Quantifying Information Modification in Developing Neural Networks via Partial Information Decomposition

Partial information decomposition for continuous variables based on shared exclusions: Analytical formulation and estimation

Describing statistical dependencies is foundational to empirical scientific research. For uncovering intricate and possibly nonlinear dependencies between a single target variable and several source variables within a system, a principled and versatile framework can be found in the theory of partial information decomposition (PID). Nevertheless, the majority of existing PID measures are restricted to categorical variables, while many systems of interest in science are continuous. In this paper, we present a novel analytic formulation for continuous redundancy-a generalization of mutual information-drawing inspiration from the concept of shared exclusions in probability space as in the discrete PID definition of I_{∩}^{sx}. Furthermore, we introduce a nearest-neighbor-based estimator for continuous PID and showcase its effectiveness by applying it to a simulated energy management system provided by the Honda Research Institute Europe GmbH. This work bridges the gap between the measure-theoretically postulated existence proofs for a continuous I_{∩}^{sx} and its practical application to real-world scientific problems.

Sampling bias corrections for accurate neural measures of redundant, unique, and synergistic information

Shannon Information theory has long been a tool of choice to measure empirically how populations of neurons in the brain encode information about cognitive variables. Recently, Partial Information Decomposition (PID) has emerged as principled way to break down this information into components identifying not only the unique information carried by each neuron, but also whether relationships between neurons generate synergistic or redundant information. While it has been long recognized that Shannon information measures on neural activity suffer from a (mostly upward) limited sampling estimation bias, this issue has largely been ignored in the burgeoning field of PID analysis of neural activity. We used simulations to investigate the limited sampling bias of PID computed from discrete probabilities (suited to describe neural spiking activity). We found that PID suffers from a large bias that is uneven across components, with synergy by far the most biased. Using approximate analytical expansions, we found that the bias of synergy increases quadratically with the number of discrete responses of each neuron, whereas the bias of unique and redundant information increase only linearly or sub-linearly. Based on the understanding of the PID bias properties, we developed simple yet effective procedures that correct for the bias effectively, and that improve greatly the PID estimation with respect to current state-of-the-art procedures. We apply these PID bias correction procedures to datasets of 53117 pairs neurons in auditory cortex, posterior parietal cortex and hippocampus of mice performing cognitive tasks, deriving precise estimates and bounds of how synergy and redundancy vary across these brain regions.

Downstream network transformations dissociate neural activity from causal functional contributions

Neuroscientists rely on distributed spatio-temporal patterns of neural activity to understand how neural units contribute to cognitive functions and behavior. However, the extent to which neural activity reliably indicates a unit's causal contribution to the behavior is not well understood. To address this issue, we provide a systematic multi-site perturbation framework that captures time-varying causal contributions of elements to a collectively produced outcome. Applying our framework to intuitive toy examples and artificial neural networks revealed that recorded activity patterns of neural elements may not be generally informative of their causal contribution due to activity transformations within a network. Overall, our findings emphasize the limitations of inferring causal mechanisms from neural activities and offer a rigorous lesioning framework for elucidating causal neural contributions.

Information-theoretic analyses of neural data to minimize the effect of researchers' assumptions in predictive coding studies

Studies investigating neural information processing often implicitly ask both, which processing strategy out of several alternatives is used and how this strategy is implemented in neural dynamics. A prime example are studies on predictive coding. These often ask whether confirmed predictions about inputs or prediction errors between internal predictions and inputs are passed on in a hierarchical neural system-while at the same time looking for the neural correlates of coding for errors and predictions. If we do not know exactly what a neural system predicts at any given moment, this results in a circular analysis-as has been criticized correctly. To circumvent such circular analysis, we propose to express information processing strategies (such as predictive coding) by local information-theoretic quantities, such that they can be estimated directly from neural data. We demonstrate our approach by investigating two opposing accounts of predictive coding-like processing strategies, where we quantify the building blocks of predictive coding, namely predictability of inputs and transfer of information, by local active information storage and local transfer entropy. We define testable hypotheses on the relationship of both quantities, allowing us to identify which of the assumed strategies was used. We demonstrate our approach on spiking data collected from the retinogeniculate synapse of the cat (N = 16). Applying our local information dynamics framework, we are able to show that the synapse codes for predictable rather than surprising input. To support our findings, we estimate quantities applied in the partial information decomposition framework, which allow to differentiate whether the transferred information is primarily bottom-up sensory input or information transferred conditionally on the current state of the synapse. Supporting our local information-theoretic results, we find that the synapse preferentially transfers bottom-up information.

