Inference of single-cell network using mutual information for scRNA-seq data analysis.

BACKGROUND: With the advance in single-cell RNA sequencing (scRNA-seq) technology, deriving inherent biological system information from expression profiles at a single-cell resolution has become possible. It has been known that network modeling by estimating the associations between genes could better reveal dynamic changes in biological systems. However, accurately constructing a single-cell network (SCN) to capture the network architecture of each cell and further explore cell-to-cell heterogeneity remains challenging. RESULTS: We introduce SINUM, a method for constructing the SIngle-cell Network Using Mutual information, which estimates mutual information between any two genes from scRNA-seq data to determine whether they are dependent or independent in a specific cell. Experiments on various scRNA-seq datasets with different cell numbers based on eight performance indexes (e.g., adjusted rand index and F-measure index) validated the accuracy and robustness of SINUM in cell type identification, superior to the state-of-the-art SCN inference method. Additionally, the SINUM SCNs exhibit high overlap with the human interactome and possess the scale-free property. CONCLUSIONS: SINUM presents a view of biological systems at the network level to detect cell-type marker genes/gene pairs and investigate time-dependent changes in gene associations during embryo development. Codes for SINUM are freely available at https://github.com/SysMednet/SINUM .

Research on Division of Labor Decision and System Stability of Swarm Robots Based on Mutual Information.

In rational decision-making processes, the information interaction among individual robots is a critical factor influencing system stability. We establish a game-theoretic model based on mutual information to address division of labor decision-making and stability issues arising from differential information interaction among swarm robots. Firstly, a mutual information model is employed to measure the information interaction among robots and analyze its influence on the behavior of individual robots. Secondly, employing the Cournot model and the Stackelberg model, we model the diverse decision-making behaviors of swarm robots influenced by discrepancies in mutual information. The intricate decision dynamics exhibited by the system under the disparity mutual information values during the game process, along with the stability of Nash equilibrium points, are analyzed. Finally, dynamic complexity simulations of the game models are simulated under the disparity mutual information values: (1) When ν1 of the game model varies within a certain range, the Nash equilibrium point loses stability and enters a chaotic state. (2) As I(X;Y) increases, the decision-making pattern of robots transitions gradually from the Cournot game to the Stackelberg game. Concurrently, the sensitivity of swarm robotics systems to changes in decision parameter decreases, reducing the likelihood of the system entering a chaotic state.

A comparative analysis of mutual information methods for pairwise relationship detection in metagenomic data.

BACKGROUND: Construction of co-occurrence networks in metagenomic data often employs correlation to infer pairwise relationships between microbes. However, biological systems are complex and often display qualities non-linear in nature. Therefore, the reliance on correlation alone may overlook important relationships and fail to capture the full breadth of intricacies presented in underlying interaction networks. It is of interest to incorporate metrics that are not only robust in detecting linear relationships, but non-linear ones as well. RESULTS: In this paper, we explore the use of various mutual information (MI) estimation approaches for quantifying pairwise relationships in biological data and compare their performances against two traditional measures-Pearson's correlation coefficient, r, and Spearman's rank correlation coefficient, ρ. Metrics are tested on both simulated data designed to mimic pairwise relationships that may be found in ecological systems and real data from a previous study on C. diff infection. The results demonstrate that, in the case of asymmetric relationships, mutual information estimators can provide better detection ability than Pearson's or Spearman's correlation coefficients. Specifically, we find that these estimators have elevated performances in the detection of exploitative relationships, demonstrating the potential benefit of including them in future metagenomic studies. CONCLUSIONS: Mutual information (MI) can uncover complex pairwise relationships in biological data that may be missed by traditional measures of association. The inclusion of such relationships when constructing co-occurrence networks can result in a more comprehensive analysis than the use of correlation alone.

Cross-scale coupling of ecosystem service flows and socio-ecological interactions in the Yellow River Basin.

