Entropy

We introduce a spectral existence criterion for the evolution of cooperation in the form of the inequality{lambda} maxb > c, where{lambda} max is the leading eigenvalue of an interaction operator encoding population structure, and b and c represent benefit and cost tradeoffs, respectively. Nowaks five rules for the evolution of cooperation correspond to cases in which the cooperation condition reduces to a scalar assortment coefficient. These results follow from the Price equation, which sheds light on a long-standing debate on the role of inclusive fitness and evolutionary dynamics in explaining the evolution of cooperation.

AO_SCPLOWBSTRACTC_SCPLOWOne of the most compelling ideas for bridging neuroscience and artificial neural networks is the establishment of a framework based on three main components: network architecture, optimization mechanism, and loss (or objective) function to be minimized. While the first two components have been extensively explored, the definition of a loss or objective function in neuroscience has been addressed less thoroughly, often from perspectives such as predictive coding. In this work, we propose an elementary loss function grounded in the comparison of neuronal responses to two signals: an external one, used for learning, and an internal one, reflecting the acquired knowledge. The loss function is thus simply the basic difference between the two, which, in terms of logical signals, corresponds to a well-known non-linearly separable function: the XOR function. We illustrate with a computational example how a binarized image recognition algorithm can be straightforwardly implemented in an autoencoder, and we show how a neuronal motif organized around an inhibitory neuron could implement such XOR operation and provide a feedback signal that makes optimization possible.

The primary objective of system neuroscience is to understand the functional mapping and its causation in the dynamics of the brain network. Some experimental and methodological studies suggest that functional modularity and its hierarchical information processing in the brain network are crucial to understanding the functional role of task-specific or state-specific information flow in the brain. However, because most of the established techniques for detecting effective network structures in the neuroscience research field are strongly based on the "Granger causality" perspective, existing causal discovery methods specified for brain network analysis cannot identify the causal hierarchy in the modular network in the brain due to spurious correlation issues and indistinguishability of causal direction under the Gaussianity of observational noise in a linear system. To address the issues, we developed a causal discovery method for synchronous neural dynamics, called the Jacobian-informed linear non-Gaussian acyclic model, "j-VAR-LiNGAM", by incorporating the information of the Jacobian matrix determined from a phase-coupled oscillator model estimated from observed neural data into the VAR-LiNGAM algorithms. The method was validated by showing that it could extract causal ordering in both synthetic data and empirical neural observed data. Moreover, by analyzing the observed neural oscillatory signals obtained from mice and humans, we confirmed that our method identified causally hierarchical structures in the brain, which aligned with the neurophysiological interpretations. These findings suggested that our proposed method can reveal the neural basis of hierarchical information processing in the brain network.

Common knowledge states that the spontaneous somatic evolution of a normal tissue may lead to a tumor. Once the tumor is formed, it naturally evolves towards a state of higher malignancy. On the other hand, perfect gene expression markers for normal tissue and tumor--the so-called N-genes and T-genes--were recently introduced. We join these two pieces of knowledge in order to argue that: 1) Only N-markers participate in the spontaneous dynamics of a normal tissue. The number of active markers decreases as the tissue approaches the transition point where it becomes a tumor. 2) Only T-markers participate in the spontaneous dynamics of tumors. The number of markers increases as the tumor becomes more malignant. 3) Both sets of genes are connected by the so-called NT-genes, i.e., genes that are simultaneously N- and T-markers. They should play a crucial role at the transition point and, possibly, when the tumor is exposed to a drug or therapy. 4) The pathways or mechanisms protecting the normal tissue from becoming a tumor may be described by a small perfect panel of N-genes. 5) The pathways or mechanisms guiding the evolution of tumors in a tissue may be described by a small perfect panel of T-genes. We illustrate the above statements with the analysis of expression data for prostate adenocarcinoma, one of the most heterogeneous tumors. In this case, there are about 1000 N-genes and 6000 T-genes, and the perfect N- and T-panels contain 11 and 8 genes, respectively. Additionally, we provide examples from lung adenocarcinoma and liver hepatocarcinoma.

In previous studies by the author on binocular vision with the asymmetric eye (AE), which models a healthy human eye with misaligned optical components, the results were primarily presented in the Rodrigues vector (RV) framework and supported by simulations and 3D visualizations in GeoGebras dynamic geometry environment. In this paper, the novel geometric kinematics of the human eye, that is, the eye with misaligned optics, and simplified assumptions about the eye rotations (the eyes translational movements are disregarded), are developed within the framework of rigid-body rotations. The originality of the analysis lies in a precise geometric decomposition of a full rotation of the eyes posture into a torsion-free rotation (the geodesic part) and a torsional rotation (the non-geodesic extension of the geodesic part). This decomposition is extended to the corresponding decomposition of the angular velocity. A novel derivation of the eyes angular velocity from the RV formulation of the eye kinematics is proposed.

