{"paper_id":"2aa1ee2b-1c2c-4ade-a12d-8e5bb3516868","body_text":"CausalXtract: a flexible pipeline to extract causal\neffects from live-cell time-lapse imaging data\nFranck Simon1,‡, Maria Colomba Comes 2,‡, Tiziana Tocci 1,2,‡, Louise Dupuis 1, Vincent Cabeli 1, Nikita Lagrange 1,\nArianna Mencattini 2, Maria Carla Parrini 3, Eugenio Martinelli 2,∗, Hervé Isambert 1,∗\n1 CNRS UMR168, Institut Curie, Université PSL, Sorbonne Université, Paris, France\n2 Department of Electronic Engineering, University of Rome Tor Vergata, Rome, Italy\n3 INSERM U830, Institut Curie, Université PSL, Paris, France\n‡ these authors contributed equally to this work\n∗ corresponding authors: herve.isambert@curie.fr, martinelli@ing.uniroma2.it\nLive-cell microscopy routinely provides massive amount of\ntime-lapse images of complex cellular systems under various\nphysiological or therapeutic conditions. However, this wealth\nof data remains difficult to interpret in terms of causal ef-\nfects. Here, we describe CausalXtract, a flexible computa-\ntional pipeline that discovers causal and possibly time-lagged\neffects from morphodynamic features and cell-cell interac-\ntions in live-cell imaging data. CausalXtract methodology\ncombines network-based and information-based frameworks,\nwhich is shown to discover causal effects overlooked by clas-\nsical Granger and Schreiber causality approaches. We show-\ncase the use of CausalXtract to uncover novel causal effects\nin a tumor-on-chip cellular ecosystem under therapeutically\nrelevant conditions. In particular, we find that cancer asso-\nciated fibroblasts directly inhibit cancer cell apoptosis, inde-\npendently from anti-cancer treatment. CausalXtract uncov-\ners also multiple antagonistic effects at different time delays.\nHence, CausalXtract provides a unique computational tool to\ninterpret live-cell imaging data for a range of fundamental and\ntranslational research applications.\nLive-cell imaging microscopy commonly produces extensive\namounts of time-lapse images of cellular systems, which can be\nsegmented to extract morphodynamic features and interactions\nof individual cells under increasingly complex and physiologi-\ncally relevant conditions. However, this wealth of information\nremains largely under-exploited due to a lack of methods and\ntools able to discover causal effects from spatio-temporal cor-\nrelations under well-controlled experimental conditions.\nCausalXtract addresses this need by integrating an advanced\nlive-cell image feature extraction tool with a reliable and scal-\nable causal discovery method, Fig. 1, in order to learn temporal\ncausal networks from live-cell time-lapse imaging data, Fig. 2.\nResults and Discussion\nCausalXtract feature extraction and causal discovery modules\nCausalXtract’s live-cell image feature extraction module (Cell-\nHunter+), Fig. 1b, is based on CellHunter software 1 and con-\nsists in three steps: detection, tracking and feature extraction\nof live cells within time-lapse video images. First, automatic lo-\ncalization/segmentation of cells (e.g. tumor and immune cells)\nis performed with the Circular Hough Transform (CHT) algo-\nrithm2 to estimate the cell centers and radii. Second, cell tra-\njectories along the frames are constructed by linking the posi-\ntions detected at the previous time step through Munkres’ algo-\nrithm for Optimal sub-pattern Assignment Problems (OAPs) 3.\nFinally, relevant descriptors related to the shape, motility, and\nstate of the cells, as well as cell-cell interactions are quantified\nfrom each cell trajectory (Methods).\nCausalXtract’s temporal causal discovery module (tMIIC),\nFig. 1c, is adapted from the causal discovery method, MIIC 4–6,\nwhich learns contemporaneous causal networks (i.e. when tem-\nporal information is not available) for a broad range of bio-\nlogical or biomedical data, from single-cell transcriptomic and\ngenomic alteration data 4,7 to medical records of patients 5,6,8.\nLive-cell time-lapse imaging data contain, however, informa-\ntion about cellular dynamics, which can in principle facilitate\nthe discovery of novel cause-effect functional processes, based\non the assumption that future events cannot cause past ones.\nTo this end, CausalXtract’s discovery module, tMIIC, recon-\nstructs time-unfolded causal networks, where each variable is\nrepresented by several nodes at different relative time points 9,\nFig. 1c. Such a time-unfolded network framework 10–13 is re-\nquired to account for the temporal correlation between succes-\nsive time steps in time series data. We benchmarked tMIIC on\nsynthetic datasets resembling the real-world data of interest an-\nalyzed in this study (i.e. number of time steps, network size and\ndegree distribution) and found that it matches or outperforms\nstate-of-the-art methods, while running order of magnitudes\nfaster on datasets of biologically relevant size including tens to\nhundreds of thousands time steps, Supplementary Figs. 1-4.\nCausalXtract’s temporal network framework goes beyond\nthe seminal concept of temporal causality originally proposed\nby Granger 14 for linear time series without reference to graph-\nical models and later extended to non-linear dynamics by\nSchreiber15,16. In particular, Granger-Schreiber causality is\nin fact too restrictive and may overlook actual causal effects,\nthat can be uncovered by graph-based causal discovery meth-\n1\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\nc\nTumor−on−chip  preparationa\nb\nCausalXtract’s  temporal  causal  discovery  module  (tMIIC)\nCausalXtract’s  live−cell  image  feature  extraction  module  (CellHunter+)\nFibroblast (CAF) cells\nHer2+ cancer cells Her2+ cancer cells\nImmune primary cells\nµ500   m\nEndothelial primary cells\nFigure 1: CausalXtract pipeline. a , Live-cell tumor ecosystem reconstituted ex vivo 1 using the tumor-on-chip technology (Methods). b,\nCausalXtract’s live-cell image feature extraction module (CellHunter+). The tracking of cancer and immune cells and of their mutual inter-\nactions is illustrated in Supplementary Movies 1-3, in absence or presence of cell division and apoptosis event. c, CausalXtract’s temporal\ncausal discovery module (tMIIC) learns a temporal causal network from the features extracted in (b). See Methods for CausalXtract’s\nimplementation details and theoretical foundations. A step-by-step notebook of CausalXtract pipeline is provided with the source code.