Aging as a Loss of Goal-Directedness: An Evolutionary Simulation and Analysis Unifying Regeneration with Anatomical Rejuvenation

So far, the approach was agnostic to the particular ANN architecture of the update function, f θ(·). Closely following,[ 46 ] here, the genuine ANN architecture deployed in our cellular agents was briefly described, which was schematically depicted in Figure 2B: Inspired by,[ 85 ] the NCA's cells' ANN was partitioned into i) a sensory part, εiν=fθ(s)(ciν(tk))∈Rs, preprocessing each neighborhood cell state separately into a respective sensor embedding, which were collected in the matrix E(Ni(tk))=εi0(tk),…,εiN(tk). These sensor embeddings are then ii) averaged across all neighbor embeddings into a single context vector si(tk)=1N+1∑ν=0Nεiν(tk)∈Rs. A subsequent iii) controller ANN, fθ(c)(·), potentially with recurrent feedback connections, eventually outputs the cell's action, ai(tk)=fθ(Ni(tk))=fθ(c)(si(tk)) based on the cell‐specific context vector s i (t k ). Notably, for the controller module, so‐called RGRN architectures were utilized[ 46 ] that were inspired by a combination of Recurrent ANNs (RNNs)[ 86 ] and Gene Regulatory Networks [ 87 ] (see Appendix A in[ 46 ] for details).

Analogous to,[ 46 ] morphogenesis was modeled by employing NCAs on a 2D N x × N y square grid with the objective that all cells of the grid assume a cell‐specific predefined target cell type, g^i, after a fixed number of t D developmental time steps, starting from an initial cell state configuration c i (0), The first N G elements, gi(tk)=ci1(tk),…,ciNG(tk), of an NCA's NC‐dimensional cell state ci(tk)=ci1(tk),…,ciNC(tk) were assigned as indicators for expressing one of 1,…,NG discrete cell types, while the remaining NH=(NC−NG) elements of the cell state represent its hidden states, hi(tk)=ci(NC−NH)(tk),…,ciNC(tk). While the full NC‐dimensional cell state vectors c i (t k ) could be utilized for intercellular communication by the NCA, a cell's “type”, g i (t k ), is now defined as the index, 1,…,NG, of the maximum element of its indicator vector g i (t k ):

For an NCA to assemble a predefined target pattern of N j = N x × N y target cell types g^={g^1,…,g^Nj}, a suitable set of NCA parameters must thus be found that minimizes the deviation of all actual cell types gi(tD) and the desired ones g^i after tD developmental time steps. Below, an evolutionary algorithm was introduced to evolve suitable sets of NCA parameters that maximize a fitness score based on comparing the “final” cell types of the NCA, gi(tD), after the developmental stage to the predefined target cell types g^i. In this context, the NCA parameters thus correspond to the virtual organism's genotype, while the grid of final cell types represent the system‐level phenotype that was seen by the selection mechanism of the evolutionary process (c.f., Figure 2C).

Evolutionary Algorithms (EAs) were heuristic optimization techniques designed to maintain and optimize a set, or population, X={x1,…,xNP}, comprising parameters or individuals, xj∈RX, over successive generations in order to maximize an objective function or fitness score, r(xj):RX→R. Drawing inspiration from the principles of natural selection and biological life's DNA‐based reproduction mechanisms, EAs predominantly select high‐fitness individuals for reproduction, and utilize crossover and mutation operations to generate new offspring by merging the genomic material of two high‐quality individuals from the current population, xo=xj⨁xk, and occasionally mutating offspring genomes, x o → x o + ξ x , by adding typically Gaussian noise to the parameters; the ⨁ symbol signifies a genuine merging operation of two genomes, which may vary according to the specific EA implementation. Thus, populations of individuals were directed toward high‐fitness regions within the parameter space RX, typically across numerous generations of successive selection and reproduction cycles.[ 46 ]

Closely following the previous work[ 46 ] and in contrast to many “traditional” EA use‐cases, a distinguish was explicitly made between the genotypic parameters, x j , and the corresponding phenotypic realizations, p j . More specifically, we utilize NCAs (c.f., Section 2.1) to model the biologically inspired developmental layer[ 23 ] in between genotypes and phenotypes, x j → p j , akin to the process of morphogenesis: Here, a genotype, x j , corresponds to a particular realization j of an NCA's parameters, comprising the set of initial cell states xj(S)={ci=(1,…,Nj)(0)}j and the corresponding ANN parameters xj(F)=θj, such that

where a distinguish was explicitly made between structural (S) and functional (F) genes. Equation (1) was then employed for t D developmental steps to successively transform the NCA's cell states from an initial state into a “final” set of cell types {g i (t D)} j on the NCA's grid. This mature phenotype,

represents a 2D tissue of cells, and is the input to the fitness score based on which the EA selects: As illustrated in Figure 2C, optimization was performed for a fitness score in the phenotype space rather than the genotype space, r(x j ) → r(p j ), while still performing genetic recombination and mutation in the genotype space.