The emergence of amino acids (AAs) and nucleobases (NBs) across meteorites, interstellar ices, and laboratory shock experiments presents a paradox: why do these specific molecular motifs--a minuscule subset of organic chemistrys combinatorial space--appear repeatedly across diverse environments, in the absence of biological selection? We identify a physical mechanism, prebiotic selection, which biases driven chemical systems toward configurations with high stationary probability p*(x) under sustained entropy flux. The bias is quantified by an information quasi-potential {Phi}I (x) = - ln p*(x), entering the overdamped Langevin dynamics O_FD O_INLINEFIG[Formula 1]C_INLINEFIGM_FD(1)C_FD where {Sigma} is the local entropy production rate (Schnakenberg 1976). {Phi}I is defined self-consistently via the full non-equilibrium stationary density, avoiding the circularity of identifying it with a scalar potential. Two central theorems underlie the framework. Theorem 1 establishes that {nabla}{Sigma} and {nabla}{Phi}I are generically linearly independent off equilibrium, so the dynamics is genuinely two-field. Theorem 2 (structural constraints on single-field gradient dynamics) shows that single-field models on compact manifolds (i) produce yield curves that are at most unimodal under linear driving, and (ii) combine disjoint perturbations additively, giving superlinearity factor S = 1 + O(||{delta} V ||2). The observed superlinear synergy of Ferris et al. (1996) lies far outside this perturbative bound and therefore requires the two-field structure of EOM-IFF; the non-monotonic peak of Blank et al. (2001) is consistent with two-field dynamics and also with single-field dynamics in the unimodal-with-peak case of Theorem 2 part 1, so it does not by itself discriminate. From these results, we: (i) define a formal substrate-minimal criterion for prebiotic selection; (ii) show consistency with the non-monotonic shock-synthesis yield of Blank et al. (2001) (R2 = 0.885, peak at P* = 28.4 {+/-} 1.4 GPa); (iii) show consistency with the superlinear clay-catalysed RNA polymerisation of Ferris et al. (1996) (synergy factor S {approx} 5.75, robust under {+/-}1-nucleotide measurement uncertainty); and (iv) state two further falsifiable predictions awaiting dedicated experimental tests. Every lemma and theorem is accompanied by explicit assumptions, regime of validity, and regime of failure; the frameworks scope is what it claims, not more. Prebiotic selection is identified as a physical process distinct from and prior to biological selection, offering a unified account of chemical convergence in carbon-nitrogen chemistry under sustained entropy flux.

Categorisation is often treated as a form of compression: a high-dimensional stimulus space is reduced to a smaller set of behaviourally or cognitively useful classes. However, compression alone does not determine whether a category map is useful. The present manuscript develops an information-theoretic framework for evaluating categorisation in terms of both category complexity and target-relevant information preservation. Across a set of synthetic demonstrations, alternative category maps over the same stimulus space are shown to preserve different target variables, including identity, action, nuisance, and hierarchical category structure. The framework is then extended to learned visual representations by analysing layer-derived category maps from a pretrained ResNet-50 network applied to CIFAR-10 images. Two scenarios are compared: a clean-only object run and a pooled nuisance run containing clean, blurred, pixelated, and noise-perturbed images. The results show that category maps can have substantial entropy while preserving information about a variable that is not aligned with the specified target, and that the value of a categorisation depends on the target variable to be preserved. The manuscript argues that categorisation should therefore be evaluated not only by compression or separability, but by the information retained about a specified cognitive, behavioural, or computational target.

Price equation and genotype dynamics are two methods for studying the fixation of one allele by natural selection in a diploid population. There are two strict monotonicity conditions that imply the fixation of one allele. The genotype dynamics is called Haldane monotone if the relative frequency of one allele strictly increases along all solutions of the genotype dynamics, so this allele is fixed. In this paper, we show that the genotype dynamics is Haldane monotone if and only if the right-hand side of the Price equation is always strictly positive. The other strict monotonicity condition requires that the relative frequency of a homozygote strictly increase according to the genotype dynamics. For example, in a model where the genotype dynamics is governed by interactions between individuals, the cost-accepting homozygote is fixed by natural selection if the other genotypes always receive a smaller average gain from all interactions than the cost-accepting homozygote. Both monotonicity conditions require that the interaction is not well-mixed in the population. These two conditions are not equivalent. In addition, we give a non-monotonicity condition, which also implies the fixation of a homozygote. The fixation of a homozygote depends on the phenotypic payoff of the interaction, the genotype-phenotype mapping, and the interaction scheme. In a sexual population, the interaction scheme of siblings depends on the mating system, and so do the conditions of fixation of the cost-accepting homozygote. We present examples showing that if we only change the monogamous mating system, assuming panmixing or mating assortativity, then the condition for the fixation of the cooperator homozygote is b > 2c and b > c, respectively.

