Estimating the directed information to infer causal relationships in ensemble neural spike train recordings

An information-theoretic framework for conditional causality analysis of brain networks

Identifying directed network models for multivariate time series is a ubiquitous problem in data science. Granger causality measure (GCM) and conditional GCM (cGCM) are widely used methods for identifying directed connections between time series. Both GCM and cGCM have frequency-domain formulations to characterize the dependence of time series in the spectral domain. However, the original methods were developed using a heuristic approach without rigorous theoretical explanations. To overcome the limitation, the minimum-entropy (ME) estimation approach was introduced in our previous work (Ning & Rathi, 2018) to generalize GCM and cGCM with more rigorous frequency-domain formulations. In this work, this information-theoretic framework is further generalized with three formulations for conditional causality analysis using techniques in control theory, such as state-space representations and spectral factorizations. The three conditional causal measures are developed based on different ME estimation procedures that are motivated by equivalent formulations of the classical minimum mean squared error estimation method. The relationship between the three formulations of conditional causality measures is analyzed theoretically. Their performance is evaluated using simulations and real neuroimaging data to analyze brain networks. The results show that the proposed methods provide more accurate network structures than the original approach.

Inference of monosynaptic connections from parallel spike trains: A review

This article presents a mini-review about the progress in inferring monosynaptic connections from spike trains of multiple neurons over the past twenty years. First, we explain a variety of meanings of "neuronal connectivity" in different research areas of neuroscience, such as structural connectivity, monosynaptic connectivity, and functional connectivity. Among these, we focus on the methods used to infer the monosynaptic connectivity from spike data. We then summarize the inference methods based on two main approaches, i.e., correlation-based and model-based approaches. Finally, we describe available source codes for connectivity inference and future challenges. Although inference will never be perfect, the accuracy of identifying the monosynaptic connections has improved dramatically in recent years due to continuous efforts.

Impaired Experience-Dependent Theta Oscillation Synchronization and Inter-Areal Synaptic Connectivity in the Visual Cortex of Fmr1 KO Mice

Fragile X syndrome (FX) is the most prevalent inheritable form of autism spectrum disorder (ASD), characterized by hypersensitivity, difficulty in habituating to new sensory stimuli, and intellectual disability. Individuals with FX often experience visual perception and learning deficits. Visual experience leads to the emergence of the familiarity-evoked theta band oscillations in the primary visual cortex (V1) and the lateromedial area (LM) of mice. These theta oscillations in V1 and LM are synchronized with each other, providing a mechanism of sensory multi-areal binding. However, how this multi-areal binding and the corresponding theta oscillations are altered in FX is not known. Using iDISCO whole brain clearing with light-sheet microscopy, we quantified immediate early gene Fos expression in V1 and LM, identifying deficits in experience-dependent neural activity in FX mice. We performed simultaneous in vivo recordings with silicon probes in V1 and LM of awake mice and channelrhodopsin-2-assisted circuit mapping (CRACM) in acute brain slices to examine the neural activity and strength of long-range synaptic connections between V1 and LM in both wildtype (WT) and Fmr1 knockout (KO) mice, the model of FX, before and after visual experience. Our findings reveal synchronized familiarity-evoked theta oscillations in V1 and LM, the increased strength of V1→LM functional and synaptic connections, which correlated with the corresponding changes of presynaptic short-term plasticity in WT mice. The LM oscillations were attenuated in FX mice and correlated with impaired functional and synaptic connectivity and short-term plasticity in the feedforward (FF) V1→LM and feedback (FB) LM→V1 pathways. Finally, using 4Pi single-molecule localization microscopy (SMLM) in thick brain tissue, we identified experience-dependent changes in the density and shape of dendritic spines in layer 5 pyramidal cells of WT mice, which correlated with the functional synaptic measurements. Interestingly, there was an increased dendritic spine density and length in naïve FX mice that failed to respond to experience. Our study provides the first comprehensive characterization of the role of visual experience in triggering inter-areal neural synchrony and shaping synaptic connectivity in WT and FX mice.

Statistically inferred neuronal connections in subsampled neural networks strongly correlate with spike train covariances

Statistically inferred neuronal connections from observed spike train data are often skewed from ground truth by factors such as model mismatch, unobserved neurons, and limited data. Spike train covariances, sometimes referred to as "functional connections," are often used as a proxy for the connections between pairs of neurons, but reflect statistical relationships between neurons, not anatomical connections. Moreover, covariances are not causal: spiking activity is correlated in both the past and the future, whereas neurons respond only to synaptic inputs in the past. Connections inferred by maximum likelihood inference, however, can be constrained to be causal. However, we show in this work that the inferred connections in spontaneously active networks modeled by stochastic leaky integrate-and-fire networks strongly correlate with the covariances between neurons, and may reflect noncausal relationships, when many neurons are unobserved or when neurons are weakly coupled. This phenomenon occurs across different network structures, including random networks and balanced excitatory-inhibitory networks. We use a combination of simulations and a mean-field analysis with fluctuation corrections to elucidate the relationships between spike train covariances, inferred synaptic filters, and ground-truth connections in partially observed networks.

Visual experience induces 4-8 Hz synchrony between V1 and higher-order visual areas

Visual perceptual experience induces persistent 4-8 Hz oscillations in the mouse primary visual cortex (V1), encoding visual familiarity. Recent studies suggest that higher-order visual areas (HVAs) are functionally specialized and segregated into information streams processing distinct visual features. However, whether visual memories are processed and stored within the distinct streams is not understood. We report here that V1 and lateromedial (LM), but not V1 and anterolateral, become more phase synchronized in 4-8 Hz after the entrainment of visual stimulus that maximally induces responses in LM. Directed information analysis reveals changes in the top-down functional connectivity between V1 and HVAs. Optogenetic inactivation of LM reduces post-stimulus oscillation peaks in V1 and impairs visual discrimination behavior. Our results demonstrate that 4-8 Hz familiarity-evoked oscillations are specific for the distinct visual features and are present in the corresponding HVAs, where they may be used for the inter-areal communication with V1 during memory-related behaviors.

