-sustainable individual. If two individuals come into contact with
one another, they fuse into one individual.
When we place the amoeba in a stellate chamber put on an agar plate
(Fig. 1B), the amoeba comes to have multiple branches and changes its shape
by expanding or shrinking the branches simultaneously. While changing its
shape, the amoeba’s total volume is conserved to be almost constant under our
experimental conditions, which include no provision of nutrients (Fig. 1C). The
amoeba stores nutrients ingested before the experiment as an internal energy
source and survives for up to about a week solely by absorbing moisture from
the agar plate containing no nutrients.
Figure 1D shows a schematic illustration of the body architecture of an
individual amoeba elongating three branches. An amoeba has a single outer gel
layer encapsulating a limited amount of intracellular sol (protoplasm), and has
no other highly differentiated structures. The amoeba’s body is homogeneous in
the sense that every part of the body is based on this simple architecture.
[2]
Microscopic level: Local contraction-relaxation oscillation
The gel layer is composed of masses of actomyosin systems 19, 20) (fibrous proteins in muscles) that are in contracting or relaxing states (Fig. 1E).
Crosslinked actomyosin systems exhibit entrained oscillatory behavior in which
a local site of the gel layer alternately contracts and relaxes at a period of about
1 ∼ 2 min, and the rhythmic contraction and relaxation lead to the repeated
decrease and increase of vertical body thickness, respectively. Although its detailed mechanism has yet to be elucidated, the contraction-relaxation oscillation
of actomyosin systems is considered to be produced through intrinsic nonlinear
oscillatory phenomena involving complicated mechanochemical reactions among
intracellular chemicals such as ATP and Ca2+
21, 22)
. Thus, the contraction-
relaxation oscillation would be considered a self-organization process at the microscopic level.
[3]
Macroscopic level: Branch’s growth-degeneration movement
Depending on the phase difference of the oscillation, the gel layer’s con-
traction tension varies from site to site. Along the pressure difference (gradient)
produced by the local contraction tension difference, the sol is led to flow horizontally
23)
(velocity=∼ 1 mm/sec).
Responding to the gel layer’s oscillation, the flow direction of the sol is
reversed. As the sol flows into and out of a branch at each oscillation period, the
branch alternately expands and shrinks by a small increment and decrement.
The dimension of the branch, therefore, changes non-monotonously (Fig. 1D).
When the accumulation of several periods of the short-term dimension
changes takes a positive value and a negative value, the amoeba’s branch undergoes long-term growth and degeneration (velocity =∼ 1 cm/h), respectively.
Therefore, to achieve rapid long-term growth or degeneration, the branch needs
to maintain coherent oscillatory behavior capable of sustaining a stationary
phase difference between its terminal and root for several periods, so that a
gap between the cumulative influx and efflux of the sol for the branch can be
widened. In other words, the long-term growth-degeneration movement would
be viewed as an outcome of a self-organization process to maintain the coherent
oscillatory behavior of the branch at a macroscopic level.
[4]
Whole individual level: Global shape-changing behavior
The amoeba changes the shape of its whole body as its multiple branches
perform the long-term growth-degeneration movements concurrently. The shapechanging behavior is considered as a self-organization process at an individual
level of the whole body, where the amoeba’s branches interact with each other
by sharing a limited amount of the sol to determine which branches to grow and
degenerate depending on relative differences in their oscillation phases 13) .
Despite its lack of a central nervous system, the amoeba exhibits sophisticated computational capabilities to optimize its shape-changing behavior
under a given environmental condition. Indeed, Nakagaki et al. showed that
the amoeba is capable of searching for a solution to a maze 5) . Inside a barrier
structure (chamber) shaped as a maze, the amoeba changes its shape in several hours into a string-like configuration that is the shortest connection path
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Kazuyuki AIHARA
between two nutrient sources placed at the entrance and exit of the maze. The
shortest path configuration maximizes the nutrient absorption efficiency under
this experimental condition. The shape-changing behavior as a self-organization
process, therefore, involves functions as a kind of computing process to optimize
the benefit of the amoeba’s whole body to its survival 6, 7) .
2.2
[1]
Implementation of Neurocomputing
Overview
Although the amoeba’s shape-changing behavior may appear to be unrelated to neurocomputing, we connect these two processes for the following reasons. 1) As noted above, the amoeba is capable of optimizing its shape-changing
behavior under a given environmental condition 5, 6, 7) . We conjectured that this
optimization capability could be exploited for solving diverse application problems, if each problem could be properly translated into a dynamic environmental
condition—e.g., optical feedback. 2) There is a one-to-one mapping between the
amoeba’s shape in a stellate chamber and the state of a recurrent neural network
model, when we identify the state of a neuron as a dimension (area or volume)
of the amoeba’s branch 13) . 3) Since extensive researches have been conducted
in the field of neurocomputing, many useful techniques for implementing various
types of applications including combinatorial optimization have already been developed in terms of neural network algorithms 24) . Using these algorithms, many
application problems are expected to be translated into environmental conditions
provided for the amoeba. 4) We considered that the fluctuations and instability
in the amoeba’s oscillatory movements could be positively exploited to explore
a broader search space 9) in terms of recurrent neural network algorithms 25) , as
well as some other metaheuristics for combinatorial optimization 26, 27, 28) .
We exploit all the above facts in implementing our neurocomputing.
That is, given the shape of the amoeba, we encode the shape as the state of a
neural network. According to a recurrent neural network model, an optical feedback unit determines an illumination pattern and feeds it back to the amoeba to
induce its shape-changing behavior. Iterating these procedures, a given problem
to be solved is translated into the optical feedback, providing a spatiotemporally
varying illumination condition. The amoeba is led to recognize and meet the
constraints of the problem by receiving optical stimulations. When the system
reaches a configuration in which the states of all neurons become unchanged, we
regard the stable configuration as an output result of the computing.
[2]
Traveling salesman problem
The traveling salesman problem (TSP), an NP-hard combinatorial optimization problem 29) , is stated as follows: Given a map of N cities defining
the travel distances from any city to any other city, search for the shortest
circular trip route for visiting each city exactly once and returning to the starting city. Figure 2A shows an example map with four cities A, B, C, and D.
Each circular route takes the total distance 12, 20, or 24, and the circular route
A → B → C → D → A (abbreviated hereafter as ABCD) is one of the shortest
(optimal) solutions. For a map of N cities, the number of all possible circular
routes N ! runs into astronomical numbers when the number of cities N becomes larger. Due to the impracticality of finding the exactly optimal solution,
many approximation algorithms to obtain a good solution quickly have been
proposed
[3]
24, 25, 26, 27, 28)
.
