Detecting Information Relays in Deep Neural Networks.

Deep learning of artificial neural networks (ANNs) is creating highly functional processes that are, unfortunately, nearly as hard to interpret as their biological counterparts. Identification of functional modules in natural brains plays an important role in cognitive and neuroscience alike, and can be carried out using a wide range of technologies such as fMRI, EEG/ERP, MEG, or calcium imaging. However, we do not have such robust methods at our disposal when it comes to understanding functional modules in artificial neural networks. Ideally, understanding which parts of an artificial neural network perform what function might help us to address a number of vexing problems in ANN research, such as catastrophic forgetting and overfitting. Furthermore, revealing a network's modularity could improve our trust in them by making these black boxes more transparent. Here, we introduce a new information-theoretic concept that proves useful in understanding and analyzing a network's functional modularity: the relay information IR. The relay information measures how much information groups of neurons that participate in a particular function (modules) relay from inputs to outputs. Combined with a greedy search algorithm, relay information can be used to identify computational modules in neural networks. We also show that the functionality of modules correlates with the amount of relay information they carry.

Emergence of functional information from multivariate correlations.

The information content of symbolic sequences (such as nucleic or amino acid sequences, but also neuronal firings or strings of letters) can be calculated from an ensemble of such sequences, but because information cannot be assigned to single sequences, we cannot correlate information to other observables attached to the sequence. Here we show that an information score obtained from multivariate (multiple-variable) correlations within sequences of a 'training' ensemble can be used to predict observables of out-of-sample sequences with an accuracy that scales with the complexity of correlations, showing that functional information emerges from a hierarchy of multi-variable correlations. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Information Fragmentation, Encryption and Information Flow in Complex Biological Networks.

Assessing where and how information is stored in biological networks (such as neuronal and genetic networks) is a central task both in neuroscience and in molecular genetics, but most available tools focus on the network's structure as opposed to its function. Here, we introduce a new information-theoretic tool-information fragmentation analysis-that, given full phenotypic data, allows us to localize information in complex networks, determine how fragmented (across multiple nodes of the network) the information is, and assess the level of encryption of that information. Using information fragmentation matrices we can also create information flow graphs that illustrate how information propagates through these networks. We illustrate the use of this tool by analyzing how artificial brains that evolved in silico solve particular tasks, and show how information fragmentation analysis provides deeper insights into how these brains process information and "think". The measures of information fragmentation and encryption that result from our methods also quantify complexity of information processing in these networks and how this processing complexity differs between primary exposure to sensory data (early in the lifetime) and later routine processing.

Information-theoretic characterization of the complete genotype-phenotype map of a complex pre-biotic world: Comment on "From genotypes to organisms: State-of-the-art and perspectives of a cornerstone in evolutionary dynamics" by Susanna Manrubia et al.

How information is encoded in bio-molecular sequences is difficult to quantify since such an analysis usually requires sampling an exponentially large genetic space. Here we show how information theory reveals both robust and compressed encodings in the largest complete genotype-phenotype map (over 5 trillion sequences) obtained to date.

Can Transfer Entropy Infer Information Flow in Neuronal Circuits for Cognitive Processing?

How cognitive neural systems process information is largely unknown, in part because of how difficult it is to accurately follow the flow of information from sensors via neurons to actuators. Measuring the flow of information is different from measuring correlations between firing neurons, for which several measures are available, foremost among them the Shannon information, which is an undirected measure. Several information-theoretic notions of "directed information" have been used to successfully detect the flow of information in some systems, in particular in the neuroscience community. However, recent work has shown that directed information measures such as transfer entropy can sometimes inadequately estimate information flow, or even fail to identify manifest directed influences, especially if neurons contribute in a cryptographic manner to influence the effector neuron. Because it is unclear how often such cryptic influences emerge in cognitive systems, the usefulness of transfer entropy measures to reconstruct information flow is unknown. Here, we test how often cryptographic logic emerges in an evolutionary process that generates artificial neural circuits for two fundamental cognitive tasks (motion detection and sound localization). Besides counting the frequency of problematic logic gates, we also test whether transfer entropy applied to an activity time-series recorded from behaving digital brains can infer information flow, compared to a ground-truth model of direct influence constructed from connectivity and circuit logic. Our results suggest that transfer entropy will sometimes fail to infer directed information when it exists, and sometimes suggest a causal connection when there is none. However, the extent of incorrect inference strongly depends on the cognitive task considered. These results emphasize the importance of understanding the fundamental logic processes that contribute to information flow in cognitive processing, and quantifying their relevance in any given nervous system.