Neuronal connected burst cascades bridge macroscale adaptive signatures across arousal states

The human brain displays a rich repertoire of states that emerge from the microscopic interactions of cortical and subcortical neurons. Difficulties inherent within large-scale simultaneous neuronal recording limit our ability to link biophysical processes at the microscale to emergent macroscopic brain states. Here we introduce a microscale biophysical network model of layer-5 pyramidal neurons that display graded coarse-sampled dynamics matching those observed in macroscale electrophysiological recordings from macaques and humans. We invert our model to identify the neuronal spike and burst dynamics that differentiate unconscious, dreaming, and awake arousal states and provide insights into their functional signatures. We further show that neuromodulatory arousal can mediate different modes of neuronal dynamics around a low-dimensional energy landscape, which in turn changes the response of the model to external stimuli. Our results highlight the promise of multiscale modelling to bridge theories of consciousness across spatiotemporal scales.

Decomposing and Tracing Mutual Information by Quantifying Reachable Decision Regions

The idea of a partial information decomposition (PID) gained significant attention for attributing the components of mutual information from multiple variables about a target to being unique, redundant/shared or synergetic. Since the original measure for this analysis was criticized, several alternatives have been proposed but have failed to satisfy the desired axioms, an inclusion-exclusion principle or have resulted in negative partial information components. For constructing a measure, we interpret the achievable type I/II error pairs for predicting each state of a target variable (reachable decision regions) as notions of pointwise uncertainty. For this representation of uncertainty, we construct a distributive lattice with mutual information as consistent valuation and obtain an algebra for the constructed measure. The resulting definition satisfies the original axioms, an inclusion-exclusion principle and provides a non-negative decomposition for an arbitrary number of variables. We demonstrate practical applications of this approach by tracing the flow of information through Markov chains. This can be used to model and analyze the flow of information in communication networks or data processing systems.

Orders between Channels and Implications for Partial Information Decomposition

The partial information decomposition (PID) framework is concerned with decomposing the information that a set of random variables has with respect to a target variable into three types of components: redundant, synergistic, and unique. Classical information theory alone does not provide a unique way to decompose information in this manner, and additional assumptions have to be made. Recently, Kolchinsky proposed a new general axiomatic approach to obtain measures of redundant information based on choosing an order relation between information sources (equivalently, order between communication channels). In this paper, we exploit this approach to introduce three new measures of redundant information (and the resulting decompositions) based on well-known preorders between channels, contributing to the enrichment of the PID landscape. We relate the new decompositions to existing ones, study several of their properties, and provide examples illustrating their novelty. As a side result, we prove that any preorder that satisfies Kolchinsky's axioms yields a decomposition that meets the axioms originally introduced by Williams and Beer when they first proposed PID.

When Neural Activity Fails to Reveal Causal Contributions

Neuroscientists rely on distributed spatio-temporal patterns of neural activity to understand how neural units contribute to cognitive functions and behavior. However, the extent to which neural activity reliably indicates a unit's causal contribution to the behavior is not well understood. To address this issue, we provide a systematic multi-site perturbation framework that captures time-varying causal contributions of elements to a collectively produced outcome. Applying our framework to intuitive toy examples and artificial neuronal networks revealed that recorded activity patterns of neural elements may not be generally informative of their causal contribution due to activity transformations within a network. Overall, our findings emphasize the limitations of inferring causal mechanisms from neural activities and offer a rigorous lesioning framework for elucidating causal neural contributions.

Pooling probability distributions and partial information decomposition

Notwithstanding various attempts to construct a partial information decomposition (PID) for multiple variables by defining synergistic, redundant, and unique information, there is no consensus on how one ought to precisely define either of these quantities. One aim here is to illustrate how that ambiguity-or, more positively, freedom of choice-may arise. Using the basic idea that information equals the average reduction in uncertainty when going from an initial to a final probability distribution, synergistic information will likewise be defined as a difference between two entropies. One term is uncontroversial and characterizes "the whole" information that source variables carry jointly about a target variable T. The other term then is meant to characterize the information carried by the "sum of its parts." Here we interpret that concept as needing a suitable probability distribution aggregated ("pooled") from multiple marginal distributions (the parts). Ambiguity arises in the definition of the optimum way to pool two (or more) probability distributions. Independent of the exact definition of optimum pooling, the concept of pooling leads to a lattice that differs from the often-used redundancy-based lattice. One can associate not just a number (an average entropy) with each node of the lattice, but (pooled) probability distributions. As an example, one simple and reasonable approach to pooling is presented, which naturally gives rise to the overlap between different probability distributions as being a crucial quantity that characterizes both synergistic and unique information.