State representation
According to the recurrent neural network model proposed by Hopfield
and Tank as an approximation algorithm25) , the N -city TSP can be solved with
N × N neurons. To implement the four-city TSP solution, we fabricated a
chamber having 16 radial lanes as shown in Fig. 1B, where each lane is called
a “neuron” to be distinguished from the amoeba’s “branch” elongating in the
lane. Each neuron is labeled with k ∈ {P n | P ∈ {A, B, C, D}, n ∈ {1, 2, 3, 4}}
indicating the city name P and its visiting order n. When the amoeba sufficiently
elongates its branch in neuron P n, this indicates that P was visited nth.
For each neuron k at time t, the state xk (t) ∈ [0.0, 1.0] is defined as the
fraction of the area occupied by the amoeba’s branch inside the corresponding
neuron (i.e., xk = the area of branch k / the area of the entire region of neuron k).
As shown in Fig. 1C, the numerical value of each state is calculated at each time
step by means of digital image processing of a transmitted light image, where
all numerical calculations are performed at double precision (16 decimals).
[4]
State transition induced by optical stimulation
When light illumination for neuron k is turned on, we represent this
status as yk (t) = 1, and otherwise we represent it as yk (t) = 0 (turned off). The
amoeba’s branch inherently grows and tends to occupy the entire region of the
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Kazuyuki AIHARA
corresponding neuron in principle when yk = 0. Namely, if no illuminations were
applied, all sixteen branches would fully elongate to fill almost all of the neuron
regions.
Recall that the amoeba’s branch alternates between short-term expansion and shrinkage at each oscillation period (Fig. 1D). The state xk , therefore,
changes in a non-monotonous manner. Only when the cumulative value of several periods of the short-term changes in xk becomes positive, the amoeba’s
branch performs the long-term growth to fill the neuron.
On the other hand, the amoeba’s branch avoids being illuminated by
optical stimulation. Although the detailed mechanism of this photoavoidance
behavior still remains unclear, it was speculated that a branch performs the
long-term degeneration when illuminated because the light-induced contraction
enhancement of the gel layer intensifies the sol efflux (extrusion) from the stimulated part at each oscillation period
30)
. Therefore, illuminating neuron k as
yk = 1, we can inhibit the long-term increase in the state xk . When xk is large,
the long-term decrease in xk can be promoted by the illumination yk = 1. As
shown in Fig. 1C, the optical feedback unit applies the illuminations with image
pattern projection by a projector connected to a PC for image processing.
[5]
Neural network dynamics for optical feedback
The optical feedback unit automatically updates each neuron’s illumination status yk ∈ {0, 1} at every interval of ∆t = 6 sec in accordance with the
following neural network dynamics. We designed the dynamics to fit our experiment by modifying the original dynamics of Hopfield and Tank 25) as follows.
Definition 2.1 (Optical feedback dynamics)
yk (t + ∆t) =
1 − f (Σj wkj σ(xj (t); a, b, c)),
σ(x; a, b, c) =
a/(1
( + exp(−b(x − c))),
0 ( if X < θ )
f (X) =
wkj =
1 ( otherwise


−α





 −β
−γ dst(P, Q)







0
),
(if k = P n ∧ j = P m ∧ n 6= m)
(if k = P n ∧ j = Qn ∧ P 6= Q)
(if k = P n ∧ j = Qm ∧ P 6= Q
∧ |n − m| = 1)
(otherwise).
We introduced the sigmoid function σ to enhance the adjustability of the system’s sensitivity, where its parameters are set as a = 1, b = 35, and c = 0.25 in
the experiment, and σ(x) > 0 even if x = 0. The step function f is defined with
a negative threshold θ = −0.5. Other parameters are set as α = 0.5, β = 0.5,
and γ = 0.025 in the experiment.
[6]
Translation of TSP conditions into coupling weights
In the above dynamics, each neuron k is connected to every neuron j
with the non-positive and symmetric coupling weight wkj (i.e., wkj = wjk ≤ 0).
The inhibitory coupling defined by the negative weight wkj (= wjk < 0) creates
a mutually exclusive relationship between the neurons k and j in which the
increase of xj results in the decrease of xk , and vice versa. Namely, when the
amoeba’s branch in neuron j elongates to take a certain threshold value xj (t),
it becomes a trigger for illuminating neuron k as yk (t + ∆t) = 1.
The inhibitory coupling weights are introduced to translate the following
three conditions that should be met by a solution of TSP into a spatiotemporallyvarying illumination condition, as defined below.
Definition 2.2 (Prohibition of revisiting a once-visited city)
If city P is visited the nth, P cannot be visited the m(6= n)th, either before
or after that. This condition is represented by the inhibitory coupling weight
wkj = −α between the neuron k = P n and j = P m. For example, as shown
in Fig. 2B, when the state of neuron A1 increases beyond the threshold value
xA1 (t) = 0.4825, mutually exclusive neurons A2, A3, and A4 are inhibited by
the illuminations.
Definition 2.3 (Prohibition of simultaneous visits to multiple cities)
If city P is visited the nth, no other city Q(6= P ) can be visited at the same time
n. The inhibitory coupling wkj = −β between neurons k = P n and j = Qn
represents this condition. For example, as shown in Fig. 2C, when neuron A1
exceeds the threshold xA1 (t) = 0.4825, mutually exclusive neurons B1, C1, and
D1 are inhibited.
Any configuration satisfying the above two conditions represents a circular route and gives a valid solution of TSP. When selecting a valid combination
of edges in the map, the amoeba can fully elongate up to four branches without
being illuminated. In other words, the optical feedback unit no longer forces the
amoeba to reshape and allows the amoeba to relax maximally as its inherent
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Kazuyuki AIHARA
growth movements of the branches inside all non-illuminated neurons are completed. Thus, we expected that all valid solutions would be more stable than
other (transient) configurations.
Definition 2.4 (Reflection of travel distance between cities)
If city P is visited the nth, and right before or right after that city Q(6= P ) is
visited the m(= n ± 1)th, the cost of traveling this route proportionally reflects
the distance between P and Q written as dst(P, Q). This rule is reflected in the
inhibitory coupling wkj = −γ dst(P, Q) between neurons k = P n and j = Qm,
where if n = 4 then m = n + 1 = 1 (mod 4), and if n = 1 then m = n − 1 = 4
(mod 4).
Under the above setting, as long as the amoeba’s branches select only
a valid combination of edges, the amoeba cannot recognize the distances of
the selected edges because the branches cannot be illuminated. To compare the
distances of shorter and longer edges, therefore, the amoeba needs to select some
invalid combination of edges by elongating some mutually exclusive branches.