\n2\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\nods, Supplementary Fig. 5 (Methods, Theorem 1). In addition,\nGranger-Schreiber causality has long been known to infer spu-\nrious causal associations based on time delays, by excluding\nthe presence of latent common causes a priori 9. CausalXtract\ncircumvents these limitations by combining graph-based and\ninformation-based approaches (Methods), while including con-\ntemporary and time-delayed effects of unobserved latent vari-\nables, that are ubiquitous in cell biology data ( e.g. the latent\neffects of cell cycle phases on cellular features and responses).\nApplication to tumor-on-chip cellular ecosystems\nWe showcase CausalXtract with the analysis of time-lapse im-\nages of a tumor ecosystem reconstituted ex vivo using the\ntumor-on-chip technology, Fig. 1a. These live-cell time-lapse\nimages come from a proof-of-concept study 1 which demon-\nstrated the effects of an anti-cancer drug (the monoclonal an-\ntibodies trastuzumab, brand name Herceptin, used to treat\nHER2+ breast cancers) on a reconstituted tumor microenvi-\nronment including cancer cells, immune cells, cancer-associated\nfibroblasts (CAF), and endothelial cells (Methods). However,\na comprehensive extraction and analysis of cellular morphody-\nnamic features and interactions remained unexplored.\nTo this end, cellular features such as cell geometry, velocity,\ndivision, apoptosis, cell-cell transient interactions and persis-\ntent contacts were first extracted from the raw images using\nCausalXtract’s feature extraction module, Fig. 1b and Supple-\nmentary Fig. 6. Then, a time-unfolded causal network, Sup-\nplementary Fig. 7, and the corresponding summary causal net-\nwork, Fig. 2a, were reconstructed between extracted cellular\nfeatures, cell-cell interactions and therapeutic conditions using\nCausalXtract’s temporal causal discovery module, Fig. 1c.\nCausalXtract inferred network, Fig. 2a, uncovers novel bi-\nologically relevant findings, in addition to confirming known\nresults from earlier studies. In particular, CausalXtract discov-\ners that CAFs directly inhibit cancer cell apoptosis, indepen-\ndently from anti-cancer treatment, Fig. 2b, while earlier stud-\nies reported that CAFs merely reduced the effect of treatment 1.\nCausalXtract also discovers that treatment increases cancer cell\nperimeter, Fig. 2c, which has not been reported so far either.\nIn addition, CausalXtract confirms known results from ear-\nlier studies. In particular, it recovers that treatment increases\ncancer cell apoptosis and the number of cancer-immune inter-\nactions, as well as decreases the division rate of cancer cells,\nFig. 2c. Likewise, CausalXtract recovers that CAFs stimulate\ncancer cell migration and increase their area, Fig. 2b.\nInterestingly, CausalXtract identifies also multiple and possi-\nbly antagonistic effects with different time delays. For instance,\nCausalXtract recovers several antagonistic relations between\nmorphodynamic features such as cell division and eccentricity\nor cell apoptosis and area, Fig. 2d. Indeed, the late phases of\ncell division are associated to a marked increase in eccentricity\n(red edge) but preceded by a net decrease in eccentricity, two to\nthree hours before cytokinesis (blue edges), once the decision to\ndivide has been made ( i.e. the probable latent cause) and the\ncell is actually duplicating its biological materials (prophase),\nFig. 2d. Likewise, the area change upon apoptosis is predicted\nto first decrease soon after apoptosis (blue edge) before eventu-\nally increasing upon cell lysis (red edge), Fig. 2d. These results\nare robust to variations in sampling rate, Supplementary Fig. 8.\nAll in all, CausalXtract is a flexible pipeline which uncovers\nnovel and possibly time-lagged causal relations between cellular\nfeatures under controlled conditions ( e.g. drug). CausalXtract\nuniquely combines live-cell feature extraction with information\ntheory and causal discovery approaches. It consists of two in-\ndependent computational modules, conceived to warrant inter-\noperability with alternative live-cell segmentation and tracking\nmethods or alternative temporal causal discovery methods.\nCausalXtract opens up new avenues to analyze live-cell\nimaging data for a range a fundamental and translational\nresearch applications, such as the use of tumor-on-chips to\nscreen immunotherapy responses on patient-derived tumor\nsamples. With the advent of virtually unlimited live-cell image\ndata, flexible hypothesis-free interpretation methods are much\nneeded17 and we believe that CausalXtract can bring unique in-\nsights based on causal discovery to interpret such information-\nrich live-cell imaging data.