Source-linked AI summary
Information-based fitness and the emergence of criticality in living systems
Jorge Hidalgo, Jacopo Grilli, Samir Suweis, Miguel A. Munoz, Jayanth R. Banavar, Amos Maritan
TL;DR
The paper addresses how living systems can dynamically approach criticality while representing complex, changing environments. Using statistical mechanics and information theory, it shows that optimal representations favor critical regions, while evolutionary and adaptive models produce stronger collective convergence when agents represent one another accurately.
Problem
A well-founded theory is lacking for how interacting living systems dynamically tune themselves near critical points while coping with diverse, changing environments.
Method
The paper combines statistical mechanics and information theory with analytical, computational evolutionary, and adaptive models of internal representations and environmental variability.
Results
Co-evolutionary and co-adaptive systems converge more accurately to criticality when individuals represent other agents with fidelity, creating a collective critical ensemble.
Takeaways & Limitations
Criticality can emerge as a general mechanism for balancing accurate representation of regular signals with flexibility for noisy signals in complex adaptive systems.
Takeaways & Limitations
Explicitly summing over 2^N network states severely limits computer simulations to approximately N ∼20.
Abstract
from arXiv · showhide
Empirical evidence suggesting that living systems might operate in the vicinity of critical points, at the borderline between order and disorder, has proliferated in recent years, with examples ranging from spontaneous brain activity to flock dynamics. However, a well-founded theory for understanding how and why interacting living systems could dynamically tune themselves to be poised in the vicinity of a critical point is lacking. Here we employ tools from statistical mechanics and information theory to show that complex adaptive or evolutionary systems can be much more efficient in coping with diverse heterogeneous environmental conditions when operating at criticality. Analytical as well as computational evolutionary and adaptive models vividly illustrate that a community of such systems dynamically self-tunes close to a critical state as the complexity of the environment increases while they remain non-critical for simple and predictable environments. A more robust convergence to criticality emerges in co-evolutionary and co-adaptive set-ups in which individuals aim to represent other agents in the community with fidelity, thereby creating a collective critical ensemble and providing the best possible trade-off between accuracy and flexibility. Our approach provides a parsimonious and general mechanism for the emergence of critical-like behavior in living systems needing to cope with complex environments or trying to efficiently coordinate themselves as an ensemble.
1 Results
The paper models living systems as probabilistic internal representations that maximize fitness by minimizing information loss from environmental sources or other agents. Analytical and computational results show that heterogeneous environments and co-evolution favor parameter distributions near criticality, whereas homogeneous environments favor specialization.
- 1.1 Mathematical framework: Living systems represent environmental cues with probability distributions Psrc(s|α) and Pint(s|β), seeking internal states that approximate relevant external information.The framework interprets these representations as efficient but potentially imperfect encodings of environmental conditions.
- 1.1 Mathematical framework: Fitness is determined by information loss, with lower Kullback-Leibler divergence from environmental sources producing greater fitness.The same comparison is used when agents represent one another in the co-evolutionary model.
- 1.3 Analytical results for the dynamical models: The fittest internal parameters lie near the peak of generalized susceptibility, where small parameter changes can accommodate diverse complex sources.This identifies criticality as the region of maximal variability and information sensitivity.
- 1.3 Analytical results for the dynamical models: Heterogeneous environmental source pools drive communities toward criticality, while sufficiently homogeneous pools produce detail-specific, specialized internal states.The approach to criticality is less precise in the environmental model than in the co-evolutionary model.
- 1.3 Analytical results for the dynamical models: The optimal internal representation matches the lowest source-distribution moments available through its free parameters and minimizes the KL divergence when a solution exists.The Hessian analysis identifies the solution as a minimum of the divergence.
- 1.3 Analytical results for the dynamical models: In co-evolution, agents represent one another through asymmetric KL divergence, so fitter agents reproduce more and the community evolves toward a critical collective state.The model uses death, reproduction, inheritance, and mutation to update the distribution of agent parameters.
2 Discussion and conclusions
The paper concludes that information-based fitness provides a mechanism by which living systems self-tune toward criticality when coping with heterogeneous environments or coordinating as an ensemble. Co-evolutionary and co-adaptive interactions produce the most accurate convergence by forming a critical collective language.
- 2 Discussion and conclusions: Internal representations lie near peaks of generalized susceptibility, providing a compromise between accommodating regular signals and responding to noisy signals.The result follows under the assumption that living systems construct approximate probability-distribution representations of complex worlds.
