Brain-like Computing Based on Distributed Representations and Neurodynamics

1

Brain-like Computing Based on Distributed
Representations and Neurodynamics
Ken YAMANE and Masahiko MORITA
University of Tsukuba
1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8573 JAPANyamane@bcl.esys.tsukuba.ac.jp, mor@bcl.esys.tsukuba.ac.jp

Received 29 May 2009
Revised manuscript received 30 October 2009

Abstract A key to overcoming the limitations of classical artificial
intelligence and to deal well with enormous amounts of information might
be brain-like computing in which distributed representations of information are processed by dynamical systems without using symbols. We
present a method for such computing. We constructed an inference system using a nonmonotone neural network, which is a kind of recurrent
neural network with continuous-time dynamics. This system deduces aconclusion according to state transitions of the network in which knowledge is embedded as trajectory attractors. It has the powerful ability ofanalogical reasoning without special treatment for exceptional knowledge.
We also propose a method of linking different neurodynamical systems
and show that two mutually interacting systems can process complex
spatiotemporal patterns.
Keywords Distributed Representations, Neurodynamics, Nonmonotone Neural Network, Selective Desensitization, Brain-like Information
Processing.

§1

Introduction

It has been noticed that classical artificial intelligence (AI) based on symbolic manipulation has the symbol grounding problem1) and the frame problem2, 3) .

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Ken YAMANE and Masahiko MORITA

For those reasons, classical AI often does not work well in the real world, which
is filled with unlimited amounts of information. In contrast, although animals
such as dogs and cats do not seem to have languages or the ability to manipulate symbols, they evidently “think” or even “reason” and can deal better withmany real-world problems than AI systems can. In their brains, it is thoughtthat information is represented distributedly by activity patterns of neurons andprocessed without being symbolized to be other patterns, of which processes canbe regarded as pattern dynamics4, 5, 6) .
If information is not represented by symbols as in the animal brain,
then the symbol grounding problem does not arise. Distributed representations7)
facilitate analogy based on the similarity between representations, which seems
to enable animals to address the frame problem. Accordingly, it is expected thatsuch processing as is done in the animal brain will overcome some limitations ofclassical AI. We designate this approach as “brain-like computing”, which aims
to complement classical AI approaches rather than to surpass or expel them.
Although many other “brain-like computing” systems have been developed, few systems can manipulate patterns to reason without using symbols or
equivalent representations. Regarding neural network models of inference, allexisting models either require local representations that correspond one-to-one
to things or concepts8, 9, 10, 11, 12) or are combined with symbol processing models to form hybrid systems13, 14, 15) because no scalable neural network has beendeveloped that is based on distributed representations and can simulate an arbitrary finite automaton. For example, the Elman network16) , in reality, cannot
learn well to simulate a given large-scale finite automaton.
We previously investigated this problem to find that it arises from anaveraging effect of synaptic weights caused by one-to-many correspondence between input and output patterns. We also found that although conventional
neural networks in general cannot escape from this averaging effect, it can be
avoided by introducing a novel method of contextual modification termed selective desensitization, and developed a neural network model that can simulate anylarge-scale finite automaton without using symbols or local representations17, 18) .
In this paper, we present a brain-like computing system that can makeinferences based solely on distributed representations using pattern dynamics ina kind of neural network. We examine its possibilities using simulation experiments. We also propose a novel method of linking different neural networks toenhance the information-processing capability of neurodynamical systems.

3

Brain-like Computing Based on Distributed Representations and Neurodynamics§2

Basic Principles

2.1

Nonmonotone Neural Network

First, we explain the nonmonotone neural network19) , which is a fullyrecurrent network of which elements have a nonmonotonic output function. Although it will be modified later for use as an inference engine, the original model
is described by the following dynamics:
τ

n

dui
= −ui +
wij yj + zi ,
dt
j=1

(1)

yi = f (ui ).