Let's recall: the goal was to achieve morphogenesis of a 2D spatial tissue of cell types, p j , that optimally resembles a predefined target pattern, {g^1,…,g^Nj}, of a total of N j cells on an N x × N y square grid of an NCA. Moreover, the aim was to derive a virtual organism with this capability by modeling a biologically inspired evolutionary process with EAs to evolve a suitable set of NCA parameters (c.f., Figure 2 (C)). Thus, the phenotype‐based fitness score was introduced, r(p j ), as[ 46 ]

where nj(G) counts the number of correctly assumed cell types after t D developmental steps (where gi(tD)=g^i), nj(T) counts the number of developmental steps at which the target pattern is realized entirely (whenever gi(tk≤tD)=g^i for all i), and nj(S) counts pairs of successive time steps, (t s , t s + 1) ⩽ t D, without cell type updates in the entire tissue (where g i (t s + 1) = g i (t s ) for all i).

Thus, an evolutionary process was designed that primarily selects for NCAs that assume the correct cell type pattern after (precisely) t D developmental steps and reinforces NCAs (by a factor of r T in the fitness score) that prematurely achieve and maintain the target pattern. To avoid developmental stagnation and increase the EA's performance, the fitness score of NCAs was specifically discounted with static behavior by r S . A fitness score of N j = N x × N y indicated that the problem is solved.

Although the above framework[ 46 ] could be utilized in conjunction with any black‐box evolutionary or genetic algorithm, the well‐established Covariance Matrix Adaptation Evolutionary Strategy (CMA‐ES)[ 80 ] to be a viable choice for our purposes.

Information theory[ 88 ] offers valuable insights into the dynamics of complex systems. In order to understand the NCA's information dynamics during aging, three key information‐theoretic metrics were used to examine the information dynamics: active information storage,[ 81 ] transfer entropy,[ 82 ] and the spatial entropy.

Active Information Storage (AIS) quantified the amount of information from an agent's past that was pertinent to predicting its future state. Specifically, AIS refers to the portion of stored information currently utilized for determining the agent's next state.[ 81 ] Mathematically, the AIS of an agent Q was expressed as the local mutual information between its semi‐infinite past qn(k) as k → ∞ and its next state q n + 1 at time step n + 1:

Here, a Q (n, k) is an approximation with history length k. The time‐averaged value, weighted by the distribution of (qn(k),qn+1), is denoted as A Q (k) = 〈a Q (n, k)〉. In this study, the local was calculated AIS across the states of the cells.

Transfer Entropy (TE) measures the information transferred from a source agent to a destination agent that was not contained in the past of the destination agent. The local TE concept introduced by Lizier was employed.[ 81 ] The local TE from a source agent Z to a destination agent Q was defined as the local mutual information between the previous state of the source z n and the next state of the destination agent q n + 1, conditioned on the semi‐infinite past of the destination qn(k) (as k → ∞):

The transfer entropy T Q (n, k) was the local TE averaged over time, denoted as T Q (k) = 〈t Q (n, k)〉, where t Q (n, k) was an approximation with history length k. Unlike mutual information, which measures static correlation, transfer entropy captures the dynamic, directional flow of information within the agent network.

Spatial Entropy (SE) measured the randomness or disorder within a spatial distribution of states. It provided insight into the complexity of spatial patterns in a system, such as an NCA with multiple states. In this study, SE was computed by evaluating the entropy of the distribution of cell states across the grid at each time step. Formally, the spatial entropy H at time step n is defined as:

where S is the set of possible states, and p(s) is the probability of state s occurring in the grid at time step n. This measure provided an indication of the level of unpredictability or complexity in the spatial arrangement of cell states. In this article, the spatial entropy for the states of the NCA was calculated across different time steps.

