Evolutionary Implications of Self-Assembling Cybernetic Materials with Collective Problem-Solving Intelligence at Multiple Scales

In recent years, the scientific community has increasingly recognized the complex multi-scale competency architecture (MCA) of biology, comprising nested layers of active homeostatic agents, each forming the self-orchestrated substrate for the layer above, and, in turn, relying on the structural and functional plasticity of the layer(s) below. The question of how natural selection could give rise to this MCA has been the focus of intense research. Here, we instead investigate the effects of such decision-making competencies of MCA agential components on the process of evolution itself, using in silico neuroevolution experiments of simulated, minimal developmental biology. We specifically model the process of morphogenesis with neural cellular automata (NCAs) and utilize an evolutionary algorithm to optimize the corresponding model parameters with the objective of collectively self-assembling a two-dimensional spatial target pattern (reliable morphogenesis). Furthermore, we systematically vary the accuracy with which the uni-cellular agents of an NCA can regulate their cell states (simulating stochastic processes and noise during development). This allows us to continuously scale the agents’ competency levels from a direct encoding scheme (no competency) to an MCA (with perfect reliability in cell decision executions). We demonstrate that an evolutionary process proceeds much more rapidly when evolving the functional parameters of an MCA compared to evolving the target pattern directly. Moreover, the evolved MCAs generalize well toward system parameter changes and even modified objective functions of the evolutionary process.

Thus, the adaptive problem-solving competencies of the agential parts in our NCA-based in silico morphogenesis model strongly affect the evolutionary process, suggesting significant functional implications of the near-ubiquitous competency seen in living matter.

Biological systems are organized in an exquisite architecture of layers, including molecular networks, organelles, cells, tissues, organs, organisms, swarms, and ecosystems. It is well recognized that life exhibits complexity at every scale. Increasingly realized, however, is the fact that those layers are not merely complex but actually active “agential matter”, which has agendas and competencies of its own [1,2]. Elsewhere, we have discussed examples of problem-solving in unconventional spaces, including transcriptional, physiological, metabolic, and anatomical space [3].

Especially interesting is the ability of these ubiquitous biological agents to deal with novel situations on the fly, which is not limited to brainy animals navigating 3D space but also occurs with respect to injury, mutations, and other kinds of external and internal perturbations (reviewed in [4]). One example of such problem-solving capabilities is the regenerative properties of some species that can regrow limbs, organs, or entire parts of their bodies when amputated, and—remarkably—stop when the precisely correct target morphology is complete [5,6,7]. This can be understood as cellular collectives navigating morphospace until the desired target shape—or the goal—is reached again. Other examples include the ability of scrambled tadpole faces to reorganize in novel ways to result in normal frog faces [8], and the normal shape and size of structures in amphibia despite drastic changes in cell number [9] and cell size [10], which are handled by exploiting different molecular mechanisms to reach correct target morphologies despite novel changes in internal components. Behavioral and morphological plasticity intersect in cases such as tadpoles made with eyes on their tails, which nevertheless can see and learn in visual assays without needing rounds of evolutionary adaptation [11].

The ability to navigate transcriptional and anatomical spaces, using perception–action loops and homeostatic setpoints, is now being increasingly targeted by biomedical and bioengineering efforts [12,13]. A fascinating body of work exists around the question of how neural and non-neural problem-solving capacities evolved, and how neuro-behavioral intelligence affects evolution [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. However, we and others have previously suggested that somatic competency pre-dates neural intelligence [32,33,34], and has a bi-directional interaction with the evolutionary and developmental process [1,3,35]. Thus, here, we address the second half of the evolution–intelligence spiral: how are evolutionary processes affected by the competency of the material? Especially important is the inclusion of the middle layer between the genotype and phenotype. Mutation operates on genomes, and selection operates on phenotypic performance, but in most organisms, the connection between them is not linear or shallow—instead, developmental physiology provides a deep reservoir of dynamics that strongly alter the process. As a contribution to the study of evolvability and developmental mechanisms potentiating it [36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53], we established a virtual embryogeny [54] system focused on anatomical morphogenesis by cells. In this minimal model of morphogenesis, we were able to study the effects of different degrees of cellular competency on the evolutionary process.