Decomposing past and future: Integrated information decomposition based on shared probability mass exclusions

A core feature of complex systems is that the interactions between elements in the present causally constrain their own futures, and the futures of other elements as the system evolves through time. To fully model all of these interactions (between elements, as well as ensembles of elements), it is possible to decompose the total information flowing from past to future into a set of non-overlapping temporal interactions that describe all the different modes by which information can be stored, transferred, or modified. To achieve this, I propose a novel information-theoretic measure of temporal dependency (I_τsx) based on the logic of local probability mass exclusions. This integrated information decomposition can reveal emergent and higher-order interactions within the dynamics of a system, as well as refining existing measures. To demonstrate the utility of this framework, I apply the decomposition to spontaneous spiking activity recorded from dissociated neural cultures of rat cerebral cortex to show how different modes of information processing are distributed over the system. Furthermore, being a localizable analysis, I_τsx can provide insight into the computational structure of single moments. I explore the time-resolved computational structure of neuronal avalanches and find that different types of information atoms have distinct profiles over the course of an avalanche, with the majority of non-trivial information dynamics happening before the first half of the cascade is completed. These analyses allow us to move beyond the historical focus on single measures of dependency such as information transfer or information integration, and explore a panoply of different relationships between elements (and groups of elements) in complex systems.

Connectivity Analysis in EEG Data: A Tutorial Review of the State of the Art and Emerging Trends

Understanding how different areas of the human brain communicate with each other is a crucial issue in neuroscience. The concepts of structural, functional and effective connectivity have been widely exploited to describe the human connectome, consisting of brain networks, their structural connections and functional interactions. Despite high-spatial-resolution imaging techniques such as functional magnetic resonance imaging (fMRI) being widely used to map this complex network of multiple interactions, electroencephalographic (EEG) recordings claim high temporal resolution and are thus perfectly suitable to describe either spatially distributed and temporally dynamic patterns of neural activation and connectivity. In this work, we provide a technical account and a categorization of the most-used data-driven approaches to assess brain-functional connectivity, intended as the study of the statistical dependencies between the recorded EEG signals. Different pairwise and multivariate, as well as directed and non-directed connectivity metrics are discussed with a pros-cons approach, in the time, frequency, and information-theoretic domains. The establishment of conceptual and mathematical relationships between metrics from these three frameworks, and the discussion of novel methodological approaches, will allow the reader to go deep into the problem of inferring functional connectivity in complex networks. Furthermore, emerging trends for the description of extended forms of connectivity (e.g., high-order interactions) are also discussed, along with graph-theory tools exploring the topological properties of the network of connections provided by the proposed metrics. Applications to EEG data are reviewed. In addition, the importance of source localization, and the impacts of signal acquisition and pre-processing techniques (e.g., filtering, source localization, and artifact rejection) on the connectivity estimates are recognized and discussed. By going through this review, the reader could delve deeply into the entire process of EEG pre-processing and analysis for the study of brain functional connectivity and learning, thereby exploiting novel methodologies and approaches to the problem of inferring connectivity within complex networks.

A Comparison of Partial Information Decompositions Using Data from Real and Simulated Layer 5b Pyramidal Cells

Partial information decomposition allows the joint mutual information between an output and a set of inputs to be divided into components that are synergistic or shared or unique to each input. We consider five different decompositions and compare their results using data from layer 5b pyramidal cells in two different studies. The first study was on the amplification of somatic action potential output by apical dendritic input and its regulation by dendritic inhibition. We find that two of the decompositions produce much larger estimates of synergy and shared information than the others, as well as large levels of unique misinformation. When within-neuron differences in the components are examined, the five methods produce more similar results for all but the shared information component, for which two methods produce a different statistical conclusion from the others. There are some differences in the expression of unique information asymmetry among the methods. It is significantly larger, on average, under dendritic inhibition. Three of the methods support a previous conclusion that apical amplification is reduced by dendritic inhibition. The second study used a detailed compartmental model to produce action potentials for many combinations of the numbers of basal and apical synaptic inputs. Decompositions of the entire data set produce similar differences to those in the first study. Two analyses of decompositions are conducted on subsets of the data. In the first, the decompositions reveal a bifurcation in unique information asymmetry. For three of the methods, this suggests that apical drive switches to basal drive as the strength of the basal input increases, while the other two show changing mixtures of information and misinformation. Decompositions produced using the second set of subsets show that all five decompositions provide support for properties of cooperative context-sensitivity—to varying extents.

