From a Swarm to a Biological Computer

Andrew Schumann

University of Information Technology and Management in Rzeszow, Sucharskiego 2, 35-225 Rzeszow, Poland

Keywords:

Physarum Polycephalum, Swarm Intelligence, Lateral Inhibition, Lateral Activation, Cognitive Biases,

Biological Computer, Behaviourism.

Abstract:

According to behaviourism, any swarm behaviour can be managed by outer stimuli: attractants (motivational

reinforcement) and repellents (motivational punishment). In the meanwhile, there are the following two main

stages in reactions to stimuli: (i) sensing (perceiving signals) and (ii) motoring (appropriate direct reactions

to signals). Hence, by placing attractants and repellents at different sites we can manage and program the

swarm behaviour. This opportunity allows us to design a biological computer – an abstract machine (i) with

inputs presented by stimuli coming from attractants and repellents and (ii) with outputs presented by the

swarm reactions to appropriate stimuli. This computer can be realized on different swarms differently. The

point is that different matters are attractants and repellents for different animals. They differ a lot even for

microorganisms. Nevertheless, their logic and mathematics are the same. Behaviourism means that (i) the

complex of swarm behavioural patterns can be reduced to a composition of some elementary swarm patterns,

(ii) if we know an appropriate attractant or repellent for each elementary pattern, then from a complex of

attractants and repellents we can deduce a complex of patterns. Nevertheless, it can be shown that both

assumptions are false. The point is that swarms are populations which behave as a distributed network, capable

of responding to a wide range of spatially represented stimuli so that in their behaviours we can observe effects

of neural networks with lateral activation and lateral inhibition mechanisms. As a result, behavioral patterns

cannot be additive. In the paper it is discussed what we can do with this feature of swarm behaviour to program

swarms.

1 INTRODUCTION

Within swarm intelligence (Bonabeau et al., 1999),

(Kennedy and Eberhart, 2001), (Zelinka and Chen,

2017) we can formalize different swarm patterns to

implement them in robotics. There are many well-

studied forms of swarm intelligence: ant colonies

(Dorigo and Stutzle, 2004), (John et al., 2008), bee

colonies (Karaboga, 2005), (Karaboga and Akay,

2009), (Michener, 1969), ﬁsh schooling (Abrahams

and Colgan, 1985), (Viscido et al., 2004), bird

ﬂocking and horse herding (Reynolds, 1987), bacte-

rial colonies with a kind of social behaviour (Ben-

Jacob, 2008), (Ingham and Ben-Jacob, 2008), (In-

gham et al., 2011), (Ivanitsky et al., 1984), (Mar-

genstern, 2011), (Passino, 2002), multinucleated gi-

ant amoebae Physarum polycephalum (Tsuda et al.,

2004), etc. The main feature of all these systems

is that their individual agents behave locally without

any centralized control, but their interactions lead to

the emergence of global behaviour of the whole group

that cannot be reduced to subsystems additively.

Each swarm such as ants, bees, some social bac-

teria, Physarum polycephalum, etc. can solve logis-

tic and transport problems very effectively (Kassaba-

lidis et al., 2001). For instance, there is a collective

navigation of bacterial swarms (Ariel et al., 2013),

(Shklarsh et al., 2012) and there is an effective path

ﬁnding by amoebae and a possibility of trafﬁc op-

timization by them (Nakagaki et al., 2007), (Naka-

gaki et al., 2001), (Shirakawa et al., 2012), (Watanabe

et al., 2011), (Whiting et al., 2015). Swarms can eas-

ily solve some complex (NP-hard) logistic problems:

(i) the travelling salesman problem can be solved by

ants (Dorigo and Gambardella, 1997) and by amoe-

bae (Zhu et al., 2013); (ii) the Steiner tree problem

can be solved by amoebae (Tero et al., 2010); (iii)

the generalized assignment problem can be solved by

bees (Ozbakir et al., 2010); (iv) mazes can be solved

by amoebae (Nakagaki et al., 2000), (Ntinas et al.,

2017), etc. As we see, even unicellular organisms

can solve logistic problems effectively. Also, they

can be involved in constructing algorithms for simu-

lating the crowd evacuation (Kalogeiton et al., 2015)

Schumann, A.

From a Swarm to a Biological Computer.

DOI: 10.5220/0007585302510258

In Proceedings of the 12th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2019), pages 251-258

ISBN: 978-989-758-353-7

251

and for simulating transport systems such as the route

systems in China (Adamatzky et al., 2013) and in the

United States of America (Adamatzky and Ilachinski,

2012).

The main characteristics of any swarm consists in

a possibility to optimising the own trafﬁc in reactions

to attractants and repellents. Attractants are things or

sites in the environment, such as food pieces and sex

pheromones, which attract individuals of swarm. Re-

pellents are things or sites in the environment, such

as predators, which repel individuals of swarm. The

same swarm behaviour is observed even among hu-

man beings in the following two cases: (i) an ad-

dictive behaviour such as the behaviour of alcoholics

(Schumann and Fris, 2017) or gamers – in this case

the role of human attractants causing addiction in-

creases strongly; (ii) an escape panic (Helbing et al.,

2000) – in this case the role of human repellents like

a terrorist act increases strongly, also.