By comparing the panels in Figs. 2D and 2E, we can confirm that the
potential risk of being suppressed by the illuminations becomes higher when selecting the longer edge A → B (dst(A, B) = 4) than the shorter edge A → C
(dst(A, C) = 1). That is, the branches A1 and B2 trying to select the longer
edge in Fig. 2D are more frequently illuminated than the branches A1 and C2 in
Fig. 2E, because the former branches A1 and B2 cause other mutually exclusive
neurons (e.g., D2) to have lower threshold values (e.g., xD2 = 0.289 < xD2 =
0.334) so that the exclusive branches can easily evoke the illuminations even by
their slight growth movements. Due to this lower tolerance for perturbations,
a longer solution is less stable than a shorter solution. Therefore, an optimal
(shortest) solution was expected to be the most stable among all solutions, because it allows the amoeba to relax maximally by maximizing its body area while
minimizing the potential risk of being suppressed.
[7]
Notes on neurocomputing for the TSP solution
Our system can be regarded as a neurocomputer for the following two
reasons. 1) The optical feedback applied according to a recurrent neural network
model guides the amoeba’s shape-changing behavior to induce the state transition of neurons. 2) The amoeba’s network-like structure coupling the branches
in the hub of the stellate chamber adjusts the strengths of interactions (e.g., sol
flow pattern) among the branches and is responsible for making the decision on
which branches to grow and degenerate during the state transition of neurons.
The original Hopfield-Tank model 25) was designed to naturally reach a
stable equilibrium in which all neurons become unchanged, and was formulated
in such a way that the stable equilibrium in principle represents a valid solution
of the TSP. A valid solution corresponds to one of the minimum energy points in
the potential landscape established by the network dynamics. Any medium that
can only relax toward equilibrium, therefore, cannot spontaneously bootstrap
itself out of a once-reached solution without an external energy supply.
§3
Results
3.1
Observation of Computing Process
[1]
Solution-searching process
Figure 3 shows an example of the computing process observed exper-
imentally. In the experiment, the map shown in Fig. 2A gives a problem to
be solved. The computing was started by placing a spherically shaped amoeba
(1 ± 0.25 mg) at the hub of the stellate chamber (Fig. 3A). In the early stage,
the spherical amoeba flattened into a disc-like shape in a circularly symmetric
manner by conserving its total volume to be almost constant.
Figure 3B shows the state transition in a period of oscillation at the time
when the amoeba came to have some growing branches that were about to reach
their own threshold values for triggering the illuminations. As noted previously,
each neuron increases its state xi (t) non-monotonously at each oscillation period with the short-term expansion-shrinkage movement of the amoeba’s branch.
Compared with the left and right panels, the center panel has the largest number of illuminated neurons, because at that moment most of the neuron states
increased to take their maximum values for that oscillation period.
At this stage, the illuminations blinked at short interval due to the nonmonotonous changes in neuron states as shown in Fig. 4. Because some mutually
exclusive branches performed the oscillatory movements by invading illuminated
neurons, a wide variety of illumination patterns were evoked within a short time.
Through a trial-and-error process to examine diverse illumination patterns, the
amoeba changed its shape in search of a less frequently illuminated configuration,
i.e., a more stable solution with a shorter total distance.
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Kazuyuki AIHARA
Figure 3C shows the state transition in a half period of the oscillation at
the time when the amoeba entered the final stage of the solution-searching process. Although the amoeba was about to reach a valid solution, the transition of
the illumination pattern was still observed. The elongated branches D1, C2, B3,
and A4 were the least frequently illuminated ones, whereas others were in the
middle of their degeneration movements.
Figure 3D shows that the amoeba reached an optimal solution DCBA
with the shortest total distance 12. A successful solution can be recognized as
a stabilization of the illumination pattern. The branches D1, C2, B3, and A4
selecting the solution sustained their elongated states for about 5 h.
[2]
Finding multiple solutions via self-destabilizations
Figure 3E shows the state transition in a half period of the oscillation
about 5 h after the situation shown in Fig. 3D. At this time, the long-maintained
stabilizing mode of the first solution was spontaneously switched to the destabilizing mode even though no explicit external perturbation was applied. Intriguingly, the amoeba spontaneously destabilized the solution DCBA, as the branch
D2 newly emerged and suddenly started to perform its long-term growth under
the illumination contrary to its photoavoidance response. The growth of the
branch D2 triggered the illuminations for neurons D1, C2, and B3. This perturbation induced some other branches to start their long-term growth-degeneration
movements again, and the solution-searching process involving the transition of
the illumination pattern was restarted.
As shown in Fig. 3F, the destabilizing mode was switched to the stabilizing mode again since the amoeba subsequently reached another shortest solution
BCDA. This solution was maintained for about 1 h.
Afterwards, self-destabilization of the solution occurred once more since
a newly emerged branch A2 invaded the illuminated region as shown in Fig. 3G,
and the solution-searching process was restarted. Figure 3H shows that the
amoeba consequently reached one more optimal solution BADC, which was
maintained for about 1.5 h.
During this 16 h experimental trial, the amoeba eventually found three
different optimal solutions. The effective running time of the computing under
the present experimental conditions was limited to within about 16 h. After the
time limit, the system became unstable enduringly (i.e., not transiently), as the
amoeba’s photoavoidance response became irrecoverably insensitive.
3.2
Statistical Results
[1]
Optimization capability
We carried out 17 experimental trials with a parameter setup almost
identical to the above one∗1 . Figure 5A (top) shows the frequencies of reaching
the optimal (shortest), the second-shortest, and the longest solutions. We certified that a solution had been reached only when the following two conditions
were met: i) the configuration calculated by inverting the illumination pattern
(i.e., 1−yk for all k) represented a valid solution, and ii) the illumination pattern
was stabilized without any change for more than 30 min. White bars indicate
the results on the first solution reached for each trial. In 12 times out of 17
trials, the amoeba reached an optimal solution first. Thus, we can confirm the
high optimization capability of our system, which exhibited a more than 70%
success rate.
In some cases, more than one solution was found in a trial, as selfdestabilization of a once-reached solution occurred several times. The amoeba
reached an average of about 2.06 solutions (the gross number including multiple
transitions into an identical solution) per a trial. Including such cases, the black
bars in Fig. 5A (top) indicate the results counting all solutions found in the
entire observations∗2. We can confirm that the optimal solution has the highest
frequency to be reached by our system.
[2]
Oscillation and information storing for enhanced optimization
Through the observation of solution-searching processes such as the one
plotted in Fig. 4, we made the following two assumptions about essential factors
enhancing our system’s optimization capability. 1) The amoeba can broaden
a search space owing to the fluctuating oscillatory movements of its branches.
This is because the non-monotonous changes in neuron states, produced through
the oscillatory movements, are capable of creating the diversity for evoking a
wide variety of illumination patterns due to their fluctuations. 2) The amoeba
can choose an optimal solution from the broadened search space because it can
∗1
The city maps used in the experimental trials were given as geometrically identical to
the map shown in Fig. 2A, but their city labels were shuffled randomly to eliminate the
possibility that the results are biased depending on whether the amoeba’s movements are
rotationally symmetric or not.