Following our previous study,[ 46 ] we utilize EAs to evolve NCAs to perform morphogenesis tasks. Each cell in our NCA implementation is equipped with a gene regulatory network‐inspired recurrent ANN, conceptually substituting the intricate internal decision‐making competencies of biological cells (see Figure 2B for a visualization and Appendix A in[ 46 ] for numerical details). In other words, each cell in the NCA is equipped with an information processing pipeline that transforms sensory input of its local environment in the NCA's grid into control instructions to regularize its own cell type. If done in a coordinated way, these local individual cell‐state updates guide the formation of a mature system‐level phenotype ‐ in our case of a target tissue ‐ over successive developmental steps.

More specifically, we utilize N x × N y = 16 × 16 cells on an NCA with fixed boundary conditions (see Section 2.1 and Appendix C in[ 46 ]) and train it analogously to[ 46 ] via evolutionary algorithms (see Section 2.2) to perform morphogenesis ‐ i.e., self‐orchestrated tissue growth ‐ of a 16 × 16 smiley‐face pattern in a predefined number of t D = 35 developmental steps (see Figure 2C for an illustration of the evolutionary process); the smiley‐face‐pattern is reminiscent of the bioelectric craniofacial prepattern that establishes the rough outlines of the amphibian face during early development.[ 89 ]

The target tissue consists of N G = 3 distinct cell types: one cell type for the background cells, one for the face cells, and a single cell type for the facial organs, i.e., for the eyes and the mouth (c.f., blue‐, magenta‐, and white‐colored cells in Figure 2A). Moreover, we allow the NCA to utilize N H = 1 hidden channel of its N C = 4 cell state channels for unconstrained intercellular communication. In cell‐state updates, given by Equation (1), we utilize a noise level of ξ c = 0.125, limit the numerical value of the proposed action a i (t k ) to l a = [− 1, 1], and clip the cell‐states post‐update c i (t k + 1) to l c = [− 3, 3] (see Appendix C and[ 46 ] for more details on our NCA implementation).

Our NCAs transform an embryonic genotype, x j , into a mature phenotype, p j , and we here quantify an NCA's performance by evaluating the corresponding fitness score, r(p j ), defined by Equation (5) with respect to the 16 × 16 smiley‐face pattern over the NCA's entire life‐span of t D developmental steps. As illustrated in Figure 3 , a successfully evolved NCA maximizes the fitness score to r max = 256, exhibiting 16 × 16 correctly expressed cell types after t k ⪆20 time steps, and then maintaining the target pattern until t k = t D (see Section 2.2 for details on the evolutionary process).

Notably, the cellular agents do not perceive any information about the current nor the final fitness score of their corresponding NCA. The fitness score is only utilized by the EA to optimize the NCA's parameters over evolutionary time‐scales. Moreover, there is no contribution either to the fitness score or to the NCA's ANN sensory inputs that would provide the system with explicit temporal information indicating the “age” of any part of the virtual organism, or even a notion of “time” in general. Thus, the self‐orchestrated pattern formation of the here ‐ and previously[ 46 ] ‐ investigated NCAs is fully driven via emergent (i.e., evolved) intercellular communication rules and intracellular decision‐making.

We aimed to assess whether the self‐regulatory behavior of biological systems that drive morphogenesis induces aging‐like phenomena at long time‐frames. To this end, we use NCAs evolved in silico to model the process of morphogenesis and then deploy their self‐regulatory dynamics for much longer times compared to the reproduction stage during evolution (i.e., when selected for reproduction). We monitor and analyze the NCAs morphological integrity throughout their lifetimes and compare such long‐term morphological trajectories with biological aging. The evolutionary process described in Sections 2.2, 3 explicitly selects for optimal fitness scores after t D developmental steps, which corresponds to the fertility age of real‐world organisms. However, this implies that the behavior of an NCA with a potentially much longer lifetime of t A > t D, and particularly t A ≫ t D, is not subject to selection and may fundamentally be considered ill‐defined or ambiguous.

To demonstrate this, we deploy the NCA solution that has been evolved to self‐assemble a 16 × 16 smiley‐face pattern during a morphogenesis phase of t D = 35 developmental steps (c.f., Figure 3) for a much longer lifetime of up to t A = 1500 time steps, notably without any further optimization considering the extended lifetime. From the results presented in Figure 4 , we learn that after an initial rapid rise in the fitness score during the developmental phase, the NCA's long‐term fitness score gradually declines, primarily due to the loss of different internal organs such as one or both eyes and, occasionally, the mouth. Moreover, this is not only a feature of the particular NCA demonstrated here, but is a systematic phenomenon of the presented framework.