Hence, by placing attractants and repellents at dif-

ferent sites we can manage and program the swarm

behaviour. This opportunity allows us to design a bio-

logical computer – an abstract machine (i) with inputs

presented by stimuli coming from attractants and re-

pellents and (ii) with outputs presented by the swarm

reactions to appropriate stimuli. This computer can

be realized on different swarms differently. The point

is that different matters are attractants and repellents

for different animals. They differ a lot even for mi-

croorganisms. Nevertheless, their logic and mathe-

matics are the same. In the biological computer we

have two main stages: (i) sensing – when a swarm

detects neighbour attractants and repellents; and (ii)

motoring – when this swarm reacts to founded attrac-

tants and repellents, e.g. to exploit attractants and to

avoid repellents.

In this paper, there are considered some basic ex-

pectations how we can design the biological computer

(Section 2). These expectations correspond to be-

haviourism. Then it is shown that these expectations

cannot be realizable at all in the behaviouristic way

(Section 3) because we observe a kind of cognitive bi-

ases even at the level of unicellular organisms. Then

there is proposed a context-based game theory which

can be used for programming the biological computer

(Section 4).

2 DESIGNING A BIOLOGICAL

COMPUTER

Designing the biological computer is based on be-

haviourism as theoretical frameworks explaining the

animal behaviour by reﬂexes produced by responses

to external stimuli with a possibility to track indi-

vidual’s history of reinforcement and punishment im-

plied by those stimuli. Each reﬂex is a direct response

of animal to a stimulus, connecting the stimulus to

a behaviour within two basic modes: either attract-

ing (e.g. reinforcement) or repelling (e.g. punishment)

coming from this stimulus. If we know what attract-

ing and repelling matters are for the animal in fact,

we can manage a control of its reactions by placing

these matters. It means, simply put that controlling

stimuli causes controlling behaviour according to be-

haviourism. So, we assume, the environment deter-

mines each animal behaviour.

In the theory of reﬂexes there are distinguished the

following two types of responses/reactions/reﬂexes:

(i) the unconditioned response to a stimulus (then

an appropriate stimulus is called unconditioned); (ii)

the conditioning response to a stimulus (then an ap-

propriate stimulus is called conditioning). The un-

conditioned stimuli are represented by biologically

active matters: either attractants (e.g. food or sex

pheromones attracting an animal) or repellents (e.g.

places evaluated automatically as dangerous by an an-

imal). These stimuli are strongly connected to some

direct responses from the very beginning, i.e. they are

given a priori, just because of chemical processes of

organisms. In the meanwhile, at the beginning, un-

conditioned stimuli are biologically neutral and as-

sume no direct reactions of organism, i.e. they are

rather ignored under standard conditions. Neverthe-

less, they can be associated with some unconditioned

responses, too. Then they become non-neutral, but

attractive (if they were associated with attractants) or

negative (if they were associated with repellents).

At the time of Pavlov, one thought that a possibil-

ity of conditioning responses is one of the basic fea-

tures of brain and nervous system. But it is known

now through experiments that it is a fundamental fea-

ture of any adaptive behaviour including the adaptive

behaviour of unicellular organisms, see (Shirakawa

et al., 2011). Conditioning responses mean a mem-

ory, when an organism was taught that a neutral stim-

ulus is a context of appearing an attractant or repel-

lent. The knowledge of this context allows the organ-

ism to behave more adaptively in the feature. Thus,

the possibility of conditioning responses is connected

to a possibility of life to have a memory and then to

be more adaptive to the environment.

Let us consider some examples of memory of uni-

cellular organisms. From (Ball, 2008), we know that

the amoeba Physarum polycephalum can learn the

patterns of shocks at regular intervals, and then it

changes its behaviour in anticipation of the next shock

to come. In (Saigusa et al., 2008), this experiment

BIOSIGNALS 2019 - 12th International Conference on Bio-inspired Systems and Signal Processing

252

is performed in the following manner. Unfavorable

conditions for Physarum polycephalum are presented

as three consecutive pulses at constant intervals. Un-

der these unfavorable conditions, the amoebae reduce

their locomotive speed in response to each episode.

When the amoebae move under the favorable con-

ditions, they spontaneously reduce their locomotive

speed at the time when the next unfavorable episode

is expected to occur. This fact shows that we deal with

the anticipation of impending environmental change.

For the amoeba the regular interval of shocks became

a conditioning stimulus.

Another experiment with the amoeba Physarum

polycephalum was performed in (Shirakawa et al.,

2011) to show that the temperature ﬂuxes can be-

come a conditioning stimulus for the amoebae. So,

we know that the amoebae avoid cold temperatures

and under these conditions they become slow. In the

experiment there was shown that when the tempera-

ture changes had stopped, the amoebae slowed down,

anticipating a cold ﬂux. It is an evidence of memory

of temperature ﬂuxes.