Early lock-in of structured and specialised information flows during neural development

The brains of many organisms are capable of complicated distributed computation underpinned by a highly advanced information processing capacity. Although substantial progress has been made towards characterising the information flow component of this capacity in mature brains, there is a distinct lack of work characterising its emergence during neural development. This lack of progress has been largely driven by the lack of effective estimators of information processing operations for spiking data. Here, we leverage recent advances in this estimation task in order to quantify the changes in transfer entropy during development. We do so by studying the changes in the intrinsic dynamics of the spontaneous activity of developing dissociated neural cell cultures. We find that the quantity of information flowing across these networks undergoes a dramatic increase across development. Moreover, the spatial structure of these flows exhibits a tendency to lock-in at the point when they arise. We also characterise the flow of information during the crucial periods of population bursts. We find that, during these bursts, nodes tend to undertake specialised computational roles as either transmitters, mediators, or receivers of information, with these roles tending to align with their average spike ordering. Further, we find that these roles are regularly locked-in when the information flows are established. Finally, we compare these results to information flows in a model network developing according to a spike-timing-dependent plasticity learning rule. Similar temporal patterns in the development of information flows were observed in these networks, hinting at the broader generality of these phenomena.

Quantifying Reinforcement-Learning Agent’s Autonomy, Reliance on Memory and Internalisation of the Environment

Intuitively, the level of autonomy of an agent is related to the degree to which the agent’s goals and behaviour are decoupled from the immediate control by the environment. Here, we capitalise on a recent information-theoretic formulation of autonomy and introduce an algorithm for calculating autonomy in a limiting process of time step approaching infinity. We tackle the question of how the autonomy level of an agent changes during training. In particular, in this work, we use the partial information decomposition (PID) framework to monitor the levels of autonomy and environment internalisation of reinforcement-learning (RL) agents. We performed experiments on two environments: a grid world, in which the agent has to collect food, and a repeating-pattern environment, in which the agent has to learn to imitate a sequence of actions by memorising the sequence. PID also allows us to answer how much the agent relies on its internal memory (versus how much it relies on the observations) when transitioning to its next internal state. The experiments show that specific terms of PID strongly correlate with the obtained reward and with the agent’s behaviour against perturbations in the observations.

Partial information decomposition reveals that synergistic neural integration is greater downstream of recurrent information flow in organotypic cortical cultures

The directionality of network information flow dictates how networks process information. A central component of information processing in both biological and artificial neural networks is their ability to perform synergistic integration–a type of computation. We established previously that synergistic integration varies directly with the strength of feedforward information flow. However, the relationships between both recurrent and feedback information flow and synergistic integration remain unknown. To address this, we analyzed the spiking activity of hundreds of neurons in organotypic cultures of mouse cortex. We asked how empirically observed synergistic integration–determined from partial information decomposition–varied with local functional network structure that was categorized into motifs with varying recurrent and feedback information flow. We found that synergistic integration was elevated in motifs with greater recurrent information flow beyond that expected from the local feedforward information flow. Feedback information flow was interrelated with feedforward information flow and was associated with decreased synergistic integration. Our results indicate that synergistic integration is distinctly influenced by the directionality of local information flow.

Networks compute information. That is, they modify inputs to generate distinct outputs. These computations are an important component of network information processing. Knowing how the routing of information in a network influences computation is therefore crucial. Here we asked how a key form of computation—synergistic integration—is related to the direction of local information flow in networks of spiking cortical neurons. Specifically, we asked how information flow between input neurons (i.e., recurrent information flow) and information flow from output neurons to input neurons (i.e., feedback information flow) was related to the amount of synergistic integration performed by output neurons. We found that greater synergistic integration occurred where there was more recurrent information flow. And, lesser synergistic integration occurred where there was more feedback information flow relative to feedforward information flow. These results show that computation, in the form of synergistic integration, is distinctly influenced by the directionality of local information flow. Such work is valuable for predicting where and how network computation occurs and for designing networks with desired computational abilities.

Bits and pieces: understanding information decomposition from part-whole relationships and formal logic

Partial information decomposition (PID) seeks to decompose the multivariate mutual information that a set of source variables contains about a target variable into basic pieces, the so-called ‘atoms of information’. Each atom describes a distinct way in which the sources may contain information about the target. For instance, some information may be contained uniquely in a particular source, some information may be shared by multiple sources and some information may only become accessible synergistically if multiple sources are combined. In this paper, we show that the entire theory of PID can be derived, firstly, from considerations of part-whole relationships between information atoms and mutual information terms, and secondly, based on a hierarchy of logical constraints describing how a given information atom can be accessed. In this way, the idea of a PID is developed on the basis of two of the most elementary relationships in nature: the part-whole relationship and the relation of logical implication. This unifying perspective provides insights into pressing questions in the field such as the possibility of constructing a PID based on concepts other than redundant information in the general n-sources case. Additionally, it admits of a particularly accessible exposition of PID theory.