Mutual generation in neuronal activity across the brain via deep neural approach, and its network interpretation

In the brain, many regions work in a network-like association, yet it is not known how durable these associations are in terms of activity and could survive without structural connections. To assess the association or similarity between brain regions with a generating approach, this study evaluated the similarity of activities of neurons within each region after disconnecting between regions. The "generation" approach here refers to using a multi-layer LSTM (Long Short-Term Memory) model to learn the rules of activity generation in one region and then apply that knowledge to generate activity in other regions. Surprisingly, the results revealed that activity generation from one region to disconnected regions was possible with similar accuracy to generation between the same regions in many cases. Notably, firing rates and synchronization of firing between neuron pairs, often used as neuronal representations, could be reproduced with precision. Additionally, accuracies were associated with the relative angle between brain regions and the strength of the structural connections that initially connected them. This outcome enables us to look into trends governing non-uniformity of the cortex based on the potential to generate informative data and reduces the need for animal experiments.

Identification of interacting neural populations: methods and statistical considerations

As improved recording technologies have created new opportunities for neurophysiological investigation, emphasis has shifted from individual neurons to multiple populations that form circuits, and it has become important to provide evidence of cross-population coordinated activity. We review various methods for doing so, placing them in six major categories while avoiding technical descriptions and instead focusing on high-level motivations and concerns. Our aim is to indicate what the methods can achieve and the circumstances under which they are likely to succeed. Toward this end, we include a discussion of four cross-cutting issues: the definition of neural populations, trial-to-trial variability and Poisson-like noise, time-varying dynamics, and causality.

Markov field models: Scaling molecular kinetics approaches to large molecular machines

With recent advances in structural biology, including experimental techniques and deep learning-enabled high-precision structure predictions, molecular dynamics methods that scale up to large biomolecular systems are required. Current state-of-the-art approaches in molecular dynamics modeling focus on encoding global configurations of molecular systems as distinct states. This paradigm commands us to map out all possible structures and sample transitions between them, a task that becomes impossible for large-scale systems such as biomolecular complexes. To arrive at scalable molecular models, we suggest moving away from global state descriptions to a set of coupled models that each describe the dynamics of local domains or sites of the molecular system. We describe limitations in the current state-of-the-art global-state Markovian modeling approaches and then introduce Markov field models as an umbrella term that includes models from various scientific communities, including Independent Markov decomposition, Ising and Potts models, and (dynamic) graphical models, and evaluate their use for computational molecular biology. Finally, we give a few examples of early adoptions of these ideas for modeling molecular kinetics and thermodynamics.

Disentangling the flow of signals between populations of neurons

Technological advances now allow us to record from large populations of neurons across multiple brain areas. These recordings may illuminate how communication between areas contributes to brain function, yet a substantial barrier remains: how do we disentangle the concurrent, bidirectional flow of signals between populations of neurons? We propose here a dimensionality reduction framework, delayed latents across groups (DLAG), that disentangles signals relayed in each direction, identifies how these signals are represented by each population and characterizes how they evolve within and across trials. We demonstrate that DLAG performs well on synthetic datasets similar in scale to current neurophysiological recordings. Then we study simultaneously recorded populations in primate visual areas V1 and V2, where DLAG reveals signatures of bidirectional yet selective communication. Our framework lays a foundation for dissecting the intricate flow of signals across populations of neurons, and how this signalling contributes to cortical computation.

Regional identification of information flow termination of electrohysterographic signals: Towards understanding human uterine electrical propagation

BACKGROUND AND OBJECTIVE

The uterine electrohysterogram (EHG) contains important information about electrical signal propagation which may be useful to monitor and predict the progress of pregnancy towards parturition. Directed information processing has the potential to be of use in studying EHG recordings. However, so far, there is no directed information-based estimation scheme that has been applied to investigating the propagation of human EHG recordings. To realize this, the approach of directed information and its reliability and adaptability should be scientifically studied.

METHODS

We demonstrated an estimation scheme of directed information to identify the spatiotemporal relationship between the recording channels of EHG signal and assess the algorithm reliability initially using simulated data. Further, a regional identification of information flow termination (RIIFT) approach was developed and applied for the first time to extant multichannel EHG signals to reveal the terminal zone of propagation of the electrical activity associated with uterine contraction. RIIFT operates by estimating the pairwise directed information between neighboring EHG channels and identifying the location where there is the strongest inward flow of information. The method was then applied to publicly-available experimental data obtained from pregnant women with the use of electrodes arranged in a 4-by-4 grid.

RESULTS

Our results are consistent with the suggestions from the previous studies with the added identification of preferential sites of excitation termination - within the estimated area, the direction of surface action potential propagation towards the medial axis of uterus during contraction was discovered for 72.15% of the total cases, demonstrating that our RIIFT method is a potential tool to investigate EHG propagation for advancing our understanding human uterine excitability.

CONCLUSIONS

We developed a new approach and applied it to multichannel human EHG recordings to investigate the electrical signal propagation involved in uterine contraction. This provides an important platform for future studies to fill knowledge gaps in the spatiotemporal patterns of electrical excitation of the human uterus.

Modeling multiscale causal interactions between spiking and field potential signals during behavior

Objective.Brain recordings exhibit dynamics at multiple spatiotemporal scales, which are measured with spike trains and larger-scale field potential signals. To study neural processes, it is important to identify and model causal interactions not only at a single scale of activity, but also across multiple scales, i.e. between spike trains and field potential signals. Standard causality measures are not directly applicable here because spike trains are binary-valued but field potentials are continuous-valued. It is thus important to develop computational tools to recover multiscale neural causality during behavior, assess their performance on neural datasets, and study whether modeling multiscale causalities can improve the prediction of neural signals beyond what is possible with single-scale causality.Approach.We design a multiscale model-based Granger-like causality method based on directed information and evaluate its success both in realistic biophysical spike-field simulations and in motor cortical datasets from two non-human primates (NHP) performing a motor behavior. To compute multiscale causality, we learn point-process generalized linear models that predict the spike events at a given time based on the history of both spike trains and field potential signals. We also learn linear Gaussian models that predict the field potential signals at a given time based on their own history as well as either the history of binary spike events or that of latent firing rates.Main results.We find that our method reveals the true multiscale causality network structure in biophysical simulations despite the presence of model mismatch. Further, models with the identified multiscale causalities in the NHP neural datasets lead to better prediction of both spike trains and field potential signals compared to just modeling single-scale causalities. Finally, we find that latent firing rates are better predictors of field potential signals compared with the binary spike events in the NHP datasets.Significance.This multiscale causality method can reveal the directed functional interactions across spatiotemporal scales of brain activity to inform basic science investigations and neurotechnologies.