All these experiments demonstrating a kind of

memory of unicellular organisms can be explained

by chemotaxis, the unconditioned response to some

chemical cues (such as pheromones), that is differ-

ent for static gradients and dynamic cues, see (Skoge

et al., 2014), where this difference is shown for the

migration of Dictyostelium cells in response to the

source of traveling waves of chemoattractant during

aggregation.

Hence, some primitive forms of conditioning re-

ﬂexes (that is a primitive memory) can be detected

even at the level of cellular proteins (including actin

ﬁlaments) responding to the changes of the environ-

ment, such as shock intervals or cold ﬂuxes. Not

only neurons can cumulate conditioning responses, as

Pavlov thought ﬁrst, but also cellular proteins. This

basic feature of chemotaxis to have a short memory is

involved into the particle swarm optimization (Wang

et al., 2011), (Zelinka and Chen, 2017) and in de-

signing organic memristors – organic devices with a

memory. For example, in (Traversa et al., ) for de-

signing circuit models there was applied an ability

of slime mould of Physarum polycephalum to both

memorize the period of temperature and humidity

variations and anticipate the next variations to come.

Some other organic memristors are proposed in (Di-

monte et al., 2014), (Erokhin, 2013), (Erokhin et al.,

2012), (Pershin et al., ), (Pershin et al., 2016).

The idea of biological computer satisﬁes the pre-

supposition of radical behaviourism: attractants, re-

pellents and conditioning stimuli (connected to at-

tractants or repellents) as environmental variables can

completely control observable behaviours of animals

(Baum, 2005), (Cheney and Ferster, 1997), (Skinner,

1976). This presupposition is used also in functional

analysis of behavioural psychology to deﬁne relation-

ships between stimuli and responses on the basis of

explicating the following four elements: motivating

operations, trigger of behaviour, behaviour itself, con-

sequence of behaviour (Bandura, 1982). Thus, the bi-

ological computer is a logic for radical behaviourism.

In other words, it deﬁnes how different Boolean com-

binations of attractants and repellents determine the

patterns of animal behaviour, ﬁrst of all their swarm

behaviour.

3 BOUNDED RATIONALITY AND

COGNITIVE BIASES

Radical behaviourism means that (i) the complex of

swarm behavioural patterns can be reduced to a com-

position of some elementary swarm patterns, (ii) if we

know an appropriate attractant or repellent for each

elementary pattern, then from a complex of attrac-

tants and repellents we can deduce a complex of pat-

terns. Hence, following this point of view we can

suppose that the animal behaviour is rational – it is

the same under the same conditions with the same

complex of attractants and repellents. Each attractant

is more or less preferable and each repellent is more

or less avoidable. Then this rationality would mean

that at each step if a swarm faces several attractants,

then the most preferable attractant is always contained

among chosen attractants; and if it faces several repel-

lents, the most avoidable repellent is always contained

among rejected repellents. In other words, the ratio-

nal animal tries to maximize its satisfaction and, at the

same time, to minimize its frustration. If preferences

affect decisions indeed, then a Boolean complex of

stimuli based on these preferences determine a com-

plex of behavioural patterns. Is it so? Can we de-

sign the biological computer as a complex of Boolean

compositions?

The assumption of rationality of agents is fun-

damental for game theory and decision theory. Ac-

cording to these theories, each rational agent al-

ways follows his or her preferences in his or her

choices of items or strategies. So, the rationality of

human beings in microeconomics is understood in

the same way of utility-maximizing consumers and

proﬁt-maximizing ﬁrms. Each rational agent of the

market follows the same decision model to maximize

their own proﬁts: I choose items to maximize my

satisfaction as a consumer and I offer something to

maximize my proﬁts as a ﬁrm. We face a zero re-

From a Swarm to a Biological Computer

253

ﬂexion of rational humans there. Everyone is trans-

parent for others, e.g. satisﬁes their expectations, if

all them are rational. This transparency in decisions

means that rational agents appeal only to facts which

are the same for everyone, therefore these agents al-

ways can agree (see the Aumann’s agreement theorem

(Aumann, 1976), (Aumann, 1989)). Preferences are

instances of facts that should be seen before meeting.

Furthermore, due to rationality and zero reﬂexion of

individuals we can always agree about long-term joint

actions.

Nevertheless, there are many empirical evidences

which contradict the assumption that, on the one

hand, consumer behaviour is reasonably character-

ized as the maximization of expected lifetime utility

subject to a budget constraint and conditional on the

available information and, on the other hand, ﬁrm be-

haviour is characterized as the maximization of ex-

pected proﬁt subject to investments and conditional

on the correct programming of consumer behaviour.