(2)Here, ui is the internal potential of the i-th element, yi is the output, wij isthe connection weight from the j-th element, zi is the external input, n is thenumber of elements, τ is a time constant, and f (u) is the output function givenas′

1 − e−cu 1 − ec (|u|−h)f (u) =·,
1 + e−cu 1 + ec′ (|u|−h)(3)where c, c′ , and h are positive constants.
The polarity of ui is important in this model. Therefore, we consider
xi ≡ sgn(ui )(sgn(u) = 1 for u > 0 and −1 for u ≤ 0) and refer to the vectorx = (x1 , . . . , xn ) as the current state of the network. The network state x at an
instant is represented as a point in state space consisting of 2n possible states. It
almost always moves to an adjacent point in the state space because xi changes
asynchronously when x changes. Consequently, a continuous trace of x is drawn
with the passage of time, which is called the trajectory of x .

2.2

Trajectory Attractors
An important feature of the nonmonotone neural network is that it canmake stable transitions along a given continuous trajectory in the state space;such a trajectory is called a trajectory attractor19) . By forming trajectory attractors from states S µ (µ = 1, . . . , m) to T µ , we can associate binary (±1)
patterns S µ with the corresponding patterns T µ : if S µ is given as the initialstate, then the network makes state transitions autonomously and recalls T µ .
We respectively refer to S µ and T µ as cue and target patterns.
To form the trajectory attractor, we train the network using a spatiotemporal pattern r (t) changing continuously from S µ to T µ as a teacher

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Ken YAMANE and Masahiko MORITA

Fig. 1

Schematic representation of the process of context-dependent recall.

signal. Specifically, we set x = S µ and feed r to each element in the form ofzi = λri (t) (ri is a component of r and λ is the input intensity), and allow the
network act according to Eqs. (1)—(3). Simultaneously, we modify connection
weights wij according toτ′dwij= −wij + αri yj ,dt

(4)where τ ′ is the time constant of learning (τ ′ ≫ τ ) and α is the learning coefficient. Although α can be a constant, we set α = α′ xi yi in this study (α′ is apositive constant) because the learning performance is improved if α decreases
concomitantly with increasing |ui |.
Intuitively, this learning decreases the energy of the network in the neighborhood of r . Accordingly, when r changes successively from S µ to T µ , a continuous groove remains in the energy landscape. In addition, because r moves
slightly ahead of x , a gentle flow from S µ to T µ is generated at the bottom of
the groove. By repeating several cycles of learning for all µ, gradually decreasing
the input intensity λ of r , the network comes to make autonomous transitions
from S µ to T µ , or trajectory attractors are formed.

2.3 Context-dependent State Transition Using Selective
Desensitization Method
The original nonmonotone neural network always recalls a fixed target
from an identical cue. For the network to recall various targets according to the5Brain-like Computing Based on Distributed Representations and Neurodynamics“context”, we modify the network dynamics using the selective desensitization
method 17, 18) .
This method desensitizes about half of the elements or renders their
output as neutral, depending on a given pattern C, which represents the context.
Specifically, assuming that the neutral value or the average output is 0, we
substitute Eq. (2) for
yi = gi f (ui ).

(5)Here gi denotes a variable gain of the element, which usually takes 1 but takes
0 when the element is desensitized. We consider the simplest case, in which
C is an n-dimensional binary pattern whose components ci take ±1 with equal
probability; the gain is given as gi = (1 + ci )/2.
Through this operation, the modified state of the network is projected
onto a subspace comprising active (undesensitized) elements. It also producestransitions according to the dynamics in the subspace. If trajectory attractors
are formed in the respective subspaces, then the network state reaches different target patterns according to context patterns, as presented schematically in
Fig. 1.

§3

Inference Based on Distributed Processing
Although our approach aims for non-linguistic thinking resembling thatdone by animals, describing how the model “thinks” and how it differs fromclassical AI would be rather difficult if we apply the model straightway to nonlinguistic information processing. Instead, here we apply it to simple linguistic
reasoning that could presumably be done by some smart animals if we were able
to convey the meaning of words to them.