The standard understanding of (Neo-Darwinian) evolution is schematized in Figure 1A: The genome of an organism encodes aspects of the organism’s cellular hardware, which together define the phenotypic traits. Given a competitive environment, natural selection then favors organisms with advantageous traits, and thus, on average, the corresponding genes tend to get passed on to the next generations more frequently. Random mutations may occur, consequently changing traits in the offspring phenotype. This affects the offspring’s reproductive success during the selection stage and, in that way, good traits prevail, and bad ones perish over time.

This view has been revised by Waddington [55,56], and more recent works [57,58,59,60,61,62,63,64,65,66], and has been the subject of vigorous debate [40,63,67,68,69,70,71,72] with respect to its capabilities for discovery, its optimal locus of control, and the degree to which various aspects are random (uncorrelated to the probability of future fitness improvements). Important open questions concern ways in which the properties of development—the layer between the mutated genotype and the selected phenotype—are evolved and in turn affect the evolutionary process [36,39,45,46,73,74,75,76,77,78]. Specifically, significant work has been conducted at the interface of evolution and learning—selectionist accounts of change and variational accounts of change respectively [30,61,62,66,79,80,81,82,83,84,85]. Significant progress has been made on the question of how evolution produces agents with behavioral competency in diverse problem spaces [17,86,87,88]. We have focused on a particular kind of competency—that of navigating anatomical morphospace [3,12,89,90]. More specifically, we here investigate in silico the evolutionary implications of the self-orchestrated process of morphogenesis, where local actions of single cells need to be aligned with a global policy of a multi-cellular collective to guide the formation of a large-scale tissue, in turn affecting the underlying evolutionary process. Work on developmental plasticity, chimeras, synthetic biobots, and the ability to overcome novel stressors has highlighted ways in which evolution seems to give rise to problem-solving machines, not fixed solutions to specific environments [91].

Thus, the problem-solving capacities of development, regeneration, and remodeling ensure that in many (perhaps most) kinds of organisms, the mapping from genotype to phenotype is not merely complex and indirect [92] (as schematized in Figure 1B) but actually enables evolution to search the space of behavior-shaping signals, not microstates, and exploit the modularity and triggers of complex downstream responses (cf. Figure 1C). We have previously argued that both evolution and human bioengineers face a range of unique problems and opportunities when dealing with the agential material of life—not passive or even just active matter but a substrate that has problem-solving competencies and agendas at many scales [93,94]. What selection sees is not the actual quality of the genome but the quality of the form and function of the flexible physiological “software” that runs on the genomically specified molecular hardware as schematically illustrated in Figure 2. This in turn suggests that the actual progress of evolution should be significantly impacted by the degree and kind of competency in the developmental architecture. Prior work has suggested a powerful feedback loop between the evolution of morphogenetic problem-solving and the effects of these competencies on the ability of evolutionary search to produce adaptive complexity [1,35,95]. Here, we construct and analyze a new model of evolving morphogenesis to study how different competency architectures within and among cells impact evolutionary metrics such as rate, robustness to noise, and transferability to new environmental challenges.

To quantitatively study the effects that different levels of competency of the decision-making centers in a multi-scale competency architecture have on the process of evolution, we here rely on tools from the research field of Artificial Life [96], which furthers computational and cybernetic models that mimic life-like behavior based on ideas taken from biology; a simple example is cellular automata (CAs) [97]. In such CAs, the (numerical) states of localized cells, organized on a discrete spatial grid, change in time via local update rules. Although typically rather simple “hardcoded” update rules are employed, CAs often display complex dynamics (cf. Conway’s Game of Life [98] or Lenia [99]) but are not known to exhibit homeostatic (closed-loop) activity. An extension of CAs, termed neural cellular automata (NCAs) [100], utilize artificial neural networks (ANNs) as more flexible trainable update rules, aiming to model the internal decision-making machinery of biological cells. Employing machine learning methods, such NCAs have been trained to perform self-orchestrated pattern formation (notably, of images from a single “seed” cell) [101] and even the co-evolution of a rigid robot’s morphology, and its controller has been demonstrated with such NCAs [102].