Moderate Amounts of Epistasis are Not Evolutionarily Stable in Small Populations.

High mutation rates select for the evolution of mutational robustness where populations inhabit flat fitness peaks with little epistasis, protecting them from lethal mutagenesis. Recent evidence suggests that a different effect protects small populations from extinction via the accumulation of deleterious mutations. In drift robustness, populations tend to occupy peaks with steep flanks and positive epistasis between mutations. However, it is not known what happens when mutation rates are high and population sizes are small at the same time. Using a simple fitness model with variable epistasis, we show that the equilibrium fitness has a minimum as a function of the parameter that tunes epistasis, implying that this critical point is an unstable fixed point for evolutionary trajectories. In agent-based simulations of evolution at finite mutation rate, we demonstrate that when mutations can change epistasis, trajectories with a subcritical value of epistasis evolve to decrease epistasis, while those with supercritical initial points evolve towards higher epistasis. These two fixed points can be identified with mutational and drift robustness, respectively.

The Surprising Creativity of Digital Evolution: A Collection of Anecdotes from the Evolutionary Computation and Artificial Life Research Communities.

Evolution provides a creative fount of complex and subtle adaptations that often surprise the scientists who discover them. However, the creativity of evolution is not limited to the natural world: Artificial organisms evolving in computational environments have also elicited surprise and wonder from the researchers studying them. The process of evolution is an algorithmic process that transcends the substrate in which it occurs. Indeed, many researchers in the field of digital evolution can provide examples of how their evolving algorithms and organisms have creatively subverted their expectations or intentions, exposed unrecognized bugs in their code, produced unexpectedly adaptations, or engaged in behaviors and outcomes, uncannily convergent with ones found in nature. Such stories routinely reveal surprise and creativity by evolution in these digital worlds, but they rarely fit into the standard scientific narrative. Instead they are often treated as mere obstacles to be overcome, rather than results that warrant study in their own right. Bugs are fixed, experiments are refocused, and one-off surprises are collapsed into a single data point. The stories themselves are traded among researchers through oral tradition, but that mode of information transmission is inefficient and prone to error and outright loss. Moreover, the fact that these stories tend to be shared only among practitioners means that many natural scientists do not realize how interesting and lifelike digital organisms are and how natural their evolution can be. To our knowledge, no collection of such anecdotes has been published before. This article is the crowd-sourced product of researchers in the fields of artificial life and evolutionary computation who have provided first-hand accounts of such cases. It thus serves as a written, fact-checked collection of scientifically important and even entertaining stories. In doing so we also present here substantial evidence that the existence and importance of evolutionary surprises extends beyond the natural world, and may indeed be a universal property of all complex evolving systems.

The Evolutionary Origin of Associative Learning.

Learning is a widespread ability among animals and, like physical traits, is subject to evolution. But how did learning first arise? What selection pressures and phenotypic preconditions fostered its evolution? Neither the fossil record nor phylogenetic comparative studies provide answers to these questions. Here, we take a novel approach by studying digital organisms in environments that promote the evolution of navigation and associative learning. Starting with a nonlearning sessile ancestor, we evolve multiple populations in four different environments, each consisting of nutrient trails with various layouts. Trail nutrients cue organisms on which direction to follow, provided they evolve to acquire and use those cues. Thus, each organism is tested on how well it navigates a randomly selected trail before reproducing. We find that behavior evolves modularly and in a predictable sequence, where simpler behaviors are necessary precursors for more complex ones. Associative learning is only one of many successful behaviors to evolve, and its origin depends on the environment possessing certain information patterns that organisms can exploit. Environmental patterns that are stable across generations foster the evolution of reflexive behavior, while environmental patterns that vary across generations but remain consistent for periods within an organism's lifetime foster the evolution of learning behavior. Both types of environmental patterns are necessary, since the prior evolution of simple reflexive behaviors provides the building blocks for learning to arise. Finally, we observe that an intrinsic value system evolves alongside behavior and supports associative learning by providing reinforcement for behavior conditioning.

