Abstract

A controller of biological or artificial organism (e.g., in bio-inspired cellular robots) consists of a number of processes that drive its dynamics. For a system of processes to perform as a successful controller, different properties can be mentioned. One of the desirable properties of such a system is the capability of generating sufficiently diverse patterns of outputs and behaviors. A system with such a capability is potentially adaptable to perform complicated tasks with proper parameterizations and may successfully reach the solution space of behaviors from the point of view of search and evolutionary algorithms. This article aims to take an early step toward exploring this capability at the levels of individuals and populations by introducing measures of diversity generation and by evaluating the influence of different types of processes on diversity generation. A reaction-diffusion-based controller called the artificial homeostatic hormone system (AHHS) is studied as a system consisting of different processes with various domains of functioning (e.g., internal or external to the control unit). Various combinations of these processes are investigated in terms of diversity generation at levels of both individuals and populations, and the effects of the processes are discussed representing different influences for the processes. A case study of evolving a multimodular AHHS controller with all the various process combinations is also investigated, representing the relevance of the diversity generation measures and practical scenarios.

1 Introduction

Both biological and artificial systems display high diversity in the patterns of outputs and behaviors that they present, such as visual patterns on the body of an animal or locomotion patterns of legs during a walk (see Figure 1 for some examples of biological and artificial organisms). In a biological organism consisting of several cells with identical genotypes, different behaviors and phenotypical traits are exhibited by the cells due to their interactions with each other and with their local environment. For example, various parts of an animal's body are different in their shapes and functionalities. During development of an animal, patterns of protein concentrations are generated along the body and lead the development of the cells. It was found in embryogenesis of Drosophila melanogaster embryos [1, 23, 27] that protein gradients govern the segmentation process of the body. In the aggregated phase of the slime mold Dictyostelium dicoideum, waves of chemical gradients organize the “body” of the aggregated pseudo-organism's slug state, and its motion is also governed by spiral chemical waves [8, 16, 19, 35]. This fact was exploited also to generate swarm robotic applications by mimicking slime mold gradient formation [36]. Complex patterns are also found on the outer skins of many animals and in the growth process of tissue structures; some of them can be described by models of self-organizing Turing processes [19, 31, 41].

Figure 1. 

Examples of unicellular and multicellular organisms in biology and robotics. The first row represents a unicellular organism (a euglena) and a multicellular organism (a cheetah). The second row represents a unicellular robot (ePuck) and a multicellular (modular) robot (symbricator). (The images of euglena and cheetah are adapted from Wikimedia Commons.)

Figure 1. 

Examples of unicellular and multicellular organisms in biology and robotics. The first row represents a unicellular organism (a euglena) and a multicellular organism (a cheetah). The second row represents a unicellular robot (ePuck) and a multicellular (modular) robot (symbricator). (The images of euglena and cheetah are adapted from Wikimedia Commons.)

Diversity can be considered differently at the individual level and at the level of a population: By the diversity at the individual level, we mean the diversity of a phenotypical trait that is generated across different cells of an organism or in different time slots. Diversity at the individual level can be rephrased as complexity of a phenotypical pattern and can be investigated both spatially and temporally. Imagine a multicellular organism, and consider a particular trait of its cells as an observable trait; for example, the level of a particular protein in every cell, or the color or shape of the cell. By taking a snapshot of the organism's body and considering the complexity of the pattern of an observable trait along the cells of the body, the spatial diversity of the trait can be computed. Similarly, in a single-cell or multicellular organism, by observing dynamics of a particular trait over time, the temporal diversity can be computed for every single cell and also for the organism as a whole. Having the capability of generating complex spatial and temporal patterns leads to various behaviors of cells across the body and over time. It may give the organism the required flexibility for performing complicated tasks and dealing with the dynamics of real-world environments for both biological and artificial organisms.

In robotics, the capability of generating diverse outputs is desirable for a controller to provide an adaptive behavior, either based on received sensory data (extrinsically forced), or based on internal dynamics (intrinsically forced) that may include internal states resulting from past behaviors and effects (memory induced). A controller in a single-module robot needs to generate output patterns that change the behavior of the module from time to time. In multimodular robots, different modules containing identical controllers usually need to behave differently according to their role and positioning in the robot. For example, in a locomotion task for a modular robot with legged-shaped or snake-shaped configuration, the organism needs to move the legs or other body segments in a rhythmic motion with different phase shifts. Central pattern generators (CPGs) have been used as coupled oscillators generating proper rhythmic motion for body segments of modular robots [22]. As the task gets more complicated, the ability of a controller to make complicated output patterns becomes more important. A system that lacks the capability of generating diversity in output patterns of the controller clearly fails to achieve complicated behaviors.

Diversity generation is also an important issue from the point of view of evolutionary robotics (ER). For example, a modular robot at the beginning of a locomotion task may be in a resting stance, and the information provided by the sensors may not be sufficient for the local controllers to generate a coordinated movement. For such a robot in early generations of evolution, controllers that make even seemingly useless movements for the modules may lead to bootstrapping due to symmetry breaking and bring the robot to proper stances where sensory information is exploitable.

At the population level, diversity is a feature of a population that is measured between different members. It has been studied in the context of evolutionary computation (EC) in areas such as optimization [42] and ER [30]. Various techniques have been investigated for measuring and preserving diversity in populations in order to prevent premature convergence of evolutionary algorithms and trapping at local optima in multimodal fitness landscapes, and also to keep the population adaptable to changes in its environment. Studies of population diversity have dealt with various aspects, for example, diversity of genotypes [17, 26, 34], of phenotypes [12], or, as in some recent works in ER, of behaviors [18, 25, 28].

From an evolutionary perspective, the capability of generating diverse phenotypes or behaviors in a population means that more areas of the solution space are potentially reachable by the evolutionary process, and it can lead to a richer exploration capability, which is a necessary condition for a successful evolutionary search. It should be noted, however, that for evolution to be successful, proper exploitation mechanisms must be considered as well.

In some areas, including ER, where behavior is a result of both genotype and environment (e.g., the physics of the body), it may not be enough for evolution to have genotypic diversity in the population, for various genotypes may lead to similar behaviors. This is of even more importance within an initial population where bootstrapping the evolutionary process requires a sufficient diversity of behaviors among individuals. In a set of processes that control the dynamics of an organism, it is desirable to be able to make phenotypically or behaviorally diverse populations with diverse parameterizations of the processes (genotypical diversity).

In this article, the capability of generating diversity at both individual and population levels is investigated in the context of a reaction-diffusion-based system called an artificial homeostatic hormone system (AHHS). The internal processes that control the dynamics of an AHHS are discussed, and the effect of inclusion or exclusion of each process on the capability of generating diversity is investigated. The investigation is performed for a number of internal processes of AHHSs that were introduced in the past and also an additional process, called tunneling, which is introduced here.