As research on the full spectrum of ecosystem service (ES) generation and utilization within coupled human and natural systems (CHANS) has expanded, many studies have shown that the spatiotemporal dynamics of ESs are managed and influenced by human activities. However, there is insufficient research on how ESs are affected by bidirectional coupling between societal and ecological factors during spatial flow, particularly in terms of cross-scale impacts. These bidirectional influences between humans and nature are closely related to the utilization and transfer of ESs and affect the perception of spatiotemporal patterns of ESs and the formulation of management strategies. To fill this research gap, this study focuses on the Yellow River Basin (YRB), using network models to track the spatial dynamics of ES flows (ESFs) and the interactions between ecosystems and socio-economic systems within the basin on an annual scale from 2000 to 2020. The results highlight cross-scale impacts and feedback processes between local subbasins and the larger regional basin: As the supply-demand ratios of freshwater ESs, soil conservation ESs, and food ESs increase within individual subbasins of the YRB, more surplus ESs flow among subbasins. This not only alleviates spatial mismatches in ES supply and demand across the entire basin but also enhances the connectivity of the basin's ESF network. Subsequently, the cascading transfer and accumulation of ESs feedback into local socio-ecological interactions, with both socio-economic factors and the capacity for ES output within subbasins becoming increasingly reliant on external ES inflows. These results underscore the crucial role of ESFs within the CHANS of the YRB and imply the importance of cross-regional cooperation and cross-scale management strategies in optimizing ES supply-demand relationships. Furthermore, this study identifies the potential risks and challenges inherent in highly coupled systems. In conclusion, this work deepens the understanding of the spatial flow characteristics of ESs and their socio-ecological interactions; the analytical methods used in this study can also be applied to research on large river basins like the YRB, and even larger regional ecosystems.

Fish communicate with water flow to enhance a school's social network.

Schooling fish rely on a social network created through signaling between its members to interact with their environment. Previous studies have established that vision is necessary for schooling and that flow sensing by the lateral-line system may aid in a school's cohesion. However, it remains unclear to what extent flow provides a channel of communication between schooling fish. Based on kinematic measurements of the speed and heading of schooling tetras (Petitella rhodostoma), we found that compromising the lateral line by chemical treatment reduced the mutual information between individuals by ∼13%. This relatively small reduction in pairwise communication propagated through schools of varying size to reduce the degree and connectivity of the social network by more than half. Treated schools additionally showed more than twice the spatial heterogeneity of fish with unaltered flow sensing. These effects were much more substantial than the changes that we measured in the nearest-neighbor distance, speed, and intermittency of individual fish by compromising flow sensing. Therefore, flow serves as a valuable supplement to visual communication in a manner that is revealed through a school's network properties.

Gene selection and cancer classification using interaction-based feature clustering and improved-binary Bat algorithm.

In high-dimensional gene expression data, selecting an optimal subset of genes is crucial for achieving high classification accuracy and reliable diagnosis of diseases. This paper proposes a two-stage hybrid model for gene selection based on clustering and a swarm intelligence algorithm to identify the most informative genes with high accuracy. First, a clustering-based multivariate filter approach is performed to explore the interactions between the features and eliminate any redundant or irrelevant ones. Then, by controlling for the problem of premature convergence in the binary Bat algorithm, the optimal gene subset is determined using different classifiers with the Monte Carlo cross-validation data partitioning model. The effectiveness of our proposed framework is evaluated using eight gene expression datasets, by comparison with other recently published algorithms in the literature. Experiments confirm that in seven out of eight datasets, the proposed method can achieve superior results in terms of classification accuracy and gene subset size. In particular, it achieves a classification accuracy of 100% in Lymphoma and Ovarian datasets and above 97.4% in the rest with a minimum number of genes. The results demonstrate that our proposed algorithm has the potential to solve the feature selection problem in different applications with high-dimensional datasets.

Improved rank-based recursive feature elimination method based ovarian cancer detection model via customized deep architecture.

BACKGROUND: Ovarian cancer is often considered the most lethal gynecological cancer because it tends to be diagnosed at an advanced stage, leading to limited treatment options and poorer outcomes. Several factors contribute to the challenges in managing ovarian cancer, namely rapid metastasis, genetic factors, reproductive history, etc. This necessitates the prompt and precise diagnosis of ovarian cancer in order to carry out efficient treatment plans and give patients who are all impacted by OC the care and support they need. METHODS: This CCLSTM model is suggested under four essential stages including preprocessing, feature extraction, feature selection and detection. Initially, the input data is preprocessed using Improved Two-step Data Normalization. Subsequently, features such as statistical, modified entropy, raw features and mutual information are extracted from the normalized data. Next, obtained features undergo the Improved Rank-based Recursive Feature Elimination method (IR-RFE) to select the most suitable features. Finally, the proposed CCLSTM model takes the selected features as input and provides a final detection outcome. RESULTS: Furthermore, the performance of the proposed CCLSTM technique is examined through a thorough assessment using diverse analyses Additionally, the CCLSTM schemes show a sensitivity value of 0.948, whereas the sensitivity ratings for ALO-LSTM + ALOCNN, Bi-GRU, LSTM, RNN, KNN, CNN, and DCNN are 0.808, 0.893, 0.829, 0.851, 0.765, 0.872, and 0.893, respectively. CONCLUSION: In the end, the development of CNN and the addition of LSTM technique have produced an ovarian cancer detection technique that is more accurate and consistent compared to other existing strategies.