Mapping the Peaks: Fitness Landscapes of the Fittest and the Flattest.

Populations exposed to a high mutation rate harbor abundant deleterious genetic variation, leading to depressed mean fitness. This reduction in mean fitness presents an opportunity for selection to restore fitness through the evolution of mutational robustness. In extreme cases, selection for mutational robustness can lead to flat genotypes (with low fitness but high robustness) outcompeting fit genotypes (with high fitness but low robustness)-a phenomenon known as survival of the flattest. While this effect was previously explored using the digital evolution system Avida, a complete analysis of the local fitness landscapes of fit and flat genotypes has been lacking, leading to uncertainty about the genetic basis of the survival-of-the-flattest effect. Here, we repeated the survival-of-the-flattest study and analyzed the mutational neighborhoods of fit and flat genotypes. We found that the flat genotypes, compared to the fit genotypes, had a reduced likelihood of deleterious mutations as well as an increased likelihood of neutral and, surprisingly, of lethal mutations. This trend holds for mutants one to four substitutions away from the wild-type sequence. We also found that flat genotypes have, on average, no epistasis between mutations, while fit genotypes have, on average, positive epistasis. Our results demonstrate that the genetic causes of mutational robustness on complex fitness landscapes are multifaceted. While the traditional idea of the survival of the flattest emphasized the evolution of increased neutrality, others have argued for increased mutational sensitivity in response to strong mutational loads. Our results show that both increased neutrality and increased lethality can lead to the evolution of mutational robustness. Furthermore, strong negative epistasis is not required for mutational sensitivity to lead to mutational robustness. Overall, these results suggest that mutational robustness is achieved by minimizing heritable deleterious variation.

Thermodynamics of evolutionary games.

How cooperation can evolve between players is an unsolved problem of biology. Here we use Hamiltonian dynamics of models of the Ising type to describe populations of cooperating and defecting players to show that the equilibrium fraction of cooperators is given by the expectation value of a thermal observable akin to a magnetization. We apply the formalism to the public goods game with three players and show that a phase transition between cooperation and defection occurs that is equivalent to a transition in one-dimensional Ising crystals with long-range interactions. We then investigate the effect of punishment on cooperation and find that punishment plays the role of a magnetic field that leads to an "alignment" between players, thus encouraging cooperation. We suggest that a thermal Hamiltonian picture of the evolution of cooperation can generate other insights about the dynamics of evolving groups by mining the rich literature of critical dynamics in low-dimensional spin systems.

Origin of life in a digital microcosm.

While all organisms on Earth share a common descent, there is no consensus on whether the origin of the ancestral self-replicator was a one-off event or whether it only represented the final survivor of multiple origins. Here, we use the digital evolution system Avida to study the origin of self-replicating computer programs. By using a computational system, we avoid many of the uncertainties inherent in any biochemical system of self-replicators (while running the risk of ignoring a fundamental aspect of biochemistry). We generated the exhaustive set of minimal-genome self-replicators and analysed the network structure of this fitness landscape. We further examined the evolvability of these self-replicators and found that the evolvability of a self-replicator is dependent on its genomic architecture. We also studied the differential ability of replicators to take over the population when competed against each other, akin to a primordial-soup model of biogenesis, and found that the probability of a self-replicator outcompeting the others is not uniform. Instead, progenitor (most-recent common ancestor) genotypes are clustered in a small region of the replicator space. Our results demonstrate how computational systems can be used as test systems for hypotheses concerning the origin of life.This article is part of the themed issue 'Reconceptualizing the origins of life'.

Evolution of drift robustness in small populations.

Most mutations are deleterious and cause a reduction in population fitness known as the mutational load. In small populations, weakened selection against slightly-deleterious mutations results in an additional fitness reduction. Many studies have established that populations can evolve a reduced mutational load by evolving mutational robustness, but it is uncertain whether small populations can evolve a reduced susceptibility to drift-related fitness declines. Here, using mathematical modeling and digital experimental evolution, we show that small populations do evolve a reduced vulnerability to drift, or 'drift robustness'. We find that, compared to genotypes from large populations, genotypes from small populations have a decreased likelihood of small-effect deleterious mutations, thus causing small-population genotypes to be drift-robust. We further show that drift robustness is not adaptive, but instead arises because small populations can only maintain fitness on drift-robust fitness peaks. These results have implications for genome evolution in organisms with small effective population sizes.