∗2
When multiple, indentical solutions were reached during a trial, these solutions were
counted as a single solution.
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Kazuyuki AIHARA
grow only the least frequently illuminated branches and can make other ones
degenerate. This is attributable to the amoeba’s capability of determining if
each branch should grow or degenerate depending on stored information on its
illuminated experiences.
To verify assumption 1, we carried out a series of control experiments
in which the interval ∆t = 6 sec to update the illumination pattern is extended.
We set the new interval as ∆t = 84 sec because it is close to an averaged period of the amoeba’s contraction-relaxation oscillation. With this degradation
in temporal resolution, the optical feedback unit becomes incapable of sensing
the non-monotonous changes in neuron states. Namely, the short-term decrements at each oscillation period are ignored, and only the long-term increments
become responsible for determining the illumination pattern. This enables us
to examine whether the system’s optimization capability is reduced if changes
in neuron states are monotonous. In other words, the control experiment simulates a virtual computing process executed by an imaginary amoeba incapable
of performing the oscillatory movements.
Figure 5B (top) shows the frequencies with which each of the three
kinds of solutions were reached in the control experiments. In only 2 of 10 trials,
the amoeba reached the optimal solution first. Comparing these results with
those in Fig. 5A (top), it is clear that the optimization capability was reduced
significantly as the effect of the amoeba’s oscillatory movements was disabled.
Next, we compare Fig. 5A (bottom) and Fig. 5B (bottom) which show
histograms of the number of illumination patterns evoked during the solutionsearching process until reaching the first solution. Obviously, a wider variety of
illumination patterns were examined in the original experiments, reflecting the
effect of the oscillatory movements (Fig. 5A [bottom]). Thus, we claim that the
amoeba’s optimization capability was enhanced by the broadening of the search
space by its oscillatory movements.
To verify assumption 2, we examined whether the amoeba’s branch
grows or degenerates, reflecting its cumulative illuminated time. As shown in
Fig. 5C, it was confirmed that less frequently illuminated branches tend to grow
longer and reach a solution whereas more frequently illuminated ones tend to degenerate. Therefore, it is likely that the amoeba can choose an optimal solution
due to its ability to store information on illuminated experiences.
§4
Time Series Analysis
4.1
Surrogate Data Analysis
We saw that the amoeba’s oscillatory movements made a significant
contribution to enhance the optimization capability of our computing system.
The oscillatory movements were observed to contain irregular fluctuations, in the
sense that their amplitude, period, and phase were changed nonuniformly with
time. To examine whether the fluctuations are generated by stochastic noise or
by deterministic chaos 31) would be an interesting subject to promote a better
understanding of essential factors to enhance the optimization capability.
In this section, we verify our hypothesis that the fluctuating oscillatory
movements are characterized as behavior generated by a set of coupled chaotic
oscillators. Thus, we need to show the following five properties—that is, the
oscillation of each branch is chaotic 31) , which means its time series has (1) serial
dependence (i.e., deterministic property implying causality), (2) nonlinearity, (3)
nonperiodicity, and (4) sensitive dependence on initial conditions; and multiple
branches are coupled together as they exhibit (5) synchronization.
We employed surrogate data analysis 32, 33, 34, 35, 36) which is a method
often used to test hypotheses in nonlinear time series analysis. The method is
similar to the Monte Carlo method in statistics. First, we set a null-hypothesis.
For example, if we evaluate the existence of a nonlinear property in time series
data, we need to set a null-hypothesis that the time series is linear. Second,
duplicating the original time series dataset in such a way that only the properties assumed in the null-hypothesis are preserved while other properties are
randomly destroyed, we generate a set of mutated datasets called “surrogates.”
Third, we compare the original dataset with the surrogates using a test statistic.
Only if the value for the original dataset is out of the interval obtained from
the surrogates, the null-hypothesis is rejected. For example, the existence of
nonlinearity becomes plausible if the null-hypothesis assuming linearity could be
rejected. We use several types of surrogates described below.
4.2
Chaos in an Oscillator
We analyzed the time series data of the area xi (t) properly reflecting
the oscillation of each branch i. It is desirable for our surrogate data analysis
that the datasets of xi (t) fed into the analysis are stationary with no trend.
Therefore, first we detrended the datasets by removing 5% of the components of
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Kazuyuki AIHARA
the Fourier transforms from the low frequencies. To extract only the phase from
the time series, we applied the Hilbert transform. In the following analysis, we
used the sine of the extracted phase. We tested whether the preprocessed data
are stationary or not by the method of Kennel 37) . We found that about 87% of
datasets were confirmed as stationary (Fig. 6A).
[1]
Determinism
To evaluate the deterministic property, we set the first null-hypothesis
that the time series does not have serial dependence. To test this, we used
Random Shuffle Surrogates 32) . A set of 199 random shuffle surrogates was
generated for the preprocessed data of each branch by randomly exchanging the
time indexes of the time series. We chose 199 for the number of surrogates
so that each individual test had a significance level of 1%. To compare the
preprocessed dataset with its surrogates, we used the Wayland statistic 38) . The
Wayland statistic takes a small value close to 0 if a time series is generated from
a deterministic system, and a larger value if not. We calculated the Wayland
statistic using embedding dimensions from 1 to 12. If the Wayland statistic for
the preprocessed dataset was out of the interval obtained from its surrogates
in more than or equal to 1 embedding space, then we regarded that the nullhypothesis was rejected. The significance level of the combined test is less than
5%. Among the preprocessed datasets for branches which were identified as
stationary, 74% of them showed the rejection of the null-hypothesis that the
time series does not have serial dependence.
[2]
Nonlinearity
To evaluate the nonlinear property, we set the second null-hypothesis
that the time series was generated by a monotonous transformation of linear
noise. We used Iterative Amplitude Adjusted Fourier Transform Surrogates 33)
for this purpose. This type of surrogates preserves the auto-correlations of the
original time series approximately and the distribution of points perfectly. We
used the Wayland statistic as a test statistic. Out of the preprocessed datasets
that have passed the previous surrogate test, 38% of them showed the rejection
for the second surrogate test.
[3]
Nonperiodicity
To evaluate the nonperiodic property, we assume in the third and fourth
null-hypotheses that the time series is a periodic orbit in its essence. The third
one assumes a periodic orbit driven by Gaussian noise. To generate surrogates
that follow this null-hypothesis, we used Pseudo-Periodic Surrogates proposed
by Small et al. 34) . We used the Wayland statistic for a test statistic. In this
case, all preprocessed data that passed the two previous surrogate tests showed
the rejection of the null-hypothesis.