The results are compared with the capability of diversity generation in continuous-time recurrent neural networks (CTRNNs), which are a well-known kind of adaptive artificial neural networks (ANNs). The CTRNN is chosen as a reference method because of its simplicity, neurobiological plausibility, and being a universal approximator of smooth dynamics [14]. It is also analytically tractable [4] and has been applied to a wide range of problems such as computer vision [10], audio applications [7], and—of most interest to us—adaptive behaviors and robotics, as in Beer and Gallagher [6], Floreano and Mondada [13], Santos and Campo [33], and Chiel et al. [9].

This study aims to provide a better understanding of quantitative and qualitative effects of the investigated processes in generating diversity at the individual and population levels. The methods of diversity investigation in this work may be usable in the future for other processes of internal dynamics in other systems in order to evaluate and predict their influence in the space of behaviors that are potentially reachable by the system; thus they may enable us to design systems suitable for generating specified behaviors with respect to the required amount of diversity.

The contributions of this article that goes beyond the state of the art are summarized as follows:

  • Dissection of internal processes of AHHSs and analysis of their properties.

  • Proposing some metrics and methods for investigating diversity on the individual and population levels from the viewpoint of EC.

  • Introducing tunneling as a new internal process for AHHSs.

2 Processes of Dynamics in AHHSs

AHHSs are inspired by the signaling network of unicellular organisms and are designed to be evolvable as decentralized controllers applicable in the control of systems that consist of several agents—for example, modular robots. An AHHS is a reaction-diffusion-based system which can be seen as a gene regulatory network (GRN) augmented by special communication processes between the units, such as diffusion. The method was originally introduced in Schmickl and Crailsheim [37] and Schmickl et al. [39], and an improved version was introduced in Hamann et al. [21]. For applications of AHHSs in single and multimodular robotic scenarios see Stradner et al. [40], Schmickl et al. [38], and Hamann et al. [20].

In this work, a restricted version of the AHHS is implemented and an additional process of communication between adjacent units is introduced into the system. As the main focus of the work, the basic processes of dynamics in the AHHS are included in the current implementation, and their influence in generating spatial and temporal diversity in the phenotypes is investigated.

An AHHS, as introduced before, consists of a set of hormones and a set of rules. Here, in order to add an extra process of communication, a set of tunnels is also introduced into the AHHS for the first time. Concentration levels of the hormones are state variables of the system, and their dynamics are controlled by several processes. These processes include base production, decay, hormone-to-hormone reaction, diffusion, and tunneling (see below for their functionality).

To each hormone in an AHHS, a base production rate and a decay rate are assigned. The base production rate indicates the constant increase rate of the hormone. The decay rate determines the decrease rate of the hormone, proportional to its current concentration level.

The process of hormone-to-hormone reaction is supported by rules. Each rule represents an influence of one hormone—called the input hormone—on the production rate of another hormone—called the target hormone. Self-influence is allowed. Also, several rules may contain the same input and target hormones. In that case the influence of the input hormone on the target hormone equals the sum of the influences specified by all such rules.

The diffusion of hormone concentration and the hormone transfer based on tunneling are the means of communication between the adjacent units in an AHHS. To every hormone in an AHHS, a diffusion rate is assigned that specifies the rate of diffusion of a hormone from a unit to its neighbors.

In an analogy to intercellular communication in biological cells, where adjacent cells connect to each other via tunnel-like junctions to transfer ions and molecules, the process of tunneling is introduced to AHHS. In contrast to diffusion, tunneling can act against hormone gradients. Tunneling is implemented by using tunnels. Each tunnel represents the influence of a hormone on the transfer rate of another hormone from a unit to one of its neighbors.

In a more formal representation of the processes above, the dynamics of hormone concentration H at time t is defined for hormone h as follows:
formula
where αh, Dh, and μh are the base production rate, diffusion rate, and decay rate of hormone h, respectively. is the influence of linear hormone-to-hormone rule i, is the influence of tunnel i executed in the unit under consideration, and is the influence of neighbor unit n to the unit under consideration when tunnel i is executed in the neighbor unit.
A linear hormone-to-hormone rule is defined as
formula
The output is applied to the hormone concentration Hh(t), and the input is Hk(t) (h = k is allowed). λi and κi are two parameters of the rule, called the dependent dose and fixed dose. Their values are allowed to be negative. The trigger function θ determines whether or not the rule is executed:
formula
for mini and maxi as parameters of the rule (between 0.0 and 1.0 in this implementation).
A tunnel is defined with an equation similar to a linear hormone-to-hormone rule:
formula
it determines the amount of hormone that is transferred from the unit to one of its neighbors. The target neighbor is represented by a parameter of the tunnel determining the direction of the neighbor.

The basis of activity differs between processes that control the concentration of a hormone. Some processes act solely upon the concentration of the same hormone, but in some processes other hormones may be involved. In addition, some processes act locally, while in other processes other units (neighboring units) are also involved. Another aspect of differences in the processes is conservation of mass in the system. Conservation of mass can be considered for individual hormones or all the hormones together. In the former case, the total amount of every particular hormone throughout the system is unaffected by the process. In the latter case, the process may transform the proteins into each other, but the total amount of all the proteins is fixed. In an AHHS, some of the processes maintain conservation of mass for every particular protein. Other processes do not maintain any conservation of mass. These processes may generate or destroy a protein within the system without compensating for this change by an opposite change in the amount of any other protein.

Based on these differences, dynamic processes of AHHSs can be placed in four groups (see Table 1 for a summary):

  • The first group consists of processes that change a hormone's concentration at a fixed rate or solely based on current value of the hormone itself in the local AHHS unit (no influence by other hormones or by the same hormone in the vicinity). Processes of base production and decay belong to this group. These processes have no conservation of mass. In an analogy with the ANN, base production and decay processes correspond to bias and to negative recurrent edges of a node in an ANN, respectively.

  • The second group consists of processes that change a hormone's concentration based on the concentrations of the same hormone and other hormones inside the containing AHHS unit. Hormone-to-hormone reaction belongs to this group. It has no conservation of mass. In an analogy with the ANN, hormone-to-hormone reaction corresponds to edges between nodes.

  • The third group contains processes of implicit communication between units such that the change in a hormone's concentration depends on the same hormone in the two units. Diffusion belongs to this group. Diffusion does not require interaction of hormones. It maintains conservation of mass, and it only works in one gradient direction, from units with higher concentration to units with lower concentration.

  • The fourth group contains processes of implicit communication between units such that the change in a hormone's concentration also depends on other hormones in the other units. Tunneling belongs to this group. Tunnels involve hormone interaction and can act against the gradient. Tunneling maintains conservation of mass, and the transferred amount of a hormone is basically determined by other hormones in the source unit but is limited by the amount of the hormone in the target unit.