The evolution of logic circuits for the purpose of protein contact map prediction.

Predicting protein structure from sequence remains a major open problem in protein biochemistry. One component of predicting complete structures is the prediction of inter-residue contact patterns (contact maps). Here, we discuss protein contact map prediction by machine learning. We describe a novel method for contact map prediction that uses the evolution of logic circuits. These logic circuits operate on feature data and output whether or not two amino acids in a protein are in contact or not. We show that such a method is feasible, and in addition that evolution allows the logic circuits to be trained on the dataset in an unbiased manner so that it can be used in both contact map prediction and the selection of relevant features in a dataset.

A genome wide dosage suppressor network reveals genomic robustness.

Genomic robustness is the extent to which an organism has evolved to withstand the effects of deleterious mutations. We explored the extent of genomic robustness in budding yeast by genome wide dosage suppressor analysis of 53 conditional lethal mutations in cell division cycle and RNA synthesis related genes, revealing 660 suppressor interactions of which 642 are novel. This collection has several distinctive features, including high co-occurrence of mutant-suppressor pairs within protein modules, highly correlated functions between the pairs and higher diversity of functions among the co-suppressors than previously observed. Dosage suppression of essential genes encoding RNA polymerase subunits and chromosome cohesion complex suggests a surprising degree of functional plasticity of macromolecular complexes, and the existence of numerous degenerate pathways for circumventing the effects of potentially lethal mutations. These results imply that organisms and cancer are likely able to exploit the genomic robustness properties, due the persistence of cryptic gene and pathway functions, to generate variation and adapt to selective pressures.

Different Evolutionary Paths to Complexity for Small and Large Populations of Digital Organisms.

A major aim of evolutionary biology is to explain the respective roles of adaptive versus non-adaptive changes in the evolution of complexity. While selection is certainly responsible for the spread and maintenance of complex phenotypes, this does not automatically imply that strong selection enhances the chance for the emergence of novel traits, that is, the origination of complexity. Population size is one parameter that alters the relative importance of adaptive and non-adaptive processes: as population size decreases, selection weakens and genetic drift grows in importance. Because of this relationship, many theories invoke a role for population size in the evolution of complexity. Such theories are difficult to test empirically because of the time required for the evolution of complexity in biological populations. Here, we used digital experimental evolution to test whether large or small asexual populations tend to evolve greater complexity. We find that both small and large-but not intermediate-sized-populations are favored to evolve larger genomes, which provides the opportunity for subsequent increases in phenotypic complexity. However, small and large populations followed different evolutionary paths towards these novel traits. Small populations evolved larger genomes by fixing slightly deleterious insertions, while large populations fixed rare beneficial insertions that increased genome size. These results demonstrate that genetic drift can lead to the evolution of complexity in small populations and that purifying selection is not powerful enough to prevent the evolution of complexity in large populations.

Evolvability Tradeoffs in Emergent Digital Replicators.

The role of historical contingency in the origin of life is one of the great unknowns in modern science. Only one example of life exists-one that proceeded from a single self-replicating organism (or a set of replicating hypercycles) to the vast complexity we see today in Earth's biosphere. We know that emergent life has the potential to evolve great increases in complexity, but it is unknown if evolvability is automatic given any self-replicating organism. At the same time, it is difficult to test such questions in biochemical systems. Laboratory studies with RNA replicators have had some success with exploring the capacities of simple self-replicators, but these experiments are still limited in both capabilities and scope. Here, we use the digital evolution system Avida to explore the interplay between emergent replicators (rare randomly assembled self-replicators) and evolvability. We find that we can classify fixed-length emergent replicators in Avida into two classes based on functional analysis. One class is more evolvable in the sense of optimizing the replicators' replication abilities. However, the other class is more evolvable in the sense of acquiring evolutionary innovations. We tie this tradeoff in evolvability to the structure of the respective classes' replication machinery, and speculate on the relevance of these results to biochemical replicators.