For example, Thaler ((Thaler, 1990), (Thaler, 1994))

shows the bounded rationality and impatience of con-

sumers and, as a result, he proposes the behavioural

life-cycle theory emphasizing self-control, mental ac-

counting, and framing. Psychologists Amos Tver-

sky and Daniel Kahneman established a research pro-

gramme of studying cognitive heuristics and biases,

i.e. the programme of studying contexts and patterns

which inﬂuence on human decisions beyond rational-

ity (Tversky and Kahneman, 1974) (i.e. beyond logi-

cal positivism and classical game theory and decision

theory).

As we see, in behavioural economics and exper-

imental psychology it was made evident through ex-

periments that classical game theory and decision the-

ory assuming the rationality of agents are too ab-

stract, because real agents are too far from rational.

The problem is that preference relations are not so

strongly linked to decisions. As a consequence, a

Boolean composition of preferences does not give a

complex preference for a complex decision. Mathe-

matically, it means that complexes of preferences as

well as complexes of behavioural patterns are not ad-

ditive.

Let us return to our task of designing the biolog-

ical computer. From these examples taken from be-

havioural economics, we can assume that logical pos-

itivism is unapplied in the logical modelling of the

swarm behaviour, as well – because we deal with a

non-additivity of all natural behaviours. As a result,

a Boolean composition of attractants and repellents

cannot give an expected complex of behavioural pat-

terns in the standard way of constructing inductive

sets.

Hence, due to recent results in behavioural eco-

nomics we know that even in human cognitions

(which evaluated as the most rational among all an-

imals) we cannot avoid cognitive biases and cognitive

heuristics which are a substantial part in our decision

making not only in everyday situations, but also in

ﬁrms. Therefore, cognitive reﬂexion and cognitive bi-

ases are evaluated now as a natural mechanism of cog-

nitions (Frederick, 2005), (Gigerenzer and Brighton,

2009).

Some primitive forms of cognitive biases and cog-

nitive heuristics of unicellular organisms were found

out by us within the project Physarum Chip: Grow-

ing Computers from Slime Mould (Adamatzky et al.,

2012). Even unicellular organisms behave differently

in stress or under favourable conditions. For instance,

under the favourable condition (meeting attractants)

bacteria behave more predictable in choosing a trajec-

tory of motion – so, if a concentration of attractant in-

creases, bacteria tumble less frequently. In the stress

(meeting repellents) bacteria try to change a trajectory

stochastically – so, if a concentration of repellent in-

creases, bacteria tumble more frequently. As we see,

we face a duality in basic reactions of bacteria to their

environment. It is a kind of the most primitive “cog-

nitive heuristics” detected at the bacterium level – to

be more or less predictable in a favourable or stress

situation, respectively.

Amoebae of Physarum polycephalum can behave

both individually and collectively. Therefore, they

can be considered a swarm, although they are unicel-

lular organisms in fact, but with myriad nuclei. We

have discovered that the main feature in perceiving

external signals even by one cell due to actin ﬁlament

networks is that the same signal can be perceived dif-

ferently – it depends on the cell shape and many other

internal circumstances. As a result, one unicellular

organism bears more outputs than inputs in possi-

ble reactions to external signals. So, even one cell

can have “cognitive biases” and behave quite unpre-

dictably and irrationally under the same conditions. It

means that amoebae of Physarum polycephalum and

Amoeba proteus can modify their own elementary ac-

tions – they possess a kind of “free will”.

4 CONTEXT-BASED GAMES

Let us start with considering preference relations on

the level of neurons. It is known from neurobiol-

ogy that there are two different synaptic effects: (i)

excitatory effect (depolarization) that increases the

membrane potential to make neuron more negative

and to decrease the likelihood of an action potential

BIOSIGNALS 2019 - 12th International Conference on Bio-inspired Systems and Signal Processing

254

and (ii) inhibitory effect (hyperpolarization) that de-

creases the membrane potential to make neuron more

positive and to increase the likelihood of an action po-

tential. So, lateral activation is the structuring of a

neural network so that neurons activate their neigh-

bours to decrease their own responses and lateral in-

hibition is the structuring of a neural network so that

neurons inhibit their neighbours in proportion to their

own excitation. In other words, the more neighbour-

ing neurons stimulated, the less strongly a neuron re-

sponds and the fewer neighbouring neurons stimu-

lated, the more strongly a neuron responds. For ex-

ample, the lateral activation decreases the contrast and

sharpness in visual response to describe more explic-

itly all the edges and regions in the image. In this case

we deal with the so-called low-level vision. The lat-

eral inhibition increases the contrast and sharpness in

visual response to perform an overall action-oriented

interpretation of the scene. It is the so-called high-

level vision.