Phytochemical, Antimicrobial, and Antioxidant Activity of Different Extracts from Frozen, Freeze-Dried, and Oven-Dried Jostaberries Grown in Moldova.

In this paper, the qualitative and quantitative profile is evaluated of the bioactive compounds, antioxidant activity (AA), microbiostatic properties, as well as the color parameters of jostaberry extracts, obtained from frozen (FJ), freeze-dried (FDJ), and oven-dried berries (DJ). The optimal extraction conditions by ultrasound-assisted extraction (UAE) and microwave-assisted extraction (MAE) were selected after determination of the total polyphenol content (TPC), total flavonoid content (TFC), total antocyanin content (TA), AA by 2,2-diphenyl-1-picrylhydrazyl-hydrate (DPPH), and the free radical cation 2,2-azinobis-3-ethylbenzothiazoline-6-sulfonates (ABTS). Non-conventional extraction methods are less destructive to anthocyanins, while drying the berries reduced TA, regardless of the extraction method. The oven-drying process reduced the concentration of TA in DJ extracts by 99.4% and of ascorbic acid by 92.42% compared to FJ. AA was influenced by the jostaberry pretreatment methods. The DPPH and ABTS tests recorded values (mg Trolox equivalent/g dry weight) between 17.60 and 35.26 and 35.64 and 109.17 for FJ extracts, between 7.50 and 7.96 and 45.73 and 82.22 for FDJ, as well as between 6.31 and 7.40 and 34.04 and 52.20 for DJ, respectively. The jostaberry pretreatment produced significant changes in all color parameters. Mutual information analysis, applied to determine the influence of ultrasound and microwave durations on TPC, TFC, TA, AA, pH, and color parameters in jostaberry extracts, showed the greatest influence on TA (0.367 bits) and TFC (0.329 bits). The DPPH and ABTS inhibition capacity of all FJ' extracts had higher values and varied more strongly, depending on pH, heat treatment, and storage time, compared to the AA values of FDJ' and DJ' extracts. A significant antimicrobial effect was observed on all bacterial strains studied for FJP. FDJP was more active on Bacillus cereus, Staphylococcus aureus, and Escherichia coli. DJP was more active on Salmonella Abony and Pseudomonas aeruginosa. The antifungal effect of DJP was stronger compared to FDJP. Jostaberry extracts obtained under different conditions can be used in food production, offering a wide spectrum of red hues.

Similar teleconnection patterns of ENSO-NAO and ENSO-precipitation in Colombia: linear and non-linear relationships.

The Central-Pacific (CP) and Eastern-Pacific (EP) types of El Niño-Southern Oscillation (ENSO) and their ocean-atmosphere effect cause diverse responses in the hydroclimatological patterns of specific regions. Given the impact of ENSO diversity on the North Atlantic Oscillation (NAO), this study aimed to determine the relationship between the ENSO-NAO teleconnection and the ENSO-influenced precipitation patterns in Colombia during the December-February period. Precipitation data from 1981 to 2023, obtained from the Climate Hazards Group (CHIRPS), were analyzed using nine ENSO and NAO indices spanning from 1951 to 2023. Using Pearson's correlation and mutual information (MI) techniques, nine scenarios were devised, encompassing the CP and EP ENSO events, neutral years, and volcanic eruptions. The results suggest a shift in the direction of the ENSO-NAO relationship when distinguishing between the CP and EP events. Higher linear correlations were observed in the CP ENSO scenarios (r > 0.65) using the MEI and BEST indices, while lower correlations were observed when considering EP events along with the Niño 3 and Niño 1.2 indices. MI show difference in relationships based on the event type and the ENSO index used. Notably, an increase in the non-linear relationship was observed for the EP scenarios with respect to correlation. Both teleconnections followed a similar pattern, exhibiting a more substantial impact during CP ENSO events. This highlights the significance of investigating the impacts of ENSO on hydrometeorological variables in the context of adapting to climate change, while acknowledging the intricate diversity inherent to the ENSO phenomenon.