NCAs exhibit a striking resemblance to the genome-based multi-scale competency architecture of biological life [102] as illustrated in Figure 2: an organism’s entire building plan is encoded in its genome (corresponding to the NCA parameters), while its cells collectively run the self-orchestrated developmental program of morphogenesis (realized by the NCA layout and ANN architecture) via perception–action cycles at the uni-cellular level (cell state updates in the NCA, cf. Figure 2A). Starting from an initial cell state configuration of the NCA, the details of a virtual organism are then, step by step, “refined” in a collective self-organizing growth phase on the cellular level, and maintained against cell state errors later on in the virtual organism’s lifetime. Thus, a single NCA, once trained, guides the growth and integrity of a virtual organism’s tissue via intracellular information processing and intercellular communication, imitating in silico the multi-scale competency-based process of morphogenesis and morphostasis. (Notably, although an NCA update function—the cells’ ANN—is trainable in principle, current approaches pre-train (or evolve, as in our case) the ANN parameters to subsequently study the NCA behavior (such as simulated morphogenesis). Thus, an NCA’s self-orchestrated (developmental) program is defined by a particular set of ANN parameters rather than being acquired during the lifetime of the NCA. Here, we investigate the efficiency at which an evolutionary process arrives at satisfying parameters under various conditions.)

Here, we deploy a swarm of virtual uni-cellular agents on the spatial grid of an NCA. As illustrated in Figure 2A, each uni-cellular agent’s internal decision-making machinery is modeled by an ANN that allows each agent to independently perceive the cell states of its adjacent neighbors on the grid and propose cell state update actions to regulate its own cell state over time. The collective of cells thereby forms a spatial pattern or tissue of cell states on the NCA via local communication rules.

We utilize evolutionary algorithms (EAs) [103] as a simulated evolutionary process to optimize the parameters of such NCAs, so the uni-cellular agents evolve to collectively self-assemble a predefined target pattern of cell states in a fixed number of developmental steps; see Figure 2B for a flow-chart of the evolutionary process. We explicitly separate the NCA parameters into a structural and a functional part. The structural parameters describe the initial cell state, and the functional parameters the weights and biases of the ANN of each agent as illustrated by the “Genome” in Figure 2B. Both the structural and functional parts of the genome are compiled into a swarm of uni-cellular phenotypes on the grid of the NCA. Thus, starting from an initial cell state configuration, given by the structural part of the genome, the uni-cellular agents of the NCA run the developmental program of morphogenesis via successive perception–action cycles (see Figure 2A) to self-assemble in successive developmental steps a system-level phenotype, i.e., a two-dimensional pattern of cell states on the NCA. The deviation of these final cell state configurations from a desired target pattern—here, a Czech flag pattern or smiley-face pattern reminiscent of that of the amphibian craniofacial pre-pattern [104]—defines the phenotypic fitness score of a particular NCA realization. Based on an entire population of NCAs, and on the corresponding fitness scores, the EA successively samples the genomes of the next generation of NCAs, which, over time, evolve to reliably self-assemble the target pattern.

The conceptually simple process of cell state updates of NCAs and the ANN-based modeling of the uni-cellular decision-making allow us to interfere with (I) the reliability of the cell state update executions, and (II) with the computational capacity of the ANNs that guide each cell’s decision-making. To vary the former (I), we introduce a decision-making probability PD that specifies the probability at which a proposed update of each individual cell is executed in the environment (or omitted otherwise). Thus, by tuning the decision-making probability from PD=0 to PD=1, we can continuously vary the behavior of the NCA from a direct-encoding scheme without competency to a multi-scale competency architecture with perfect reliability in cell state update executions.

To systematically vary the computational capacity of the involved ANNs (II), we introduce independent copies of a particular sub-module of the uni-cellular agents’ ANNs, i.e., of the policy module illustrated in Figure 2A (see Section 3.1 and Section 4.2 and Appendix A for details on the ANN architectures). This increases the number of evolvable parameters of the ANNs, which are responsible for performing the same operation, namely, interpreting the cell’s local environment and proposing a cell state update action. Thus, by taking the average output of all redundant policy modules of a single agent, a cell’s decision-making can be biased by the several redundant paths through which signals are transmitted in the ANN, inspired by error-correcting codes [105,106,107]. We explicitly define a redundancy number R that specifies how many redundant copies of the policy module are used in the ANNs of the cells of the NCA.

The decision-making probability (I) and the redundancy number (II) represent two levels of competency in our system (schematically illustrated by the red arrowin Figure 1C and Figure 2B), which we can scale continuously (I) or discretely (II) to systematically tune the behavior of an NCA. Throughout the manuscript, we refer to these two parameters as “competency levels”, but we would like to stress that many more options would have been possible to vary the competency in our system. For instance, the particular ANN architecture can have large effects on the competency of the uni-cellular agents; a systematic investigation thereof is out of the scope of this work. Here, we utilize two particular ANN architectures, one based on a Feedforward (FF) and one based on a recurrent ANN architecture [108] that is inspired by gene regulatory networks (GRNs) [109], which we thus term recurrent gene regulatory network (RGRN), see Appendix A for details.