Table 1. 

Processes can be grouped based on the hormones involved and whether their activity is internal or external to the unit.


Self-sufficient
Other hormones involved
Internal to the unit (local) and no conservation of mass Base production, decay Hormone-to-hormone reaction 
 
External to the unit and with conservation of mass Diffusion Tunneling 

Self-sufficient
Other hormones involved
Internal to the unit (local) and no conservation of mass Base production, decay Hormone-to-hormone reaction 
 
External to the unit and with conservation of mass Diffusion Tunneling 

3 Investigating Diversity

The diversity brought by a controller is a result of processes governing the dynamics of the system and inputs from the environment. Diversity can be investigated at two different levels: population level and individual level. For each level a proper evaluation metric is required.

In the following sections, we design experiments in order to get an evaluation of diversity at the two levels by using the patterns of generated outputs by AHHS organisms. At the population level, we compute the diversity over a population of independent organisms. Each organism contains its own genotype. The diversity at the population level is considered as the number of distinct behaviors generated in the population by different organisms. At the individual level, diversity is an internal property of the output pattern generated by a single organism. We define the individual diversity as the complexity of the output pattern generated by an organism across the organism's body and over time. A number of qualitative types are defined for output patterns, considering the complexity in the spatial or the temporal dimension.

The experiments are performed for different combinations of processes of the AHHS, with the aim of investigating the effect of each process in generating the diversity for the system.

The results at the individual level are compared with the diversity generated by a CTRNN system as a reference method.

3.1 Diversity at Population Level

In this section diversity is studied at the population level over randomly generated individuals. We claim that from the perspective of evolutionary algorithms perspective, a high capability of generating diversity at the population level is a desirable property of a system of processes that control the behavior of an organism. A diverse initial population means a diverse set of starting points for the searching procedure and is a better initial covering over the search space. Also, the capability of making a broad range of phenotypical traits and behaviors means a broad reachable area in search space and increases the probability of reaching the solution space. It should be noted that for a system aiming to evolve toward a solution space it is not enough to maximize the rates of diversity. A controller that generates a purely random behavior over time can generate high diversity, but that is not a controller space that we would like to search in. There needs to be a tradeoff between the exploration capability provided by the high potential of generating diversity and exploitation capabilities of the system to limit the search. The difference between diversity generation by a purely random generation controller and what we are searching for is that the dynamics of our systems are bound and controlled by a set of restricted processes defined in the system and therefore the randomness is constrained.

In order to evaluate the diversity, first we define a behavior. A behavior is a sequence of consecutive states of an arbitrary phenotypical trait of an organism that is observed during a limited time. Figure 2 represents example behaviors for three AHHS organisms in the first 100 time steps. In a population of organisms, phenotypical diversity is computed based on the number of different behaviors observed in the population.

Figure 2. 

Example behaviors for three AHHS organisms in the first 100 time steps. The observed phenotypical trait in this case is a hormone concentration.

Figure 2. 

Example behaviors for three AHHS organisms in the first 100 time steps. The observed phenotypical trait in this case is a hormone concentration.

We set up the experiments with all combinations of AHHS processes of dynamics and with two levels of population size: First, a large population of randomly parameterized AHHSs, namely the overall population, is generated. By a large population, we mean a population with a size several times greater than the size of the behavior space. In order to assess the influence of different AHHS processes in generating diversity in the population, for every combination of AHHS processes, the processes are activated in the system and the diversity of the population is evaluated.

At the other level of population size, the overall population is randomly divided into small subpopulations with a fixed size, and the same process of diversity evaluation for combinations of processes is repeated for every subpopulation. A subpopulation is taken as a typical population that is required to evolve in an evolutionary task, and its size is chosen accordingly.

The diversity of the overall population is a representation of the fraction of behavior space that is reachable by the system of processes. The maximum diversity in this case is reached if every behavior in the behavior space is presented by at least one of the individuals in the population. On the other hand, diversity of subpopulations represents diversity of a randomly generated population in a typical evolutionary task. The maximum diversity within a subpopulation is reached if every individual exhibits a unique behavior that is different from the behaviors of the others.

In addition to finding the overall and subpopulation diversities, performing a comparison between the two diversities might also be useful. Comparing the diversity in the overall population and its subpopulations gives a measure to assess phenotypical similarity between the subpopulations. For example, if a set of dynamical processes leads to a set of subpopulations with high internal diversity but the overall diversity (the diversity in the overall population) is comparatively low, it means that the subpopulations are phenotypically similar to each other, although they are diverse internally. On the other hand, with the same internal diversity for the subpopulations and a high diversity for the overall population, we can conclude that the subpopulations are phenotypically variant. By applying this analysis, the efficiency of using island models of evolution for evolving the system can be assessed. From the point of view of island models in evolutionary algorithms [43], having a high similarity between the subpopulations (islands) means similar starting points (initial conditions) for individual islands. That might prevent the occurence of a set of islands each of which follows a unique search trajectory, so that the system fails to benefit from using island models.

3.2 Simulation at Population Level

An AHHS organism is set up consisting of two adjacent AHHS units, say unit0 and unit1. Then unit1 is observed for a phenotypical trait. Both units contain an identical instance of an AHHS genome and maintain three hormones. The concentration levels of the hormones change in the interval of [0.0, 1.0] under the control of dynamic processes of the AHHS. The initial concentration levels are set to 0.5 in unit0 and to zero in unit1. One of the three dynamic hormones is arbitrarily chosen, and its concentration levels are observed as a phenotypical trait.

The number of rules is 30 and is chosen based on preliminary experiments for maximum diversity generation (data is not shown) and also matches previous experiments with AHHSs [21]. The number of tunnels is chosen arbitrarily as three.

3.3 Results and Discussion at Population Level

A behavior is defined as a sequence of phenotypical traits (concentration levels of a chosen hormone) of unit1 in five consecutive time steps (time steps 95 to 100). The value range of the trait is discretized into eight bins. Therefore, the size of the behavior space is 85, that is, 32,768. Note that the number of bins (eight) and the size of the sequence (five) are chosen arbitrarily, taking computational convenience into account.

A large population of 1,000,000 randomly generated AHHSs are created. The size of the population is chosen to be reasonably higher than the size of behavior space, so that the computed value for overall diversity is statistically meaningful. The population is then divided into 10,000 subpopulations of size 100. The size of the subpopulations is again arbitrarily chosen, based on a typical population size of AHHSs in an evolutionary robotic task.