Information Load from Neuromediator Diffusion to Extrasynaptic Space: The Interplay between the Injection Frequency and Clearance.

In our study, we simulate the release of glutamate, a neurotransmitter, from the presynaptic cell by modeling the diffusion of glutamate into both synaptic and extrasynaptic space around the synapse. We have also incorporated a new factor into our model: convection. This factor represents the process by which the body clears glutamate from the synapse. Due to this process, the physiological mechanisms that typically prevent glutamate from spreading beyond the synapse are altered. This results in a different distribution of glutamate concentrations, with higher levels outside the synapse than inside it. The variety of biological effects that occur in response to this extrasynaptic glutamate highlights the importance of preventing neurotransmitters from spreading beyond the synapse. We aim to explain the physical reasons behind these biological effects, which are observed as excitotoxicity. Our results show that preventing the spread of glutamate outside the synapse increases the amount of information exchanged within the synapse and its surroundings for frequencies of glutamate release up to 30-50 Hz, followed by a decrease. Additionally, we find that the rate at which glutamate is cleared from the synapse is effective at relatively low levels (≤0.5 nm/µs in our calculation grid) and remains constant at higher levels.

Identification of Tumor Budding-Associated Genes in Breast Cancer through Transcriptomic Profiling and Network Diffusion Analysis.

Breast cancer has the highest diagnosis rate among all cancers. Tumor budding (TB) is recognized as a recent prognostic marker. Identifying genes specific to high-TB samples is crucial for hindering tumor progression and metastasis. In this study, we utilized an RNA sequencing technique, called TempO-Seq, to profile transcriptomic data from breast cancer samples, aiming to identify biomarkers for high-TB cases. Through differential expression analysis and mutual information, we identified seven genes (NOL4, STAR, C8G, NEIL1, SLC46A3, FRMD6, and SCARF2) that are potential biomarkers in breast cancer. To gain more relevant proteins, further investigation based on a protein-protein interaction network and the network diffusion technique revealed enrichment in the Hippo signaling and Wnt signaling pathways, promoting tumor initiation, invasion, and metastasis in several cancer types. In conclusion, these novel genes, recognized as overexpressed in high-TB samples, along with their associated pathways, offer promising therapeutic targets, thus advancing treatment and diagnosis for breast cancer.

Towards a rigorous analysis of mutual information in contrastive learning.

Contrastive learning has emerged as a cornerstone in unsupervised representation learning. Its primary paradigm involves an instance discrimination task utilizing InfoNCE loss where the loss has been proven to be a form of mutual information. Consequently, it has become a common practice to analyze contrastive learning using mutual information as a measure. Yet, this analysis approach presents difficulties due to the necessity of estimating mutual information for real-world applications. This creates a gap between the elegance of its mathematical foundation and the complexity of its estimation, thereby hampering the ability to derive solid and meaningful insights from mutual information analysis. In this study, we introduce three novel methods and a few related theorems, aimed at enhancing the rigor of mutual information analysis. Despite their simplicity, these methods can carry substantial utility. Leveraging these approaches, we reassess three instances of contrastive learning analysis, illustrating the capacity of the proposed methods to facilitate deeper comprehension or to rectify pre-existing misconceptions. The main results can be summarized as follows: (1) While small batch sizes influence the range of training loss, they do not inherently limit learned representation's information content or affect downstream performance adversely; (2) Mutual information, with careful selection of positive pairings and post-training estimation, proves to be a superior measure for evaluating practical networks; and (3) Distinguishing between task-relevant and irrelevant information presents challenges, yet irrelevant information sources do not necessarily compromise the generalization of downstream tasks.

Decoding MRI-informed brain age using mutual information.