\nMaterials and Methods\nTumor-on-chip preparation and live-cell microscopy\nVideos analyzed in the present study refer to biological experiments\nemulating a 3D breast tumor ecosystem 1. All tumor-on-chip experi-\nments have a central endothelium compartment containing endothe-\nlial cells (primary human umbilical vein endothelial cells, HUVECs)\nand two lateral chambers filled with biomimetic hydrogel (collagen\ntype I at 2.3 mg/mL) seeded with cancer cells (HER2+ breast can-\ncer BT474 cell line) and immune cells (peripheral blood mononu-\nclear cells, PBMCs) from healthy donors, Fig. 1a. Four experimen-\ntal conditions were considered depending on the presence or absence\nof breast cancer-associated fibroblasts (CAF cell line Hs578T) and\ndrug treatment (trastuzumab, Herceptin). Videos were acquired by\ninverted motorized Leica microscopes with a frame rate of 2 minutes\nfor up to 48h (1440 frames). Fig. 1b shows a crop frame with can-\ncer cells, PBMCs and CAFs. Each video was cropped into multiple\nsmall 300×300 pixel videos (referred to as crops in the following),\neach of which represented a field of view at subsequent time frames\ncontaining a “main” cancer cell (MCC) initially placed at the cen-\nter of the image, some PBMC immune cells, other cancer cells and\npossibly CAFs within the surrounding of the MCC depending on the\nexperimental conditions. 36 video crops of up to 1440 frames were\nanalyzed (46,935 frames in total) corresponding to 9 video crops per\nexperimental conditions.\nCausalXtract’s live-cell image feature extraction module\nThe live-cell image feature extraction module (CellHunter+),\nFig. 1b, extends the CellHunter software 1 and consists in three steps:\ndetection, tracking and feature extraction of live cells within time-\nlapse video images. First, cell detection is based on the segmentation\nof circular-shaped objects using CHT 2 with radii set around the the-\noretical radii of the two cell populations ( rim = 4 px for immune cells\nand rca = 14 px for MCCs with a pixel resolution 1 px = 0.645 µm1).\nThen, cell tracking is performed by linking cells detected at the ith\nframe to cells located at the (i + 1)th frame within a maximum dis-\ntance from the detected cell candidate. While the motions of both\nMCCs and immune cells ressemble random walks with time-varying\ndrift and volatility, these two cell types exhibit different motility\ncharacteristics1. Hence, different maximum distances are considered\nfor the two cell populations: it was set to 40 px for MCCs and to\n20 px for immune cells. For each cell population, an OAP using\nthe Munkres algorithm 3 is solved: the globally best possible pairing\namong located objects is based on an assignment cost equal to the\ninverse of the distance between pairs of cell candidates at the ith\nand (i + 1)th frames. Cell appearing/disappearing and cell overlaps\ndue to projection errors of the 3D scene in the 2D domain are also\nhandled. Finally, cellular morphodynamic features and cell-cell in-\nteraction features are extracted at successive positions along each\n3\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\ndc\nba\ndivision\n500\ntime  steps\n0 1000 eccentricity area\napoptosis\nFigure 2: Application of CausalXtract to time-lapse images of tumor ecosystems reconstituted ex vivo 1. a, Summary causal network inferred\nby CausalXtract. The underlying time-unfolded causal network is shown on Supplementary Fig. 7. Red (resp. blue) edges correspond to\npositive (resp. negative) associations. Bidirected dashed edges represent the effect of unobserved (latent) common causes. Annotations on\nedges correspond to time delays in time-steps (1 ts = 2 min). The inferred network is largely robust to variations in sampling rate (δτ ) and\nmaximum lag (τ ), Supplementary Fig. 8. Here δτ = 7 ts and τ = 84 ts are chosen automatically by CausalXtract, Supplementary Fig. 8b.\nb, The CAF presence subnetwork highlighting the direct causal effects of CAFs on cancer cells. In particular, CausalXtract uncovers that\nCAFs directly inhibit cancer cell apoptosis independently from treatment, which has not been reported so far. c, The treatment subnetwork\nhighlighting the direct causal effects of treament on cancer cells. In particular, CausalXtract uncovers that treatment increases cancer cell\nperimeter, which has not been reported either. d, The eccentricity-area subnetwork highlighting multiple direct and possibly antagonistic\ntime-lagged effects, notably, between cell division and eccentricity and between cell apoptosis and area, as discussed in main text.\n4\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\ntrajectory. For each MCC, 15 descriptors were extracted, Supple-\nmentary Fig. 6, and classified into four main categories: cell shape,\nmotility, state, and interaction descriptors.\nShape descriptors. The active contour algorithm implemented in\nMatlab18 was used to segment the MCC boundaries on each video\ncrop frame. Taking as input a frame representing the ith snapshot\nof the tth MCC, it returns a binary image, where the MCC is rep-\nresented by a white region. From the binary image, the shape prop-\nerties of the region occupied by each MCC were extracted using the\nMatlab regionprops algorithm. The resulting descriptors of the ex-\ntracted shape are listed below:\n• area indicates the number of pixels composing the region. The\nequivalent diameter of the tth MCC in the ith frame is defined\nas dt\ni =\n√\n4 · area/π.\n• perimeter represents the distance along the MCC boundary.\n• circularity is defined as 4 · area · π/perimeter 2, which is equal\nto 1 when the region is perfectly circular.\n• eccentricity denotes the eccentricity of the ellipse with the same\nsecond moments as the region. The value is equal to 1 when\nthe region is a line and to 0 when the region is a circle.\n• instantaneous shape change is defined as, |dt\ni − dt\ni−1|, corre-\nsponding to the difference in absolute value of the equivalent\ndiameters between the ith and (i − 1)th frames of the tth MCC.\nMotility descriptors. The positions pt\ni = ( xt\ni, yt\ni ) and pt\ni−1 of the\ntth MCC in the ith and (i − 1)th frames were compared using the\nEuclidean distance d(·) to define the following motility parameters:\n• instantaneous cancer velocity 19 is defined as d(pt\ni, pt\ni−1)/∆t,\nwhere ∆t is the time interval between two consecutive frames.