causal inference neural network
2010). α-functions, bi-exponential functions) (Brette et al. In the biophysics-based simulation, we generated simulated data for a set of integrate-and-fire neurons with noise. For an edge YBt → YAt + 1, the edge strength was measured by the frequency of this edge appearing in the model ensemble. In International Conference on Machine Learning, (pp. model_utils.py contains functions for generating artificial time series from different autoregressive models with known causal structures. YBt + 1 is driven by YBt and YDt in condition 1, while YBt + 1 is driven by YAt, YBt and YCt in condition 2. Nature, 370(6485), 140–143. For different cluster similarity levels, CAIM consistently detected the corrected number of clusters and identified the correct cluster structure. We will introduce latent variables which represent unmeasured time series, then use the expectation maximization (EM) to infer properties of partially observed Markov processes (Geiger et al. In neural data generation, the trajectory of a neuron in cluster i is generated by flipping the binary state of Yi1:T with a probability λ. λ represented noise level and 1-λ characterized the within-cluster homogeneity. CAS Vancouver: MIT Press. For a node YAt + 1, we construct a random forest model to predict YAt + 1 based on variables in Yt = [YA(t), …, YZ(t)]. This project is stable and being incubated for long-term support. Neural Networks from Scratch with Python by Sentdex, Model based controlled learning of MDP policies with an application to lost sales inventory control, Phase Reduction and Synchronization of Coupled Noisy Oscillators, Anytime Ellipsoidal Over approximation of Forward Reach Sets of Uncertain Linear Systems, Human Machine Adaptive Shared Control for Safe Automated Driving under Automation Degradation, On Joint Reconstruction of State and Input Output Injection Attacks for Nonlinear Systems, Implicit Linear Algebra and Basic Circuit Theory II port behaviour of rigid multiports. 2015) between neuron i and cluster j. Both AUCs and execution times of BNSR and CAIM were similar, although the AUC of CAIM was consistently higher. J Comput Neurosci, 23(3), 349–398. PubMed Neuroimage, 155, 605–611. For causal discovery, we compared our causal network discovery algorithm to Bayesian network structure learning (BNS), Bayesian network structure learning with resampling (BNSR) (Chen et al. J Neurophysiol, 102(2), 1315–1330. Neural Netw, 16(9), 1325–1352. The first 100 observations of all neurons for noise level 0.1 are depicted in Fig. Article Brette, R., Rudolph, M., Carnevale, T., Hines, M., Beeman, D., Bower, J. M., Diesmann, M., Morrison, A., Goodman, P. H., Harris Jr., F. C., Zirpe, M., Natschläger, T., Pecevski, D., Ermentrout, B., Djurfeldt, M., Lansner, A., Rochel, O., Vieville, T., Muller, E., Davison, A. P., el Boustani, S., & Destexhe, A. The walktrap algorithm finds densely connected subgraphs based on random walks. Weak pairwise correlations imply strongly correlated network states in a neural population. Causal discovery aims to detect causal relationships among variables based on observational data. Google Scholar. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. Regularization paths for generalized linear models via coordinate descent. San Francisco: Morgan Kaufmann Publishers Inc. Meyer, P. E., Kontos, K., Lafitte, F., & Bontempi, G. (2007). This neuron cannot generate a second spike for a brief time after the first one (refractoriness). Using remote-sensing to estimate TP concentration is useful, as it provides a synoptic view of the entire water region; however, the weak optical characteristics of TP lead to difficulty in accurately estimating TP concentration. Validation experiments based on simulated data and a real-world reaching task dataset demonstrated that CAIM accurately revealed causal relationships among neural clusters. 2016). Huynh-Thu, V. A., Irrthum, A., Wehenkel, L., & Geurts, P. (2010). Gütig, R., & Sompolinsky, H. (2006). Correlation, partial correlation, and mutual information have been used to measure the association between a pair of neurons. An HTM node abstracts space as well as time. Chen, R., Hillis, A. E., Pawlak, M., & Herskovits, E. H. (2008). On the use of dynamic Bayesian networks in reconstructing functional neuronal networks from spike train ensembles. Causal inference is the process of drawing a conclusion about a causal connection based on the conditions of the occurrence of an effect. We compared CAIM with other methods. BNS generated an unweighted graph. The model ensemble included 1000 models. An interactive example : here 8 Meek, C. (1995). K-means was randomly initialized 100 times. Such a neuron model can represent virtually all postsynaptic potentials or currents described in the literature (e.g. Neuroinform (2021). 