A morphospace is an abstract space of theoretically possible biological traits, shapes, or property values. It is interesting to explore which parts of a morphospace life occupies, as compared to those parts which could be occupied, but are not. Comparing random and natural non-coding (nc) RNA secondary structures is an established approach to studying morphospace occupation for RNA structures. Most earlier studies have focused on the minimum free energy (MFE) structure, while relatively few have looked at the Boltzmann distribution, describing the ensemble of energetically suboptimal RNA folds. These suboptimal structures may have important roles and functions, and hence should be examined carefully. Here we compare random and natural ncRNA in terms of their Boltzmann distributions, finding that natural RNA tend to have very similar profiles to random RNA, with the main difference being that natural RNA are slightly more energetically stable, except for very short sequences (20 to 30 nucleotides) which tend to be slightly less stable. We infer that natural ncRNA occupy similar parts of the morphospace that random RNA do, indicating that the biophysics of the genotype-phenotype map largely determines the ensemble properties of ncRNA.

Protein structure is controlled by a high-dimensional energy landscape, which is a function of all of the atomic coordinates of the protein. Can this landscape be accurately described by a low-dimensional representation? We find that residue core identity, a binary N-dimensional encoding indicating whether each of the N amino acids in a protein is buried in the core or not, can predict the proteins backbone conformation more efficiently than all other representations that we tested. Core identity is 4 times more efficient than previous estimates of the bits per residue needed to encode a proteins native fold, 2 times more efficient than the C contact map, and 1.5 times more efficient than the machine-learned embeddings from FoldSeeks 3Di. Even when the folded structure is unavailable, predicting each residues burial from sequence yields a more accurate estimate of fold quality than predicting pairwise contacts from the same sequence information. Thus, this work emphasizes that the problem of determining a proteins native fold can be re-framed as predicting each residues core identity.

At the fundamental conceptual level, two alternatives have traditionally been considered for how mutations arise and how evolution happens: 1) random mutation and natural selection, and 2) Lamarckism. Recently, the theory of Interaction-based Evolution (IBE) has been proposed, according to which mutations are neither random nor Lamarckian, but are influenced by information accumulating internally in the genome over generations. Based on the estimation-of-distribution algorithms framework, we present a simulation model that demonstrates nonrandom, non-Lamarckian mutation concretely while capturing indirectly several aspects of IBE: selection, recombination, and nonrandom, non-Lamarckian mutation interact in a complementary fashion; evolution is driven by the interaction of parsimony and fit; and random bits do not directly encode improvement but enable generalization by the manner in which they connect with the rest of the evolutionary process. Connections are drawn to Darwins observations that changed conditions increase the rate of production of heritable variation; to the causes of bell-shaped distributions of traits and how these distributions respond to selection; and to computational learning theory, where analogizing evolution to learning in accord with IBE casts individuals as examples and places the learned hypothesis at the population level. The model highlights the importance of incorporating internal integration of information through heritable change in both evolutionary theory and evolutionary computation.

Oscillatory behaviour is important in multiple biological contexts. However, the inherent nonlinearity and high dimensionality of mathematical models in biology makes proving the existence and the localization of limit cycle oscillations challenging. Here, we provided an elementary proof for the existence and a method for rigorously localizing the oscillatory solutions in a class of benchmark biomolecular oscillators. To construct the proof, we used a geometric approach based on Brouwers Fixed Point theorem. We constructed a toroidal-like manifold within a positively invariant set by removing the hypervolume containing the fixed point and the trajectories converging to it along its stable manifold. We showed that the vector field describing the system dynamics maps a cross section of the toroidal-like manifold onto itself. The existence of a limit cycle solution in this manifold was guaranteed by Brouwers Fixed Point theorem. For different sets of initial conditions in these cross-sections, we used an interval-based Reachability Analysis to localize the oscillatory behaviour that complements the Brouwers Fixed Point theorem approach. These results add a simple and elegant approach to demonstrating the existence of limit cycles in biomolecular systems as well as a method for rigorous localization of the region of existence.