\n• net displacement 19 indicates the resultant distance between the\ninitial and current positions of the tth MCC, d(pt\n1, pt\ni).\n• directionality19 is defined as the ratio of net displacement,\nd(pt\n1, pt\ni), and curvilinear distance, ∑i\nk=2 d(pt\nk, pt\nk−1). It mea-\nsures the persistence of motion and ranges from 0 for confined\ncells to 1 for cells moving perfectly straight in one direction.\nState descriptors. They record apoptosis or division events:\n• apoptosis indicates if the MCC has died during the experiment.\nIt is set to ‘No’ as long as the cell has not died and becomes ‘Yes’\nfor the remaining frames after the cell undergoes apoptosis.\n• division indicates if the MCC has divided during the experi-\nment. It is set to ‘No’ as long as the cell has not divided and\nbecomes ‘Yes’ for the remaining frames after the cell divides.\nInteraction descriptors . Interactions between MCCs and immune\ncells were defined with respect to two radii around each MCC, r1 =\nrim + rca + 2 = 20 px and r2 = 2 × (rim + rca) = 36 px1. Hence,\nr1 refers to MCC and immune cells in actual physical contact, while\nr2 refers to MCC and immune cells in close vicinity. Then, for each\nsample the following interaction features were defined:\n• number of cancer-immune interactions (r 2) corresponds to the\nnumber of immune cells within the interaction radius r2 around\nthe MCC on that frame.\n• number of cancer-immune interactions (r 1) corresponds to the\nnumber of immune cells in close contact with the MCC on that\nframe.\n• minimal cancer-immune distance (r 2) is the minimum distance\nbetween the MCC and the immune cells within a radius r2.\n• mean immune velocity (r 2) is the mean instantaneous veloc-\nity norm of the immune cells within the interaction radius r2\naround the MCC.\n• mean immune velocity ( r1) is the mean instantaneous velocity\nnorm of the immune cells in close contact with the MCC.\nOverview of causal discovery methods for non-temporal data\nTraditional causal discovery methods 20,21 aim to learn causal net-\nworks from datasets of independent samples by proceeding through\nsuccessive steps. They first learn structural constraints in the form\nof unconditional or conditional independence between variables and\nremove the corresponding edges from an initial fully connected net-\nwork. The second step then consists in orienting some of the re-\ntained edges based on the signature of causality in observational\ndata. This corresponds to orienting three-variable “v-structure” mo-\ntifs as, X → Z ← Y , whenever the edge X − Y has been removed\nwithout conditioning on the variable Z, which implies that Z can-\nnot be a cause of X nor Y . This does not guarantee, however,\nthat X (or Y ) is an actual cause of Z, which also requires to rule\nout the possibility that the edge between X and Z (or Y and Z)\nmight originate from a latent common cause, L, unobserved in the\ndataset, i.e. X L99 L 99K Z. In addition, classical causal discov-\nery methods are prone to spurious conditional independences, which\nlead to many false negative edges and limit the accuracy of inferred\norientations. The recent causal discovery method, MIIC 4–6, which\ncombines constraint-based and information-based principles, learns\nmore robust causal graphical models by first collecting iteratively sig-\nnificant information contributors before assessing conditional inde-\npendences. In practice, MIIC’s strategy limits spurious conditional\nindependences which improves its edge sensitivity and orientation\nreliability compared to traditional constraint-based methods 4–6. In\naddition, MIIC can handle missing data 5 and also heterogeneous\nmultimodal data, by analyzing continuous and categorical variables\non the same footing, based on a mutual information supremum prin-\nciple for finite dataset 5,6. Last, MIIC distinguishes genuine causal\nrelations from putative and latent causal effects 6, that are ubiquitous\nin real-world applications.\nCausalXtract’s causal discovery module for time series data\nIn order to analyze time series datasets, CausalXtract’s causal dis-\ncovery module (tMIIC) aims to learn a time-unfolded graph, Gt,\nwhere each variable is represented by a series of nodes associated\nto its value at different relative time points, Fig. 1c. Such a time-\nunfolded network framework 10–13 is required to account for the tem-\nporal correlation between successive samples in time series data. As-\nsuming that the dynamics can be considered stationary (see Bench-\nmarking of CausalXtract’s causal discovery module section, below),\nthe time-unfolded graph, Gt, should be translationally invariant over\ntime and can be assigned a periodic structure a priori. In addition,\nGt can be restricted to a few time steps from the running time, t,\nback to a maximum time lag, t − τ , since nodes at future time points\n(t′ > t) cannot a priori influence the observed data at current or\nprevious time points (t ′ ⩽ t), Fig. 1c. The maximum time lag τ\nshould be chosen so as to have little effect on the final graphical\nmodel, which can be achieved for instance by setting τ to twice the\naverage relaxation time of the variables of the dataset. In practice,\nwe may also limit the number of time points ν in Gt by introducing\na time increment δτ between consecutive time points, which leads to\nν = τ /δτ time-lagged layers in Gt.\nSuch a compact periodic graphical representation over a sliding\ntemporal window is learned with tMIIC, which extends MIIC causal\ndiscovery method to analyze time series data. First, tMIIC identifies\nall necessary edges involving at least one contemporaneous node at\ntime t, Fig. 1c. Once these time-lagged and contemporaneous nec-\nessary edges have been identified, they are simply duplicated at ear-\nlier time points to enforce the translational invariance of Gt skeleton.