An Information-Theoretic Framework to Measure the Dynamic Interaction Between Neural Spike Trains

OBJECTIVE

While understanding the interaction patterns among simultaneous recordings of spike trains from multiple neuronal units is a key topic in neuroscience, existing methods either do not consider the inherent point-process nature of spike trains or are based on parametric assumptions. This work presents an information-theoretic framework for the model-free, continuous-time estimation of both undirected (symmetric) and directed (Granger-causal) interactions between spike trains.

METHODS

The framework computes the mutual information rate (MIR) and the transfer entropy rate (TER) for two point processes X and Y, showing that the MIR between X and Y can be decomposed as the sum of the TER along the directions X → Y and Y → X. We present theoretical expressions and introduce strategies to estimate efficiently the two measures through nearest neighbor statistics.

RESULTS

Using simulations of independent and coupled point processes, we show the accuracy of MIR and TER to assess interactions even for weakly coupled and short realizations, and demonstrate the superiority of continuous-time estimation over the standard discrete-time approach. We also apply the MIR and TER to real-world data, specifically, recordings from in-vitro preparations of spontaneously-growing cultures of cortical neurons. Using this dataset, we demonstrate the ability of MIR and TER to describe how the functional networks between recording units emerge over the course of the maturation of the neuronal cultures.

CONCLUSION AND SIGNIFICANCE

the proposed framework provides principled measures to assess undirected and directed spike train interactions with more efficiency and flexibility than previous discrete-time or parametric approaches, opening new perspectives for the analysis of point-process data in neuroscience and many other fields.

Introducing a differentiable measure of pointwise shared information

Partial information decomposition of the multivariate mutual information describes the distinct ways in which a set of source variables contains information about a target variable. The groundbreaking work of Williams and Beer has shown that this decomposition cannot be determined from classic information theory without making additional assumptions, and several candidate measures have been proposed, often drawing on principles from related fields such as decision theory. None of these measures is differentiable with respect to the underlying probability mass function. We here present a measure that satisfies this property, emerges solely from information-theoretic principles, and has the form of a local mutual information. We show how the measure can be understood from the perspective of exclusions of probability mass, a principle that is foundational to the original definition of mutual information by Fano. Since our measure is well defined for individual realizations of random variables it lends itself, for example, to local learning in artificial neural networks. We also show that it has a meaningful Möbius inversion on a redundancy lattice and obeys a target chain rule. We give an operational interpretation of the measure based on the decisions that an agent should take if given only the shared information.

Estimating Transfer Entropy in Continuous Time Between Neural Spike Trains or Other Event-Based Data

Transfer entropy (TE) is a widely used measure of directed information flows in a number of domains including neuroscience. Many real-world time series for which we are interested in information flows come in the form of (near) instantaneous events occurring over time. Examples include the spiking of biological neurons, trades on stock markets and posts to social media, amongst myriad other systems involving events in continuous time throughout the natural and social sciences. However, there exist severe limitations to the current approach to TE estimation on such event-based data via discretising the time series into time bins: it is not consistent, has high bias, converges slowly and cannot simultaneously capture relationships that occur with very fine time precision as well as those that occur over long time intervals. Building on recent work which derived a theoretical framework for TE in continuous time, we present an estimation framework for TE on event-based data and develop a k-nearest-neighbours estimator within this framework. This estimator is provably consistent, has favourable bias properties and converges orders of magnitude more quickly than the current state-of-the-art in discrete-time estimation on synthetic examples. We demonstrate failures of the traditionally-used source-time-shift method for null surrogate generation. In order to overcome these failures, we develop a local permutation scheme for generating surrogate time series conforming to the appropriate null hypothesis in order to test for the statistical significance of the TE and, as such, test for the conditional independence between the history of one point process and the updates of another. Our approach is shown to be capable of correctly rejecting or accepting the null hypothesis of conditional independence even in the presence of strong pairwise time-directed correlations. This capacity to accurately test for conditional independence is further demonstrated on models of a spiking neural circuit inspired by the pyloric circuit of the crustacean stomatogastric ganglion, succeeding where previous related estimators have failed.