2012), which is the marginal likelihood or evidence P(G | D), where D is the observed data. The causal sufficiency assumption is widely used in causal discovery in order to make the causal discovery process computationally tractable. Litwin-Kumar, A., & Doiron, B. That is, B→ is a two-slice temporal Bayesian network (2TBN). Chen, Y., Bressler, S. L., & Ding, M. (2006). This framework considers the dependence between two variables X and Y given a set of variables Z. There are many studies of causal discovery from multiple time series from problem domains which are not neuroscience-related, such as inferring gene regulatory networks using time-series gene expression data (Bar-Joseph et al. A microcircuit lies at the heart of the information processing capability of the brain. Noise level is 0.1. 2007 , Li et al. Randomized experiments are the gold standard for causal inference because the treatment assignment is random and physically manipulated: one … Using the CAIM framework, we can detect clusters that have neurons with zero-lag synchrony; then model information propagation in a pathway and focus on the pattern that activation of cluster A at time point t leads to activation of cluster B at time point t + 1. For all noise levels, CAIM achieved the best clustering performance. Neural activities can be recorded by calcium imaging or electrophysiology with electrodes. 2007), spatial maps in the entorhinal cortex (Hafting et al. Each node is associated with an updating rule that specifies how its state changes over time due to the activation of the parent set. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. (a) Inference network, q(z;t;yjx). Studying and modelling dynamic biological processes using time-series gene expression data. Causal inference using deep neural networks. b and c are DBNs for condition 1 and 2. (1969). (2012). Ko, H., Cossell, L., Baragli, C., Antolik, J., Clopath, C., Hofer, S. B., & Mrsic-Flogel, T. D. (2013). We generated three datasets: low similarity (Hamming distance = 2696), middle similarity (Hamming distance = 1461), and high similarity (Hamming distance = 862). 2006; Hu et al. 2018). Under these conditions, the causal relationship can be discovered by machine learning algorithms. Let X ⫫ Y | Z denote that X and Y are conditionally independent given Z. X is not the cause of Y if Xt ⫫ Yt + 1 | Zt. PubMed For a single tree, the importance of a variable is computed by summing the variance reduction values of all tree nodes where this variable is used to split. Google Scholar. The edge weights of YAt → YBt + 1, YAt → YCt + 1, YBt → YDt + 1, and YCt → YDt + 1 were 0.90, 0.90, 0.50, and 0.47, respectively. CAS In this clustering problem, P is about several hundred and T is several thousand. PubMed Let Yi1:T be the trajectory of cluster i. Microstructure of a spatial map in the entorhinal cortex. This weighted graph was robust to the threshold to infer binary cluster states and remained stable for the threshold in [0.3 0.7]. Nature, 440(7087), 1007–1012. Recent work has made it possible to approximate this problem as … CAIM aims to infer causal relationships based on observational calcium imaging or electrophysiology time series. We establish novel nonasymptotic high probability bounds for deep feedforward neural nets. In contrast to the experimental advances in neural recording techniques, computational analysis of ensemble neural activities is still emerging. For most longitudinal image data, the number of visits for each subject is small, often less than ten. Each cluster is associated with a latent variable (the cluster state variable) which represents whether the cluster is activated or not. Chen, R. Causal Network Inference for Neural Ensemble Activity. (2012). The proposed method, called Causal Inference for Microcircuits (CAIM), aims to reconstruct causal mesoscopic-scale networks from observational calcium imaging or electrophysiology time series. 2007). Introduction An important task constantly being carried out by the brain is determining whether inputs from different sensory modalities originate from the same cause or from separate causes. The walktrap algorithm uses the results of random walks to merge separate communities in a bottom-up manner and creates a dendrogram. Proc Natl Acad Sci U S A, 104(1), 347–352. Cellular resolution functional imaging in behaving rats using voluntary head restraint. (2004). b The spike trains of cluster states (the first 200 frames). We train a convolutional neural network on a normalized empirical probability density distribution (NEPDF)matrix . Investigating causal relations by econometric models and cross-spectral methods. Hu, M., Li, W., & Liang, H. (2018). Noise level was 0.2. For example, calcium imaging can observe ensemble neural activity of hundreds of neurons. 