\nTime-lagged edges are then pre-oriented with a first arrowhead point-\ning towards the future, considering that current time points cannot\ncause earlier events. Then, contemporaneous and time-lagged edges\ncan be further oriented using MIIC orientation probability scores ap-\nplied to Gt, which may also uncover a second arrowhead (backward in\ntime) for time-lagged edges. This corresponds to time-lagged latent\ncausal effects from unobserved common causes, Fig. 1c.\nLearning such structural models including latent variables from\n5\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\ntime series data was first proposed for time-lagged effects 10 and sub-\nsequently extended to contemporaneous effects 11 by adapting the\nconstraint-based FCI method allowing for latent variables 21. While\ntraditional constraint-based methods suffer from poor recall, the re-\ncent PCMCI 12 / PCMCI+ 22 method improves recall by introducing\nad hoc conditioning rules for auto-correlated time series. By con-\ntrast, tMIIC does not require any ad hoc conditioning rules, as it\nrelies on the same robust information-theoretic strategy as MIIC\nto limit spurious independence and improve edge recall. tMIIC also\ncaptures time-lagged and contemporaneous effects due to latent vari-\nables.\nRelation to Granger-Schreiber temporal causality\nThe concept of temporal causality was originally formulated by\nGranger14 without reference to any graphical model by compar-\ning linear autoregression with or without past values of possible\ncausal variables. This was later extended to non-linear relations by\nSchreiber15,16 using the notion of Transfer Entropy, TX→Y , which\ncan be expressed in terms of multivariate conditional information,\nTX→Y = I(Yt; Xt′<t|Yt′<t) (1)\nwhere Xt′<t and Yt′<t denote the sets of variables, Xt′ and Yt′ ,\ntaken at earlier time points t′ than t.\nWhile Eq. 1 is asymmetric upon X/Y permutation, a simple com-\nparison of Transfer Entropy asymmetry (e.g. TX→Y > T Y →X ⩾ 0)\ndoes not necessarily translate into causal direction as this asymmetry\nis also expected for non-causal relations. Interestingly, this is in fact\nthe absence of Transfer Entropy in one direction ( e.g. TZ→X ≈ 0)\nwhich suggests the possibility of a causal relation in the opposite\ndirection, X → Z, as in the case of v-structures in graph-based\ncausal discovery methods, provided that a latent common cause can\nbe excluded between the two variables (as discussed above).\nWe clarify in Theorem 1 below this relation between temporal\ncausality without reference to any structural model (Eq. 1) and\nstructural causality entailed by time-unfolded causal graphical mod-\nels ( Gt). This highlights the common foundations of temporal and\nstructural causalities beyond their seemingly unrelated definitions.\nTheorem 1. [TY →X = 0 implies temporal (2 var + t) v-structures]\nIf Xt is adjacent to Yt in Gt and TY →X = I(Xt; Yt′<t|Xt′<t) = 0 ,\nthen for all Yt′ adjacent to Yt in Gt, with t′ < t, there is a temporal\n(2 var +t) v-structure, Yt′ → Yt ← Xt, in Gt, Supplementary Fig. 5a.\nProof : if TY →X = I(Xt; Yt′<t|Xt′<t) = 0 , then all pairs (Xt, Yt′ )\nshould be unconnected (assuming ‘faithfulness’, i.e. no coincidental\ncancellation of effects) and all unshielded triples Yt′\nYt Xt should\nbe temporal v-structures, Yt′ → Yt ← Xt, as Yt /∈ Xt′<t in TY →X □\nNote, however, that the converse of Theorem 1 is not true: a\ntemporal v-structure does not imply a vanishing Transfer Entropy,\nas shown with the counterexample in Supplementary Fig. 5b. As a\nresult, the presence of a temporal v-structure, Yt′ → Yt ← Xt in Gt,\ndoes not necessarily imply a vanishing transfer entropy, TY →X = 0,\nas long as there remains an edge between any Yt′′ and Xt, as in\nthe example in Supplementary Fig. 5b. Hence, Granger-Schreiber\ncausality is in fact too restrictive and may miss actual causal effects,\nwhich can be uncovered by structural causal discovery methods like\ntMIIC. In addition, Granger-Schreiber causality is also known to in-\nfer spurious causal associations by excluding the presence of latent\ncommon causes a priori . By constrast, CausalXtract’s causal dis-\ncovery module includes time-delayed as well as synchronous effects\noriginating from unobserved latent variables, as discussed above.\nBenchmarking of CausalXtract’s causal discovery module\nThe performance of CausalXtract’s causal discovery module (tMIIC)\nhas been assessed using Tigramite package 22, which provides differ-\nent methods to learn temporal causal networks from time series data.\nWe compared tMIIC to two methods capable of orienting contem-\nporaneous edges (PC and PCMCI+) and tested three different ker-\nnels for estimating mutual information (Parcorr, GPDC and KNN).\nBenchmark networks and datasets have been chosen to resemble the\nreal-world data analyzed in this study ( i.e. similar number of time\nsteps, network size and degree distribution) and include a large range\nof linear and non-linear relations between variables.\nA first series of datasets was generated for a 15 node benchmark\nnetwork (Supplementary Fig. 1a) with linear combinations of contri-\nbutions inspired by the Tigramite package, Supplementary Table 1.\nRunning times and scores (Precision, Recall, F-score) have been aver-\naged over 10 datasets (Supplementary Fig. 1b) and show that tMIIC\nscores are at par with PC and PCMCI+ using GPDC or KNN ker-\nnels but that tMIIC runs orders of magnitude faster, which enables\nto use tMIIC on much larger datasets of biological interest including\na few tens or hundreds of thousands samples. Only PC or PCMCI+\nusing ParCorr kernel match tMIIC running speed but with signifi-\ncantly lower scores, as Fscores level off around 0.6-0.7 at large sample\nsize, while tMIIC Fscore exceeds 0.9 (Supplementary Fig. 1b).