Degeneracy measures in biologically plausible random Boolean networks

Background

Degeneracy—the ability of structurally different elements to perform similar functions—is a property of many biological systems. Highly degenerate systems show resilience to perturbations and damage because the system can compensate for compromised function due to reconfiguration of the underlying network dynamics. Degeneracy thus suggests how biological systems can thrive despite changes to internal and external demands. Although degeneracy is a feature of network topologies and seems to be implicated in a wide variety of biological processes, research on degeneracy in biological networks is mostly limited to weighted networks. In this study, we test an information theoretic definition of degeneracy on random Boolean networks, frequently used to model gene regulatory networks. Random Boolean networks are discrete dynamical systems with binary connectivity and thus, these networks are well-suited for tracing information flow and the causal effects. By generating networks with random binary wiring diagrams, we test the effects of systematic lesioning of connections and perturbations of the network nodes on the degeneracy measure.

Results

Our analysis shows that degeneracy, on average, is the highest in networks in which ~ 20% of the connections are lesioned while 50% of the nodes are perturbed. Moreover, our results for the networks with no lesions and the fully-lesioned networks are comparable to the degeneracy measures from weighted networks, thus we show that the degeneracy measure is applicable to different networks.

Conclusions

Such a generalized applicability implies that degeneracy measures may be a useful tool for investigating a wide range of biological networks and, therefore, can be used to make predictions about the variety of systems’ ability to recover function.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12859-022-04601-5.

Inferring network structure with unobservable nodes from time series data

Network structures play important roles in social, technological, and biological systems. However, the observable nodes and connections in real cases are often incomplete or unavailable due to measurement errors, private protection issues, or other problems. Therefore, inferring the complete network structure is useful for understanding human interactions and complex dynamics. The existing studies have not fully solved the problem of the inferring network structure with partial information about connections or nodes. In this paper, we tackle the problem by utilizing time series data generated by network dynamics. We regard the network inference problem based on dynamical time series data as a problem of minimizing errors for predicting states of observable nodes and proposed a novel data-driven deep learning model called Gumbel-softmax Inference for Network (GIN) to solve the problem under incomplete information. The GIN framework includes three modules: a dynamics learner, a network generator, and an initial state generator to infer the unobservable parts of the network. We implement experiments on artificial and empirical social networks with discrete and continuous dynamics. The experiments show that our method can infer the unknown parts of the structure and the initial states of the observable nodes with up to 90% accuracy. The accuracy declines linearly with the increase of the fractions of unobservable nodes. Our framework may have wide applications where the network structure is hard to obtain and the time series data is rich.

Macrophage calcium reporter mice reveal immune cell communication in vitro and in vivo

Cell communication underlies emergent functions in diverse cell types and tissues. Recent evidence suggests that macrophages are organized in communicating networks, but new tools are needed to quantitatively characterize the resulting cellular conversations. Here, we infer cell communication from spatiotemporal correlations of intracellular calcium dynamics that are non-destructively imaged across cell populations expressing genetically encoded calcium indicators. We describe a hematopoietic calcium reporter mouse (Csf1rCreGCaMP5fl) and a computational analysis pipeline for inferring communication between reporter cells based on "excess synchrony." We observed signals suggestive of cell communication in macrophages treated with immune-stimulatory DNA in vitro and tumor-associated immune cells imaged in a dorsal window chamber model in vivo. Together, the methods described here expand the toolkit for discovery of cell communication events in macrophages and other immune cells.

Embedding optimization reveals long-lasting history dependence in neural spiking activity

Information processing can leave distinct footprints on the statistics of neural spiking. For example, efficient coding minimizes the statistical dependencies on the spiking history, while temporal integration of information may require the maintenance of information over different timescales. To investigate these footprints, we developed a novel approach to quantify history dependence within the spiking of a single neuron, using the mutual information between the entire past and current spiking. This measure captures how much past information is necessary to predict current spiking. In contrast, classical time-lagged measures of temporal dependence like the autocorrelation capture how long-potentially redundant-past information can still be read out. Strikingly, we find for model neurons that our method disentangles the strength and timescale of history dependence, whereas the two are mixed in classical approaches. When applying the method to experimental data, which are necessarily of limited size, a reliable estimation of mutual information is only possible for a coarse temporal binning of past spiking, a so-called past embedding. To still account for the vastly different spiking statistics and potentially long history dependence of living neurons, we developed an embedding-optimization approach that does not only vary the number and size, but also an exponential stretching of past bins. For extra-cellular spike recordings, we found that the strength and timescale of history dependence indeed can vary independently across experimental preparations. While hippocampus indicated strong and long history dependence, in visual cortex it was weak and short, while in vitro the history dependence was strong but short. This work enables an information-theoretic characterization of history dependence in recorded spike trains, which captures a footprint of information processing that is beyond time-lagged measures of temporal dependence. To facilitate the application of the method, we provide practical guidelines and a toolbox.

Neural Estimator of Information for Time-Series Data with Dependency

Novel approaches to estimate information measures using neural networks are well-celebrated in recent years both in the information theory and machine learning communities. These neural-based estimators are shown to converge to the true values when estimating mutual information and conditional mutual information using independent samples. However, if the samples in the dataset are not independent, the consistency of these estimators requires further investigation. This is of particular interest for a more complex measure such as the directed information, which is pivotal in characterizing causality and is meaningful over time-dependent variables. The extension of the convergence proof for such cases is not trivial and demands further assumptions on the data. In this paper, we show that our neural estimator for conditional mutual information is consistent when the dataset is generated with samples of a stationary and ergodic source. In other words, we show that our information estimator using neural networks converges asymptotically to the true value with probability one. Besides universal functional approximation of neural networks, a core lemma to show the convergence is Birkhoff’s ergodic theorem. Additionally, we use the technique to estimate directed information and demonstrate the effectiveness of our approach in simulations.

Directed Data-Processing Inequalities for Systems with Feedback

We present novel data-processing inequalities relating the mutual information and the directed information in systems with feedback. The internal deterministic blocks within such systems are restricted only to be causal mappings, but are allowed to be non-linear and time varying, and randomized by their own external random input, can yield any stochastic mapping. These randomized blocks can for example represent source encoders, decoders, or even communication channels. Moreover, the involved signals can be arbitrarily distributed. Our first main result relates mutual and directed information and can be interpreted as a law of conservation of information flow. Our second main result is a pair of data-processing inequalities (one the conditional version of the other) between nested pairs of random sequences entirely within the closed loop. Our third main result introduces and characterizes the notion of in-the-loop (ITL) transmission rate for channel coding scenarios in which the messages are internal to the loop. Interestingly, in this case the conventional notions of transmission rate associated with the entropy of the messages and of channel capacity based on maximizing the mutual information between the messages and the output turn out to be inadequate. Instead, as we show, the ITL transmission rate is the unique notion of rate for which a channel code attains zero error probability if and only if such an ITL rate does not exceed the corresponding directed information rate from messages to decoded messages. We apply our data-processing inequalities to show that the supremum of achievable (in the usual channel coding sense) ITL transmission rates is upper bounded by the supremum of the directed information rate across the communication channel. Moreover, we present an example in which this upper bound is attained. Finally, we further illustrate the applicability of our results by discussing how they make possible the generalization of two fundamental inequalities known in networked control literature.