The fourth one assumes that the times series is generated from a noisy
periodic orbit. We used the method of Luo et al. 35) for generating a set of 39
surrogates that are consistent with this null-hypothesis. We chose 39 for the
number of surrogates since the datasets were too short. We used the Wayland
statistic as a test statistic. Hence, the significance level of the individual test
became 5%. To make the significance level of the combined test 5%, we regarded
that the null-hypothesis was rejected if in more than 1 embedding dimension out
of 12, the individual test was rejected. Then, 90% of datasets which passed the
previous 3 surrogate tests showed the rejection of the null-hypothesis. Summarizing the above results, in about 22% of datasets tested, the time series data
were likely to be generated by chaotic dynamics (Fig. 6A).
[4]
Sensitive dependence on initial conditions
In some of the branches, when we evaluated the time evolution of the
distance between points in a phase space and their closest neighbors, the distance increased exponentially (Fig. 6B). That is, we could recognize the sensitive
dependence on initial conditions. This also gives evidence suggesting that the
branch’s behavior is characterized as chaotic.
4.3
Synchronization of Oscillators
We used Twin Surrogates 36) for evaluating the phase synchronization
between two branches. Twin surrogates are random datasets that preserve the
dynamics of the original system approximately, except for initial conditions.
Therefore, the null-hypothesis here is that there is no interaction between the
dynamics of the branches.
We generated 199 sets of twin surrogates for each preprocessed dataset.
Then we used the Hilbert transform to extract the phase θj (t) for branch j. We
calculated the following quantity as a test statistic:
|
N
1 X
exp i(θj (t) − θk (t))|.
N t=1
18
Kazuyuki AIHARA
This quantity takes a large value close to 1 if the difference between the two
phases are always close to a constant, and a small value close to 0 otherwise.
Let sj,k be the test statistic for a pair of branches j and k obtained from
the preprocessed data. In addition, denote by mj,k and σj,k the mean and the
standard deviation of the test statistic for a pair of branches j and k obtained
from the twin surrogates. Then we can obtain the z-score of the pair of branches
j and k by
−
sj,k − mj,k
.
σj,k
When searching for a solution, the amoeba’s branches were phase synchronized with almost uniform strength (Fig. 6C). The synchronization was inphase. When the solution became stabilized, the phase synchronization became
weak and asymmetric. Especially in four branches elongated to select the solution, the phase synchronization became weaker compared with other shorter
branches.
We also investigated the relationship between the strength of phase synchronization and area occupied by the branches. We found that weakly synchronized branches tend to grow larger (Fig. 6D). Conversely, most of the strongly
synchronized branches degenerated properly in response to the illuminations.
Therefore, Figs. 6C and 6D are pieces of evidence suggesting that our system
exploits phase synchronization among oscillating branches to suppress undesirable growth movements of the branches and to enhance its controllability.
4.4
Summary
Our time series analysis showed the following facts. 1) A significant
proportion of the amoeba’s branches could be characterized as chaotic oscillators.
However, some portion of the branches were not verified as chaos. We think that
in these branches, either dynamical noise was strong or their dynamics were highdimensional systems. 2) The branches were phase-synchronized strongly and
uniformly while in the solution-searching process, but when stabilizing a solution
the synchronization among longer branches selecting the solution became weaker.
§5
Discussion
5.1
Chaos
It has already been shown theoretically that coupled chaotic oscillators
can fluctuate wildly, synchronize coherently, and desynchronize spontaneously
in a nonperiodic but nonrandom manner 42) . All these properties seem to be
exploited in our computing scheme as essential factors to enhance its optimization capability. In this section, we will discuss the implications of chaos in our
computing scheme.
[1]
Chaos creates diversity
We saw that the amoeba’s oscillatory movements have the effect of
broadening a search space and contribute to the enhancement of our system’s
optimization capability (Figs. 5A and 5B). Indeed, the oscillatory movements
producing non-monotonous changes in neuron states were capable of evoking a
wider variety of illumination patterns compared with monotonous movements
without oscillations. In addition, compared with a case in which the amoeba’s
branches can only perform regular oscillations (e.g., sine waves), it seems obvious
that chaotic oscillations capable of fluctuating wildly would be more powerful
in evoking diverse illumination patterns. This is because chaotic oscillations
can create irregular gaps in their amplitude, period, and phase that lead the
amoeba’s branches to take diverse combinations of neuron states. However, it
remains to be seen whether the power of chaotic oscillations to create the diversity is stronger than that of stochastic noise under the settings of our computing
scheme.
[2]
Chaos provides controllability
Chaos that involves its own deterministic dynamics allows itself to be
controlled rapidly to reach a desirable orbit by suitable external forcing 41) . This
controllability would be an advantage of chaotic systems, because it cannot be
achieved in stochastic systems having no such dynamics.
Our system can reach a valid solution, since the amoeba’s branches,
which elongate moderately to evoke some illuminations, can be subdivided into
those degenerated by the illuminations and those which grow further without
being illuminated (Fig. 5C). The degenerated branches tend to exhibit strongly
synchronized oscillations, whereas the grown ones tend to be weakly synchro-
20
Kazuyuki AIHARA
nized (Fig. 6D). It seems that the strongly synchronized branches tend not to
grow further, as their oscillations are restricted not to fluctuate freely. That is,
although the amoeba’s oscillations are originally chaotic, they are considered to
be controllable through the application of illuminations so that strongly synchronized regular oscillations can be performed. In other words, the controllability
of chaos would enable the amoeba’s branches to be clearly differentiated into
weakly synchronized ones and strongly synchronized ones. We consider that the
amoeba can select a solution more rapidly because its branches can be split more
distinctively into grown ones and degenerated ones owing to their controllability.
[3]
Chaos produces self-destabilization
One of our system’s notable features is its capability of finding multiple solutions via several iterations of self-destabilization. Self-destabilization is
a spontaneous behavior of the amoeba in which its branch starts the long-term
growth movement under the illuminated condition, contrary to its photoavoidance response. This induces a once-reached stable solution to be destabilized
even without explicit external perturbation.
Self-destabilization occurs nonperiodically at stochastically distributed
sites, when observed macroscopically. Although its mechanism is under investigation, we consider that it would be attributable to the existence of chaotic
dynamics capable of exponentially amplifying intrinsic fluctuations at the microscopic level to influence the destabilization at the macroscopic level. Indeed, the
system’s behavior is chaotic as its time evolution is unstable and unreproducible.
The capability of self-destabilization, however, is robustly maintained and qualitatively reproducible. This resembles the robustness of strange attractors of
chaotic systems. Possible sources of the intrinsic fluctuations would be thermal
noise, local nonuniformity of chemical distributions, and local unevenness of the
gel layer’s stiffness distribution. Some form of positive feedback effect produced
by the coupling of chemical and hydrodynamic processes may be responsible for
expanding the tiny fluctuations into the extensive and sustained destabilization
movement.