Swarms are populations which behave as a dis-

tributed network, capable of responding to a wide

range of spatially represented stimuli, for example,

colonies of ants or fungi have such a behaviour. In

their behaviours we can observe effects of neural net-

works with lateral activation and lateral inhibition

mechanisms. It was shown experimentally by us

in our project Physarum Chip: Growing Computers

from Slime Mould that effects of lateral activation and

lateral inhibition are detected in the plasmodium of

Physarum polycephalum, the supergroup Amoebozoa,

phylum Mycetozoa, class Myxogastria. This means

that we can perform experiments how the plasmod-

ium prefers items in the two different modes: within

lateral inhibition and lateral activation. Let us re-

call that the plasmodium is an active feeding stage of

Physarum polycephalum that moves by protoplasmic

streaming and can switch its direction or even multi-

ply in accordance with appropriate attractants (chem-

ical signals which attract the organism) and repellents

(chemical signals which repel it). This behaviour is

intelligent and can be controlled by different locations

of chemical signals attracting and repelling the plas-

modium.

The true slime mould (plasmodium) of Physarum

polycephalum has the following two distinct stages in

responding to signals: ﬁrst, the sensory stage (per-

ceiving signals) and, second, the motor stage (action

as responding). The effect of lateral activation in the

plasmodium is to decrease contrast between spatial

environmental stimuli at the sensory stage and to split

protoplasmic tubes towards two or more attractants at

the motor stage. The effect of lateral inhibition is to

increase contrast between spatial environmental stim-

uli at the sensory stage and to fuse protoplasmic tubes

towards one attractant at the motor stage. Hence, in

the lateral activation the plasmodium prefers items by

splitting tubes and in lateral inhibition the plasmod-

ium prefers items by fusing the same tubes.

To program the biological computer we need a

context-based game theory we are working on. Its

main presuppositions:

1. Each game can be assumed inﬁnite, because its

rules can change depending on context – to be lat-

erally activated or laterally inhibited.

2. Players can have different levels of mood or reﬂex-

ion: at the level of lateral activation they are more

unintentional (zero reﬂexion) and at the level of

lateral inhibition they are much more intentional

and a degree of intentionality depends on a degree

of inhibition.

3. Some utilities can have proto-symbolic meanings

or symbolic meanings. These meanings are results

of accepting (proto-)symbolic values by some

players, e.g. swarms can consider the same item

being laterally inhibited or laterally activated. The

higher symbolism of payoffs, the higher level of

reﬂexion of appropriate players. On the zero level

of reﬂexion, the payoffs do not have symbolic

meanings at all. For consensus the players are

looking for joint symbolic meanings.

4. Resistance points for players are reduced to the

payoffs of the zero level of reﬂexion.

5. The joint (proto-)symbolic meanings can change

through the game if a player increases his/her

level of reﬂexion.

6. For any game there is performative efﬁciency,

when all (proto-)symbolic meanings of one player

are shared by other players.

In the case of these new game-theoretic as-

sumptions we can calculate some aspects of (proto-

)symbolic interactions by probabilistic tools in non-

Archimedean numbers (Schumann, 2014). These new

assumptions correspond to bio-inspired game theory

as well as to theory of reﬂexive games. So, on the one

hand, in bio-inspired game theory all the game moves

are performed under the lateral inhibition or lateral

activation conditions, therefore the swarm behaviour

is not forecasted by additive measures. On the other

hand, in reﬂexive games players can lie to each other,

therefore their behaviours are not predictable by addi-

tive measures, too.

The context-based game is deﬁned as follows. It

is a tuple

G = h(States

)

t∈N

, Agt, (Act

t,n

)

t,n∈N

, (Mov

)

t,n∈N

(Tab

)

t,n∈N

i, where

From a Swarm to a Biological Computer

255

• States

is a (ﬁnite) set of states at time t =

0, 1, 2, . . .;

• Agt = {1, . . . , k} is a ﬁnite set of players;

• Act

t,n

is a non-empty set of concurrent actions

with radius n at t = 0, 1, 2, . . ., an element of Act

Agt

t,n

is called a move at time t = 0, 1, 2, . . .;

• Mov

: States

×Act

Agt

t,n

→ 2

Act

0} is a mapping

indicating the available sets of actions to a given

player in a given set of states, n > 0 is said to be a

radius of hybrid actions, a move m

Agt

= (m

)

A∈Agt

is legal at hs

, . . . , s

i if m

∈ Mov

(s, A) for all

A ∈ Agt, where s = hs

, . . . , s

• Tab

: States

× Act

Agt

t,n

→ States

t+1

is the transi-

tion table which associates, with a given set of

states at t and a given move of the players at t,

the set of states at t + 1 resulting from that move.

Let us consider an example of context-based game

for Physarum polycephalum. Let States

consist of

attractants a

, a

, . . . , a

and zero repellents, States

consist of attractants a

, a

, . . . , a

and repellents

, r

, . . . , r

, etc. Suppose, Agt = {1, 2}, i.e. we

have just two players. At t = 0 there are no repellents.

It means that both players are laterally activated and

try to occupy all visible attractants at once. In our

case, there are supposed m

visible attractants. So,

at t = 0 we deal with concurrent actions with radius

. Let l attractants be neighbours for player 1 and

− l attractants be neighbours for player 2. Then

we have l available actions for 1 at t = 0 and m

− l

available actions for 2 at the same time. At the end of

step t = 0, l attractants are occupied by player 1 and

− l attractants are occupied by player 2.