9, Elsevier, pp. These results demonstrated good cluster separation. CAIM, BNSR, GLMNET generated weighted directed graphs. For threshold = 0.5, CAIM’s AUCs were 1 for all scenarios. For example, D1- and D2-medium spiny neurons (MSNs) in the dorsal striatum are grouped into spatially compact clusters (Barbera et al. The simulation modeled interactions among a set of integrate-and-fire (I&F) neurons with noise. Construction of training datasets for the GC-MLP neural network. CAIM and BNSR consistently achieved higher AUCs than did BNS and GLMNET. We'll have a loss function R. We'll think of training a deep neural network with N hidden layers, and we'll aim to find the network parameters W that minimize the expected loss. If a set of nodes, πi, causally affects the activity of node i, then there exists a link from the nodes in πi to node i. πi is referred to as the parent set of node i. The Silhouette score has a range of [−1, 1]. It carries out a specific computation of a region. The interactions among clusters were described by a ground-truth DBN G*. Ma, W. J., Beck, J. M., Latham, P. E., & Pouget, A. Higher relative mutual information indicates a stronger association between two binary random variables. Neural Network; Also referred to as Artificial Neural Network (ANN)or just neural net, it’s obvious the concept was inspired from human biology and the way neurons of the human brain function together to understand inputs from human senses. This is one of the limitations of CAIM. operator is an intervention. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Google Scholar. 2010). CAIM uses graph-based clustering. Stability investigations of multivariable regression models derived from low- and high-dimensional data. In some real-world applications, we need to determine a threshold on this ranking to obtain a binary causal graph. Higher similarity is more challenging for neuron clustering. Under these assumptions, a remarkable result according to Geiger and Pearl (Geiger and Pearl 1990) and Meek (Meek 1995) is the Markov completeness theorem: for linear Gaussian and for multinomial causal relations, an algorithm that identifies the Markov equivalent class is complete (that is, it extracts all information about the underlying causal structure). Therefore, CAIM does not assume that the brain network model is invariant across subjects. Computing Communities in Large Networks Using Random Walks. Miniaturized integration of a fluorescence microscope. The gold standard of establishing a causal relationship is performing planned or randomized experiments (Fisher 1970). Your email address will not be published. Such a pattern was changed in condition 2. Department of Diagnostic Radiology and Nuclear Medicine, University of Maryland School of Medicine, 22 South Greene Street, Baltimore, MD, 21201, USA, You can also search for this author in 411–418). Parameters in GLMNET were tuned based on internal cross-validation. For each node Yit + 1, we used the algorithm in (Chen et al. PLoS One, 5(9), 1–10. Input to neuron clustering is s1:T. Clustering generates a partition of the variable space. If the dependencies in the underlying process change over time, the generated model is an average over different temporal dependency structures. (2006). Then it uses the modularity score to select where to cut the dendrogram. Fine-scale specificity of cortical networks depends on inhibitory cell type and connectivity. Neurons often form clusters and neurons in the same cluster have similar functional profiles. This de nition can be made precise in the context of a causal graphical model [56] (Figure 1A, see (2005). 2a. Yoshimura, Y., Dantzker, J. L. M., & Callaway, E. M. (2005). Second, constructing a model from such high-dimensional data with a cluster structure often leads to overfitting (Hastie et al. We present a general supervised deep learning framework that transforms input vectors to an image-like representation for every pair of inputs . Nature, 496(7443), 96–100. AUCs of BNS, BNSR, CAIM, and GLMNET for the simulated spike train data with varying noise levels, AUCs of BNS, BNSR, CAIM, and GLMNET for the simulated spike train data with varying cluster similarity levels, AUCs of BNS, BNSR, CAIM, and GLMNET for the simulated spike train data with varying cluster numbers. The transition probability table for node 5 is depicted in Fig. 2b. Analysis of microcircuits provides a system-level understanding of the neurobiology of health and disease. Causal inference by identification of vector autoregressive processes with hidden components. a The loading matrix for neuron clustering. An example of a growth-shrink based method is MRNET (Meyer et al. Recurrent neural networks (RNNs) are widely used in computational neuroscience and machine learning applications. 8b and c which are the networks for two different conditions, respectively. We prove valid Mahwah: Lawrence Erlbaum Associates, Inc.. Yoshimura, Y., & Callaway, E. M. (2005). 2009). Microcircuits have been shown to encode sensory input (Luczak et al. Cortical travelling waves: Mechanisms and computational principles. Overall, CAIM achieved the highest Silhouette score and Rand index. CAIM achieved the highest AUC in most combinations of experimental setups and thresholds. Bayesian methods have been used to model neural activity data. Curr Opin Neurobiol, 17(5), 609–618. We present a general supervised deep learning framework that transforms input vectors to an image-like representation for every pair of inputs . We chose threshold = 0.5. 