3.1

Method of Inference
Assume that a cue pattern S 1 is given to the model in a context C 1 andthat the network makes state transitions from S 1 via S 2 to T 1 . If the pattern S 1
represents Sparrow, S 2 Bird and T 1 can fly, then we can regard this transition
as a reasoning process “Sparrow is a Bird, and therefore can fly” (Fig. 2). In thiscase, C 1 is regarded as representing the context in which <Flying Ability>
is asked. Under this interpretation, we ask a question “Can Sparrow fly?” bygiving the cue pattern S 1 and the context pattern C 1 to the model, and themodel makes an answer “Sparrow can fly.” Here, we emphasize that S 1 , S 2 , T 1 ,and C 1 are not symbols but patterns between which the distance or similarity

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Ken YAMANE and Masahiko MORITA

Fig. 2

Method of inference.

can be defined naturally. Consequently, we can include the relation of similarity
between things or contexts in the representations.
We must give knowledge to the model to construct an actual inference
system. That knowledge is given by forming a trajectory attractor, for example,
from S 1 (Sparrow ) via S 2 (Bird ) to T 1 (can fly) in the state subspace specified
by context pattern C 1 (Flying Ability), as presented in Fig. 2. If another trajectory attractor from S 3 (Horse) via S 4 (Mammal ) to T 2 (cannot fly) is formedin the same subspace, then the system has gained another piece of knowledge“Horse is a Mammal and therefore cannot fly.” Similarly, a trajectory attractor
from S 1 (Sparrow ) via S 5 (Animal ) to T 3 (move) in another subspace specified
as C 2 (Mobility) corresponds to a piece of knowledge “Sparrow is an Animal,
and moves.”
After acquiring knowledge, the system can infer a conclusion deductively by state transitions along a learned trajectory. Moreover, because of the
distributed representation of knowledge and powerful generalization ability ofthe model, it is expected that the system can infer plausible conclusions even if
novel questions are asked.

3.2

Encoding
The reasoning ability of the model described above depends largely on

encoding: how to represent information using distributed patterns. In the case of
the brain, not only concrete objects but also abstract concepts are represented as
patterns of neuronal activity. These patterns are considered to be structured, i.e.,
related things are represented by similar patterns. For example, it is suggested
that the categorical structure of visual objects is represented by the pattern ofBrain-like Computing Based on Distributed Representations and Neurodynamics

Fig. 3

7

Hierarchical structure of similarities among code patterns.

activity distributed over neurons in the monkey’s visual cortex and it is organizedhierarchically 20) . However, detailed representations and encoding mechanisms
in the brain are unclear, so that we cannot apply them directly to this model.
In addition, developing a system in which suitable encoding is achieved by selforganization is an interesting but difficult subject. We therefore must reservethat issue for future study. Accordingly, we used a convenient method described
below, which would not be the best.
First, we generated context and target patterns randomly under thecondition that each component took the value +1 or −1 with equal probability.
Accordingly, similarities (direction cosines) between these patterns were nearlyzero, except that context patterns <Flying Ability> and <Wing> were set
to have a similarity of 0.5, and target patterns <can fly> and <have wings>
were set as identical so that we can investigate the case in which context patterns
are similar.
Second, we constructed cue patterns based on categories. Specifically,
the code pattern of an object was generated by adding a certain amount ofnoise to (flipping a certain number of components of) a pattern representing
its category. Patterns representing categories were generated to form a tree
structure, as shown in Fig. 3, where the numerical values denote similarities
between patterns.
This structure was also used for training; that is, each piece of knowledge
was represented as generally as possible using a superordinate concept becausesuch generalized knowledge has widely various applications. It is noteworthy

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Ken YAMANE and Masahiko MORITA

Table 1

Originally given knowledge.

Context

Knowledge

Flying Ability

Sparrow ⇒Bird ⇒can fly, Bat ⇒can fly, Horse⇒Mammal ⇒cannot fly,
Cherry⇒Plant⇒cannot fly, Airplane⇒can fly, Helicopter ⇒can fly,
Automobile⇒cannot fly, Bike⇒cannot fly, Boat ⇒cannot fly,
Cup⇒Tableware⇒cannot fly.

Breeding

Swallow ⇒Bird⇒egg, Dog⇒Mammal ⇒young, Lily⇒Seed plant ⇒seed.

Breathing

Pigeon⇒Bird⇒lung, Bear ⇒Mammal⇒lung.

Mobility

Eagle⇒Animal⇒move, Dandelion⇒Plant ⇒not move,
Bike⇒Vehicle⇒move, Fork ⇒nonliving⇒not move.

Feeding

Lion⇒Animal⇒feed, Azalea⇒Plant ⇒not feed,Spoon⇒nonliving⇒not feed.

Chloroplast

Giraffe⇒Animal ⇒not have, Morning glory⇒Plant ⇒have,Pot⇒nonliving⇒not have.