At time t = 1, there are attractants a

, a

, . . . , a

and repellents r

, r

, . . . , r

. It means that both play-

ers are laterally inhibited. Let l repellents be neigh-

bours for player 1 and k

− l repellents be neighbours

for player 2. Then we have l available actions for 1 to

avoid l places at once and k

− l available actions for

2 to avoid k

− l places at once, too. After avoiding

all the dangerous places, both player 1 and player 2

decide whether there are some attractants from a

, a

. . . , a

at free places. It can be a situation that there

are no attractants which can be occupied (they are

too close to repellents). In this case, player 1 moves

to one free place and player 2 moves to another fee

place.

Hence, at time t = 0, 1 the rules of game changed.

At t = 0 it was a lateral activation and t = 1 it was

a lateral inhibition. Therefore, at t = 0, we have a

zero reﬂexion of both players, but at t = 1 the level

of reﬂexion increased. It means, while at t = 0 there

were no (proto-)symbolic meanings for both players,

at t = 1 some (proto-)symbolic meanings appeared.

5 CONCLUSIONS

In accordance with behaviourism, any animal be-

haviour based on unconditioned and conditioning re-

ﬂexes can be controlled or even managed by stim-

uli in the environment: attractants (motivational rein-

forcement) and repellents (motivational punishment).

In the meanwhile, there are the following two main

stages in reactions to stimuli: sensing (perceiving

signals) and motoring (appropriate direct reactions

to signals). In our research, the strict limits of be-

haviourism have been studied from the point of view

of symbolic logic and algebraic mathematics: How

far can animal behaviours be controlled by the topol-

ogy of stimuli? In other words, how far can we design

unconventional computers on the basis of animal re-

actions to stimuli?

On the one hand, we can try to design reversible

logic gates in which the number of inputs is the same

as the number of outputs. In our case, the behaviouris-

tic stimuli for swarms are inputs and appropriate reac-

tions of swarms at the motoring stage are outputs. It

means that behaviourism can hold true, indeed. Nev-

ertheless, on the other hand, at the sensing stage the

same signal can be perceived so differently. The prob-

lem is that at the sensing stage even each unicellu-

lar organism can be regarded as a logic gate in which

the number of outputs (means of perceiving signals)

greatly exceeds the number of inputs (signals). It is

connected to actin ﬁlament networks and other sub-

cellular protein mechanisms perceiving the stimuli to

react to them in various ways in accordance with the

needs and tasks of the cell at the current moment. As

a consequence, even one cell can resist outside inﬂu-

ences. Hence, we face some strict biological limits

in applying behaviourism. From the standpoint of

symbolic logic and algebraic mathematics, this means

that we cannot examine animal behaviours as con-

ventional spatial algorithms, such as Kolmogorov-

Uspensky machines. The mathematics of animal be-

haviours is much more complicated. The matter is

that we should know how logical-mathematically we

can design logic gates in which the number of inputs

is far exceeded by the number of outputs. The uni-

verse of such incorrect mathematical “functions” is

known for mathematicians well and called by them

non-well-founded. In our research, some new mathe-

matical tools for studying the non-well-founded uni-

verse of animal behaviours were proposed: context-

based games and reﬂexive games.

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REFERENCES

Abrahams, M. and Colgan, P. (1985). Risk of predation, hy-

drodynamic efﬁciency, and their inﬂuence on school

structure. Environmental Biol. of Fishes, 13(3):195–

202.

Adamatzky, A., Erokhin, V., Grube, M., Schubert, T., and

Schumann, A. (2012). Physarum chip project: Grow-

ing computers from slime mould. International Jour-

nal of Unconventional Computing, 8(4):319–323.

Adamatzky, A. and Ilachinski, A. (2012). Slime mold imi-

tates the united states interstate system. Complex Sys-

tems, 21(1).

Adamatzky, A., Yang, X., and Zhao, Y. (2013). Slime

mould imitates transport networks in china. Interna-

tional Journal of Intelligent Computing and Cybernet-

ics, 6(3):232–251.

Ariel, G., Shklarsh, A., Kalisman, O., Ingham, C., and Ben-

Jacob, E. (2013). From organized internal trafﬁc to

collective navigation of bacterial swarms. New Jour-

nal of Physics, 15:12501.

Aumann, R. J. (1976). Agreeing to disagree. The Annals of

Statistics, 4(6):1236–1239.

Aumann, R. J. (1989). Notes on Interactive Epistemology.

Mimeo. Hebrew University of Jerusalem, Jerusalem.

Ball, P. (2008). Slime mould displays remarkable rhythmic

recall. Nature, 451:385.

Bandura, A. (1982). Self-efﬁcacy mechanism in human

agency. American Psychologist, 37:122–147.

Baum, W. (2005). Understanding behaviorism : Behavior,

culture, and evolution. Blackwell Pub.