- 2 Discussion and conclusions: Computational evolutionary and adaptive models show that communities exposed to broadly different, changing environments cluster near a critical state.The supplied conclusion identifies this clustering as the main model-level outcome.
- 2 Discussion and conclusions: Co-evolutionary and co-adaptive systems converge more accurately to criticality when individuals represent other agents with fidelity, creating a critical collective language.The agents’ mutual representations turn the community itself into the relevant collective entity.
- 2 Discussion and conclusions: The framework is presented as applicable to genetic and neural networks and more broadly to complex adaptive systems.The paper specifically connects this scope to understanding why neural activity may be tuned to criticality.
- 2 Discussion and conclusions: Phenotypic diversification in bacterial communities is proposed as a possible example of criticality-related bet hedging.The paper compares this diversification with distributing assets among multiple phenotypes to reduce long-term extinction risk.
3 Materials
The materials define the information-theoretic and evolutionary machinery used to compare environmental sources with internal representations and to evolve agent populations. The framework combines KL divergence, Fisher information, and genetic-algorithm updates.
- Information-theoretic measures: Kullback-Leibler divergence quantifies information loss when an approximating distribution Q(s) is used for a source distribution P(s).Maximizing likelihood is equivalent to minimizing this divergence in the large-sample limit.
- Information-theoretic measures: Fisher information measures information about parameters encoded in states and corresponds here to generalized susceptibility.Generalized susceptibility is described as the response to parameter variation and is known to peak at critical points.
- Co-evolutionary model: The internal model is specified by a probability distribution Pint(s|βk) proportional to exp{−Hint(s|βk)}.This parameterized distribution defines the internal state used by each agent.
- Co-evolutionary model: The co-evolutionary model selects pairs of agents, compares their asymmetric KL divergences, and reproduces the fitter agent while removing the other.Offspring inherit parental parameters except when mutation changes them by a Gaussian perturbation.
- Evolutionary model: Environmental sources are sampled from ρsrc(α), while each agent represents them with Pint(s|β), and fitness decreases with average KL divergence.Agents with larger divergence are more likely to be removed and replaced by inherited offspring.
Supplementary Information: Information-based fitness and the emergence of
The supplied passages identify the paper’s subject as criticality in living systems and name its authors. No substantive section content is included beyond these identifiers.
- The supplied section is titled “criticality in living systems.”
- The supplied passages provide bibliographic identification rather than methodological or empirical findings.
- The paper is authored by J. Hidalgo, J. Grilli, S. Suweis, M.A. Muñoz, J.R. Banavar, and A. Maritan.
S1 Brief primer on critical phenomena
Criticality marks the transition between ordered and disordered behavior, where systems become highly sensitive to perturbations. Statistical mechanics characterizes this regime through susceptibility and information-theoretic tools such as KL divergence.
- Critical points separate ordered and disordered phases and exhibit collective, scale-invariant behavior in many-body systems.Their properties depend on a few essential attributes, such as dimensionality and symmetry.
- At criticality, susceptibility measures the system’s differential response to an infinitesimal external perturbation.For finite systems, the susceptibility maximum provides an excellent estimate of the critical-point location.
- The KL divergence quantifies information loss when one probability distribution approximates another and is nonnegative, vanishing only for identical distributions.It is asymmetric and therefore is not a properly defined distance.
- Maximizing the likelihood of a trial distribution is equivalent, in the large-sample limit, to minimizing its KL divergence from the original distribution.This connection is identified with Sanov’s theorem in large deviations theory.
S3 Representing the external world
The paper models environmental sources and internal representations as probability distributions linked by external and internal parameters. It then shows analytically that optimizing representation across variable environments drives internal parameters toward regions of high Fisher information and criticality.
- Modeling sources and representations: Environmental sources are probability distributions over N binary variables, parameterized by environmental variables α, while agents use internal parameters β to generate response distributions.The framework can represent environmental conditions and gene-network states as probabilistic source and internal distributions.
- Modeling sources and representations: Minimizing KL divergence provides the optimal mapping from source parameters α to internal parameters β, while environmental variability is represented by ρsrc(α).A single optimized source is insufficient for coping with a complex and changing environment.
- Quenched and annealed choices: In the quenched choice, each source receives its own KL-optimal internal response, and the preferred β maximizes the resulting distribution of internal states.This selects the internal parameter able to cope with the greatest number of sources.
- Quenched and annealed choices: In the annealed choice, β minimizes the average KL divergence across environmental parameters, producing one representation that describes the varying environment on average.The annealed formulation is the principal case studied in the main text.