\nImportantly, increasing the number of time-lagged layers from\nτ = 2 (as in the actual model, Supplementary Fig. 1a) to 5 or 10\nlayers in the inferred time-unfolded network (Supplementary Fig. 2)\nleads to very similar network reconstructions for simulated station-\nary data. This demonstrates tMIIC insensitivity to an overestimated\nmaximum lag for the reconstituted network. Interestingly, however,\nwhen the generated data is no longer stationary, increasing the num-\nber of layers leads to multiple self-loops at non-stationary variables,\nwhilst the rest of the network remains relatively unaffected (Supple-\nmentary Fig. 3). It demonstrates that CausalXtract’s causal discov-\nery module is robust to the presence of non-stationary variables but\nrequires long-time range interactions, and therefore multiple time-\nlagged layers, to account for these non-stationary dynamics at spe-\ncific variables. This striking observation on benchmark networks is\nalso consistent with the multiple self-loops observed for a number\nof non-stationary variables in the real-world application on cellular\necosystems, Fig. 2a and Supplementary Fig. 6.\nA second series of more complex datasets was also generated\nfor another 15 node benchmark network (Supplementary Fig. 4a)\nwith non-linear combinations of contributors, Supplementary Ta-\nble 2. Here, tMIIC tends to outperform both PC and PCMCI+,\nin terms of Recall and Fscores, while remaining orders of magnitude\nfaster compared to GPDC and KNN kernels. Only PC or PCMCI+\nusing ParCorr kernel match tMIIC running speed but with signifi-\ncantly lower scores ( i.e. Fscores level off around 0.4-0.5 at large sam-\nple size, while tMIIC Fscore exceeds 0.8). This demonstrates that\nCausalXtract’s causal discovery module (tMIIC) is both a reliable\nand scalable method to discover complex temporal causal relations\nin very large time series datasets including a few hundred thousand\nsamples.\nData availability\nThe original live-cell time-lapse image data and extracted crops are\navailable at: https://doi.org/10.5281/zenodo.7755699.\nCode availability\nThe source code of the CausalXtract pipeline is available at:\nhttps://github.com/miicTeam/CausalXtract. It includes a demo\nR markdown notebook of CausalXtract pipeline, which reproduces\nstep-by-step the results reported in the manuscript, Fig. 2, start-\ning from the original live-cell time-lapse images of the tumor-on-chip\necosystem, Fig. 1a. Tigramite package used for benchmark compar-\nison is available at: https://github.com/jakobrunge/tigramite\nAcknowledgements\nThis work was supported by ITMO Cancer (grant No 20CM106)\nand the European Union’s Horizon 2020 research and innovation\nprogramme under the Marie Skłodowska-Curie grant agreement No\n847718. LD acknowledges support from AMX PhD fellowship, VC\nfrom ARC foundation and NL from CNRS-Imperial College joint\nPhD programme.\n6\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\nReferences\n1. Nguyen, M. et al. Dissecting Effects of Anti-cancer Drugs and\nCancer-Associated Fibroblasts by On-Chip Reconstitution of Im-\nmunocompetent Tumor Microenvironments. Cell Reports 25,\n3884–3893.e3 (2018).\n2. Davies, E. Machine vision 3rd ed. (Morgan Kaufmann, 2004).\n3. Munkres, J. Algorithms for the assignment and transportation\nproblems. J. Soc. Ind. Appl. Math. 5, 32–38 (1957).\n4. Verny, L., Sella, N., Affeldt, S., Singh, P. P. & Isambert, H. Learn-\ning causal networks with latent variables from multivariate infor-\nmation in genomic data. PLoS Comput. Biol. 13, e1005662 (2017).\n5. Cabeli, V. et al. Learning clinical networks from medical records\nbased on information estimates in mixed-type data. PLoS Com-\nput. Biol. 16, e1007866 (2020).\n6. Da Câmara Ribeiro-Dantas, M. et al. Learning interpretable\ncausal networks from very large datasets, application to\n400,000 medical records of breast cancer patients. Preprint at\nhttps://doi.org/10.48550/arXiv.2303.06423 (2023).\n7. Desterke, C. et al. Inferring Gene Networks in Bone Marrow\nHematopoietic Stem Cell-Supporting Stromal Niche Populations.\niScience 23, 101222 (2020).\n8. Sella, N. et al. Interactive exploration of a global clinical network\nfrom a large breast cancer cohort. npj Digital Med 5, 113 (2022).\n9. Assaad, C., Devijver, E. & Gaussier, E. Survey and Evaluation of\nCausal Discovery Methods for Time Series. Journal of Artificial\nIntelligence Research 73, 767–819 (2022).\n10. Entner, D. & Hoyer, P. On Causal Discovery from Time Series\nData using FCI. Proceedings of the 5th European Workshop on\nProbabilistic Graphical Models, PGM 2010 (2010).\n11. Malinsky, D. & Spirtes, P. Causal Structure Learning from Mul-\ntivariate Time Series in Settings with Unmeasured Confounding\nin Proceedings of 2018 ACM SIGKDD Workshop on Causal Dis-\ncovery, CD@KDD 2018 (eds Le, T. D., Zhang, K., Kiciman, E.,\nHyvärinen, A. & Liu, L.) 92 (2018), 23–47.\n12. Runge, J., Nowack, P., Kretschmer, M., Flaxman, S. & Sejdinovic,\nD. Detecting and quantifying causal associations in large nonlinear\ntime series datasets. Science Advances 5 (2019).\n13. Runge, J. et al. Inferring causation from time series in Earth sys-\ntem sciences. en. Nat. Commun. 10, 2553 (2019).\n14. Granger, C. W. J. Investigating causal relations by econometric\nmodels and cross-spectral methods. Econometrica 37, 424 (1969).\n15. Schreiber, T. Measuring Information Transfer. Physical Review\nLetters 85, 461–464 (2000).\n16. Barnett, L., Barrett, A. B. & Seth, A. K. Granger Causality and\nTransfer Entropy Are Equivalent for Gaussian Variables. Physical\nReview Letters 103 (2009).