Ben-Jacob, E. (2008). Social behavior of bacteria: from

physics to complex organization. European Physical

Journal B, 65(3):315–322.

Bonabeau, E., Dorigo, M., and Theraulaz, G. (1999).

Swarm Intelligence: From Natural to Artiﬁcial Sys-

tems. Oxford University Press.

Cheney, C. and Ferster, C. (1997). Schedules of Reinforce-

ment. Copley Publishing Group.

Dimonte, A., Berzina, T., Pavesi, M., and Erokhin,

V. (2014). Hysteresis loop and cross-talk of or-

ganic memristive devices. Microelectronics Journal,

45(11):1396–1400.

Dorigo, M. and Gambardella, L. M. (1997). Ant colonies

for the travelling salesman problem. Biosystems,

43(2):73–81.

Dorigo, M. and Stutzle, T. (2004). Ant Colony Optimiza-

tion. MIT Press.

Erokhin, V. (2013). On the learning of stochastic networks

of organic memristive devices. International Joural

Unconventional Compiting, 9(3-4):303–310.

Erokhin, V., Howard, G. D., and Adamatzky, A. (2012). Or-

ganic memristor devices for logic elements with mem-

ory. International Journal of Bifurcation and Chaos,

22(11).

Frederick, S. (2005). Cognitive reﬂection and decision mak-

ing. 19(4):24–42.

Gigerenzer, G. and Brighton, H. H. H. (2009). Why bi-

ased minds make better inferences. Topics in Cogni-

tive Scince, 1(1):107–143.

Helbing, D., Farkas, I., and Vicsek, T. (2000). Simu-

lating dynamical features of escape panic. Nature,

407(6803):487–490.

Ingham, C. J. and Ben-Jacob, E. (2008). Swarming and

complex pattern formation in paenibacillus vortex

studied by imaging and tracking cells. BMC Micro-

biology, 36:8.

Ingham, C. J., Kalisman, O., Finkelshtein, A., , and Ben-

Jacob, E. (2011). Mutually facilitated dispersal be-

tween the nonmotile fungus aspergillus fumigatus and

the swarming bacterium paeni bacillus vortex. Pro-

ceedings of the National Academy of Sciences of the

United States of America, 108(49):19731–19736.

Ivanitsky, G. R., Kunisky, A. S., and Tzyganov, M. A.

(1984). Study of ‘target patterns’ in a phage-bacterium

system. In Krinsky, V., editor, Self-organization: Au-

towaves and Structures Far From Equilibrium, pages

214–217. Springer, Heidelberg.

John, A., Schadschneider, A., Chowdhury, D., and Nishi-

nari, K. (2008). Characteristics of ant-inspired trafﬁc

ﬂow. Swarm Intelligence, 2(1):25–41.

Kalogeiton, V. S., Papadopoulos, D. P., Georgilas, I., Sir-

akoulis, G. C., and Adamatzky, A. (2015). Cellu-

lar automaton model of crowd evacuation inspired by

slime mould. International Journal of General Sys-

tems, 44(3):354–391.

Karaboga, D. (2005). An idea based on honey bee swarm

for numerical optimization. Technical report-tr06,

Engineering Faculty, Computer Engineering Depart-

ment, Erciyes University.

Karaboga, D. and Akay, B. (2009). A comparative study of

artiﬁcial bee colony algorithm. Applied Mathematics

and Computation, 214(1):108–132.

Kassabalidis, I., El-Sharkawi, M. A., Marks, R. J., Arab-

shahi, P., and Gray, A. A. (2001). Swarm intelligence

for routing in communication networks. In Global

Telecommunications Conference, 2001. GLOBECOM

’01. IEEE, volume 6, pages 3613–3617.

Kennedy, J. and Eberhart, R. (2001). Swarm Intelligence.

Morgan Kaufmann Publishers, Inc.

Margenstern, M. (2011). Bacteria inspired patterns grown

with hyperbolic cellular automata. HPCS, pages 757–

763.

Michener, C. (1969). Comparative social behavior of bees.

Annu. Rev. Entomol., 14:299–342.

Nakagaki, T., Iima, M., Ueda, T., Nishiura, Y., Saigusa, T.,

Tero, A., Kobayashi, R., and Showalter, K. (2007).

Minimum-risk path ﬁnding by an adaptive amoeba

network. Physical Review Letters, 99:68–104.

Nakagaki, T., Yamada, H., and Toth, A. (2000). Maze-

solving by an amoeboid organism. Nature, 407:470–

470.

Nakagaki, T., Yamada, H., and Tothm, A. (2001). Path ﬁnd-

ing by tube morphogenesis in an amoeboid organism.

Biophysical Chemistry, 92:47–52.

Ntinas, V. G., Vourkas, I., Sirakoulis, G. C., and Adamatzky,

A. (2017). Oscillation-based slime mould electronic

From a Swarm to a Biological Computer

257

circuit model for maze-solving computations. IEEE

Trans. on Circuits and Systems, 64-I(6):1552–1563.