To study the effects of different competency levels of the decision-making centers in a multi-scale competency architecture on the underlying evolutionary process of a morphogenesis task, we systematically vary in large-scale simulations the decision-making probability (I) and the redundancy number (II) of NCAs with FF and RGRN ANN architectures. Furthermore, we expose the corresponding NCAs to different noise conditions during cell state updates (III) and perform several statistically independent evolutionary searches at each parameter combination (I–III) to investigate the performance of the evolutionary process of finding solutions to such noisy pattern formation tasks.

The manuscript is organized as follows: In Section 3, we describe the numerical and computational methods applied herein. More specifically, we introduce NCAs in Section 3.1, and describe the neuroevolution approach used to optimize the NCAs ANN parameters based on ideas of evolution and natural selection via EAs in Section 3.2. We specify the particular morphogenetic problem we primarily focused on—the 8×8 Czech flag task—in Section 4.1, and compare in Section 4.2 the efficiency of evolving the target pattern via a direct encoding scheme and a multi-scale competency architecture. In Section 4.3, we functionally define and systematically vary the different tunable competency levels in our system to illustrate the evolutionary implications of utilizing a multi-scale competency architecture rather than a direct encoding scheme for morphogenesis tasks. We then study the effects of allowing the evolutionary process to afford competency as a gene during optimization in Section 4.4, and eventually investigate our multi-scale competency approach for robustness and generalizability regarding system parameter changes in Section 4.5, and for transferability to modified target patterns in Section 4.6. We conclude in Section 5, and attach an appendix.

Biological systems are composed of layers of organization, each level providing the foundation for the next higher level of abstraction: membranes, DNA, and proteins form cells, which then collectively organize into tissue and, in further hierarchical steps, into tissues, organs, bodies, swarms, ecosystems, etc. Each of these layers has a degree of ability to adapt in real-time to new conditions to establish and maintain specific outcomes in terms of physiological, metabolic, transcriptional, and anatomical spaces. In other words, evolution works with material that is not passive matter but rather has a degree of competency—an agential material that forms the layer between the genotype and the phenotype. Many scientific studies have been dedicated to investigating how evolution gives rise to such intriguing problem-solving machines we call organisms. In this study, we ask the reverse question: what is it like to evolve over such a material vs. one that passively maps genotypes into the form and function that selection operates over—how does it affect the process of evolution itself? We test this in silico by utilizing evolutionary algorithms to adapt the behavior of a swarm of virtual uni-cellular agents in large-scale simulations of virtual embryos. In our minimal model, the cells collectively self-assemble a predefined target tissue on a neural cellular automaton. We find that competency at the cellular level of our multi-scale model system strongly affects the resulting evolutionary process, as well as the generalizability, evolvability, and transferability of the evolved solutions, suggesting the profound evolutionary implications of the highly intricate multi-scale competency architecture of biological life.

Cellular automata (CAs) have been introduced by von Neumann to study self-replicating machines [97] and are simple models for Artificial Life [96]. In CAs, a discrete spatial grid of cells is maintained over time, each cell i being attributed a binary, integer, real, or even vector-valued state ci(tk) at each step in time tk. The cell states evolve over time via local updated rules ci(tk+1)=fu(Ni(tk)) as a function of its own ci(tk) and the numerical states ciν(tk) of its iν=1,⋯,N neighboring cells on the grid that we collect in the matrix Ni(tk)=ci0(tk),⋯,ciN(tk) with i0≡i. Although typically rather simple “hardcoded” (i.e., predefined) update rules fu(·) are employed, CAs often display complex dynamics and can even be utilized for universal computation (cf. Conway’s Game of Life [98] or Wolfram’s rule 110 [110,111]).

Neural cellular automata (NCAs) [100] extend CAs by replacing the local update rule with more flexible [112] artificial neural networks (ANNs) fu(·)→fθ(·), where θ denotes the set of trainable parameters of the ANN (see Appendix A for details). Employing Machine Learning, such NCAs have been trained to perform self-orchestrated pattern formation [101] (notably, of RGB images from a single “seed” pixel) and even the co-evolution of a rigid robot’s morphology, and its controller has been demonstrated recently with NCAs in silico [102]. Such self-orchestrated pattern formation is reminiscent of the self-regulated development of a biological organism, from a single fertilized egg cell to a complex anatomical form. Thus, NCAs have been proposed as toy models for morphogenesis [101].