Table 2

Added knowledge.

Context

Knowledge

Flying Ability

Penguin⇒cannot fly, Tuna⇒Fish⇒cannot fly.

Wing

Helicopter ⇒not have.

Breeding

Trout⇒Fish⇒egg, Grayfish⇒young.

Breathing

Salmon⇒Fish⇒branchi.

Death

Duck ⇒living⇒will die, Cow ⇒living⇒will die, Apple⇒living⇒will die,Boat ⇒nonliving⇒will not die, Knife⇒nonliving⇒will not die.

that some exceptions can exist (we can give “Sparrow is a Bird and therefore
can fly” to the system although some birds cannot fly), and that the superordinate concept can differ in respective contexts (e.g., “Sparrow is an Animal andtherefore moves”).

3.3

Simulation Experiment

We simulated the model with n = 2000 elements and constructed thesystem on a computer to examine the reasoning ability of the model. The parameters were c = 50, c′ = 10, h = 0.5, τ = 5000τ ′ , and α′ = 2.
For actual operation of the system, some points were noted. First, it isdifficult for the system to learn exceptional knowledge because the corresponding
trajectory attractors are affected strongly by those corresponding to common
knowledge. They are therefore difficult to form. To cope with this problem, we
increased training sessions for each piece of exceptional knowledge by five times
so that all trajectory attractors were formed securely.
Second, conventional neural networks must generally relearn acquired

Brain-like Computing Based on Distributed Representations and Neurodynamics

9knowledge when they learn new knowledge21) . Such relearning is inefficient,however, if the system requires relearning even for a small addition of knowledge.
We tested the system before and after additional learning of new knowledge toinvestigate this point.
The kind of knowledge that was given originally to the system and thatwhich was added afterward are shown respectively in Tables 1 and 2. Initially,
we trained the network 10 times on average for each piece of the original knowledge until the system acquired all. Then we asked the system various questions
about both learned and unlearned knowledge. Next, we trained the network forthe additional knowledge. The number of training iterations was 10 times on
average. Then we asked various questions again.
Table 3 shows results in abbreviated form. In this table, shaded regions
indicate the part related to additional learning (the question was newly askedor the answer changed); bold face indicates the answer that was given directlyin training. We can say that the system gives a suitable answer to unlearned
questions if we consider the limited knowledge given to the system.

3.4

Discussion

The inferential system described above has the following brain-like features that most existing systems do not have.
(a) Analogy based on similarity between cue patterns. A production
system model22) , a typical inference engine of classical AI, basically cannot makean inference unless all the knowledge necessary for leading to the conclusion
have been given as “if—then” rules. By contrast, the present system can reason
analogically using other knowledge. For example, the system answers “Eagle
can fly” to a novel question “Can Eagle fly?” because Eagle is represented usinga code pattern similar to <Sparrow >. Therefore, the network state is attracted
to the trajectory attractor <Sparrow → Bird → can fly>.
(b) Analogy based on similarity between context patterns.

The system

can perform analogical reasoning when the context is different but similar to a
familiar context. For example, if the system knows that X can fly, then it isgenerally inferred that X has wings, even though it learned nothing about wings.
This is because contexts <Flying Ability> and <Wing> are represented bysimilar patterns and trajectory attractors formed in the former context produce
flows parallel to them in the subspace corresponding to the latter context.

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Ken YAMANE and Masahiko MORITA

Table 3

Reasoning results.

Flying Wing Breeding Breathing Mobility Feeding Chloroplast Death
Sparrow
Eagle
Swallow
Duck
Penguin

have
have
have
have
have
not
have
.
.
.

egg
egg
egg
egg
egg

lung
lung
lung
lung
lung

move
move
move
move
move

feed
feed
feed
feed
feed

not
not
not
not
not

Pigeon
.
.
.

fly
fly
fly
fly
fly
not
fly
.
.
.

die
die
die
die

egg
.
.
.

lung
.
.
.

move
.
.
.

feed
.
.
.

not
.
.
.

die
die
.
.
.