\n17. Driscoll, M. K. & Zaritsky, A. Data science in cell imaging. Journal\nof Cell Science 134, jcs254292 (2021).\n18. Chan, T. F. & Vese, L. A. Active contours without edges. IEEE\nTrans. Image Process. 10, 266–277 (2001).\n19. Masuzzo, P., Van Troys, M., Ampe, C. & Martens, L. Taking aim\nat moving targets in computational cell migration. Trends Cell\nBiol. 26, 88–110 (2016).\n20. Pearl, J. Causality (Cambridge university press, 2009).\n21. Spirtes, P., Glymour, C. N., Scheines, R. & Heckerman, D. Cau-\nsation, prediction, and search (MIT press, 2000).\n22. Runge, J. Discovering contemporaneous and lagged causal rela-\ntions in autocorrelated nonlinear time series datasets in Proceed-\nings of the 36 th Conference on Uncertainty in Artificial Intelli-\ngence (eds Peters, J. & Sontag, D.) 124 (2020), 1388–1397.\n7\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\nSupplementary Fig. 1: Benchmark assessment of CausalXtract’s causal discovery module (tMIIC) using generated time series datasets.\na, Example of a 15 node causal network to generate benchmark time series datasets based on linear combinations of contributions,\nSupplementary Table 1. Examples of temporal causal networks reconstructed by tMIIC based on 100, 1,000 or 10,000 simulated time\nsteps. b, Running times and scores (Precision, Recall, Fscore) averaged over 10 datasets and compared to PC and PCMCI+ methods using\ndifferent kernels (GPDC, KNN, ParCorr); tMIIC is at par with PC and PCMCI+ scores using GPDC and KNN kernels but runs orders of\nmagnitude faster. Only ParCorr kernel matches tMIIC running speed but with significantly lower scores at large sample size, see Methods.\n8\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\nSupplementary Fig. 2: CausalXtract insensitivity to an overestimated maximum lag τ . a, Example of a temporal causal network model\nwith a maximum lag τ = 2 . Corresponding temporal causal networks inferred by CausalXtract’s causal discovery module (tMIIC), from\n1,000 time step stationary time series (Supplementary Table 1), while assuming different maximum lags τ = 2 , 5 or 10. b, Running times\nand scores (Precision, Recall, Fscore) of tMIIC temporal causal network reconstructions for τ = 2 , 5 or 10, averaged over ten stationary\ntime series of 10 to 105 time steps. Overestimating the maximum lag τ has little impact on the reconstructed networks, as long as the time\nseries are stationary, as demonstrated in Supplementary Fig. 3.\n9\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\nSupplementary Fig. 3: CausalXtract sensitivity to non-stationary variables. a, Example of a temporal causal network model (τ = 2) with\na low frequency periodic input (T = 100 ) applied to X8 and a time-linear trend applied to X13. Corresponding temporal causal networks\ninferred by tMIIC from 1,000 time step time series (Supplementary Table 1) including non-stationary inputs to X8 and X13. Increasing\nthe maximum lag from τ = 2 to τ = 5 or 10 leads to the appearence of multiple self-loops, which result from the non-stationary dynamics\nof X8 and X13, whilst the rest of the network remains largely unaffected. b, Running times and scores (Precision, Recall, Fscore ignoring\nX8 and X13 self-loops) of tMIIC causal network reconstructions for τ = 2, 5 or 10, averaged over ten time series of 10 to 105 time steps.\n10\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\nSupplementary Fig. 4: Benchmark assessment of CausalXtract’s causal discovery module (tMIIC) using more complex time series datasets.\na, Example of a 15 node causal network to generate more complex benchmark time series datasets based on non-linear combinations of\ncontributions, Supplementary Table 2. Examples of temporal causal networks reconstructed by tMIIC based on 100, 1,000 or 10,000\nsimulated time steps. b, Running times and scores (Precision, Recall, Fscore) averaged over 10 datasets and compared to PC and PCMCI+\nmethods using different kernels (GPDC, KNN, ParCorr); tMIIC outperforms both PC and PCMCI+, in terms of Recall and Fscores, while\nrunning orders of magnitude faster, except for the ParCorr kernel, which leads, however, to significantly lower scores at large sample size.\n11\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\nYt\nX t Yt’<t X t’<tI(     ;       |        ) = 0\nX X tt’<t\nY Ytt’<t\nYt\nX t\nX t Yt’<t X t’<tI(     ;       |        ) = 0\nYt’\nY Y\nX X\nt\ntt’<t\nt’<t\nX t\nYt’\na b\nSupplementary Fig. 5: Time-unfolded causal network framework and relation to Granger-Schreiber temporal causality. a , A vanishing\nTransfer Entropy, i.e. TY →X = I(Xt; Yt′<t|Xt′<t) = 0 , implies i) the absence of (dashed) edge between Xt and any Yt′ , with t′ < t, and\nii) if Xt is adjacent to Yt, the presence of temporal (2-variable + time) v-structures, Yt′ → Yt ← Xt, for all Yt′ adjacent to Yt, with t′ < t\n(Methods, Theorem 1). These results can be readily extended to include the presence of other observed variables, Vt′⩽t, by redefining\nTransfer Entropy as, TY →X = I(Xt; Yt′<t|Xt′<t, Vt′⩽t), which discards contributions from indirect paths through other observed variables,\nVt′⩽t. b, By contrast, the presence of a temporal (2-variable + time) v-structure, Yt′ → Yt ← Xt does not imply a vanishing Transfer\nEntropy, as long as there remains an edge between any Yt′′<t and Xt. It implies that Granger-Schreiber temporal causality is in fact too\nrestrictive and may overlook actual causal effects, which can be uncovered by graph-based causal discovery methods like CausalXtract’s\ncausal discovery module (tMIIC). Hence, CausalXtract’s time-unfolded network framework, combining graph-based and information-based\napproaches, sheds light on the common foundations of the seemingly unrelated graph-based causality and Granger-Schreiber temporal\ncausality, while clarifying their actual differences and limitations.