Constructing Brain Connectivity Model Using Causal Network Reconstruction Approach

Studying brain function is a challenging task. In the past, we could only study brain anatomical structures post-mortem, or infer brain functions from clinical data of patients with a brain injury. Nowadays technology, such as functional magnetic resonance imaging (fMRI), enable non-invasive brain activity observation. Several approaches have been proposed to interpret brain activity data. The brain connectivity model is a graphical tool that represents the interaction between brain regions, during certain states. It depicts how a brain region cause changes to other parts of the brain, which can be implied as information flow. This model can be used to help interpret how the brain works. There are several mathematical frameworks that can be used to infer the connectivity model from brain activity signals. Granger causality is one such approach and is one of the first that has been applied to brain activity data. However, due to the concept of the framework, such as the use of pairwise correlation, combined with the limitation of brain activity data such as low temporal resolution in case of fMRI signal, makes the interpretation of the connectivity difficult. We therefore propose the application of the Tigramite causal discovery framework on fMRI data. The Tigramite framework uses measures such as causal effect to analyze causal relations in the system. This enables the framework to identify both direct and indirect pathways or connectivities. In this paper, we applied the framework to the Human Connectome Project motor task-fMRI dataset. We then present the results and discuss how the framework improves interpretability of the connectivity model. We hope that this framework will help us understand more complex brain functions such as memory, consciousness, or the resting-state of the brain, in the future.

Discovering Higher-Order Interactions Through Neural Information Decomposition

If regularity in data takes the form of higher-order functions among groups of variables, models which are biased towards lower-order functions may easily mistake the data for noise. To distinguish whether this is the case, one must be able to quantify the contribution of different orders of dependence to the total information. Recent work in information theory attempts to do this through measures of multivariate mutual information (MMI) and information decomposition (ID). Despite substantial theoretical progress, practical issues related to tractability and learnability of higher-order functions are still largely unaddressed. In this work, we introduce a new approach to information decomposition—termed Neural Information Decomposition (NID)—which is both theoretically grounded, and can be efficiently estimated in practice using neural networks. We show on synthetic data that NID can learn to distinguish higher-order functions from noise, while many unsupervised probability models cannot. Additionally, we demonstrate the usefulness of this framework as a tool for exploring biological and artificial neural networks.

Partial Information Decomposition and the Information Delta: A Geometric Unification Disentangling Non-Pairwise Information

Information theory provides robust measures of multivariable interdependence, but classically does little to characterize the multivariable relationships it detects. The Partial Information Decomposition (PID) characterizes the mutual information between variables by decomposing it into unique, redundant, and synergistic components. This has been usefully applied, particularly in neuroscience, but there is currently no generally accepted method for its computation. Independently, the Information Delta framework characterizes non-pairwise dependencies in genetic datasets. This framework has developed an intuitive geometric interpretation for how discrete functions encode information, but lacks some important generalizations. This paper shows that the PID and Delta frameworks are largely equivalent. We equate their key expressions, allowing for results in one framework to apply towards open questions in the other. For example, we find that the approach of Bertschinger et al. is useful for the open Information Delta question of how to deal with linkage disequilibrium. We also show how PID solutions can be mapped onto the space of delta measures. Using Bertschinger et al. as an example solution, we identify a specific plane in delta-space on which this approach's optimization is constrained, and compute it for all possible three-variable discrete functions of a three-letter alphabet. This yields a clear geometric picture of how a given solution decomposes information.

Correlated activity favors synergistic processing in local cortical networks in vitro at synaptically relevant timescales

Neural information processing is widely understood to depend on correlations in neuronal activity. However, whether correlation is favorable or not is contentious. Here, we sought to determine how correlated activity and information processing are related in cortical circuits. Using recordings of hundreds of spiking neurons in organotypic cultures of mouse neocortex, we asked whether mutual information between neurons that feed into a common third neuron increased synergistic information processing by the receiving neuron. We found that mutual information and synergistic processing were positively related at synaptic timescales (0.05-14 ms), where mutual information values were low. This effect was mediated by the increase in information transmission-of which synergistic processing is a component-that resulted as mutual information grew. However, at extrasynaptic windows (up to 3,000 ms), where mutual information values were high, the relationship between mutual information and synergistic processing became negative. In this regime, greater mutual information resulted in a disproportionate increase in redundancy relative to information transmission. These results indicate that the emergence of synergistic processing from correlated activity differs according to timescale and correlation regime. In a low-correlation regime, synergistic processing increases with greater correlation, and in a high-correlation regime, synergistic processing decreases with greater correlation.