OBJECTIVE: We aimed to develop a standardized method to investigate the relationship between estimated brain age and regional morphometric features, meeting the criteria for simplicity, generalization, and intuitive interpretability. METHODS: We utilized T1-weighted magnetic resonance imaging (MRI) data from the Cambridge Centre for Ageing and Neuroscience project (N = 609) and employed a support vector regression method to train a brain age model. The pre-trained brain age model was applied to the dataset of the brain development project (N = 547). Kraskov (KSG) estimator was used to compute the mutual information (MI) value between brain age and regional morphometric features, including gray matter volume (GMV), white matter volume (WMV), cerebrospinal fluid (CSF) volume, and cortical thickness (CT). RESULTS: Among four types of brain features, GMV had the highest MI value (8.71), peaking in the pre-central gyrus (0.69). CSF volume was ranked second (7.76), with the highest MI value in the cingulate (0.87). CT was ranked third (6.22), with the highest MI value in superior temporal gyrus (0.53). WMV had the lowest MI value (4.59), with the insula showing the highest MI value (0.53). For brain parenchyma, the volume of the superior frontal gyrus exhibited the highest MI value (0.80). CONCLUSION: This is the first demonstration that MI value between estimated brain age and morphometric features may serve as a benchmark for assessing the regional contributions to estimated brain age. Our findings highlighted that both GMV and CSF are the key features that determined the estimated brain age, which may add value to existing computational models of brain age. CRITICAL RELEVANCE STATEMENT: Mutual information (MI) analysis reveals gray matter volume (GMV) and cerebrospinal fluid (CSF) volume as pivotal in computing individuals' brain age. KEY POINTS: Mutual information (MI) interprets estimated brain age with morphometric features. Gray matter volume in the pre-central gyrus has the highest MI value for estimated brain age. Cerebrospinal fluid volume in the cingulate has the highest MI value. Regarding brain parenchymal volume, the superior frontal gyrus has the highest MI value. The value of mutual information underscores the key brain regions related to brain age.

Dynamic Cooperative Communications with Mutual Information Accumulation for Mobile Robots in Industrial Internet of Things.

Mobile robots play an important role in the industrial Internet of Things (IIoT); they need effective mutual communication between the cloud and themselves when they move in a factory. By using the sensor nodes existing in the IIoT environment as relays, mobile robots and the cloud can communicate through multiple hops. However, the mobility and delay sensitivity of mobile robots bring new challenges. In this paper, we propose a dynamic cooperative transmission algorithm with mutual information accumulation to cope with these two challenges. By using rateless coding, nodes can reduce the delay caused by retransmission under poor channel conditions. With the help of mutual information accumulation, nodes can accumulate information faster and reduce delay. We propose a two-step dynamic algorithm, which can obtain the current routing path with low time complexity. The simulation results show that our algorithm is better than the existing heuristic algorithm in terms of delay.

Differentiable self-supervised clustering with intrinsic interpretability.

Self-supervised clustering has garnered widespread attention due to its ability to discover latent clustering structures without the need for external labels. However, most existing approaches on self-supervised clustering lack of inherent interpretability in the data clustering process. In this paper, we propose a differentiable self-supervised clustering method with intrinsic interpretability (DSC2I), which provides an interpretable data clustering mechanism by reformulating clustering process based on differentiable programming. To be specific, we first design a differentiable mutual information measurement to explicitly train a neural network with analytical gradients, which avoids variational inference and learns a discriminative and compact representation. Then, an interpretable clustering mechanism based on differentiable programming is devised to transform fundamental clustering process (i.e., minimum intra-cluster distance, maximum inter-cluster distance) into neural networks and convert cluster centers to learnable neural parameters, which allows us to obtain a transparent and interpretable clustering layer. Finally, a unified optimization method is designed, in which the differentiable representation learning and interpretable clustering can be optimized simultaneously in a self-supervised manner. Extensive experiments demonstrate the effectiveness of the proposed DSC2I method compared with 16 clustering approaches.

A Partial Information Decomposition for Multivariate Gaussian Systems Based on Information Geometry.

There is much interest in the topic of partial information decomposition, both in developing new algorithms and in developing applications. An algorithm, based on standard results from information geometry, was recently proposed by Niu and Quinn (2019). They considered the case of three scalar random variables from an exponential family, including both discrete distributions and a trivariate Gaussian distribution. The purpose of this article is to extend their work to the general case of multivariate Gaussian systems having vector inputs and a vector output. By making use of standard results from information geometry, explicit expressions are derived for the components of the partial information decomposition for this system. These expressions depend on a real-valued parameter which is determined by performing a simple constrained convex optimisation. Furthermore, it is proved that the theoretical properties of non-negativity, self-redundancy, symmetry and monotonicity, which were proposed by Williams and Beer (2010), are valid for the decomposition Iig derived herein. Application of these results to real and simulated data show that the Iig algorithm does produce the results expected when clear expectations are available, although in some scenarios, it can overestimate the level of the synergy and shared information components of the decomposition, and correspondingly underestimate the levels of unique information. Comparisons of the Iig and Idep (Kay and Ince, 2018) methods show that they can both produce very similar results, but interesting differences are provided. The same may be said about comparisons between the Iig and Immi (Barrett, 2015) methods.

Non-Equilibrium Enhancement of Classical Information Transmission.