Bat
Horse
Dog
Cow
Bear
Lion
Giraffe
.
.
.

fly
not
not
not
not
not
not
.
.
.

have
not
not
not
not
not
not
.
.
.

young
young
young
young
young
young
young
.
.
.

lung
lung
lung
lung
lung
lung
lung
.
.
.

move
move
move
move
move
move
move
.
.
.

feed
feed
feed
feed
feed
feed
feed
.
.
.

not
not
not
not
not
not
not
.
.
.

die
die
die
die
die
die
die
.
.
.

Tuna
Trout
GrayfishSalmon
.
.
.

not
not
not
not
.
.
.

not
not
not
not
.
.
.

egg
egg
young
egg
.
.
.

branchibranchibranchibranchi
.
.
.

move
move
move
move
.
.
.

feed
feed
feed
feed
.
.
.

not
not
not
not
.
.
.

die
die
die
die
.
.
.

CherryDandelion
Lily
Apple
AzaleaMorning glory
.
.
.

not
not
not
not
not
not
.
.
.

not
not
not
not
not
not
.
.
.

seed
seed
seed
seed
seed
seed
.
.
.

not
not
not
not
not
not
.
.
.

not
not
not
not
not
not
.
.
.

have
have
have
have
have
have
.
.
.

die
die
die
die
die
die
.
.
.

Airplane
Helicopter

fly
fly

move
move

not
not

not
not

not

Car
Bike
Boat
.
.
.

not
not
not
.
.
.

have
have
not
not
not
not
.
.
.

move
move
move
.
.
.

not
not
not
.
.
.

not
not
not
.
.
.

not
not
not
not
.
.
.

Cup
Fork
Knife
Spoon
Pot
.
.
.

not
not
not
not
not
.
.
.

not
not
not
not
not
.
.
.

not
not
not
not
not
.
.
.

not
not
not
not
not
.
.
.

not
not
not
not
not
.
.
.

not
not
not
not
not
.
.
.

Brain-like Computing Based on Distributed Representations and Neurodynamics

11

This feature is unique compared with other neural network models suchas mixture of local experts models, in which each local network or “expert” isspecialized to a particular context and generalization across different contexts
does not occur.
(c) Nonmonotonic reasoning. In general, inference systems with high ability of analogical reasoning suffer from exceptional knowledge. In stark contrast,
systems that deal excellently with exceptional knowledge require numerous detailed rules and have difficulty using analogy. Although various methods havebeen proposed23, 24, 25, 26) to cope with this dilemma, it seems impossible to solve
it using classical AI approaches only because this problem is deeply related to
the frame problem. However, the present system is not troubled by exceptional
knowledge that causes a logical inconsistency because logical inferences are notused in this system.
For example, although Bat is a Mammal and is therefore represented by asimilar code pattern to <Horse>, <Dog>, etc., the system replies “Bat can fly”
to the question “Can Bat fly?” because exceptional knowledge has been given.
Nevertheless, it performs analogical reasoning to other questions about Bat similarly as it does to questions about Horse and Dog (e.g. “Bat bears young”). In
other words, the trajectory attractor <Bat → can fly> in the context <Flying
Ability> does not much affect the flow from <Bat > toward <Mammal > inother contexts because the influence of trajectory attractors toward <Mammal >in various contexts, such as <Horse → Mammal > in <Flying Ability> and<Dog → Mammal > in <Breed>, is stronger.
For the same reason, the influence of exceptional knowledge in the samecontext is limited and does not impair the ability of analogical reasoning as longas the knowledge is exceptional. For example, when a novel cue pattern withequal similarities to <Bat> and to <Dog> is given in the context <Flying
Ability>, the network state moves to <cannot fly> through the neighborhoodof <Mammal >. Such an inference can be regarded as common-sense reasoningof a kind.
(d) Analogical reasoning using a structure of similarities among code
patterns. A crucial advantage of distributed representations over symbolic
representations is that the relation among objects, like that shown in Fig. 3, canbe expressed implicitly by similarities or distances between code patterns. The
present system can make good use of this advantage for inference.

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Ken YAMANE and Masahiko MORITA

Fig. 4

Process of reasoning using a structure of code patterns.