Diversity in a population is computed as the number of distinct behaviors that are achieved by at least one individual of the population divided by the maximum number of possible behaviors in the population. Note that if the size of the population is greater than the size of the behavior space, the maximum possible number of behaviors in the population equals the size of the behavior space; otherwise it equals the population size. This means, with a behavior space of size 32,768, the maximum number of possible behaviors in a subpopulation of size 100 is 100, but for the overall population of size 1,000,000 it is 32,768.

The diversity is evaluated for all the different combinations of AHHS processes in all of the subpopulations and the overall population. In order to do that, for every combination of AHHS processes, a randomly generated population (of size 1,000,000) is created, and the diversities of the overall population as well as its subpopulations are computed (as described above).

Figure 3 represents the average diversity for subpopulations, the diversity of overall population, and a number called ratio. This number represents the ratio between the overall and subpopulation diversities and is used to facilitate the comparison between the two diversities in order to assess phenotypical similarity between the subpopulations and consequently to assess the usefulness of applying island models for the system. Its value is calculated as the number of distinct behaviors achieved in the overall population divided by the averaged number of behaviors achieved in the subpopulations and scaled by a factor of 100. Although this ratio is not an accurate measure of similarity between the subpopulations, it gives an impression of that relation, especially when two combinations of processes show a significant difference between their ratios.

Figure 3. 

Diversity of behaviors averaged over all subpopulations (solid line), diversity in the whole population (dashed line), and the ratio between the number of distinct behaviors achieved in the whole population and the average number achieved in the subpopulations (dotted line). Variables r, c, a, d, and t indicate hormone-to-hormone reaction, decay, base production, diffusion, and tunneling, respectively.

Figure 3. 

Diversity of behaviors averaged over all subpopulations (solid line), diversity in the whole population (dashed line), and the ratio between the number of distinct behaviors achieved in the whole population and the average number achieved in the subpopulations (dotted line). Variables r, c, a, d, and t indicate hormone-to-hormone reaction, decay, base production, diffusion, and tunneling, respectively.

An ANOVA test is executed on the data for diversity generation of subpopulations. The results, shown in Table 2, suggest a high importance for hormone-to-hormone reaction and base-production processes, lower importance for diffusion and decay, and lowest importance for tunnels in producing diversity at the population level.

Table 2. 

An ANOVA test is used to statistically compare the influence of every underlying process on diversity generation of the system at the population level. The significantly high values of r and a suggest a high importance for these processes. The values suggest the next level of importance for d and c, and the lowest importance for t.


DOF
Sum sq.
Mean sq.
F-value
Pr(> F)
Reaction (r25,625,873.17 25,625,873.17 3,549,429.14 0.0000 
Decay (c1,363,111.01 1,363,111.01 188,803.94 0.0000 
Production (a13,772,313.32 13,772,313.32 1,907,597.45 0.0000 
Diffusion (d2,131,198.84 2,131,198.84 295,191.47 0.0000 
Tunnels (t90,217.96 90,217.96 12,496.05 0.0000 
Residuals 319,968 2,310,078.34 7.22   

DOF
Sum sq.
Mean sq.
F-value
Pr(> F)
Reaction (r25,625,873.17 25,625,873.17 3,549,429.14 0.0000 
Decay (c1,363,111.01 1,363,111.01 188,803.94 0.0000 
Production (a13,772,313.32 13,772,313.32 1,907,597.45 0.0000 
Diffusion (d2,131,198.84 2,131,198.84 295,191.47 0.0000 
Tunnels (t90,217.96 90,217.96 12,496.05 0.0000 
Residuals 319,968 2,310,078.34 7.22   

The combinations of processes are arranged in Figure 3 in increasing order of overall diversity. For clarity in the figures, the processes hormone-to-hormone reaction, decay, base-production, diffusion, and tunneling are encoded as r, c, a, d, and t, respectively.

With respect to diversity in overall population, three groups of process combinations can be detected in Figure 3. Process combinations that show no diversity and basically make a single behavior of “doing nothing” make up the no-diversity group. The rest of process combinations are grouped into low-overall-diversity and high-overall-diversity generators. A jump in overall diversity representing an increase in the reachable behavior space starts from the process combination {rct} and distinguishes the two groups.

The following observations are implied by Figure 3:

  • Process combinations of {c}, {r}, {rc}, and no process are in the no-diversity group.

  • The process combinations in the high-overall-diversity group also represent high ratio values in comparison with the low-overall-diversity group, which means there is low similarity between the subpopulations.

  • Since the observed unit is initially empty, in order to generate some diversity, the process combinations need to include a communication process enabling transfer of the hormones from outside the unit (d or t), or an internal process that initiates production of hormones out of nothing (a). The combinations that do not have any of these processes produce no diversity and belong to the no-diversity group.

  • Processes a, d, and t can produce some diversity on their own:

    • For {a}, the concentration of the observable hormone independently increases in unit1 at a constant rate.1 Due to the randomized parameterizations of individuals in the population, the rate of increase may be different in different individuals, leading to diversity in the population.

    • For {d}, the hormone is diffused from unit0 to unit1 at various rates until the concentrations of the hormone become equal in the two units. Therefore, again, increase of the hormone may be detectable in unit1, with different rates for different individuals.2

    • For {t}, the hormone is first transferred from unit0 according to the parameters of tunnels in the individual. At the same time other hormones may have also been transferred from unit0 to unit1, which would lead to a back transfer of the observable hormone from unit1 to unit0. Diversity is generated according to the parameters of the tunnels in every individual.

  • The value of the ratio, in Figure 3, shows a higher value for {t} than for {a} and {d}, indicating less similarity between the subpopulations controlled only by {t}.

  • All the combinations in the group of high overall diversity contain r. Since r is an internal process with no ability to generate hormones out of nothing, in the process combinations in this group, a communication process (d or t) or a process that is able to independently initiate hormone production (a) is included. As seen in the figure, all combinations that include r together with such a process are in the group of high overall diversity, except for {rt}, which still represents a high ratio compared to other combinations in its group (low overall diversity), representing comparatively low similarity between subpopulations.

  • In the group of high overall diversity, the combinations that include d lead to higher diversity in both subpopulations and overall population than do their counterpart combinations with no d. These combinations also show lower similarity between subpopulations (high ratio), which implies the usability of island models of evolution in searching the behavior space. This is not necessarily the case for the combinations in the low-overall-diversity group.

  • Similar points to the above can be made for the combinations that include c.

  • Another point that is implied by focusing on subpopulation diversity is that combinations with a lead to higher subpopulation diversities than do their counterparts with no a, except for {adt}, which creates slightly less diversity.

3.4 Diversity at the Individual Level

Diversity at the individual level is defined as a property of every individual organism and is computed for an observed phenotypical trait. Diversity of a phenotypical trait (e.g., output of a controller), which is generated by a system over time and over units of the organism, is investigated by defining some measures of complexity for the spatiotemporal pattern of the trait.