An NCA basically represents a grid of cells that are equipped with identical ANNs, each perceiving the numerical cell states of its host’s local environment, Ni(tk), and proposing actions, ai(tk)=fθ(Ni(tk)), to regulate its own cell state (1)ci(tk+1)=ci(tk)+ai(tk)+ξc, and, in turn, the cell states of its neighbors—where we also account for potential noise ξc in the environment during the process of morphogenesis. Thus, each cellular agent can only perceive the numerical states of its direct neighbors Ni(tk) at an instant of time tk and, in turn, communicate with these neighbors via cell state updates ci(tk+1), following a policy π(Ni(tk))≈fθ(Ni(tk)) that is approximated by an ANN with parameters θ. Through the lens of Reinforcement Learning [113], an NCA can thus be understood as a trainable, locally communicating multi-agent system that can be utilized such that the collective of cells achieves a target system-level outcome (see Appendix B for details).

In contrast to previous contributions of in silico morphogenesis experiments in NCAs [101], we here do not use standard convolutional filters in our ANN architectures but utilize permutation-invariant ANNs with respect to a cell’s neighbors, Ni(tk) (see Figure 2A for an illustration). Inspired by Ref. [114], this is achieved by partitioning a cell’s ANN into (i) a sensory part fθ(s)(·), preprocessing its own, and the state of each neighboring cell separately into a respective sensor embedding εiν(tk)=fθ(s)(ciν(tk))∈Rs, for iν=0,⋯,N. These neighbor-wise sensor embeddings are (ii) averaged into a cell’s context vector si(tk)=1N+1∑ν=0Nεiν(tk)∈Rs of fixed size s, which is then used as the input of (iii) a controller ANN fθ(c)(·), potentially with recurrent feedback connections, that eventually outputs the cell’s action ai(tk)=fθ(c)(si(tk)); for details we refer to Appendix A.

Due to the mean aggregation of a cell’s sensory embedding, each cell completely loses its ability to spatially distinguish between neighboring (and even its own) state inputs and thus fully integrates into the tissue locally. We would like to stress the close relation of our approach to the concept of breaking down the computational boundaries of a cell’s “Self” via forgetting [93] and to the scaling of goals from a single agent’s to a system-level objective [95].

To model the developmental process of morphogenesis, we here employ NCAs on a two-dimensional Nx×Ny square grid with the objective that all cells of the grid assume their correct, predefined target cell type g^i after a fixed number of tD developmental time steps, starting from an initial cell state configuration ci(0). We attribute a number of NG elements gi(tk)∈RNG of the NC-dimensional cell state ci(tk)∈RNC of an NCA as indicators for expressing one of 1,⋯,NG discrete cell types such that ci(tk)=gi(tk)∪hi(tk); the remaining NH=(NC−NG) elements of the cell state represent hidden states hi(tk)∈RNH of a cell that can be utilized by the NCA for intercellular communication. We explicitly define each cell’s type gi(tk) as the argument (i.e., the index) of the maximum element of the NG-dimensional indicator vector gi(tk):(2)gi(tk)=argmaxg∈RNGgi(tk).

Training an NCA to assemble a predefined target pattern (realized by a set of Nj=Nx×Ny target cell types {g^1,⋯,g^Nj} for the entire grid) thus boils down to finding a suitable set of NCA parameters (cf. “Genotype” in Figure 2B) that minimizes the deviation of each cell i’s type gi(tD) from g^i after tD developmental time steps, i.e., after the developmental stage of the virtual organism (cf. “System-level Phenotype” in Figure 2B, from left to right, and details below). Here, we are interested in the evolutionary implications of biologically inspired multi-scale competency architectures, the latter being modeled by our morphogenetic NCA implementation. We thus introduce in Section 3.2, and utilize in Section 4, evolutionary algorithms to evolve suitable sets of NCA parameters that maximize the fitness score based on comparing the “final” cell types of the NCA, gi(tD), with the predefined target cell types g^i.