For example, although the system has learned nothing about Fish in
the context <Mobility> and has not learned explicitly that they belong to
Animal in any context, it drew the conclusion that “Tuna moves.” The processof inference is shown in Fig. 4(a), in which the time course of similarities between
the network state x and individual code patterns is shown (the abscissa is scaledby the time constant τ ). As this graph shows, the network state moves initiallyfrom <Tuna> toward <Fish> and <Animal > as a result of the influence ofvarious trajectory attractors to them. Then it is attracted to the trajectory
attractor <Animal → move>, which was formed when the system learned “Eagle
is an Animal; consequently, moves.”
Similarly, the system infers “Tuna will die”, as depicted in Fig. 4(b),where the network state is carried by a flow along <Tuna → Fish → Animal >.
It is then attracted to a trajectory attractor formed when the system learned“Duck, Cow and Apple are living. Consequently, will die.”(e) Addition of knowledge. Conventional multilayer neural networks confront the serious problem that previously acquired knowledge is disrupted suddenly in the process of learning a new set of knowledge, which is called catastrophic interference21) . The present system is, however, robust to additional
learning of new knowledge.

Brain-like Computing Based on Distributed Representations and Neurodynamics

Fig. 5

13

Interference to learned knowledge by additional learning.

In the experiment described above, for example, addition of knowledge
about Fish caused no error in inference as to acquired knowledge because <Fish>is distant from other code patterns. Similarly, additional learning in a novel
context <Death> has little influence on acquired knowledge.
However, if completely conflicting knowledge such as “Horse can fly” isadded, some previous knowledge (“Horse cannot fly”) is lost. Furthermore, theaddition of exceptional knowledge can interfere with related general knowledge.
In particular, addition of much exceptional knowledge of the same kind canchange some general knowledge. Some examples are that if very many flightlessbirds are newly learned, the system will infer “it cannot fly” for an unknown bird.
Nevertheless, the interference is actually limited in most cases. For example, evenafter the system additionally learned “Helicopter has no wings,” it can infer byanalogy that “Eagle has wings” and “Car has no wings.”
To examine the interference effects more specifically, we gradually increased the amount of additional knowledge in the experiment described above
while maintaining the ratio of exceptional knowledge around 20%. Figure 5
shows the result in which the percentage of correct inferences as to original
knowledge is shown against the ratio of added knowledge to the original. That
figure shows that the rate does not decrease rapidly; it remains higher than 80%,even when a double amount of knowledge is added. This result indicates thatthe system can accumulate knowledge merely through occasional relearning.

§4Interaction Between Neurodynamical Systems
The objects and concepts involved in the inference system presented

above are represented by static patterns and can be symbolized, but we believe

14

Ken YAMANE and Masahiko MORITA

Fig. 6Structure of a model of interaction using dynamic modification.

that a critical advantage of brain-like computing will be revealed when we deal
with objects that are difficult to represent as symbols such as motions of the
body.
However, the system described above is insufficient for dealing with suchobjects because they should often be represented by spatiotemporal patterns andbecause a single neurodynamical system cannot process complex spatiotemporal
patterns. Considering that the actual brain comprises many areas with specific functions working together, we describe in this section a method of linking
multiple neurodynamical systems to construct a modular system.

4.1

Dynamic Modification

The basic idea of our method is to modify a neurodynamical systemusing the state of another system as the context. However, we cannot apply
the previous selective desensitization directly because the context pattern has
been assumed static so that each trajectory attractor can be formed in a certain
subspace.
Nevertheless, if the pattern of desensitized elements varies rather slowly,
then it is thought that short trajectory attractors formed in a series of subspaces
substantially comprise a long trajectory attractor. Accordingly, to induce interactions between neurodynamical systems, we need only transform the state of aBrain-like Computing Based on Distributed Representations and Neurodynamics

Fig. 715

Two-joint robot arm.

dynamical system into a pattern that does not change sharply. We refer to sucha pattern as the modification pattern instead of the context pattern.
Figure 6 portrays a simple model of mutual interaction, where two nonmonotone neural networks of the same size are mutually linked through a twolayer network consisting of normal binary neurons. Each layered network transforms the output pattern of one neurodynamical system (nonmonotone neuralnetwork) into the modification pattern for the other system. In the following
experiment, however, it has the sole function of retrieving the state vector (consisting of binary components) from the output pattern with many zero components (corresponding to desensitized elements) because we performed encoding
so that the state vector would not change sharply.