The word complexity has been used in scientific works with many different meanings and in different applications (for a review of complexity measures see Badii and Pliti [2], Daw et al. [11]). Complexity can be measured based on entropy with its information theory definition [32]. The metrics can combine aspects of entropy with properties of dynamical systems [3]. Other methods are related to Kolmogorov complexity [24] and measure algorithmic complexity of the pattern.

Here we are interested in measuring irregularities of an observed trait along both spatial (organism's body) and temporal axes. With some similarity to a work by Fusco and Minelli [15] in measuring morphological complexity, the number of consecutive monotonic subsequences with nonzero slopes is used as an estimate of irregularity.

For a sequence X of observed traits,
formula
where Xi is a monotonic subsequence of X and the signs of the slope for every Xi and Xi+1 are different, the complexity is defined as
formula

In this work, an organism is set up as a row of adjacent units with identical controllers and is examined for an arbitrary trait during a fixed period of time. A spatiotemporal matrix is generated out of the observed values, where every row in the matrix represents the state of the observed trait for the organism in a single time step. In the same way, every column of the matrix represents the state of the trait in a single unit over time (see Figure 4 for some examples).

Figure 4. 

Examples of different qualitative types of spatiotemporal patterns. Horizontal coordinate represents the units along the organism's body, and vertical coordinate represents time.

Figure 4. 

Examples of different qualitative types of spatiotemporal patterns. Horizontal coordinate represents the units along the organism's body, and vertical coordinate represents time.

Note that in this study the system is not interacting with the environment and therefore there is no input from the outside. Therefore, all the generated diversity comes from inside and demonstrates the intrinsic capability of the system for diversity generation.

3.4.1 Qualitative Types of Spatiotemporal Patterns

We define a set of qualitative types for spatiotemporal patterns. The behavior of the complexities along both spatial and temporal coordinates are considered in these definitions.

  • Static-flat: No diversity in any direction, that is, identical value for the trait all over the organism and during the whole time (Figure 4a).

  • Dynamic-flat-monotone: No spatial complexity and low temporal complexity, that is, identical value for the trait along the organism's body, but the value changes monotonically over time (Figure 4b).

  • Dynamic-flat-nonmonotone: No spatial complexity and high temporal complexity, that is, no difference between the units of the organism along its body, but the behavior of the units has nonmonotone dynamics and shows some ups and downs in the traits value (Figure 4c).

  • Static-complex-monotone: Low spatial complexity; the value of the trait changes monotonically along the body. The body pattern does not change over time (no temporal complexity) (Figure 4e).

  • Static-complex-nonmonotone: Spatial complexity; the change is nonmonotone along the body. The body pattern does not change over time (no temporal complexity) (Figure 4f).

  • Dynamic-complex: A complex body-pattern that changes over time and does not lose its complexity (Figure 4d).

  • Transient-flat: Very low spatial and temporal complexity, with monotonic change over time and along the body. It eventually converges to either a flat pattern or a static-complex-monotone pattern (Figure 4h).

  • Transient-complex: Very low temporal complexity, with monotonic change, but high spatial complexity with nonmonotonic change along the body. It eventually ends up in either a monotone or a nonmonotone static-complex pattern (Figure 4i).

  • Vanishing: Spatial complexity decreases over time. It may eventually converge to zero and represent a flat pattern, or may end up in a dynamic complex pattern with a spatial complexity that does not change over time (Figure 4g).

As mentioned, a vanishing pattern may eventually converge to other pattern types. The same is true of a dynamic-flat-monotone pattern: Since the trait values are bounded, a monotonic change (either increasing or decreasing) is a transient pattern that may end up as a static-flat pattern, or may be a part of a very slow dynamic-flat-nonmonotone pattern. We have chosen the number of observation time steps to be large in comparison with the number of bins (20 versus 8; see Section 3.6). Therefore, we suspect that most of the dynamic-flat-monotone patterns are transient patterns ending up as static-flat.

In order to gain an impression of the usefulness of different types of patterns, it will be interesting to think of some ways of exploiting the patterns in a simple application. The non-transient patterns in limited locomotion of a modular robot may be a good example.

  • Based on the definition of the dynamic-complex pattern type, it is composed of many different patterns, including traveling waves. Modular robots can benefit from traveling waves along the body (as exploited in CPGs [22]), for example, a modular robot in a snake configuration or a legged organism.

  • Static complex patterns (either monotone or nonmonotone) are similar to body patterns in biological embryos [1, 23, 27]. As in embryos, these patterns can be used for structuring and discrimination between different modules of a robot and assigning different roles to different modules. For example, by having an oscillation generator in the modules, a static pattern can be used to assign different phase shifts to different modules so that a traveling wave is generated along the body.

  • A nonmonotone dynamic flat pattern basically means an oscillation over time where all the units perform the same behavior. By externally assigning different phase shifts to different modules, a traveling wave can be produced, generating a proper locomotion for the modular robot.

In order to assign a type to an spatiotemporal pattern that is represented by a matrix A, complexity is measured for every row and every column of the matrix by using Equation 5. Let Cs(ai, ∗) be the measured complexity of ith row of A, and Cs(a∗, j) be the measured complexity of the jth column of A. Table 3 describes the method for assigning a type to a spatiotemporal pattern.

Table 3. 

Type assignment to a spatiotemporal pattern by using Equation 5.