2017; Chen and Herskovits 2015). CAIM is designed to generate a network model from data streams which include thousands of data points. The goal of neuron clustering is to group P neurons into K homogeneous clusters. It’s capable of revealing causal interactions among neural dynamics. Therefore, neuron clustering focuses on examining the instantaneous synchrony (the zero-lag synchrony) between neuron pairs. "Method" provides the CAIM algorithm, including neuron clustering and causal network inference. Assumptions are beliefs that allow movement from statistical associations to causation. Our new rates are sufficiently fast (in some cases minimax optimal) to allow us to establish valid second-step inference after first-step estimation with deep learning, a result also new to the literature. Voxelwise Bayesian lesion-deficit analysis. The number of clusters was determined by the gap statistic. In the sampling step, we sampled G* and generated simulated data for cluster states. For the second step of causal inference, we develop a novel autoencoding architecture that applies generative moment-matching neural-networks Zhao et al. Excluding these low firing neurons from causal discovery doesn’t exclude the possibility that they contributed to the observed ensemble activity. In synchrony analysis, an undirected graph is generated. CAS One method is based on the likelihood function. Neurons in group B had two or three neurons in group A as parent nodes. Deep Neural Networks for Estimation and Inference: Application to Causal E ects and Other Semiparametric Estimands Max H. Farrell Tengyuan Liang Sanjog Misra University of Chicago, Booth School of Business February 1, 2019 Abstract We study deep neural networks and their use in semiparametric inference. A diffusion-conv olution recurrent neural network (DCRNN) is designed to integrate diffusion con volution, a sequence-to-sequence architecture and a scheduled sampling technique [61]. J. Graph Algorithms Appl, 10, 191–218. Eldawlatly, S., Zhou, Y., Jin, R., & Oweiss, K. G. (2010). In the step function model, the neural network estimator performs slight better than the misspecified OLS, but neither appears to work well. Figure 1 shows the architecture of CAIM. The loading matrix of neuron clustering for subtask 1. Article Google Scholar. 2001). The computational framework in CAIM can be used for other applications such as modeling cortical traveling waves (Muller et al. In the FCM-based method, we first calculated a P× P distance matrix where the (i, j) element of this matrix is the Manhattan distance between neurons i and j. Therefore, the number of clusters is automictically determined by the algorithm. Google Scholar. Our algorithm generates a directed weighted graph G modeling the linear/nonlinear interactions among cluster state variables. Network dynamics are determined by these updating rules. For a set of variables V = {X1, …, Xp}, a causal graphical model is G = (V, E), where an edge Xi → Xj represents Xi is a direct cause of Xj relative to variables in V, and G is a directed acyclic graph. A copula-based Granger causality measure for the analysis of neural spike train data. In subtask 3, we evaluated CAIM with different cluster numbers. Neurons in group A had no parent nodes. Nat Neurosci, 9(3), 420–428. A. Simulation with causal model presented in Figure 1b. Causal inference is a statistical tool that enables our AI and machine learning algorithms to reason in similar ways. In a DBN, nodes are variables of interest, and edges (links) represent interactions among variables. Wiwie, C., Baumbach, J., & Röttger, R. (2018). For example, Pr(YB(t + 1) = active | YA(t) = active, YC(t) = active) = 0.88 represents activation of cluster A and activation of cluster C at time point t result in the activation of cluster B at time point t + 1 with probability 0.88. Nature, 484(7392), 62–68. CAIM combines neural recording, Bayesian network modeling, and neuron clustering. Cambridge: MIT Press. The Rand index determines the similarity between the estimated label and the ground-truth label as a function of positive and negative agreements in pairwise cluster assignments; when two labels agree perfectly, the Rand index is 1. Neuroimage, 40(4), 1633–1642. PubMed Harvey, C. D., Coen, P., & Tank, D. W. (2012). Pregowska, A., Szczepanski, J., & Wajnryb, E. (2015). Ye, N. (2003). Article Dynamic network analysis is designed to generate a network model from longitudinal MR data. CAS (1990), On the logic of causal models. New York: Cambridge University Press. Proc Annu Int Conf IEEE Eng Med Biol Soc EMBS, 10(6), 4607–4610. The neuron model (Gütig and Sompolinsky 2006) is as follows: where V is the membrane potential, Vrest is the rest potential, ε is a Gaussian random variable with mean 0 and standard deviation 1, τ is the membrane time constant, and σ is a parameter controlling the noise term. Mutual information against correlations in binary communication channels. We investigated the causal inference under the potential outcome framework to analyze the sensitivity of each Causal inference from observation data is a core problem in many scientific fields . This generates a sequence of sensory input spikes S t (vertical black lines) with Poisson statistics.