- Analytical optimization: The KL minimum occurs when the first I moments of the internal distribution match those of the source, and the positive-definite Hessian makes this solution a minimum.This establishes the moment-matching interpretation of the optimal representation.
- Analytical optimization: The induced internal-parameter distribution is proportional to generalized susceptibility, which diverges at criticality and peaks there for large finite systems.Thus, even when external parameters are not centered at criticality, internal representations accumulate near the critical point.
- Analytical optimization: For annealed representations, the averaged objective has a maximum at criticality, and positive-definiteness implies a drift of internal parameters toward that region.The projection argument shows that the internal parameters move closer to criticality than the average external parameters.
- Numerical illustration: The numerical example uses a mean-field Ising-inspired model whose internal representation is related to Boltzmann learning.The model is used to examine how optimized internal parameters alter distance from criticality.
S4 Computational models
The paper tests evolutionary and co-evolutionary models in which agents adapt internal probabilistic representations through selection, mutation, and interactions. Across model variants, stationary parameter distributions converge toward criticality, with convergence shaped by system size, community size, mutation, and environmental averaging.
- Model design: The Evolutionary Model evolves individuals exposed to varying external stimuli, whereas the Co-evolutionary Model treats the community itself as the external environment.In the co-evolutionary setting, individuals respond to the states of other agents, which also act as sources.
- Model design: At each evolutionary step, two agents are selected, relative fitness is computed from complementary KL divergences, and the fitter agent reproduces while the other is removed.Offspring mutate with probability ν by adding a Gaussian-distributed parameter perturbation, and time advances by 1/M.
- Co-evolutionary results: Different initial parameter distributions converge to the same stationary distribution in the co-evolutionary simulations.The reported linear-quadratic results compare multiple initial conditions and transient periods before the common stationary distribution is reached.
- Parameter dependence: Larger communities sharpen the stationary distribution, while increasing the number of interacting individuals K slows stationarity and weakens drift toward criticality.A larger K makes the effective source more homogeneous because it averages over more individuals.
- Networked models: For networked internal topologies, stationary parameter distributions still peak around the critical point, although explicit state summation limits simulations to approximately N ∼20.The network architecture is fixed and shared across individuals in these simulations.
- Evolutionary results: The stationary distribution becomes closer to the critical point as N increases and sharper as mutation probability ν or mutation variance σ decreases.The distribution also approaches an asymptotic shape as ensemble size and the number of external sources increase.
S5 Effective criticality and heterogeneity of the environment
The paper measures environmental heterogeneity through the entropy of moment distributions and finds that increasingly heterogeneous environments constrain optimal internal representations toward criticality, even when the averaged environment itself is not critical.
- Minimizing divergence to the averaged environment is equivalent to minimizing mean KL divergence across sources, causing agents facing complex averaged environments to become critical.
- Only two of six averaged environments are critical by Zipf’s law, while two non-critical averaged environments still produce optimal internal distributions near criticality.
- Environmental heterogeneity is quantified using the continuous Shannon entropy of the distribution of observables’ moments.
- Because the entropy is defined in the continuum limit, it can take negative values.
- When environmental entropy is low, optimal representations can occur across parameter space; as entropy increases, they become confined to the critical region.
S6 Adaptive and Co-adaptive models
The supplementary models show that adaptive and co-adaptive dynamics reproduce the evolutionary tendency toward criticality. Heterogeneous external environments and interactions among agents drive convergence, whereas specific sources do not.
- Criticality also emerges under adaptive dynamics rather than only evolutionary dynamics.
- Co-adaptive model: In the co-adaptive model, individuals iteratively choose small parameter changes that minimize their mean KL divergence to the community, with occasional random fluctuations.
- Co-adaptive model: Individuals cluster near the critical point independently of their initial conditions, reaching the same stationary solution as the corresponding co-evolutionary model.
- Adaptive model: In the adaptive model, individuals independently adjust parameters to minimize mean KL divergence to externally generated sources, with stochastic adaptation noise.
- Adaptive model: Heterogeneous environments tune adaptive populations toward criticality, whereas very specific sources do not.
Supplementary Videos
The supplementary videos visualize evolutionary dynamics in communities and environments, showing agents’ internal representations, parameter evolution, and criticality-related information landscapes.
- Videos are available for download in the source archive on arXiv.
- In co-evolving communities, agents map other individuals in a two-parameter space, and better representations increase reproductive probability while mutations alter inherited parameters.
- A second video compares community dynamics for N = 10 and N = 100 binary variables while displaying Fisher Information for the two-parameter map.
- A third video shows agents evolving internal maps of sources generated from different environments, with reproductive success determined by representation quality.