Static-flat i : Cs(ai, ∗) = 0 
and ∀j : Cs(a∗, j) = 0 
Dynamic-flat-monotone i : Cs(ai, ∗) = 0 
and ∀j : Cs(a∗, j) = 1 
Dynamic-flat-nonmonotone i : Cs(ai, ∗) = 0 
and ∀j : Cs(a∗, j) > 1 
Static-complex-monotone i : Cs(ai, ∗) ≤ 1 ∧ ∃i : Cs(ai, ∗) = 1 
and ∀j : Cs(a∗, j) = 0 
Static-complex-nonmonotone i : Cs(ai, ∗) > 1 
and ∀j : Cs(a∗, j) = 0 
Dynamic-complex Cs(ai, ∗) is increasing or is not monotonic over i
or Cs(ai, ∗) is fixed and ∃j : Cs(a∗, j) > 0 
Transient-flat i : Cs(ai, ∗) ≤ 1 ∧ ∃i : Cs(ai, ∗) = 1 
and ∀j : Cs(a∗, j) ≤ 1 ∧ Cs(a∗, j) = 1 
Transient-complex i : Cs(ai, ∗) > 1 
and ∀j : Cs(a∗, j) ≤ 1 ∧ Cs(a∗, j) = 0 
Vanishing Cs(ai, ∗) is decreasing over i 
Static-flat i : Cs(ai, ∗) = 0 
and ∀j : Cs(a∗, j) = 0 
Dynamic-flat-monotone i : Cs(ai, ∗) = 0 
and ∀j : Cs(a∗, j) = 1 
Dynamic-flat-nonmonotone i : Cs(ai, ∗) = 0 
and ∀j : Cs(a∗, j) > 1 
Static-complex-monotone i : Cs(ai, ∗) ≤ 1 ∧ ∃i : Cs(ai, ∗) = 1 
and ∀j : Cs(a∗, j) = 0 
Static-complex-nonmonotone i : Cs(ai, ∗) > 1 
and ∀j : Cs(a∗, j) = 0 
Dynamic-complex Cs(ai, ∗) is increasing or is not monotonic over i
or Cs(ai, ∗) is fixed and ∃j : Cs(a∗, j) > 0 
Transient-flat i : Cs(ai, ∗) ≤ 1 ∧ ∃i : Cs(ai, ∗) = 1 
and ∀j : Cs(a∗, j) ≤ 1 ∧ Cs(a∗, j) = 1 
Transient-complex i : Cs(ai, ∗) > 1 
and ∀j : Cs(a∗, j) ≤ 1 ∧ Cs(a∗, j) = 0 
Vanishing Cs(ai, ∗) is decreasing over i 

The spatiotemporal patterns of each type are counted in populations of organisms with randomly parameterized controllers. Independent experiments are performed for controllers of all combinations of AHHS processes and also for CTRNN controllers (see below), and the results are compared.

Figure 5 represents some patterns generated by AHHSs exhibiting traveling waves. As another example of spatiotemporal patterns generated by AHHSs, Figure 6 represents patterns with maximum total complexity that is measured along both spatial and temporal dimensions.

Figure 5. 

Four example patterns evolved for generating a traveling wave with an AHHS.

Figure 5. 

Four example patterns evolved for generating a traveling wave with an AHHS.

Figure 6. 

Four example patterns evolved for maximum overall complexity with an AHHS.

Figure 6. 

Four example patterns evolved for maximum overall complexity with an AHHS.

3.4.2 CTRNNs

CTRNNs are Hopfield continuous networks with an unrestricted weight matrix. They are networks of biologically inspired neurons that have been shown to be universal approximators of smooth dynamics and to exhibit complicated dynamical behaviors [14]. A neuron i in the network is of the following general form:
formula
where yi is the state of the ith neuron, τi is the neuron's time constant, wji is the weight of the connection from the jth to the ith neuron, θj is a bias term, gj is a gain term, Ii is an external input, and σ(x) = 1/(1 + ex) is the standard logistic output function.

3.5 Simulation at the Individual Level

3.5.1 AHHS Settings for the Experiment

An AHHS organism is created that is composed of six adjacent AHHS units, say unit0 to unit5. All units contain an identical instance of an AHHS genotype and maintain three hormones. The hormones' concentration levels may change in the interval of [0.0, 1.0]. Initially, the concentration levels of all the three hormones are set to 0.5 in unit0 and to zero in the other units. One of the three hormones is observed as the phenotypical trait in unit1 to unit5.

3.5.2 CTRNN Settings for the Experiment

In the same way as for the AHHS organism, an organism of six adjacent CTRNN units is created, say unit0 to unit5. Every CTRNN unit consists of three nodes, which assimilate the three hormones of the AHHS implementation. Every unit is connected to its immediate neighbors by an incoming and outgoing edge for every peer node in the two units. The initial value of each node is set to 0.5 in unit0 and set to zero in other units. Ii is set to zero for all the units. Weights are randomly initialized. By considering the study of the parameter space structure of CTRNN in [5], the values of the θs are set, based on the weights, so that the richest possible dynamics is achieved. One of the three nodes is observed as the phenotypical trait in unit1 to unit5.

3.6 Results and Discussion at the Individual Level

For every combination of the AHHS dynamical processes as well as the CTRNN, 10,000 randomly generated organisms are independently generated and run for 100 time steps. The arbitrarily chosen phenotypical traits are observed in time steps 80 to 100. Similar to the experiments at the population level, the value range of the observed trait is discretized into eight bins. A spatiotemporal pattern is formed as a 20 × 5 matrix representing the state of the trait observed in the five units of the organism at 20 consecutive time steps. Figure 7 demonstrates an example spatiotemporal pattern.

Figure 7. 

An example spatiotemporal pattern generated by an AHHS organism of size 5 in 20 consecutive time steps.

Figure 7. 

An example spatiotemporal pattern generated by an AHHS organism of size 5 in 20 consecutive time steps.

Figure 8a plots the number of patterns of different types for the different AHHS combinations of processes and for the CTRNN. The numbers are proportional to the size of populations. The values for the static-flat patterns are omitted for the sake of clarity.

Figure 8. 

Pattern types for all combinations of processes of the AHHS and for the CTRNN. Static-flat patterns are not shown (the rest of each bar up to 1.0). r, c, a, d, t indicate hormone-to-hormone reaction, decay, base production, diffusion, and tunneling processes, respectively.

Figure 8. 

Pattern types for all combinations of processes of the AHHS and for the CTRNN. Static-flat patterns are not shown (the rest of each bar up to 1.0). r, c, a, d, t indicate hormone-to-hormone reaction, decay, base production, diffusion, and tunneling processes, respectively.

In order to evaluate the influence of the number of tunnels in generating of different spatiotemporal patterns, the experiments are repeated for AHHS combinations of processes while the number of tunnels is set to 30 and the numbers of hormones and rules stay unchanged (3 and 30, respectively). Figure 8b represents the results.

The following observations are implied by Figure 8a and Figure 8b:

  • Only a negligible number of patterns generated by the CTRNN exhibit the transient pattern types (dynamic-flat-monotone, vanishing, transient-flat, and transient-complex). Most of the patterns of static-complex types are nonmonotone, which means an static oscillatory pattern over the body.

  • Process combinations {d}, {c}, {cd}, and no process do not generate any interesting pattern (only the static-flat pattern, which is omitted from the figure, or the dynamic-flat-monotone, which is a transient pattern).

  • Since all the units of the organism except unit0, which is not observed, start with the same initial hormone concentration levels of zero, a communication process is necessary to transfer some hormones from unit0 in order to initiate symmetry breaking. Therefore, process combinations with no d or t only generate patterns with no spatial diversity (flat patterns).

Due to that, process combinations {ca}, {a}, {r}, {rc}, {ra}, {rca}, generate dynamic-flat-monotone and dynamic-flat-nonmonotone patterns. Since dynamic-flat-monotone is a transient pattern, {ca} and {a} eventually end up as static-flat patterns. That is because a and c respectively increase and decrease the value of the observed hormone by a constant rate. If the rate of decrease is higher than that of increase, the hormone level never rises from zero. If the rate of increase is lower, the hormone concentration eventually ends up at saturation level (1.0) for all the units making a static-flat pattern.