We cite a series of previous experimental studies suggesting that selfdestabilization is caused through chaotic dynamics. Takamatsu et al. observed
the spatiotemporal patterns of the amoeba’s thickness oscillation under simple boundary conditions 1, 2, 3, 4) . To keep the horizontal shape unchanged, the
amoeba was placed inside a chamber that was shaped as a ring network with
several nodes. This system was viewed as a coupled oscillator system in which
the amoeba’s parts inside the network nodes corresponded to oscillators, and
these oscillators were connected through tubular channels of the amoeba itself.
Depending on the number and connection geometry of the oscillators, there
existed a number of quasi-stable oscillation modes capable of sustaining stationary differences among the oscillators’ phases for several periods. We point out
that in this system the amoeba performed spontaneous transitional behavior
among the multiple quasi-stable modes even without external perturbation 4) .
Although the mechanism of this behavior has not been clarified, Takamatsu et
al. suggested the existence of a dynamical process involving chaotic dynamics to
destabilize the quasi-stable modes through spontaneous desynchronization. Indeed, similar self-destabilizing transitional behavior, called “chaotic itinerancy,”
has been shown to be produced by some theoretical models of coupled chaotic
oscillators
42)
.
On the other hand, the efficiency of chaotic dynamics for combinatorial
optimization has already been demonstrated theoretically with chaotic neural
network models 43, 44) . It was confirmed that these models exhibit high optimization capabilities as the self-destabilizing transitional behavior assists in the
search for a global optimum solution without being stuck at local optima.
5.2
[1]
Multilevel Optimization Processes
Self-destabilization and conflicted optimization processes
What does self-destabilization of the once-reached solution imply for the
amoeba’s survival? We conjecture that self-destabilization reflects the existence
of multiple optimization processes involving conflicting interests at different levels.
As long as the amoeba’s branches can only behave in a manner that is
beneficial to their own partial domains, they cannot always provide long-term
benefits for the whole body. It seems reasonable to consider that the amoeba’s
branch optimizes its benefits when it grows under a favorable non-illuminated
condition or degenerates under a stressful illuminated condition. Therefore,
when the amoeba selects a valid solution, it appears that all branches correctly
completed the optimization processes of their own benefits. These benefits,
however, should be considered as no more than short-term benefits of body
parts, as described below.
22
Kazuyuki AIHARA
The amoeba’s branches are assumed to perform their growth movements
inherently in order to search for nutrient sources, because they produce the
shape-changing behavior resulting in the locomotion of the whole body. When
selecting a valid solution, further growth movements of the branches are blocked
by the stellate chamber and illuminations. Because the agar plate in our experiment contains no nutrients, as long as the amoeba stably maintains the solution
without changing its shape, there is no chance of nutrient acquisitions. The
amoeba can only live on its own stored energies ingested before the experiment.
Therefore, permanently maintaining the solution would not be beneficial to the
whole body. Rather, such maintenance would imply a stalemated situation leading to death from starvation.
To optimize the long-term benefit of the whole body, the amoeba has
to change its shape again to increase the chance of finding nutrient sources.
To restart the shape-changing behavior, however, the amoeba needs to spontaneously destabilize the solution by growing its branch under the illuminated condition, contrary to its photoavoidance response. Clearly, this self-destabilization
behavior conflicts with the benefit of the branch averting the illuminated condition. Thus, we can recognize the existence of two optimization processes involving conflicting interests at different levels, i.e., the lower-level benefit of
the branches gained by pursuing their short-term favorable conditions and the
higher-level benefit of the whole body gained by increasing the long-term possibilities of nutrient acquisitions.
The amoeba thus faces a trade-off between the lower-level benefit and the
higher-level benefit. Self-destabilization of the solution, therefore, may be viewed
as behavior representing the amoeba’s self-disciplined decision to give preference
temporarily to the higher-level benefit over the lower-level benefit. That is, we
may be able to consider that the amoeba took the risk of being illuminated
and chose to invest its effort to grow under the stressful illuminated condition
in future nutrient acquisitions. The capability of this kind of self-disciplined
decision-making would be essential for the survival of the amoeba that is required
to search for nutrient sources in harsh environments, because it enables the
amoeba surrounded by aversive stimuli to break through the stalemated situation
with enterprising responses in expectation of future nutrient acquisitions.
[2]
Self-disciplined computing
Conventional digital computers can only operate as instructed in ad-
vance by their programs, and cannot create new criteria for their operations on
their own. This weakness stems from the fact that the digital computation was
designed to be executed only on a single physical level with the goal of processing of binary sequences, such that all functions can be translated into binary
operations. The fundamental unit of information is a binary digit, which cannot be divided further into smaller units, and a number of the units cannot be
assembled together to form a larger unit. Therefore, the single physical level at
which the binary sequence is processed has no access to the flows of information
from lower and higher physical levels. This inaccessibility to the multilevel information flows is what makes it impossible for conventional digital computers
to produce useful information on their own—i.e., without a priori programs.
In contrast, a biological organism is a hierarchically structured system in
which multiple optimization processes run concurrently at different levels. Often
these multilevel optimization processes have conflicting interests, such as a tradeoff between short-term and long-term benefits. Despite the lack of a supervisory
decision program, some organisms are capable of making a reasonable decision to
mediate the trade-off in a self-disciplined manner, as hinted at in our experiment.
In other words, the multilevel optimization processes can decide what to optimize
by determining reasonable criteria for making judgments about what is good and
bad on their own.
The self-disciplined decision capability would be attributed to the fact
that the multiple optimization processes in the organism have high accessibilities
to the multilevel information flows because they run concurrently by sharing
common computational resources such as energies and structured substances.
As mentioned previously, chaotic dynamics capable of amplifying information at
a lower level to influence a higher level would be responsible for producing the
multilevel information flows with high accessibilities.
If we could develop a computing system capable of making self-disciplined
decisions, the system would have practical advantages. For example, if such a
system could mediate the trade-off between the exploitation of programmed
knowledge and the exploration of unprogrammed knowledge, it would be able to
cope with hard decision-making problems of how to select a reasonable solution
from among several options in expectation of future benefit.
If the essential dynamics of the multilevel optimization processes involving the self-disciplined decision capability could be extracted, our computing scheme would be implemented not only by the amoeba but also by other
24
Kazuyuki AIHARA
faster oscillatory media capable of multilevel self-organization such as chemicals 45, 46, 47) , nanoparticles 48) , biomolecules 49, 50) and optical devices 51) . These
self-organizing oscillatory media may be able to perform massively parallel computing without introducing elaborate microscopic control techniques to regulate
precisely the molecular-scale elements. That is, all we have to do is guide the
self-organization processes loosely by some macroscopic feedback techniques.
§6
Conclusion
In this paper, we demonstrated a neurocomputing system that incorpo-
rates a photosensitive amoeboid organism as a computing substrate to perform
combinatorial optimization with the assistance of optical feedback applied according to a recurrent neural network model. We showed our system’s high
optimization capability in solving the four-city traveling salesman problem.