Addressing indirect frequency coupling via partial generalized coherence

Distinguishing between direct and indirect frequency coupling is an important aspect of functional connectivity analyses because this distinction can determine if two brain regions are directly connected. Although partial coherence quantifies partial frequency coupling in the linear Gaussian case, we introduce a general framework that can address even the nonlinear and non-Gaussian case. Our technique, partial generalized coherence (PGC), expands prior work by allowing pairwise frequency coupling analyses to be conditioned on other processes, enabling model-free partial frequency coupling results. By taking advantage of recent advances in conditional mutual information estimation, we are able to implement our technique in a way that scales well with dimensionality, making it possible to condition on many processes and produce a partial frequency coupling graph. We analyzed both linear Gaussian and nonlinear simulated networks. We then performed PGC analysis of calcium recordings from mouse olfactory bulb glomeruli under anesthesia and quantified the dominant influence of breathing-related activity on the pairwise relationships between glomeruli for breathing-related frequencies. Overall, we introduce a technique capable of eliminating indirect frequency coupling in a model-free way, empowering future research to correct for potentially misleading frequency interactions in functional connectivity analyses.

Inferring functional connectivity through graphical directed information

Objective.

Accurate inference of functional connectivity is critical for understanding brain function. Previous methods have limited ability distinguishing between direct and indirect connections because of inadequate scaling with dimensionality. This poor scaling performance reduces the number of nodes that can be included in conditioning. Our goal was to provide a technique that scales better and thereby enables minimization of indirect connections.

Approach.

Our major contribution is a powerful model-free framework, graphical directed information (GDI), that enables pairwise directed functional connections to be conditioned on the activity of substantially more nodes in a network, producing a more accurate graph of functional connectivity that reduces indirect connections. The key technology enabling this advancement is a recent advance in the estimation of mutual information (MI), which relies on multilayer perceptrons and exploiting an alternative representation of the Kullback–Leibler divergence definition of MI. Our second major contribution is the application of this technique to both discretely valued and continuously valued time series.

Main results.

GDI correctly inferred the circuitry of arbitrary Gaussian, nonlinear, and conductance-based networks. Furthermore, GDI inferred many of the connections of a model of a central pattern generator circuit in Aplysia, while also reducing many indirect connections.

Significance.

GDI is a general and model-free technique that can be used on a variety of scales and data types to provide accurate direct connectivity graphs and addresses the critical issue of indirect connections in neural data analysis.

Statistical methods for dissecting interactions between brain areas

The brain is composed of many functionally distinct areas. This organization supports distributed processing, and requires the coordination of signals across areas. Our understanding of how populations of neurons in different areas interact with each other is still in its infancy. As the availability of recordings from large populations of neurons across multiple brain areas increases, so does the need for statistical methods that are well suited for dissecting and interrogating these recordings. Here we review multivariate statistical methods that have been, or could be, applied to this class of recordings. By leveraging population responses, these methods can provide a rich description of inter-areal interactions. At the same time, these methods can introduce interpretational challenges. We thus conclude by discussing how to interpret the outputs of these methods to further our understanding of inter-areal interactions.

Modeling statistical dependencies in multi-region spike train data

Neural computations underlying cognition and behavior rely on the coordination of neural activity across multiple brain areas. Understanding how brain areas interact to process information or generate behavior is thus a central question in neuroscience. Here we provide an overview of statistical approaches for characterizing statistical dependencies in multi-region spike train recordings. We focus on two classes of models in particular: regression-based models and shared latent variable models. Regression-based models describe interactions in terms of a directed transformation of information from one region to another. Shared latent variable models, on the other hand, seek to describe interactions in terms of sources that capture common fluctuations in spiking activity across regions. We discuss the advantages and limitations of each of these approaches and future directions for the field. We intend this review to be an introduction to the statistical methods in multi-region models for computational neuroscientists and experimentalists alike.

Inferring neural information flow from spiking data

The brain can be regarded as an information processing system in which neurons store and propagate information about external stimuli and internal processes. Therefore, estimating interactions between neural activity at the cellular scale has significant implications in understanding how neuronal circuits encode and communicate information across brain areas to generate behavior. While the number of simultaneously recorded neurons is growing exponentially, current methods relying only on pairwise statistical dependencies still suffer from a number of conceptual and technical challenges that preclude experimental breakthroughs describing neural information flows. In this review, we examine the evolution of the field over the years, starting from descriptive statistics to model-based and model-free approaches. Then, we discuss in detail the Granger Causality framework, which includes many popular state-of-the-art methods and we highlight some of its limitations from a conceptual and practical estimation perspective. Finally, we discuss directions for future research, including the development of theoretical information flow models and the use of dimensionality reduction techniques to extract relevant interactions from large-scale recording datasets.

Visual Experience-Dependent Oscillations and Underlying Circuit Connectivity Changes Are Impaired in Fmr1 KO Mice

Fragile X syndrome (FX), the most common inherited form of autism and intellectual disability, is a condition associated with visual perceptual learning deficits. We recently discovered that perceptual experience can encode visual familiarity via persistent low-frequency oscillations in the mouse primary visual cortex (V1). Here, we combine this paradigm with a multifaceted experimental approach to identify neurophysiological impairments of these oscillations in FX mice. Extracellular recordings reveal shorter durations, lower power, and lower frequencies of peak oscillatory activity in FX mice. Directed information analysis of extracellularly recorded spikes reveals differences in functional connectivity from multiple layers in FX mice after the perceptual experience. Channelrhodopsin-2 assisted circuit mapping (CRACM) reveals increased synaptic strength from L5 pyramidal onto L4 fast-spiking cells after experience in wild-type (WT), but not FX, mice. These results suggest differential encoding of visual stimulus familiarity in FX via persistent oscillations and identify circuit connections that may underlie these changes.