Ozbakir, L., Baykasoglu, A., and Tapkan, P. (2010). Bees

algorithm for generalized assignment problem. Ap-

plied Mathematics and Computation, 215(11):3782–

3795.

Passino, K. M. (2002). Biomimicry of bacterial foraging for

distributed optimization and control. Control Systems,

22(3):52–67.

Pershin, Y. V., Fontaine, L., S., D. V., and M. Memris-

tive model of amoeba learning. Physical Review E,

80(2):021926.

Pershin, Y. V., Ventra, D., and M. (2016). Memristive

and memcapacitive models of Physarum learning. In

Advances in Physarum Machines, pages 413–422.

Springer.

Reynolds, C. W. (1987). Flocks, herds, and schools: A

distributed behavioral model. Computer Graphics,

21:25–34.

Saigusa, T., Tero, A., Nakagak, T., and Kuramoto, Y.

(2008). Amoebae anticipate periodic events. Phys.

Rev. Lett., 100(1):018101.

Schumann, A. (2014). Reﬂexive games and non-

archimedean probabilities. P-Adic Numbers, Ultra-

metric Analysis, and Applications, 6(1):66–79.

Schumann, A. and Fris, V. (2017). Swarm intelligence

among humans – the case of alcoholics. In Proceed-

ings of the 10th International Joint Conference on

Biomedical Engineering Systems and Technologies -

Volume 4: BIOSIGNALS, (BIOSTEC 2017), pages 17–

25. ScitePress.

Shirakawa, T., p. Gunji, Y., and Miyake, Y. (2011). An asso-

ciative learning experiment using the plasmodium of

Physarum polycephalum. Nano Communication Net-

works, 2:99–105.

Shirakawa, T., Yokoyama, K., Yamachiyo, M., Y-p, G., and

Miyake, Y. (2012). Multi-scaled adaptability in motil-

ity and pattern formation of the Physarum plasmod-

ium. 4:131–138.

Shklarsh, A., Finkelshtein, A., Ariel, G., Kalisman, O., Ing-

ham, C., and Ben-Jacob, E. (2012). Collective nav-

igation of cargo-carrying swarms. Interface Focus,

2:689–692.

Skinner, B. F. (1976). About Behaviorism. Random House,

Inc, New York.

Skoge, M., Yue, H., Erickstad, M., Bae, A., Levine, H.,

Groisman, A., Loomis, W. F., and Rappel, W.-J.

(2014). Cellular memory in eukaryotic chemotaxis.

Proceedings of the National Academy of Sciences of

the United States of America, 111(40):14448–14453.

Tero, A., Nakagaki, T., Toyabe, K., Yumiki, K., and

Kobayashi, R. (2010). A method inspired by

Physarum for solving the Steiner problem. In-

ernational Joural of Unconventional Computing,

6(2):109–123.

Thaler, R. (1990). Saving, fungibility, and mental accounts.

Journal of Economic Perspectives, 4(1):193–205.

Thaler, R. (1994). Psychology and saving policies. Ameri-

can Economic Review, 84(2):186–192.

Traversa, F. L., Pershin, Y. V., and Di Ventra, M. Memory

models of adaptive behavior. IEEE transactions on

neural networks and learning systems, 24(9):1437–

1448.

Tsuda, S., Aono, M., and Gunji, Y. P. (2004). Robust and

emergent Physarum-computing. BioSystems, 73:45–

55.

Tversky, A. and Kahneman, D. (1974). Judgments un-

der uncertainty: Heuristics and biases. Science,

185(4157):1124–1131.

Viscido, S., Parrish, J., and Grunbaum, D. (2004). Individ-

ual behavior and emergent properties of ﬁsh schools:

a comparison of observation and theory. In Marine

Ecology Progress Series, vol. 273, pages 239–249.

Wang, Y., Li, B., Weise, T., Wang, J., Yuan, B., and Tian, Q.

(2011). Self-adaptive learning based particle swarm

optimization. Information Sciences, 181(20):4515–

4538.

Watanabe, S., Tero, A., Takamatsu, A., and Nakagaki, T.

(2011). Trafﬁc optimization in railroad networks us-

ing an algorithm mimicking an amoeba-like organism,

Physarum plasmodium. Biosystems, 105(3):225–232.

Whiting, J. G. H., de Lacy Costello, B., and Adamatzky,

A. (2015). Transfer function of protoplasmic tubes of

Physarum polycephalum. Biosystems, 128:48–51.

Zelinka, I. and Chen, G., editors (2017). Evolutionary Al-

gorithms, Swarm Dynamics and Complex Networks:

Methodology, Perspectives and Implementation, vol-

ume 26. Springer.

Zhu, L., Aono, M., Kim, S.-J., and Hara, M. (2013).

Amoeba-based computing for traveling salesman

problem: Long-term correlations between spatially

separated individual cells of Physarum polycephalum.

Biosystems, 112(1):1–10.

BIOSIGNALS 2019 - 12th International Conference on Bio-inspired Systems and Signal Processing

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