We pointed out that the fluctuating oscillatory movements of the amoeba’s
branches are essential for examining a broad search space and for enhancing our
system’s optimization capability. This means that our computing scheme cannot
be effective when implemented by some other photosensitive media incapable of
performing spatiotemporal oscillatory behavior. Additionally, it was suggested
that the amoeba can correctly choose an optimal solution from the search space
because its branches can store information on illuminated experiences.
Applying several surrogate data analyses, we showed that the fluctuating oscillatory movements can be characterized as behavior generated by coupled
chaotic oscillators. We emphasized the importance of the existence of chaotic
dynamics in our computing scheme by pointing out that chaos creates the diversity to broaden the search space, provides the controllability to quickly reach
a solution, and produces the self-destabilization of the once-reached solution to
find a number of better solutions without the help of external perturbations.
Our system presents a model of an unconventional computing scheme
implemented by a particular oscillatory medium in which multiple optimization
processes run concurrently on different spatiotemporal scales. In our experiment,
it was hinted that the multilevel optimization processes producing the amoeba’s
oscillatory movements have the capability of making self-disciplined decisions,
such as the decision to take a short-term risk in expectation of a long-term
future benefit. In exploring the potential of our computing scheme to perform
some unique functions that are difficult for conventional digital computers to
reproduce, the self-disciplined decision capability will be a key feature.
Acknowledgment
The authors thank Prof. Koichiro Matsuno, Prof.
Yukio-Pegio Gunji and Dr. Asaki Nishikawa for their support in discussion.
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ACM 28, 5, pp. 464-480, 1985.
51)
Christodoulides, N., Lederer, F. and Silberberg, Y., “Discritizing Light Behaviour in Linear and Nonlinear Waveguide Lattices,” Nature 424, pp. 817-823,
2003.
§7
Commun.
Appendix: Experimental Setups
The amoeba was fed oat flakes (Quaker Oats, Snow Brand Co.) on
a 1% agar gel at 25◦ C in the dark. The stellate container structure (thickness approximately 0.2 mm) is made from an ultrathick photoresist resin (SU-8
3050, Kayaku MicroChem Corp.) by a photolithography technique, and was
coated with Au using a magnetron sputterer (MSP-10, Shinkuu Device Co.,
Ltd.). The experiments were conducted in a dark thermostat and humidistat chamber (27±0.3 ◦ C, relative humidity 96±1%, THG062PA, Advantec Toyo
Kaisha, Ltd.). For transmitted light imaging, the sample was placed on a surface
light guide (MM80-1500, Sigma Koki Co., Ltd.) connected to a halogen lamp
light source (PHL-150, Sigma Koki Co., Ltd.) equipped with a band-pass filter
(46159-F, Edmund Optics Inc.), which was illuminated with light (intensity 2
µW/mm2 ) at a wavelength of 600±10 nm, which does not affect the amoeba’s behavior 30) . The intensity of the white light (monochrome color R255:G255:B255)
illuminated from the projector (3000 lm, contrast ratio 2000:1, U5-232, PLUS
Vision Corp.) was 123 µW/mm2 . The outer edge of the circuit (the border
between the structure and the agar region) was always illuminated to prevent
the amoeba from moving beyond the edge.
§8
Author’s Profile
Masashi Aono, Ph.D.: He is a research scientist at Advanced Science
Institute (ASI) in RIKEN, Japan. After graduated from Faculty of Environment and Information Studies in Keio University (1999), he received his M.A.
(2001) and Ph.D. (2004) from Department of Information Media Science, Kobe
University. He is interested in mathematical modeling of complex systems and
experimental development of bio-computers.
Yoshito Hirata, Ph.D.: He received B.E. and M.E. from Department
of Mathematical Engineering and Information Physics, The University of Tokyo
in 1998 and 2000, respectively. He obtained Ph.D. from School of Mathematics
and Statistics, The University of Western Australia in 2004. He is now a project
research associate at Institute of Industrial Science, The University of Tokyo.
He is specialized in analysis, modeling and prediction based on time series data.
Masahiko Hara, Ph.D.: He received the Dr. degree in organic materials engineering in 1988 from Tokyo Institute of Technology (TITech), Japan,
and studied soft matter physics in Univ. of Manchester, UK during his graduate
course. Currently, he is Professor in TITech, and Director of Global Collaboration Research Group in RIKEN. His research interests include self-assembly,
spatio-temporal and flucto-order functions for emergent functions.
Kazuyuki Aihara, Ph.D.: He received the B.E. degree in electrical
engineering in 1977 and the Ph.D. degree in electronic engineering 1982 from
the University of Tokyo, Tokyo, Japan. Currently, he is Professor at Institute
of Industrial Science, Graduate School of Information Science and Technology,
and Graduate School of Engineering in the University of Tokyo. He is also
Director of Aihara Complexity Modelling Project, ERATO, Japan Science and
Technology Agency. His research interests include mathematical modelling of
complex systems, parallel distributed processing with complex networks, and
time series analysis of real-world data.
30
Kazuyuki AIHARA
A
D
Oscillation Period: 1-2min
Gel Layer
contracting
relaxing
Sol
E Actomyosins
in Gel Layer
Relaxing
Sol
Gel
expanding
B
relaxing
contracting
Contracting
Sol
Sol
Gel
shrinking
C
contracting
relaxing
PJ
Relaxing
VC
Sol
SM
PC
LS
Fig. 1
Sol
Gel
expanding
(A) An individual unicellular amoeba of the true slime mold Physarum polycephalum
(scale bar = 7 mm). (B) An Au-coated plastic chamber on an agar plate (scale bar = 7 mm).
The amoeba acts only inside the chamber where agar is exposed, because of its aversion to
metal surfaces. (C) Experimental Setup. For transmitted light imaging using a video camera
(VC), a surface light source (LS) beneath the sample amoeba (SM) was employed to emit light
of a specific wavelength known to have no significant effect on the amoeba’s behavior
30)
. The
recorded image was digitally processed using a PC to update the high-intensity monochrome
image (visible white light) for illumination with a projector (PJ). See the Appendix for details.
(D) Schematic illustration of the amoeba’s body architecture. (E) Schematic illustration of
the contraction-relaxation oscillation of actomyosin systems in the gel layer.