Subthreshold neuronal membranes exhibit resonant, band-pass impedance characterised by an effective inductance arising from voltage-gated channel kinetics--principally Ih. This paper presents a six-layer computational framework that builds from this single-neuron RLC description to a complete account of how coupled neural oscillator networks compute. Layer 1 establishes the RLC neuron as a frequency-selective bandpass filter. Layer 2 shows that coupled RLC neurons encode relational information in phase differences (binding). Layer 3 demonstrates that networks of coupled oscillators form attractor landscapes supporting memory and pattern completion, with fixed-point, limit-cycle, and chaotic attractor classes. Layer 4 identifies the synaptic coupling matrix as a learned impedance network whose topology determines attractor geometry. Layer 5 maps neuromodulatory systems to bias controls that sweep RLC parameters (resonant frequency, quality factor, gain) without modifying stored memories. Layer 6 assembles the full system with cross-frequency multiplexing and homeostatic stabilisation. The framework is grounded in measurable electrical quantities and generates testable predictions distinguishing it from rate-coding and RC integrate-and-fire models. We explicitly address the linearisation gap between the subthreshold regime where the RLC description is rigorous and the nonlinear regime where attractor dynamics operate, the noise and precision limits of analog neural computation ([~] 3.3 effective bits per neuron, compensated by massive parallelism), and the distinction between causal and correlative evidence for oscillation-based coding claims. The framework does not replace existing models; it extends them by showing that rate coding is one (coarse) description of the output of an analog computation whose richer dynamics-- resonance, phase, temporal fine structure--may carry additional computational content.

We describe an approach for analyzing biological networks using rows of the Krylov subspace of the adjacency matrix. Specifically, we explore the scenario where the Krylov subspace matrix is computed via power iteration using a non-random and potentially non-uniform initial vector that captures a specific biological state or perturbation. In this case, the rows the Krylov subspace matrix (i.e., Krylov trajectories) carry important functional information about the network nodes in the biological context represented by the initial vector. We demonstrate the utility of this approach for community detection and perturbation analysis using the C. Elegans neural network.

A key component of intraneuronal communication is the modulation of postsynaptic firing frequencies by stochastic transmitter release from presynaptic neurons. The time interval between successive postsynaptic firings is called the inter-spike interval (ISI), and understanding its statistics is integral to neural information processing. We start with a model of an excitatory chemical synapse with postsynaptic neuron firing governed as per a classical integrate-and-fire model. Using a first-passage time framework, we derive exact analytical results for the ISI statistical moments, revealing parameter regimes driving precision in postsynaptic action potential timing. Next, we extended this analysis to include both an excitatory and an inhibitory presynaptic connection onto the same postsynaptic neuron. We consider both a fixed postsynaptic-firing threshold and a threshold that adapts based on the postsynaptic membrane potential history. Our analysis shows that the latter adaptive threshold can result in scenarios where increasing the inhibitory input frequency increases the postsynaptic firing frequency. Moreover, we characterize parameter regimes where ISI noise is hypo-exponential or hyperexponential based on its coefficient of variation being less than or higher than one, respectively.

In this study, resource-constrained learning methods were developed as a model for the learning behavior of the fly brain, specifically the mushroom body. Recent research on the mushroom bodies of flies shows that unfamiliar odors activate certain output neurons (MBONs); however, these effects are rapidly suppressed upon repeated exposure to the same odor. Such MBON behaviors appear to reflect odor learning. We investigated how flies continue learning about odors throughout their lives despite their small brains. Researchers have suggested that learning about new odors can help flies forget existing memories. Therefore, we hypothesized that the main reason for continual learning is that it serves as a strategy for forgetting. To test the validity of this hypothesis, we designed three models using a kernel perceptron. This approach is suitable for estimating ongoing learning capacity within a budget. According to the results of computer simulations and theoretical analysis, the model demonstrated the importance of forgetting mechanisms for two reasons: first, to prepare for subsequent learning sessions, and second, to reduce the negative effects of deleting memories. Author summaryDrosophila mushroom body output neurons (MBONs) in the 3 compartment of the fruit fly brain are highly activated by novel odors, and their activation triggers alerting behavior. Interestingly, these specific neurons react only to unfamiliar odor information, suggesting they constantly undergo incremental learning of new odors. This study was aimed at constructing three incremental learning models of the MBON 3 neurons. Although there have been numerous studies on complex circuit designs to reproduce activation waveforms, herein we constructed a fundamental learning model based on a kernelized learning method. Since kernelized learning models interpret Hebbian learning as the addition or subtraction of kernel functions, the model is easy to analyze theoretically. Consequently, we conclude that the forgetting property observed in the MBON 3 neurons is essential for reducing error when learning occurs within a brain of limited capacity.