Dynamic representations in networked neural systems

A group of neurons can generate patterns of activity that represent information about stimuli; subsequently, the group can transform and transmit activity patterns across synapses to spatially distributed areas. Recent studies in neuroscience have begun to independently address the two components of information processing: the representation of stimuli in neural activity and the transmission of information in networks that model neural interactions. Yet only recently are studies seeking to link these two types of approaches. Here we briefly review the two separate bodies of literature; we then review the recent strides made to address this gap. We continue with a discussion of how patterns of activity evolve from one representation to another, forming dynamic representations that unfold on the underlying network. Our goal is to offer a holistic framework for understanding and describing neural information representation and transmission while revealing exciting frontiers for future research.

Control of criticality and computation in spiking neuromorphic networks with plasticity

The critical state is assumed to be optimal for any computation in recurrent neural networks, because criticality maximizes a number of abstract computational properties. We challenge this assumption by evaluating the performance of a spiking recurrent neural network on a set of tasks of varying complexity at - and away from critical network dynamics. To that end, we developed a plastic spiking network on a neuromorphic chip. We show that the distance to criticality can be easily adapted by changing the input strength, and then demonstrate a clear relation between criticality, task-performance and information-theoretic fingerprint. Whereas the information-theoretic measures all show that network capacity is maximal at criticality, only the complex tasks profit from criticality, whereas simple tasks suffer. Thereby, we challenge the general assumption that criticality would be beneficial for any task, and provide instead an understanding of how the collective network state should be tuned to task requirement.

Contextual Modulation in Mammalian Neocortex is Asymmetric

Neural systems are composed of many local processors that generate an output given their many inputs as specified by a transfer function. This paper studies a transfer function that is fundamentally asymmetric and builds on multi-site intracellular recordings indicating that some neocortical pyramidal cells can function as context-sensitive two-point processors in which some inputs modulate the strength with which they transmit information about other inputs. Learning and processing at the level of the local processor can then be guided by the context of activity in the system as a whole without corrupting the message that the local processor transmits. We use a recent advance in the foundations of information theory to compare the properties of this modulatory transfer function with that of the simple arithmetic operators. This advance enables the information transmitted by processors with two distinct inputs to be decomposed into those components unique to each input, that shared between the two inputs, and that which depends on both though it is in neither, i.e., synergy. We show that contextual modulation is fundamentally asymmetric, contrasts with all four simple arithmetic operators, can take various forms, and can occur together with the anatomical asymmetry that defines pyramidal neurons in mammalian neocortex.

Generalised Measures of Multivariate Information Content

The entropy of a pair of random variables is commonly depicted using a Venn diagram. This representation is potentially misleading, however, since the multivariate mutual information can be negative. This paper presents new measures of multivariate information content that can be accurately depicted using Venn diagrams for any number of random variables. These measures complement the existing measures of multivariate mutual information and are constructed by considering the algebraic structure of information sharing. It is shown that the distinct ways in which a set of marginal observers can share their information with a non-observing third party corresponds to the elements of a free distributive lattice. The redundancy lattice from partial information decomposition is then subsequently and independently derived by combining the algebraic structures of joint and shared information content.

Topological Information Data Analysis

This paper presents methods that quantify the structure of statistical interactions within a given data set, and were applied in a previous article. It establishes new results on the k-multivariate mutual-information (Ik) inspired by the topological formulation of Information introduced in a serie of studies. In particular, we show that the vanishing of all Ik for 2≤k≤n of n random variables is equivalent to their statistical independence. Pursuing the work of Hu Kuo Ting and Te Sun Han, we show that information functions provide co-ordinates for binary variables, and that they are analytically independent from the probability simplex for any set of finite variables. The maximal positive Ik identifies the variables that co-vary the most in the population, whereas the minimal negative Ik identifies synergistic clusters and the variables that differentiate–segregate the most in the population. Finite data size effects and estimation biases severely constrain the effective computation of the information topology on data, and we provide simple statistical tests for the undersampling bias and the k-dependences. We give an example of application of these methods to genetic expression and unsupervised cell-type classification. The methods unravel biologically relevant subtypes, with a sample size of 41 genes and with few errors. It establishes generic basic methods to quantify the epigenetic information storage and a unified epigenetic unsupervised learning formalism. We propose that higher-order statistical interactions and non-identically distributed variables are constitutive characteristics of biological systems that should be estimated in order to unravel their significant statistical structure and diversity. The topological information data analysis presented here allows for precisely estimating this higher-order structure characteristic of biological systems.

MAXENT3D_PID: An Estimator for the Maximum-Entropy Trivariate Partial Information Decomposition

Partial information decomposition (PID) separates the contributions of sources about a target into unique, redundant, and synergistic components of information. In essence, PID answers the question of “who knows what” of a system of random variables and hence has applications to a wide spectrum of fields ranging from social to biological sciences. The paper presents MaxEnt3D_Pid, an algorithm that computes the PID of three sources, based on a recently-proposed maximum entropy measure, using convex optimization (cone programming). We describe the algorithm and its associated software utilization and report the results of various experiments assessing its accuracy. Moreover, the paper shows that a hierarchy of bivariate and trivariate PID allows obtaining the finer quantities of the trivariate partial information measure.