Information transmission plays a crucial role across various fields, including physics, engineering, biology, and society. The efficiency of this transmission is quantified by mutual information and its associated information capacity. While studies in closed systems have yielded significant progress, understanding the impact of non-equilibrium effects on open systems remains a challenge. These effects, characterized by the exchange of energy, information, and materials with the external environment, can influence both mutual information and information capacity. Here, we delve into this challenge by exploring non-equilibrium effects using the memoryless channel model, a cornerstone of information channel coding theories and methodology development. Our findings reveal that mutual information exhibits a convex relationship with non-equilibriumness, quantified by the non-equilibrium strength in transmission probabilities. Notably, channel information capacity is enhanced by non-equilibrium effects. Furthermore, we demonstrate that non-equilibrium thermodynamic cost, characterized by the entropy production rate, can actually improve both mutual information and information channel capacity, leading to a boost in overall information transmission efficiency. Our numerical results support our conclusions.

A Survey on Error Exponents in Distributed Hypothesis Testing: Connections with Information Theory, Interpretations, and Applications.

A central challenge in hypothesis testing (HT) lies in determining the optimal balance between Type I (false positive) and Type II (non-detection or false negative) error probabilities. Analyzing these errors' exponential rate of convergence, known as error exponents, provides crucial insights into system performance. Error exponents offer a lens through which we can understand how operational restrictions, such as resource constraints and impairments in communications, affect the accuracy of distributed inference in networked systems. This survey presents a comprehensive review of key results in HT, from the foundational Stein's Lemma to recent advancements in distributed HT, all unified through the framework of error exponents. We explore asymptotic and non-asymptotic results, highlighting their implications for designing robust and efficient networked systems, such as event detection through lossy wireless sensor monitoring networks, collective perception-based object detection in vehicular environments, and clock synchronization in distributed environments, among others. We show that understanding the role of error exponents provides a valuable tool for optimizing decision-making and improving the reliability of networked systems.

Inversion Method for Transformer Winding Hot Spot Temperature Based on Gated Recurrent Unit and Self-Attention and Temperature Lag.

The hot spot temperature of transformer windings is an important indicator for measuring insulation performance, and its accurate inversion is crucial to ensure the timely and accurate fault prediction of transformers. However, existing studies mostly directly input obtained experimental or operational data into networks to construct data-driven models, without considering the lag between temperatures, which may lead to the insufficient accuracy of the inversion model. In this paper, a method for inverting the hot spot temperature of transformer windings based on the SA-GRU model is proposed. Firstly, temperature rise experiments are designed to collect the temperatures of the entire side and top of the transformer tank, top oil temperature, ambient temperature, the cooling inlet and outlet temperatures, and winding hot spot temperature. Secondly, experimental data are integrated, considering the lag of the data, to obtain candidate input feature parameters. Then, a feature selection algorithm based on mutual information (MI) is used to analyze the correlation of the data and construct the optimal feature subset to ensure the maximum information gain. Finally, Self-Attention (SA) is applied to optimize the Gate Recurrent Unit (GRU) network, establishing the GRU-SA model to perceive the potential patterns between output feature parameters and input feature parameters, achieving the precise inversion of the hot spot temperature of the transformer windings. The experimental results show that considering the lag of the data can more accurately invert the hot spot temperature of the windings. The inversion method proposed in this paper can reduce redundant input features, lower the complexity of the model, accurately invert the changing trend of the hot spot temperature, and achieve higher inversion accuracy than other classical models, thereby obtaining better inversion results.

Time-scale invariant contingency yields one-shot reinforcement learning despite extremely long delays to reinforcement.

Reinforcement learning inspires much theorizing in neuroscience, cognitive science, machine learning, and AI. A central question concerns the conditions that produce the perception of a contingency between an action and reinforcement-the assignment-of-credit problem. Contemporary models of associative and reinforcement learning do not leverage the temporal metrics (measured intervals). Our information-theoretic approach formalizes contingency by time-scale invariant temporal mutual information. It predicts that learning may proceed rapidly even with extremely long action-reinforcer delays. We show that rats can learn an action after a single reinforcement, even with a 16-min delay between the action and reinforcement (15-fold longer than any delay previously shown to support such learning). By leveraging metric temporal information, our solution obviates the need for windows of associability, exponentially decaying eligibility traces, microstimuli, or distributions over Bayesian belief states. Its three equations have no free parameters; they predict one-shot learning without iterative simulation.