The light absorption of the protein cryptochrome and its chromophore FAD is important for the regulation of circadian rhythms and in some species for sensing magnetic fields. To compute the absorption spectrum of chromophore, typically only a small region is treated quantum-mechanically due the high computational cost of spectroscopic calculations. We present a formalism that allows a quantum-mechanical treatment of not only the chromophore but also the neighbouring amino acids which differ from species to species. This is achieved by using the real-time time-dependent Hartree-Fock method. This method allows extending the quantum domain from typically only a few dozen atoms up to around 1,200 atoms for the largest calculations. The presented framework allows the treatment of neighbouring tryptophan residues or the cofactor molecule MTHF in the same calculation and allows to extract information of which regions absorb light depending on wavelength. The presented results also show that the environment around the chromophore FAD amplifies the light absorption in cryptochrome.

Mitochondria often form branching membrane networks distributed throughout the cell interior. In many, though not all, cell types, these networks are observed to consist of one large connected component together with many smaller fragments. Why does this pattern arise? Does it reflect a specific biological function, an external biophysical constraint, or something simpler? Using results from extremal graph theory, we prove a new theorem which suggests that, under a sufficiently broad sampling of the space of mitochondria-like graphs, the predominance of three-way junctions makes the appearance of a large component likely. This suggests that, in some settings, a large component may serve as a useful null model for mitochondrial network structure rather than requiring a dedicated explanation. More broadly, our result points towards testable predictions, since systematic deviations from this baseline may help reveal additional constraints or mechanisms shaping mitochondrial morphology.

Curvature evolution on a deforming surface is governed by the full change in the surface metric, but on biological surfaces captured by serial three-dimensional imaging, only the local area change is observable. The loss of the shear component leaves prediction of curvature evolution underdetermined from imaging alone. On the thoracic aorta, where curvature change marks disease progression, we derive a closed-form equation that predicts the change in integrated Gaussian curvature from the area dilation and initial geometry. The equation combines a conformal term in the area dilation with a leading anisotropy correction from the initial geometry. These two analytic levels, augmented by multi-scale spatial features at neighboring regions and a graph neural network trained on residuals, form a four-level nested predictor. On a synthetic aortic geometry under prescribed isotropic expansion, the equation recovers the analytic coefficient exactly. Across a continuum from pure expansion to pure shear, it holds R2 [≥] 0.71. On 236 paired thoracic aortic surfaces spanning dissection, aneurysm, traumatic injury, and non-pathologic controls, the equation recovers within-surface curvature change patterns with per-patient median Pearson [Formula] and pooled R2 = +0.238 [+0.225, +0.250], matching the graph neural network on the same inputs. The residual is a direct measurement of how far the observed growth field departs from conformality. HighlightsO_LIClosed-form equation predicts aortic curvature change from paired computed tomography scans. C_LIO_LIRecovers analytic predictions exactly on synthetic aortic geometries. C_LIO_LIAnisotropy proxy holds R2 [≥] 0.71 from pure expansion to pure shear. C_LIO_LICoefficients tie to geometric mechanisms ensuring interpretability. C_LIO_LIAnisotropy term, computable from one CT, is twice as large on diseased aortas. C_LI

Neural population activity typically evolves on low-dimensional manifolds and can be described as trajectories in attractor-like state spaces, including metastable switching among quasi-stable assembly states. Here we develop a unified definition of clustered neural networks with local excitatory-inhibitory balance in which enhanced within-cluster effective coupling can be realized by connection probability (structural clustering), synaptic efficacy (weight clustering), or any mixture of both. We introduce a single mixing parameter{kappa} [isin] [0, 1] that redistributes a defined clustering contrast between connection probabilities and synaptic efficacies while preserving the mean input of a balanced random network. Using mean-field theory and network simulations, we show that metastable dynamics are supported across the full{kappa} continuum. Shifting contrast between structural and weight clustering changes higher-order input structure, reshaping multistable regimes, neuronal correlations, and the balance between single- and multi-cluster episodes. Because real nervous systems jointly organize topology and synaptic strength, our approach provides a biologically realistic assembly definition and a basis for future models combining structural and functional plasticity. In practical terms,{kappa} offers a translation axis for neuromorphic and other constrained substrates, clarifying trade-offs between routing resources and synaptic weight resolution when implementing attractor-based computational primitives such as winner-take-all decisions and working-memory states for artificial agents.