4.2

Computer Simulation

As a simple example of application, we trained this model to learn
the movement of a computer-simulated two-joint robot arm without actuators(Fig. 7). Specifically, each neurodynamical system with 2000 elements represents the normalized state (θi , ωi ) of one arm joint (−1 ≤ θi , ωi ≤ 1, i = 1, 2).
It learned 10 trajectories for the corresponding joint, undergoing selective desensitization by the modification pattern that depends on the state of the othersystem. The initial states of the learned trajectories of the arm are listed inTable 4.
After 10 cycles of training, we gave various initial states to the model.
Figure 8 presents an example of the behavior of the model, where the actual
trajectory of the arm and the trajectory recalled by the model are displayed,

16

Ken YAMANE and Masahiko MORITA

Table 4

Initial states of the robot arm given to the model.

θ1

ω1

θ2

ω2

θ1

ω1

θ2

ω2

1
2

−0.20
−0.15

0.69
0.70

−0.21
0.28

−0.33
0.55

6
7

0.00
0.00

0.00
0.00

0.00
0.50

0.90
−0.90

3
4

0.00
0.00

0.00
0.00

−0.50
−0.50

−0.90
0.90

8
9

0.00
0.47

0.00
−0.68

0.50
0.41

0.90
0.95

5

0.00

0.00

0.00

−0.90

10

1.00

0.61

−0.13

−0.62

respectively, in the upper and lower panels. Furthermore, Fig. 9 shows an example in which the initial state is different only for θ2 . The figure shows that thearm trajectory is well reproduced by the model in both cases, and that it is well
produced for all learned trajectories. Results show that each neurodynamicalsystem makes state transitions depending on the state of the other system, sothat the model can learn and recall complex spatiotemporal patterns.
Although this robot arm task is very simple and although the advantagesof the model would be insufficiently demonstrated, we think that the model isimportant for additional development of brain-like computing because, by linking through modification patterns, many neurodynamical systems can mutuallyinteract while preserving modularity (effective connections can be switched on
and off easily). They might learn the dynamics of a more complex system froma few sample trajectories. Exploring this possibility remains as a subject for
future study.

§5Conclusion

We have presented a neurodynamical system that forms inferences according to context-dependent state transitions. We also demonstrated its many
advantages over existing inference systems, such as a powerful capability of analogical reasoning and simple treatment of exceptions. We have also proposed amethod of linking multiple neurodynamical systems and showed that two dynamical mutually interacting systems can reproduce complex movements of a
two-joint robot arm.
Although these models do not directly model the brain, they are similar
to the brain in that information is not represented by symbols or local representations but is represented distributedly by patterns. Furthermore, their basic
principles, namely trajectory attractors and selective desensitization, are sug-Brain-like Computing Based on Distributed Representations and Neurodynamics

Fig. 8

State transitions from the initial state (θ1 , ω1 , θ2 , ω2 ) = (0.0, 0.0, −0.5, 0.9).

Fig. 9State transitions from the initial state (θ1 , ω1 , θ2 , ω2 ) = (0.0, 0.0, 0.0, 0.9).

17

18

Ken YAMANE and Masahiko MORITA

gested to be used in the brain 5, 6, 27) . This fact, together with the results of thisstudy, suggests the great potential of such brain-like computing.
However, the present models leave much room for improvement andmany problems remain. For example, the mode of encoding objects to patterns
has not been examined, particularly for abstract concepts of which information
is not detected by sensors. Comparison with humans or animals will also benecessary for evaluation and further improvement of the models. Nevertheless,we believe that future study of our brain-like computing models will bring amajor breakthrough in intelligent information processing that is required not
only to address real-world problems but also to address important problems in
the era of information explosion.

Acknowledgment

This work was partly supported by a Grant-in-Aid

for Scientific Research on Priority Areas (18049008, 19024010, and 21013007)
from the Ministry of Education, Culture, Sports, Science and Technology ofJapan.

References
1)

Harnad, S., “The symbol grounding problem,” Physica D, 42, pp. 335—346,
1990.

2)

McCarty, J. and Hayes, P. J., “Some philosophical problems from the standpoint of artificial intelligence,” Machine Intelligence, 4, pp. 463—502, 1969.

3)

Sandewall, E., “An approach to the frame problem and its implementation,”
Machine Intelligence, 7, pp. 195—204, 1972.