Improving on transfer entropy-based network reconstruction using time-delays: Approach and validation

Transfer entropy constitutes a viable model-free tool to infer causal relationships between two dynamical systems from their time-series. In an information-theoretic sense, transfer entropy associates a cause-and-effect relationship with directed information transfer, such that one may improve the prediction of the future of a dynamical system from the history of another system. Recent studies have proposed the use of transfer entropy to reconstruct networks, but the inherent dyadic nature of this metric challenges the development of a robust approach that can discriminate direct from indirect interactions between nodes. In this paper, we seek to fill this methodological gap through the cogent integration of time-delays in the transfer entropy computation. By recognizing that information transfer in the network is bound by a finite speed, we relate the value of the time-delayed transfer entropy between two nodes to the number of walks between them. Upon this premise, we lay out the foundation of an alternative framework for network reconstruction, which we illustrate through closed-form results on three-node networks and numerically validate on larger networks, using examples of Boolean models and chaotic maps.

Reconfiguring Motor Circuits for a Joint Manual and BCI Task

Designing brain-computer interfaces (BCIs) that can be used in conjunction with ongoing motor behavior requires an understanding of how neural activity co-opted for brain control interacts with existing neural circuits. For example, BCIs may be used to regain lost motor function after stroke. This requires that neural activity controlling unaffected limbs is dissociated from activity controlling the BCI. In this study we investigated how primary motor cortex accomplishes simultaneous BCI control and motor control in a task that explicitly required both activities to be driven from the same brain region (i.e. a dual-control task). Single-unit activity was recorded from intracortical, multi-electrode arrays while a non-human primate performed this dual-control task. Compared to activity observed during naturalistic motor control, we found that both units used to drive the BCI directly (control units) and units that did not directly control the BCI (non-control units) significantly changed their tuning to wrist torque. Using a measure of effective connectivity, we observed that control units decrease their connectivity. Through an analysis of variance we found that the intrinsic variability of the control units has a significant effect on task proficiency. When this variance is accounted for, motor cortical activity is flexible enough to perform novel BCI tasks that require active decoupling of natural associations to wrist motion. This study provides insight into the neural activity that enables a dual-control brain-computer interface.

ITGH: Information-Theoretic Granger Causal Inference on Heterogeneous Data

Granger causality for time series states that a cause improves the predictability of its effect. That is, given two time series x and y, we are interested in detecting the causal relations among them considering the previous observations of both time series. Although, most of the algorithms are designed for causal inference among homogeneous processes where only time series from a specific distribution (mostly Gaussian) are given, many applications generate a mixture of various time series from different distributions. We utilize Generalized Linear Models (GLM) to propose a general information-theoretic framework for causal inference on heterogeneous data sets. We regard the challenge of causality detection as a data compression problem employing the Minimum Description Length (MDL) principle. By balancing the goodness-of-fit and the model complexity we automatically find the causal relations. Extensive experiments on synthetic and real-world data sets confirm the advantages of our algorithm ITGH (for Information-Theoretic Granger causal inference on Heterogeneous data) compared to other algorithms.

Identifying genuine protein-protein interactions within communities of gene co-expression networks using a deconvolution method

Direct relationships between biological molecules connected in a gene co-expression network tend to reflect real biological activities such as gene regulation, protein-protein interactions (PPIs), and metabolisation. As correlation-based networks contain numerous indirect connections, those direct relationships are always 'hidden' in them. Compared with the global network, network communities imply more biological significance on predicting protein function, detecting protein complexes and studying network evolution. Therefore, identifying direct relationships in communities is a pervasive and important topic in the biological sciences. Unfortunately, this field has not been well studied. A major thrust of this study is to apply a deconvolution algorithm on communities stemming from different gene co-expression networks, which are constructed by fixing different thresholds for robustness analysis. Using the fifth Dialogue on Reverse Engineering Assessment and Methods challenge (DREAM5) framework, the authors demonstrate that nearly all new communities extracted from a 'deconvolution filter' contain more genuine PPIs than before deconvolution.

Causal Discovery with Attention-Based Convolutional Neural Networks

Having insight into the causal associations in a complex system facilitates decision making, e.g., for medical treatments, urban infrastructure improvements or financial investments. The amount of observational data grows, which enables the discovery of causal relationships between variables from observation of their behaviour in time. Existing methods for causal discovery from time series data do not yet exploit the representational power of deep learning. We therefore present the Temporal Causal Discovery Framework (TCDF), a deep learning framework that learns a causal graph structure by discovering causal relationships in observational time series data. TCDF uses attention-based convolutional neural networks combined with a causal validation step. By interpreting the internal parameters of the convolutional networks, TCDF can also discover the time delay between a cause and the occurrence of its effect. Our framework learns temporal causal graphs, which can include confounders and instantaneous effects. Experiments on financial and neuroscientific benchmarks show state-of-the-art performance of TCDF on discovering causal relationships in continuous time series data. Furthermore, we show that TCDF can circumstantially discover the presence of hidden confounders. Our broadly applicable framework can be used to gain novel insights into the causal dependencies in a complex system, which is important for reliable predictions, knowledge discovery and data-driven decision making.

An Information-Theoretic Measure of Multiscale Causality for Spike-Field Activity

Simultaneous recordings of spikes and fields could enable analyses of functional connectivity in the brain at multiple spatiotemporal scales. However, these analyses require developing novel methods to assess causality between binary-valued spikes and continuous-valued fields, which have fundamentally different statistical profiles and time-scales. Thus classical measures of causality cannot be directly applied in multiscale networks. We develop a novel parametric method to assess causality for multiscale spike-field activities by computing directed information. Directed information is an information theoretic measure of causality but is in general hard to estimate. Our method estimates the causality in two steps. First, we construct point process generalized linear models (GLM) for each neuron's spiking activity to estimate its firing rate using the history of both spikes and fields and compute the directed information to spike nodes from any node. Second, we construct regression models for fields using the history of the estimated firing rates and the history of fields, and then compute the directed information to each field node from any node. In both steps, we estimate model parameters using maximum likelihood and devise statistical tests to assess the significance of the causality. Using simulated data from basic three-node structures and a ten-node network, we show that our method can asymptotically identify the true causality. This method could help uncover functional connectivity in the brain at multiple spatiotemporal scales.