Quantifying high-order interdependencies via multivariate extensions of the mutual information

This paper introduces a model-agnostic approach to study statistical synergy, a form of emergence in which patterns at large scales are not traceable from lower scales. Our framework leverages various multivariate extensions of Shannon's mutual information, and introduces the O-information as a metric that is capable of characterizing synergy- and redundancy-dominated systems. The O-information is a symmetric quantity, and can assess intrinsic properties of a system without dividing its parts into "predictors" and "targets." We develop key analytical properties of the O-information, and study how it relates to other metrics of high-order interactions from the statistical mechanics and neuroscience literature. Finally, as a proof of concept, we present an exploration on the relevance of statistical synergy in Baroque music scores.

Multivariate information processing characterizes fitness of a cascaded gene-transcription machinery

We report that a genetic two-step activation cascade processes diverse flavors of information, e.g., synergy, redundancy, and unique information. Our computations measuring reduction in Shannon entropies and reduction in variances produce differently behaving absolute magnitudes of these informational flavors. We find that similarity can be brought in if these terms are evaluated in fractions with respect to corresponding total information. Each of the input signal and final gene-product is found to generate common or redundant information fractions (mostly) to predict each other, whereas they also complement one another to harness synergistic information fraction, predicting the intermediate biochemical species. For an optimally growing signal to maintain fixed steady-state abundance of activated downstream gene-products, the interaction information fractions for this cascade module shift from net-redundancy to information-independence.

Electrophysiological Brain Connectivity: Theory and Implementation

We review the theory and algorithms of electrophysiological brain connectivity analysis. This tutorial is aimed at providing an introduction to brain functional connectivity from electrophysiological signals, including electroencephalography (EEG), magnetoencephalography (MEG), electrocorticography (ECoG), stereoelectroencephalography (SEEG). Various connectivity estimators are discussed, and algorithms introduced. Important issues for estimating and mapping brain functional connectivity with electrophysiology are discussed.

Information Decomposition of Target Effects from Multi-Source Interactions: Perspectives on Previous, Current and Future Work

The formulation of the Partial Information Decomposition (PID) framework by Williams and Beer in 2010 attracted a significant amount of attention to the problem of defining redundant (or shared), unique and synergistic (or complementary) components of mutual information that a set of source variables provides about a target. This attention resulted in a number of measures proposed to capture these concepts, theoretical investigations into such measures, and applications to empirical data (in particular to datasets from neuroscience). In this Special Issue on “Information Decomposition of Target Effects from Multi-Source Interactions” at Entropy, we have gathered current work on such information decomposition approaches from many of the leading research groups in the field. We begin our editorial by providing the reader with a review of previous information decomposition research, including an overview of the variety of measures proposed, how they have been interpreted and applied to empirical investigations. We then introduce the articles included in the special issue one by one, providing a similar categorisation of these articles into: i. proposals of new measures; ii. theoretical investigations into properties and interpretations of such approaches, and iii. applications of these measures in empirical studies. We finish by providing an outlook on the future of the field.

Pointwise Partial Information Decomposition Using the Specificity and Ambiguity Lattices

What are the distinct ways in which a set of predictor variables can provide information about a target variable? When does a variable provide unique information, when do variables share redundant information, and when do variables combine synergistically to provide complementary information? The redundancy lattice from the partial information decomposition of Williams and Beer provided a promising glimpse at the answer to these questions. However, this structure was constructed using a much criticised measure of redundant information, and despite sustained research, no completely satisfactory replacement measure has been proposed. In this paper, we take a different approach, applying the axiomatic derivation of the redundancy lattice to a single realisation from a set of discrete variables. To overcome the difficulty associated with signed pointwise mutual information, we apply this decomposition separately to the unsigned entropic components of pointwise mutual information which we refer to as the specificity and ambiguity. This yields a separate redundancy lattice for each component. Then based upon an operational interpretation of redundancy, we define measures of redundant specificity and ambiguity enabling us to evaluate the partial information atoms in each lattice. These atoms can be recombined to yield the sought-after multivariate information decomposition. We apply this framework to canonical examples from the literature and discuss the results and the various properties of the decomposition. In particular, the pointwise decomposition using specificity and ambiguity satisfies a chain rule over target variables, which provides new insights into the so-called two-bit-copy example.