Thus, we argue that the biologically‐ubiquitous self‐regulatory behavior of a substrates agential parts may induce system‐level dynamics that resemble the process of biological aging if applied on timescales that exceed development. The organisms learned during evolution with development as the primary goal; once they reached that goal, we observed that they don't regenerate by themselves and slowly show signs of morphological deterioration, e.g., aging. In this sense, aging can be seen primarily as a loss of goal‐directedness, the goal of development being different from the goal of maintenance of the anatomy over timescales exceeding development. This view on aging is related to programmatic theories of aging like antagonistic pleiotropy or hyperfunction theory[ 11 , 16 , 17 ] that state that what is good for development could ultimately cause aging. In our framework, the lack of goal‐directedness after development may give rise to informational antagonistic pleiotropy (or hyperfunction) as cells may lose some developmental policies after reaching the target morphology. However, our framework goes beyond antagonistic pleiotropy by enabling the deliberate re‐targeting of the morphogenetic setpoint in an aging tissue toward new anatomical goals. In turn, a continually regenerative setpoint – e.g., by reactivating developmental pathways – may reverse aging, as we demonstrate in silico in Section 4.4.

To determine how the perception‐action cycles of the unicellular agents of a collective multi‐cellular organism influence, at least in part, the rate of morphological decline, we here investigate the effects of an NCA's system parameters on its long‐term morphological integrity. Based on the NCA system defined in Section 3, we identify four key mechanisms involved in the corresponding uni‐cellular perception‐action cycles affecting morphological integrity: a) cell‐specific differentiation reliablility, b) reliable state updates, c) reliable communication between neighbors, and d) persistent genetic encoding. Our framework allows us to selectively control these corresponding processes (a–d) and explicitly corrupt them independently over the lifetime of the virtual organism. Thereby we demonstrate the severe effects of these processes on the rate of morphological aging.

In the following sections, we therefore study the factors accelerating aging, not the cause of aging per se that we see primarily as a loss of goal‐directedness. We introduce these known aging mechanisms (a–d), respectively, based on and deployed to the biologically motivated unicellular competencies in our NCA system. We then demonstrate and discuss the corresponding implications on enhanced rates of morphological aging in long‐term simulations of such virtual organisms. We illustrate the respective results in a compact form in Figure 5A–D, and continue below with an information‐theoretic analysis of these processes.

One critical parameter in our system is the noise level ξ c utilized to corrupt the NCA's update step given by Equation (1). While a moderate noise level can help an evolutionary process to identify robust NCA parameters that solve a specific morphogenesis task; but if the selective noise level is too low, the evolutionary process will in our case most likely identify a direct encoding of the target pattern, largely neglecting cellular competencies in the pattern formation task.[ 46 ] In turn, a too‐large noise level will not only prevent the evolutionary process from finding a good solution to the morphogenesis task but, in general, will drastically limit the cell specification capacities of the NCAs cell states. Therefore, noisy cell state updates in our system could be seen as the accumulation of misdifferentiated cells, ultimately corrupting the state perceptions of neighboring cells on the NCA and thereby creating a default of cellular communication (a hallmark of aging) similarly to.[ 6 , 90 ] This mechanism accelerates aging but is not a cause per se.

We argue that an increasing noise level ξ c → ξ c (t k ) over the lifetime of such an NCA‐akin organism successively affects its morphological integrity. To demonstrate this, we thus utilize the pre‐evolved NCA from above (c.f., Figures 3, 4) and deploy it for an extended lifetime of t A = 1000 timesteps while successively increasing the noise level linearly from ξ c (t k = 0) to ξ c (t A ) = 1. The results depicted in Figure 5A show that while the correspondingly affected virtual organisms initially follow their unaffected cousins (c.f., Figures 3, 4 and black‐dashed line in Figure 5A), after a particular duration (specific to the particular NCA solution and noise schedule ξ c (t k )) the fitness score rapidly drops at t k ≈ 380 − 400, emphasizing an enhanced rate of degradation of the target tissue compared to the baseline with constant noise ξ c = 0.125.