4)

Morita, M. and Suemitsu, A., “Computational modeling of pair-association
memory in inferior temporal cortex,” Cogn. Brain Res., 13, pp. 169—178, 2002.

5)

Suemitsu, A., Morokami, S., Murata, K. and Morita, M., “Computational
examination on the dynamics of recall activity in the inferior temporal cortex,”in Proc. of the 2002 IJCNN, pp. 136-141, 2002.

6)

Suemitsu, A. and Morita, M., “A neural network model of context-dependentneuronal activity in the inferotemporal cortex,” in Proc. of the 2006 IJCNN,
pp. 685—690, 2006.

7)

Rumelhart, D. E., McClelland, J. L. and the PDP Research Group, Parallel
distributed processing: Explorations in the microstructure of cognition volume
1: Foundations, MIT Press, Cambridge, 1986.

8)

Blelloch, G. E., “CIS: A massively concurrent rule-based system,” in Proc. of
the AAAI-86, pp. 735—741, 1986.

9)

D’Avila Garcez, Artur S., Lamb, Luis C. and Gabbay, Dov M., Neural-symbolic
cognitive reasoning, Springer-Verlag, Berlin, 2009.

Brain-like Computing Based on Distributed Representations and Neurodynamics

19

10)

Touretzky, D. S. and Hinton, G. E., “Symbols among the neurons: details ofa connectionist inference architecture,” in Proc. of the IJCAI-85, pp. 238—243,
1985.

11)

Touretzky, D. S. and Hinton G. E., “A distributed connectionist production
system,” Cogn. Sci., 12, pp. 423—466, 1988.

12)

Samad, T., “Towards connectionist rule-based systems,” in Proc. of the IEEE
ICNN-88, 2, pp.525—532, 1988.

13)

Barnden, J. A. and Pollack, J. B., Advances in connectionist and neural computation theory, 1, High level connectionist models, Ablex, Norwood, 1991.

14)

Holyoak, K. J. and Barnden J. A., Advances in connectionist and neural computation theory, 2, Analogical connections, Ablex, Norwood, 1994.

15)

Gallant, S. I., Neural network learning and expert systems, MIT Press, Cambridge, 1993.

16)

Elman, J. L., “Finding structure in time,” Cogn. Sci., 14, pp. 179—211, 1990.

17)

Morita, M., Murata, K. and Morokami, S., “Context-dependent sequential
recall by a trajectory attractor network with selective desensitization,” in Proc.
of the 3rd ICNNAI, pp. 235—238, 2003.

18)

Morita, M., Matsuzawa, K. and Morokami, S., “A Model of context-dependentassociation using selective desensitization of nonmonotonic neural elements,”
Systems and Computers in Japan, 6, pp. 73—83, 2005.

19)

Morita, M., “Memory and learning of sequential patterns by nonmonotone
neural networks,” Neural Netw., 9, pp. 1477—1489, 1996.

20)

Kiani, R., Esteky, H., Mirpour, K. and Tanaka, K., “Object category structure
in response patterns of neural population in monkey inferior temporal cortex,”
J. Neurophysiol., 97, pp. 4296—4309, 2007.

21)

McCloskey, M. and Cohen, N., “Catastrophic interference in connectionist
networks: The sequential learning problem,” The Psychology of Learning andMotivation, 24, pp. 109—164, 1989.

22)

Klar, D., Langley, P. and Neches, R., Production system models of learning anddevelopment, MIT Press, Cambridge, 1987.

23)

McDermott, D. and Doyle, J., “Non-monotonic logic I,” Artif. Intell., 13, pp.
41—72, 1980.

24)

McCarthy, J., “Circumscription; A form of non-monotonic reasoning,” Artif.
Intell., 13, pp. 27—39, 1980.

25)

Reiter, R., “A logic for default reasoning,” Artif. Intell., 13, pp. 81—132, 1980.

26)

Hanks, S. and McDermott, D., “Default reasoning, non-monotonic logic, andthe frame problem,” in Proc. of the AAAI-86, pp. 328—333, 1986.

27)

Suemitsu, A., Miyazawa, Y. and Morita, M., “A model of the activity of the
hippocampal neurons based on the theory of selective desensitization,” Neural
Information Processing (Part I), Lecture Notes in Computer Science, 5506, pp.
383—390, 2009.