Nonlinear Modeling of Neural Interaction for Spike Prediction Using the Staged Point-Process Model

Neurons communicate nonlinearly through spike activities. Generalized linear models (GLMs) describe spike activities with a cascade of a linear combination across inputs, a static nonlinear function, and an inhomogeneous Bernoulli or Poisson process, or Cox process if a self-history term is considered. This structure considers the output nonlinearity in spike generation but excludes the nonlinear interaction among input neurons. Recent studies extend GLMs by modeling the interaction among input neurons with a quadratic function, which considers the interaction between every pair of input spikes. However, quadratic effects may not fully capture the nonlinear nature of input interaction. We therefore propose a staged point-process model to describe the nonlinear interaction among inputs using a few hidden units, which follows the idea of artificial neural networks. The output firing probability conditioned on inputs is formed as a cascade of two linear-nonlinear (a linear combination plus a static nonlinear function) stages and an inhomogeneous Bernoulli process. Parameters of this model are estimated by maximizing the log likelihood on output spike trains. Unlike the iterative reweighted least squares algorithm used in GLMs, where the performance is guaranteed by the concave condition, we propose a modified Levenberg-Marquardt (L-M) algorithm, which directly calculates the Hessian matrix of the log likelihood, for the nonlinear optimization in our model. The proposed model is tested on both synthetic data and real spike train data recorded from the dorsal premotor cortex and primary motor cortex of a monkey performing a center-out task. Performances are evaluated by discrete-time rescaled Kolmogorov-Smirnov tests, where our model statistically outperforms a GLM and its quadratic extension, with a higher goodness-of-fit in the prediction results. In addition, the staged point-process model describes nonlinear interaction among input neurons with fewer parameters than quadratic models, and the modified L-M algorithm also demonstrates fast convergence.

Estimation of neural connections from partially observed neural spikes

Plasticity is one of the most important properties of the nervous system, which enables animals to adjust their behavior to the ever-changing external environment. Changes in synaptic efficacy between neurons constitute one of the major mechanisms of plasticity. Therefore, estimation of neural connections is crucial for investigating information processing in the brain. Although many analysis methods have been proposed for this purpose, most of them suffer from one or all the following mathematical difficulties: (1) only partially observed neural activity is available; (2) correlations can include both direct and indirect pseudo-interactions; and (3) biological evidence that a neuron typically has only one type of connection (excitatory or inhibitory) should be considered. To overcome these difficulties, a novel probabilistic framework for estimating neural connections from partially observed spikes is proposed in this paper. First, based on the property of a sum of random variables, the proposed method estimates the influence of unobserved neurons on observed neurons and extracts only the correlations among observed neurons. Second, the relationship between pseudo-correlations and target connections is modeled by neural propagation in a multiplicative manner. Third, a novel information-theoretic framework is proposed for estimating neuron types. The proposed method was validated using spike data generated by artificial neural networks. In addition, it was applied to multi-unit data recorded from the CA1 area of a rat's hippocampus. The results confirmed that our estimates are consistent with previous reports. These findings indicate that the proposed method is useful for extracting crucial interactions in neural signals as well as in other multi-probed point process data.

The Effect of Common Signals on Power, Coherence and Granger Causality: Theoretical Review, Simulations, and Empirical Analysis of Fruit Fly LFPs Data

When analyzing neural data it is important to consider the limitations of the particular experimental setup. An enduring issue in the context of electrophysiology is the presence of common signals. For example a non-silent reference electrode adds a common signal across all recorded data and this adversely affects functional and effective connectivity analysis. To address the common signals problem, a number of methods have been proposed, but relatively few detailed investigations have been carried out. As a result, our understanding of how common signals affect neural connectivity estimation is incomplete. For example, little is known about recording preparations involving high spatial-resolution electrodes, used in linear array recordings. We address this gap through a combination of theoretical review, simulations, and empirical analysis of local field potentials recorded from the brains of fruit flies. We demonstrate how a framework that jointly analyzes power, coherence, and quantities based on Granger causality reveals the presence of common signals. We further show that subtracting spatially adjacent signals (bipolar derivations) largely removes the effects of the common signals. However, in some special cases this operation itself introduces a common signal. We also show that Granger causality is adversely affected by common signals and that a quantity referred to as “instantaneous interaction” is increased in the presence of common signals. The theoretical review, simulation, and empirical analysis we present can readily be adapted by others to investigate the nature of the common signals in their data. Our contributions improve our understanding of how common signals affect power, coherence, and Granger causality and will help reduce the misinterpretation of functional and effective connectivity analysis.

A Tutorial for Information Theory in Neuroscience

Understanding how neural systems integrate, encode, and compute information is central to understanding brain function. Frequently, data from neuroscience experiments are multivariate, the interactions between the variables are nonlinear, and the landscape of hypothesized or possible interactions between variables is extremely broad. Information theory is well suited to address these types of data, as it possesses multivariate analysis tools, it can be applied to many different types of data, it can capture nonlinear interactions, and it does not require assumptions about the structure of the underlying data (i.e., it is model independent). In this article, we walk through the mathematics of information theory along with common logistical problems associated with data type, data binning, data quantity requirements, bias, and significance testing. Next, we analyze models inspired by canonical neuroscience experiments to improve understanding and demonstrate the strengths of information theory analyses. To facilitate the use of information theory analyses, and an understanding of how these analyses are implemented, we also provide a free MATLAB software package that can be applied to a wide range of data from neuroscience experiments, as well as from other fields of study.

Extracting neuronal functional network dynamics via adaptive Granger causality analysis

Quantifying the functional relations between the nodes in a network based on local observations is a key challenge in studying complex systems. Most existing time series analysis techniques for this purpose provide static estimates of the network properties, pertain to stationary Gaussian data, or do not take into account the ubiquitous sparsity in the underlying functional networks. When applied to spike recordings from neuronal ensembles undergoing rapid task-dependent dynamics, they thus hinder a precise statistical characterization of the dynamic neuronal functional networks underlying adaptive behavior. We develop a dynamic estimation and inference paradigm for extracting functional neuronal network dynamics in the sense of Granger, by integrating techniques from adaptive filtering, compressed sensing, point process theory, and high-dimensional statistics. We demonstrate the utility of our proposed paradigm through theoretical analysis, algorithm development, and application to synthetic and real data. Application of our techniques to two-photon Ca²⁺ imaging experiments from the mouse auditory cortex reveals unique features of the functional neuronal network structures underlying spontaneous activity at unprecedented spatiotemporal resolution. Our analysis of simultaneous recordings from the ferret auditory and prefrontal cortical areas suggests evidence for the role of rapid top-down and bottom-up functional dynamics across these areas involved in robust attentive behavior.