On the other hand, the process combinations {r}, {rc}, {rca}, and {ra} are capable of generating dynamic-flat-nonmonotone patterns due to inclusion of r, while {rca} and {ra} generate a rather high number of dynamic-flat-nonmonotone patterns due to inclusion of a along with r, which enables more possibilities for the values by raising the hormone levels.

  • The only process that is able to generate static-complex and dynamic-complex patterns on its own is {t}, according to both Figure 8a and Figure 8b.

  • The variable r represents an important effect in generating dynamic-complex patterns, although it is required to be combined with a communication process (d or t). The only exception is the process combination {rt}, which represents a low number of dynamic-complex patterns, although the number of static-complex patterns is rather high.

  • The low number of dynamic-complex patterns in comparison with static-complex patterns in {rt} is also seen in Figure 8b, where the number of tunnels is much higher.

  • The highest amount of diversity generation is associated with the process combinations that include all of r, a, and d, while the combinations that include all of r, a, and t generate considerably high diversity, especially if the number of tunnels is high (Figure 8b).

  • Comparison of Figure 8a and Figure 8b implies that an increase in the number of tunnels makes a considerable impact in the diversity generation of the system in favor of dynamic-complex patterns.

4 Evolving for a Task

In this section, different settings of AHHS are evolved for a multimodular controller in order to give an impression of the relevance of the diversity measures described in the previous sections to evolutionary tasks in practice. The scenario is for a special conveyor consisting of a 3 × 3 grid. Every cell of the grid has a value that can be considered the height of the cell (or a potential level). Every cell's height is controlled by an AHHS controller. At the beginning of the task, one or more boxes stand on a specific cell(s) of the grid. The task is to convey the box from the starting point to two target points, one after the other (Figure 9). The box is moved to one of the Von Neumann neighboring cells if the height of the neighbor is lower than the height of the current cell (the difference needs to be more than a specified threshold). If there are several neighbors with sufficiently low height, the box moves to the neighbor with the lowest height.

  • AHHS configuration. All the AHHS controllers are genetically identical, with 3 hormones, 3 tunnels, and 30 rules. One of the hormones is chosen to directly determine the controller's output value. Hormone concentrations are variable between zero and one. At the beginning of an evaluation, all the hormone concentrations are set to 0.5 in the cell (0,0) and to zero in the other cells.

  • Evolutionary algorithm configuration. For every experiment, a population of 30 randomly initialized AHHSs are evolved for 1000 generations. Elitism is applied for the three best individuals. The rest of the population are selected by linear proportional selection and mutated to make the next generation (for the details of evolutionary operators of AHHSs see [20]).

Figure 9. 

Demonstration of the conveyor experiments.

Figure 9. 

Demonstration of the conveyor experiments.

4.1 Experiments

AHHS is evolved for generating controllers for two experiments with the special conveyor. A difference of +0.02 between the output value of a cell holding the box and one of its Von Neumann neighbors is enough to push the box to the neighboring cell. In the first experiment, a single box has to be transported. The box starts at cell (0,0) and is supposed to move to (2,2) (as its first target) and to move back to (0,0) (as the second target) in a maximum of 10 time steps overall. Figure 9a represents the first experiment, and Figure 9b represents a potential path of the box generated by the conveyor. Note that in every move of the box, the value of the current cell has to be more than that of the next cell (by 0.02), and therefore the output pattern has to be changed over time.

In the second experiment, two boxes are placed at cells (0,0) and (2,2) and have to be conveyed at the same time. The boxes are supposed to move to the first targets at (2,0) and (0,2), respectively, and then to their second targets at (2,2) and (0,0). The process is allowed to take six time steps overall. Figure 9c demonstrates the second experiment.

Fitness values are calculated at the end of each experiment as follows:
formula
where n is the number of boxes, and di is defined as follows:
formula
where t1, t2, and c are the coordinates of the first and second targets and the final position of the box, and dist(a, b) is the distance between a and b, defined as the minimum number of steps necessary for reaching b starting from a.

In both experiments, the AHHS controllers need to make dynamic and complex patterns to fulfill the tasks. Note that the system we have here is a 2D system, which is different from the experiments on diversity measures.

Both experiments are repeated for 100 independent runs for all possible combinations of AHHS processes, and the results are demonstrated in Figure 10a and Figure 10b. The order of the results for the demonstrated combinations of processes is that in Figure 11.

Figure 10. 

Fitness values of the conveyor experiments for all combinations of processes of AHHS. Boxplots indicate median and quartiles; whiskers indicate minimum and maximum; circles indicate outliers (values are collected from 100 independent runs in each case).

Figure 10. 

Fitness values of the conveyor experiments for all combinations of processes of AHHS. Boxplots indicate median and quartiles; whiskers indicate minimum and maximum; circles indicate outliers (values are collected from 100 independent runs in each case).

Figure 11. 

Results from both population level and individual level plotted together. The setting is three hormones, 30 rules, and three tunnels. r, c, a, d, t indicate hormone-to-hormone reaction, decay, base production, diffusion, and tunneling processes, respectively.

Figure 11. 

Results from both population level and individual level plotted together. The setting is three hormones, 30 rules, and three tunnels. r, c, a, d, t indicate hormone-to-hormone reaction, decay, base production, diffusion, and tunneling processes, respectively.

By comparison of Figure 10a and b with Figure 11, we conclude that process combinations that are able to generate high numbers of complex-dynamic patterns are more successful in producing proper behavior for the two investigated tasks. On the other hand, the combinations that were detected as poor diversity generators fail to produce the desired behaviors. Another interesting result of these comparisons concerns the process combination {rt}. In both experiments, this combination is successful while, for example, {at} and {dt} with higher numbers of complex-dynamic patterns failed to evolve for the desired behaviors. In a similar way, {cdt} seems to be more evolvable toward the desired behaviors than its neighboring process combinations. This might be explained by looking at the ratio factor of diversity in the population level (in Figure 11). The ratio values are considerably higher for {rt} and {cdt} than their neighboring process combinations, showing that evolution is more successful in finding the desired behaviors (if they exist), due to the intrinsic diversity of the populations with these settings.

5 Conclusion

In this article, the capability of generating diversity is investigated as a desirable property of controller systems with nontrivial behaviors, for example, in evolutionary robotics, where a known problem of bootstrapping of the evolutionary process can be reduced by having behavioral diversity in the population [29].