A
D
4
A
2 1
2
D
4
C3
D2
B1
=.4825
C4
B2
C3
D2
C4
B2
B3
C3
C2 C1 B4
Fig. 2
A1 = C2
= .3
B1
B2 C4
B3
B2
C2 C1 B4
D4 A1 A2
A3
A4
D2
D2
= .289
D1
B3
C3
D4 A1 A2
B1
A3
A4
D2
B1 D1
C4
D3
A1
=.4825
D1
D4 A1 A2
D3
C2 C1 B4
A3
A4
B3
C2 C1 B4
A1 = B2
= .3
D1
C3
D4 A1 A2
B1
B2
C3
A3
A4
D2
B3
D3
B3
D3
C2 C1 B4
C
A1 = C2
= .5
B1 D1
D4 A1 A2
A1
D1
A3
A4
D2
C2 C1 B4
A3
A4
D4 A1 A2
D3
B2 C4
C4
D4 A1 A2
D3
A1 = B2
= .5
D1
C
15
B
A3
A4
D2
B
E
D4 A1 A2
D3
D3
A3
A4
D2
D2
= .334
B1 D1
B1
B2 C4
C4
B3
C3
C2 C1 B4
B2
B3
C3
C2 C1 B4
(A) The map of four cities for the TSP solution. (B) and (C) Optical feedback
representing conditions 1 and 2 of TSP. See main text for details. B: Prohibition of revisit to
a once-visited city. The parameters are set as α = 0.5 and β = γ = 0.0 so that only condition
1 can be emphasized. C: Prohibition of simultaneous visits to more than one city. Parameters
β = 0.5 and α = γ = 0.0 are set to emphasize condition 2. (D) and (E) A simulation
demonstrating how condition 3, reflecting the difference between the shorter and longer routes,
creates the difference in the transitions of illumination patterns in D and E. Parameters α =
0.5, β = 0.5, and γ = 0.025 are set as identical to the values in the experiment. Time advances
from the top (t) to the bottom (t′′ ), where t < t′ < t′′ . D: Branches xA1 (t) = 0.5 and xB2 (t) =
0.5 trying to select the longer route A → B with distance 4 (top). Branch B2 is inhibited by
illumination yB2 (t′′ ) = 1 when branch D2 appearing as a perturbator elongates beyond the
threshold value xD2 (t′′ ) = 0.289 (bottom). E: Branches xA1 (t) = 0.5 and xC2 (t) = 0.5 trying
the shorter route A → C with distance 1 (top). Branch C2 is inhibited since the perturbation
by branch D2 exceeded xD2 (t′′ ) = 0.334 (bottom). The difference in the route distances is
reflected in the difference in the threshold values of the perturbator D2, i.e., selecting the longer
route A → B, the amoeba is inhibited more easily by a smaller perturbation xD2 (t′′ ) = 0.289
(< xD2 (t′′ ) = 0.334). Because the amoeba can correctly recognize information on the distances
only when inhibited by illuminations, it is necessary for some mutually exclusive branches
(e.g., D2) to expand in order to act as perturbators for evoking the illuminations. In general,
a branch (e.g., D2) becomes free to expand while the illumination for inhibiting its expansion
is canceled (e.g., yD2 (t′ ) = 0) due to short-term shrinkage movements of some other branches
(e.g., xA1 (t′ ) = xB2 (t′ ) = 0.3 and xA1 (t′ ) = xC2 (t′ ) = 0.3 shown in D and E [middle],
respectively). This is why the amoeba’s oscillatory movements producing non-monotonous
changes in neuron states via the short-term shrinkage movements (Fig. 1D) are essential in
our computing scheme.
32
Fig. 3
Kazuyuki AIHARA
Computing process to solve the four-city TSP. (A) Initial configuration recorded as
a transmitted light image before digital image processing. (B) Early stage of the solutionsearching process. The three panels show the time evolution within a period of oscillation.
The phase of vertical thickness oscillation is binarized into the relaxing (thickness-increasing)
and contracting (-decreasing) states, represented by the black and gray pixels, respectively. (C)
Final stage of the solution-searching process. The two panels show the time evolution within
a half period of oscillation. (D) The first-reached solution DCBA with the shortest travel
distance 12 (duration ≃ 5 h). (E) Self-destabilization of the solution shown in D. The newly
emerged branch D2 started to invade the illuminated region, contrary to its photoavoidance
response. (F) The second-reached solution BCDA with the shortest distance (duration ≃ 1
h). (G) Self-destabilization of the solution shown in F. The newly emerged branch A2 invaded
the illuminated region. (H) The third-reached solution BADC with the shortest distance
(duration ≃ 1.5 h).
Fig. 4
Spatiotemporal plot of oscillatory movements of the amoeba’s branches in the solution-
searching process. Non-monotonous changes in neuron states xi (t) are represented by jagged
profiles. Each profile xi (t) is colored with dark gray at periods when the corresponding neuron
is illuminated. Within the time window between about 15 min and 30 min, various illumination
patterns were examined, since each illumination blinked frequently. The background color is
darker after about 30 min to indicate that the system reached a valid solution of the TSP.
Note that the time series data in this figure were taken from an experimental trial different
from that shown in Fig. 3.
34
Fig. 5
Kazuyuki AIHARA
Statistical results. (A) and (B) Frequency distribution of reached solutions (top) and
histogram of the number of examined illumination-patterns before reaching the first solution
(bottom). The results shown in A and B were obtained from 17 trials of the original experiments (∆t = 6 sec) and 10 trials of the control experiments (∆t = 84 sec), respectively. In the
top panels, the white bars indicate the results only for the first-reached solutions, whereas the
black bars are for all solutions reached in the entire observations. (C) Comparison between
the cumulative illuminated time and the dimension (area) xi of the amoeba’s branch when
reaching the first solution. The cumulative time was calculated as a ratio of the illuminated
time to the elapsed time required to reach the first solution. The solid line and broken lines
show that the median and quartile points are dependent on the illuminated time.
Fig. 6
Summary of results for time series analysis. (A) Classifications of branches based on
results of surrogate data analysis. The stationarity was tested by the method of Kennel
37)
.
Surrogates used here are random shuffle surrogates (RSS), iterative amplitude adjusted Fourier
transform surrogates (IAAFT), Small’s pseudo-periodic surrogates (Small’s PPS), and Luo’s
pseudo-periodic surrogates. (B) The distance with the closest point increased exponentially.
Here we used the method of Ref.
39)
implemented in TISEAN package
40)
for the calculation.
(C) Network diagrams of phase synchronization. The time advances from i) to ii). Panel i)
shows synchronization pattern among the amoeba’s branches in solution-searching process, and
panel ii) shows that of when the amoeba stabilized the first-reached solution. The width of each
line indicates the strength of phase synchronization between the corresponding pair of branches.
Black labels represents branches elongating to select a solution, whereas other shorter branches
are indicated by gray labels. Note that these panels show results of data taken from another
experimental trial different from the one shown in Fig. 3.(D) Relationship between the strength
of phase synchronization and the area occupied by a branch. The horizontal axis represents
the synchronization level, which is the largest z-score of synchronization index among all pairs
related to a branch. The vertical axis represents the dimension of branch when reaching the
first solution. The solid line and broken lines show that the median and quartile points have
dependences on the strength of phase synchronization.

Amoeba-Based Chaotic Neurocomputing: Combinatorial

Transcription

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