Function-specific and Enhanced Brain Structural Connectivity Mapping via Joint Modeling of Diffusion and Functional MRI

A joint structural-functional brain network model is presented, which enables the discovery of function-specific brain circuits, and recovers structural connections that are under-estimated by diffusion MRI (dMRI). Incorporating information from functional MRI (fMRI) into diffusion MRI to estimate brain circuits is a challenging task. Usually, seed regions for tractography are selected from fMRI activation maps to extract the white matter pathways of interest. The proposed method jointly analyzes whole brain dMRI and fMRI data, allowing the estimation of complete function-specific structural networks instead of interactively investigating the connectivity of individual cortical/sub-cortical areas. Additionally, tractography techniques are prone to limitations, which can result in erroneous pathways. The proposed framework explicitly models the interactions between structural and functional connectivity measures thereby improving anatomical circuit estimation. Results on Human Connectome Project (HCP) data demonstrate the benefits of the approach by successfully identifying function-specific anatomical circuits, such as the language and resting-state networks. In contrast to correlation-based or independent component analysis (ICA) functional connectivity mapping, detailed anatomical connectivity patterns are revealed for each functional module. Results on a phantom (Fibercup) also indicate improvements in structural connectivity mapping by rejecting false-positive connections with insufficient support from fMRI, and enhancing under-estimated connectivity with strong functional correlation.

Spatio-Temporal Patterning in Primary Motor Cortex at Movement Onset

Voluntary movement initiation involves the engagement of large populations of motor cortical neurons around movement onset. Despite knowledge of the temporal dynamics that lead to movement, the spatial structure of these dynamics across the cortical surface remains unknown. In data from 4 rhesus macaques, we show that the timing of attenuation of beta frequency local field potential oscillations, a correlate of locally activated cortex, forms a spatial gradient across primary motor cortex (MI). We show that these spatio-temporal dynamics are recapitulated in the engagement order of ensembles of MI neurons. We demonstrate that these patterns are unique to movement onset and suggest that movement initiation requires a precise spatio-temporal sequential activation of neurons in MI.

Windowed Granger causal inference strategy improves discovery of gene regulatory networks

Accurate inference of regulatory networks from experimental data facilitates the rapid characterization and understanding of biological systems. High-throughput technologies can provide a wealth of time-series data to better interrogate the complex regulatory dynamics inherent to organisms, but many network inference strategies do not effectively use temporal information. We address this limitation by introducing Sliding Window Inference for Network Generation (SWING), a generalized framework that incorporates multivariate Granger causality to infer network structure from time-series data. SWING moves beyond existing Granger methods by generating windowed models that simultaneously evaluate multiple upstream regulators at several potential time delays. We demonstrate that SWING elucidates network structure with greater accuracy in both in silico and experimentally validated in vitro systems. We estimate the apparent time delays present in each system and demonstrate that SWING infers time-delayed, gene-gene interactions that are distinct from baseline methods. By providing a temporal framework to infer the underlying directed network topology, SWING generates testable hypotheses for gene-gene influences.

Granger Causality and Jensen-Shannon Divergence to Determine Dominant Atrial Area in Atrial Fibrillation

Atrial fibrillation (AF) is already the most commonly occurring arrhythmia. Catheter pulmonary vein ablation has emerged as a treatment that is able to make the arrhythmia disappear; nevertheless, recurrence to arrhythmia is very frequent. In this study, it is proposed to perform an analysis of the electrical signals recorded from bipolar catheters at three locations, pulmonary veins and the right and left atria, before to and during the ablation procedure. Principal Component Analysis (PCA) was applied to reduce data dimension and Granger causality and divergence techniques were applied to analyse connectivity along the atria, in three main regions: pulmonary veins, left atrium (LA) and right atrium (RA). The results showed that, before the procedure, patients with recurrence in the arrhythmia had greater connectivity between atrial areas. Moreover, during the ablation procedure, in patients with recurrence in the arrhythmial both atria were more connected than in patients that maintained sinus rhythms. These results can be helpful for procedures designing to end AF.

Model-free inference of direct network interactions from nonlinear collective dynamics

The topology of interactions in network dynamical systems fundamentally underlies their function. Accelerating technological progress creates massively available data about collective nonlinear dynamics in physical, biological, and technological systems. Detecting direct interaction patterns from those dynamics still constitutes a major open problem. In particular, current nonlinear dynamics approaches mostly require to know a priori a model of the (often high dimensional) system dynamics. Here we develop a model-independent framework for inferring direct interactions solely from recording the nonlinear collective dynamics generated. Introducing an explicit dependency matrix in combination with a block-orthogonal regression algorithm, the approach works reliably across many dynamical regimes, including transient dynamics toward steady states, periodic and non-periodic dynamics, and chaos. Together with its capabilities to reveal network (two point) as well as hypernetwork (e.g., three point) interactions, this framework may thus open up nonlinear dynamics options of inferring direct interaction patterns across systems where no model is known.

Network dynamical systems can represent the interactions involved in the collective dynamics of gene regulatory networks or metabolic circuits. Here Casadiego et al. present a method for inferring these types of interactions directly from observed time series without relying on their model.

A study of problems encountered in Granger causality analysis from a neuroscience perspective

Granger causality methods were developed to analyze the flow of information between time series. These methods have become more widely applied in neuroscience. Frequency-domain causality measures, such as those of Geweke, as well as multivariate methods, have particular appeal in neuroscience due to the prevalence of oscillatory phenomena and highly multivariate experimental recordings. Despite its widespread application in many fields, there are ongoing concerns regarding the applicability of Granger causality methods in neuroscience. When are these methods appropriate? How reliably do they recover the system structure underlying the observed data? What do frequency-domain causality measures tell us about the functional properties of oscillatory neural systems? In this paper, we analyze fundamental properties of Granger-Geweke (GG) causality, both computational and conceptual. Specifically, we show that (i) GG causality estimates can be either severely biased or of high variance, both leading to spurious results; (ii) even if estimated correctly, GG causality estimates alone are not interpretable without examining the component behaviors of the system model; and (iii) GG causality ignores critical components of a system's dynamics. Based on this analysis, we find that the notion of causality quantified is incompatible with the objectives of many neuroscience investigations, leading to highly counterintuitive and potentially misleading results. Through the analysis of these problems, we provide important conceptual clarification of GG causality, with implications for other related causality approaches and for the role of causality analyses in neuroscience as a whole.