The controller system that is investigated here is the AHHS. The implemented AHHS system includes the processes of hormone-to-hormone-reaction, decay, base production, and diffusion, which were introduced in [37, 39], and a newly defined process called tunneling. These processes that drive dynamics in an organism that is controlled by AHHS are described and discussed based on their scope of functioning and their influence on generating diversity.

Metrics of diversity are introduced on different levels: the population level and individual level. The metrics on the population level evaluate diversity of behaviors generated by different organisms in a random population. The evaluation is performed for populations of both big and small sizes with respect to the size of the behavior space. For a big population (namely, the overall population) the results demonstrate to what extent a particular combination of processes covers the behavior space. For a small population they demonstrate how much of phenotypical variance is achieved by a random initialization that in turn influences the efficiency of starting an evolutionary algorithm in the population. By comparing results from the two sizes of populations, we have a measure of similarity between different random small populations, which can be used as an indicator of whether or not employing island models in evolutionary algorithms is proper for the system.

A metric is also introduced for diversity evaluation on the individual level, where the spatiotemporal patterns generated by organisms are considered. On this level several qualitatively different types are defined such that every spatiotemporal pattern belongs to one of the types. For all different combinations of AHHS processes that are involved in controlling a random organism and for a large number of controllers, generated spatiotemporal patterns are classified into these types, and the number of patterns in every type is calculated. The experiment is performed for randomly initialized controllers consisting of different combinations of AHHS processes and also by a CTRNN, and the calculated numbers are compared in these different cases.

As a summary of the evaluated capabilities of diversity generation, Figure 11 represents the diversity values on both population and individual levels in the same graph. Although there are some differences in the configurations for the two levels due to the requirement of computability in the study of population level and spatial dimension in spatiotemporal patterns on the individual level, the comparison of results may still be interesting.

All in all, the effects of the investigated internal processes of AHHSs can be summarized as follows.

  • Hormone-to-hormone reaction is a process that may generate or denature a hormone, depending on its own concentration and the concentrations of other hormones in the same unit. This process is similar to feedback loops in recurrent neural and regulatory networks. This process is intuitively expected to generate high diversity, and as is shown by the results, it is the most important process in producing diversity on both population and individual levels. All the process combinations that make high population diversity (including the ratio of overall population to subpopulation) contain hormone-to-hormone reactions, and the same combinations generate comparatively high degrees of diversity at the individual level.

  • By combining the base-production process with the hormone-to-hormone reaction process without adding any communication process, a comparatively high diversity can be generated. For example, in the process combinations {hormone-to-hormone reaction, base-production} and {hormone-to-hormone reaction, decay, base production} a rather high number of dynamic-flat-nonmonotone patterns, which are spatiotemporal patterns with no spatial complexity and a rhythmic change in temporal dimension, are generated.

  • Combining the diffusion process as an implicit communication process with the hormone-to-hormone reaction process produces a proper complexity in the temporal dimension. This effect is not detectable for tunneling as another process of implicit communication.

  • The hormone-to-hormone reaction, base-production, and diffusion processes together make the highest number of non-static-flat patterns in general and dynamic-complex patterns in particular. They also demonstrate the highest achievability of behavior space, as shown by high diversity in overall population on the population level.

  • The effect of the diffusion process can be also investigated by comparing the process combinations {decay, base production, diffusion} and {base production, diffusion} with their counterparts without diffusion, {decay, base-production} and {base production}. In the cases where diffusion is included, a spatial dimension of complexity is added to the produced spatiotemporal patterns that changes the type of the patterns, while no significant change in the population diversity is generated either in subpopulations or in the overall population.

  • Inclusion of decay process along with hormone-to-hormone reaction, base production, and diffusion in a process combination also increases both individual and population diversities.

  • Tunneling is the only process that is able to generate static-complex and dynamic-complex patterns on its own {t}. On the other hand, although the tunneling process seems to have some similarities with hormone-to-hormone reaction in the influence of other hormones on the target hormone and with diffusion in being an implicit communication process between neighbor units with conservation of mass, it contributed differently in the dynamics of the system. This can be clearly observed by comparing the process combinations {hormone-to-hormone reaction, tunneling} and {hormone-to-hormone reaction, diffusion}. The difference is visible on both population and individual levels; the individual level possesses large proportions of static-complex patterns, implying that tunneling has a high spatial influence that can dominate the temporal influence of hormone-to-hormone reaction. The high spatial influence of tunneling is also observable in the combinations {hormone-to-hormone reaction, tunneling} and {hormone-to-hormone reaction} in Figure 8b. We suspect that the high spatial influence of tunneling dominates its temporal influence and reduces its overall effect on the dynamics of the system.

In order to get an impression of the relevance between the described metrics of diversity generation and evolutionary tasks in practice, an evolutionary task for a multimodular system of AHHSs is investigated. The evolutionary behavior of all the possible combinations of processes is demonstrated, revealing a correlation between the capability of diversity generation and evolvability of the system for the given task.

Similar methods and metrics to those used in this work might be applicable for achieving a deeper understanding of processes that are considered to be included in AHHS or other control systems and investigating their effects in generating target behaviors.

In the future, the generated spatiotemporal patterns, especially dynamic-complex patterns, will be further partitioned qualitatively into subtypes and investigated in more detail in order to provide more information about the type of behaviors that can be generated by the controllers and the processes that need to be included in a system for producing particular target behaviors. Similarly, some processes can be investigated in more abstract or more detailed way; for example, by considering the r process as a feedback loop, it can be abstracted into two different processes with positive and negative feedback effects, and their influences on diversity generation for the system can be studied separately.

The diversity generation capabilities of the processes will also be investigated in the future in closer relation with evolutionary algorithms. The effects of these capabilities of the system can be investigated in more detail and for more variations of tasks, while evolutionary algorithms (both regular and island models) are applied in order to evolve for specific types of behaviors and patterns in real tasks, especially in the context of evolutionary robotics.

Acknowledgments

This work is supported by: EU-IST-FET project SYMBRION, No. 216342; EU-ICT project REPLICATOR, No. 216240; the Austrian Federal Ministry of Science and Research (BM.W F); EU-ICT project CoCoRo, No. 270382; and EU-ICT project ASSISI_bf, No. 601074.

Notes

1 

The concentration ends up at the maximum level (1.0), but in the interval of observation it may no longer be saturated for some values of the constant production rate.

2 

The diversity is lower than for {a}, because the maximum possible value for the concentration level in this case is half of the initial concentration level of the hormone in unit0 (0.5/2).

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Author notes

Contact author.

∗∗

Artificial Life Lab of the Department of Zoology, Universitätsplatz 2, Karl-Franzens University Graz, 8010 Graz, Austria. E-mail: payam.zahadat@uni-graz.at (P.Z.); thomas.schmickl@uni-graz.at (T.S.)