\n12\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\nSupplementary Fig. 6: Time series of cellular features extracted from the tumor ecosystems. Example of time series of cellular features\nextracted by CausalXtract’s feature extraction module (CellHunter+) from the tumor ecosystems analyzed in this study, Fig. 1a. It includes\ntwo experimental control parameters ( i.e. treatment and CAF presence) and 15 cellular features extracted every 2 minutes over a period\nof two days. Continuous features are highlighted for one trajectory (traj.18), while categorical features are shown for all trajectories.\n13\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\na\nb\nSupplementary Fig. 7: Time-unfolded causal network inferred by CausalXtract. a , Time-unfolded causal network assuming stationary\ndynamics of cellular ecosystems implying translational time invariance of the inferred causal network. b, Only edges involving at least\none contemporaneous variables (i.e. at time t) need to be tested for conditional independence by tMIIC and the remaining edges are then\nduplicated at all previous time steps before assigning orientations when time-lagged latent variables are taken into account, Fig. 1c. Variables\nretaining multiple self-loops with different time-delays correspond to non-stationary variables in Supplementary Fig. 6, in agreement with\nbenchmarks from simulated data including non-stationary variables, Supplementary Fig. 3.\n14\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\nb\na\nc\nSupplementary Fig. 8: Robustness of CausalXtract’s temporal causal networks to variations in sampling rate. Summary causal networks\ninferred by CausalXtract using different sampling rates (δτ ). a, δτ = 8 ts and τ = 80 ts, in time step units (1 ts = 2 min). b, δτ = 7 ts, and\nτ = 84 ts, as chosen automatically by CausalXtract based on the average relaxation time across the 15 monitored variables, τR = 40 ts,\nwhich defines a maximum lag τ = 2 τR = 80 ts. Given a total number of (time-lagged and -unlagged) nodes, chosen to be around 200 nodes\nfor computational efficiency, it leads to 13 temporal layers (ν + 1 = 200 /15 ≃ 13) and a lag increment δτ = τ /ν ≃ 7 ts. This summary\ncausal network corresponds to Fig. 2a. c, δτ = 5 ts and τ = 60 ts, corresponding to τ = ν · δτ with ν + 1 = 13 temporal layers, as in (b).\n15\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\nSupplementary Table 1: 15 nodes model.\nNodes\nX 1\nt ← −0.47 f2(X 1\nt−1) + 0.29 f3(X 2\nt−1) × η1\nX 2\nt ← 0.49 f2(X 2\nt−1) + 0.4 f1(X 1\nt−2) + η2\nX 3\nt ← 0.56 f1(X 3\nt−1) + 0.44 f4(X 4\nt−2) − 0.26 f2(X 10\nt−2) + 0.56 f2(X 4\nt ) + η3\nX 4\nt ← 0.24 f3(X 4\nt−1) − 0.24 f2(X 6\nt−2) − 0.12 f4(X 14\nt−1) × η4\nX 5\nt ← −0.39 f3(X 5\nt−1) − 0.42 f3(X 5\nt−2) − 0.39 f3(X 11\nt ) + η5\nX 6\nt ← −0.32 f2(X 6\nt−1) + η6\nX 7\nt ← −0.17 f4(X 7\nt−1) − 0.17 f1(X 7\nt−2) + η7\nX 8\nt ← 0.39 f4(X 8\nt−1) − 0.46 f4(X 7\nt−1) − 0.39 f3(X 1\nt−1) − 0.4 f3(X 12\nt−2) + η8\nX 9\nt ← −0.34 f1(X 9\nt−1) + 0.43 f3(X 12\nt−2) + η9\nX 10\nt ← 0.2 f1(X 10\nt−1) + 0.18 f4(X 9\nt−2) + 0.17 f1(X 9\nt−1) + 0.48 f3(X 7\nt−1) − 0.26 f4(X 4\nt−1) + η10\nX 11\nt ← 0.41 f2(X 11\nt−1) + 0.54 f3(X 2\nt ) − 0.55 f2(X 12\nt ) + η11\nX 12\nt ← −0.45 f2(X 12\nt−1) − 0.43 f4(X 3\nt−2) − 0.17 f4(X 9\nt−2) × η12\nX 13\nt ← 0.45 f3(X 13\nt−1) + η13\nX 14\nt ← 0.28 f2(X 14\nt−1) + 0.37 f1(X 12\nt−2) × η14\nX 15\nt ← 0.52 f3(X 15\nt−1) + η15\nF unctions\nf1(x) = x\nf2(x) = x (1 − 4 e− x2\n2 )\nf3(x) = x (1 − 4 x3 e− x2\n2 )\nf4(x) = cos(x)\nNoises\nThe η are white noises generated for each node or contribution using a normal distribution:\nη ∼ N (0, 1)\n16\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint \n\nSupplementary Table 2: 15 nodes model with combinations.\nNodes\nX 1\nt ← η − 0.7 f6(u(η + X 1\nt−1)) − 0.87 f5(u(η + (X 14\nt−1 × X 1\nt−2)))\nX 2\nt ← η + 0.65 f1(u(η + X 2\nt−1)) − 0.63 f3(u(η + X 2\nt−2)) + 0.79 f3(u(η + X 5\nt−1))\nX 3\nt ← η − 0.76 f5(u(η + X 3\nt−1)) − 0.59 f6(u(η + X 7\nt−1)) − 0.85 f2(u(η + X 15\nt−1))\n−0.89 f5(u(η + (X 13\nt−2 × X 7\nt−1)))\nX 4\nt ← η − 0.7 f6(u(η + X 5\nt−1)) − 0.86 f2(u(η + X 8\nt−2)) + 0.53 f1(u(η + (X 4\nt−1 × X 9\nt−2)))\nX 5\nt ← η + 0.54 f2(u(η + (X 14\nt−1 × X 6\nt−2)))\nX 6\nt ← η − 0.85 f2(u(η + X 6\nt−1)) − 0.79 f3(u(η + X 3\nt−2)) + 0.59 f1(u(η + X 4\nt−1))\n+0.75 f3(u(η + X 1\nt )) + 0.57 f2(u(η + X 14\nt−1))\nX 7\nt ← η + 0.74 f1(u(η + X 7\nt−1)) + 0.54 f6(u(η + X 9\nt−1)) − 0.53 f2(u(η + (X 9\nt−1 × X 7\nt−1)))\nX 8\nt ← η × (−0.63 f1(u(η + X 6\nt−1)) + 0.81 f5(u(η + X 13\nt )) + 0.53 f6(u(η + (X 6\nt−2 × X 6\nt−1)))\n−0.69 f6(u(η + (X 13\nt × X 6\nt−1))))\nX 9\nt ← η + 0.79 f3(u(η + X 4\nt−2)) + 0.69 f6(u(η + (X 9\nt−1 × X 15\nt−1)))\nX 10\nt ← η + 0.54 f6(u(η + X 10\nt−1))\nX 11\nt ← η + 0.83 f6(u(η + X 11\nt−1)) − 0.76 f4(u(η + X 13\nt−1)) − 0.73 f3(u(η + X 2\nt−1))\n+0.74 f2(u(η + X 4\nt )) − 0.87 f2(u(η + X 10\nt−2)) + 0.72 f4(u(η + X 12\nt−1))\n−0.73 f1(u(η + (X 10\nt−2 × X 13\nt−1)))\nX 12\nt ← η + 0.7 f3(u(η + X 10\nt−1)) − 0.55 f5(u(η + X 9\nt )) − 0.54 f5(u(η + (X 12\nt−1 × X 10\nt−1)))\nX 13\nt ← η − 0.62 f3(u(η + X 14\nt−2)) − 0.61 f1(u(η + (X 13\nt−1 × X 14\nt−2)))\nX 14\nt ← η − 0.78 f6(u(η + X 14\nt−1))\nX 15\nt ← η − 0.68 f4(u(η + X 15\nt−1)) + 0.85 f4(u(η + X 15\nt−2)) − 0.6 f5(u(η + X 10\nt−2))\n+0.68 f6(u(η + X 14\nt−1)) + 0.81 f4(u(η + (X 14\nt−1 × X 10\nt−2)))\nF unctions\nu(x) = max(−1, min(1, x))\nf1(x) = x\nf2(x) = x (1 − 4 e− x2\n2 )/1.52387\nf3(x) = 4 x2\nf4(x) = 8 x3\nf5(x) = 16 x4\nf6(x) = cos(πx)\nNoises\nThe η are white noises generated for each node or contribution using a normal distribution:\nη ∼ N (0, 0.1)\n17\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted February 8, 2024. ; https://doi.org/10.1101/2024.02.06.579177doi: bioRxiv preprint","source_license":"CC-BY-4.0","license_restricted":false}