Inferring neuronal network functional connectivity with directed information

A major challenge in neuroscience is to develop effective tools that infer the circuit connectivity from large-scale recordings of neuronal activity patterns. In this study, context tree maximizing (CTM) was used to estimate directed information (DI), which measures causal influences among neural spike trains in order to infer putative synaptic connections. In contrast to existing methods, the method presented here is data driven and can readily identify both linear and nonlinear relations between neurons. This CTM-DI method reliably identified circuit structures underlying simulations of realistic conductance-based networks. It also inferred circuit properties from voltage-sensitive dye recordings of the buccal ganglion of Aplysia. This method can be applied to other large-scale recordings as well. It offers a systematic tool to map network connectivity and to track changes in network structure such as synaptic strengths as well as the degrees of connectivity of individual neurons, which in turn could provide insights into how modifications produced by learning are distributed in a neural network.NEW & NOTEWORTHY This study brings together the techniques of voltage-sensitive dye recording and information theory to infer the functional connectome of the feeding central pattern generating network of Aplysia. In contrast to current statistical approaches, the inference method developed in this study is data driven and validated by conductance-based model circuits, can distinguish excitatory and inhibitory connections, is robust against synaptic plasticity, and is capable of detecting network structures that mediate motor patterns.

A statistical framework for neuroimaging data analysis based on mutual information estimated via a gaussian copula

We begin by reviewing the statistical framework of information theory as applicable to neuroimaging data analysis. A major factor hindering wider adoption of this framework in neuroimaging is the difficulty of estimating information theoretic quantities in practice. We present a novel estimation technique that combines the statistical theory of copulas with the closed form solution for the entropy of Gaussian variables. This results in a general, computationally efficient, flexible, and robust multivariate statistical framework that provides effect sizes on a common meaningful scale, allows for unified treatment of discrete, continuous, unidimensional and multidimensional variables, and enables direct comparisons of representations from behavioral and brain responses across any recording modality. We validate the use of this estimate as a statistical test within a neuroimaging context, considering both discrete stimulus classes and continuous stimulus features. We also present examples of analyses facilitated by these developments, including application of multivariate analyses to MEG planar magnetic field gradients, and pairwise temporal interactions in evoked EEG responses. We show the benefit of considering the instantaneous temporal derivative together with the raw values of M/EEG signals as a multivariate response, how we can separately quantify modulations of amplitude and direction for vector quantities, and how we can measure the emergence of novel information over time in evoked responses. Open-source Matlab and Python code implementing the new methods accompanies this article. Hum Brain Mapp 38:1541-1573, 2017. © 2016 Wiley Periodicals, Inc.

T-type calcium channel blocker Z944 restores cortical synchrony and thalamocortical connectivity in a rat model of neuropathic pain

Oscillations are fundamental to communication between neuronal ensembles. We previously reported that pain in awake rats enhances synchrony in primary somatosensory cortex (S1) and attenuates coherence between S1 and ventral posterolateral (VPL) thalamus. Here, we asked whether similar changes occur in anesthetized rats and whether pain modulates phase-amplitude coupling between VPL and S1. We also hypothesized that the suppression of burst firing in VPL using Z944, a novel T-type calcium channel blocker, restores S1 synchrony and thalamocortical connectivity. Local field potentials were recorded from S1 and VPL in anesthetized rats 7 days after sciatic chronic constriction injury (CCI). In rats with CCI, low-frequency (4-12 Hz) synchrony in S1 was enhanced, whereas VPL-S1 coherence and theta-gamma phase-amplitude coupling were attenuated. Moreover, Granger causality showed decreased informational flow from VPL to S1. Systemic or intrathalamic delivery of Z944 to rats with CCI normalized these changes. Systemic Z944 also reversed thermal hyperalgesia and conditioned place preference. These data suggest that pain-induced cortical synchrony and thalamocortical disconnectivity are directly related to burst firing in VPL.

Learning Minimal Latent Directed Information Polytrees

We propose an approach for learning latent directed polytrees as long as there exists an appropriately defined discrepancy measure between the observed nodes. Specifically, we use our approach for learning directed information polytrees where samples are available from only a subset of processes. Directed information trees are a new type of probabilistic graphical models that represent the causal dynamics among a set of random processes in a stochastic system. We prove that the approach is consistent for learning minimal latent directed trees. We analyze the sample complexity of the learning task when the empirical estimator of mutual information is used as the discrepancy measure.

Causal Inference Based on the Analysis of Events of Relations for Non-stationary Variables

The main concept behind causality involves both statistical conditions and temporal relations. However, current approaches to causal inference, focusing on the probability vs. conditional probability contrast, are based on model functions or parametric estimation. These approaches are not appropriate when addressing non-stationary variables. In this work, we propose a causal inference approach based on the analysis of Events of Relations (CER). CER focuses on the temporal delay relation between cause and effect, and a binomial test is established to determine whether an "event of relation" with a non-zero delay is significantly different from one with zero delay. Because CER avoids parameter estimation of non-stationary variables per se, the method can be applied to both stationary and non-stationary signals.

Empirical Bayesian significance measure of neuronal spike response

Background

Functional connectivity analyses of multiple neurons provide a powerful bottom-up approach to reveal functions of local neuronal circuits by using simultaneous recording of neuronal activity. A statistical methodology, generalized linear modeling (GLM) of the spike response function, is one of the most promising methodologies to reduce false link discoveries arising from pseudo-correlation based on common inputs. Although recent advancement of fluorescent imaging techniques has increased the number of simultaneously recoded neurons up to the hundreds or thousands, the amount of information per pair of neurons has not correspondingly increased, partly because of the instruments’ limitations, and partly because the number of neuron pairs increase in a quadratic manner. Consequently, the estimation of GLM suffers from large statistical uncertainty caused by the shortage in effective information.

Results

In this study, we propose a new combination of GLM and empirical Bayesian testing for the estimation of spike response functions that enables both conservative false discovery control and powerful functional connectivity detection. We compared our proposed method’s performance with those of sparse estimation of GLM and classical Granger causality testing. Our method achieved high detection performance of functional connectivity with conservative estimation of false discovery rate and q values in case of information shortage due to short observation time. We also showed that empirical Bayesian testing on arbitrary statistics in place of likelihood-ratio statistics reduce the computational cost without decreasing the detection performance. When our proposed method was applied to a functional multi-neuron calcium imaging dataset from the rat hippocampal region, we found significant functional connections that are possibly mediated by AMPA and NMDA receptors.

Conclusions

The proposed empirical Bayesian testing framework with GLM is promising especially when the amount of information per a neuron pair is small because of growing size of observed network.

Electronic supplementary material

The online version of this article (doi:10.1186/s12868-016-0255-x) contains supplementary material, which is available to authorized users.