Evolutionary algorithms (EAs) are heuristic optimization algorithms that maintain and optimize a set, i.e., a population X={x1,⋯,xNP} of parameters xj∈RX, also termed individuals, over successive generations to maximize an objective function, or a fitness score r(xj):RX→R. Inspired by the ideas of natural selection and the DNA-based reproduction machinery of biological life, EAs (i) predominantly select the high-fitness individuals of a given population for reproduction, and utilize (ii) crossover and (iii) mutation operations to generate new offspring by (ii) merging the genomic material of two high-quality individuals from the current population xo=xj⨁xk, and (iii) occasionally mutating the offspring genomes xo→xo+ξx by adding (typically Gaussian) noise to the parameters; the ⨁ symbol indicates a genuine merging operation of two genomes, which may depend on the particular EA implementation. In that way, a population X of individuals is guided towards high-fitness regions in the parameter space RX, typically over many generations of successive selection and reproduction cycles (i)–(iii).

In contrast to biological life, many use cases of EAs do not require a distinction between individuals in the parameter space, i.e., genotypes xj, and the corresponding organisms in their natural environment, i.e., phenotypes, pj: while the genetic crossover and mutation operations of biological reproduction rely on bio-molecular mechanisms at the level of RNA and DNA, i.e., are performed in the genotype space, selection typically happens at the much more abstract level of an organism’s natural environment, i.e., in the phenotype space. Carrying this through computationally can be resource-demanding, depending on the complexity of a simulated environment. Nevertheless, to address the asymmetry between genotypes and phenotypes in multi-scale competency architectures, it is essential to evaluate the fitness score of the EA in the phenotype space instead of the genotype space r(xj)→r(pj).

We explicitly separate the genotype and phenotype representations of individuals by introducing a biologically inspired developmental layer [1] in between genotypes and phenotypes xj→LayerDev.pj as illustrated in Figure 2. More precisely, we follow Section 3.1 and model the developmental process of morphogenesis in silico by utilizing NCAs: we treat an NCA j’s parameters, such as the set of i=1,⋯,Nj initial cell states xj(S)={ci(0)}j and the corresponding ANN parameters xj(F)=θj, as the (virtual) organism’s genome, (3)xj=xj(S)∪xj(F)={ci(0)}j,θj, explicitly partitioning the genome into a structural (S) and a functional (F) part, as indicated by the superscripts. We then perform a fixed number of tD developmental steps employing Equation (1) and interpret the corresponding set of “final” cell types {gi(tD)}j of the entire NCA, cf. Equation (2), as the mature phenotype, (4)pj={gi(tD)}j, representing a two-dimensional tissue of cells.

In an effort to evolve the parameters xj of an NCA j to achieve the morphogenesis of a two-dimensional spatial pattern of cell types pj that resembles a pattern of predefined target cell types {g^1,⋯,g^Nj} of a total of Nj cells on an Nx×Ny square grid (see Section 3.1), we define the phenotype-based fitness score r(pj) as (5)r(pj)=(2nj(G)−Nj)+rTnj(T)−rSnj(S), where (i) nj(G) is the number of correctly assumed cell types gi(tD)=g^i after tD developmental steps, (ii) nj(T) is the number of time steps at which the entire target cell type pattern is correctly assumed, i.e., whenever gi(tk≤tD)=g^i for all i, and (iii) nj(S) is the number of successive time steps ts and ts+1≤tD, where all cell types stagnate, i.e., where gi(ts+1)=gi(ts) for all i. With Equation (5), we thus reward the entire NCA j by counting all correctly assumed cell types after tD developmental steps (while discounting all incorrect cell types gi(tD)≠g^i), we reward maintaining the target pattern over time with a factor of rT, and discount a stagnation of a suboptimal pattern over time by a factor of rS. We consider the problem solved if a final fitness score of Nj=Nx×Ny is reached. Notably, there is no explicit fitness or reward feedback at the level of the uni-cellular agents in our system; the fitness score is solely used as the selection criterion for sampling the next evolutionary generations, so the cellular collective needs to evolve an intrinsic signaling mechanism to successfully perform the requested morphogenesis task.

The here proposed setting of genotypes xj, corresponding phenotypes pj, and associated fitness scores r(pj), given by Equations (3)–(5), respectively, can be used in combination with any black-box evolutionary or genetic algorithm. We rely on the well-established Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) [103] to simultaneously evolve the set of initial cell state configurations (i.e., structural genes, xj(S)) and the set of corresponding ANN parameters of an NCA (i.e., functional genes, xj(F)) with the objective of the purely self-orchestrated formation of two-dimensional spatial tissue as illustrated by Figure 2A and described by Equation (1).