ManyLands: A Journey Across 4D Phase Space of Trajectories

Mathematical models of ordinary differential equations are used to describe and understand biological phenomena. These models are dynamical systems that often describe the time evolution of more than three variables, i.e., their dynamics take place in a multi-dimensional space, called the phase space. Currently, mathematical domain scientists use plots of typical trajectories in the phase space to analyze the qualitative behavior of dynamical systems. These plots are called phase portraits and they perform well for 2D and 3D dynamical systems. However, for 4D, the visual exploration of trajectories becomes challenging, as simple subspace juxtaposition is not sufficient. We propose ManyLands to support mathematical domain scientists in analyzing 4D models of biological systems. By describing the subspaces as Lands, we accompany domain scientists along a continuous journey through 4D HyperLand, 3D SpaceLand, and 2D FlatLand, using seamless transitions. The Lands are also linked to 1D TimeLines. We offer an additional dissected view of trajectories that relies on small-multiple compass-alike pictograms for easy navigation across subspaces and trajectory segments of interest. We show three use cases of 4D dynamical systems from cell biology and biochemistry. An informal evaluation with mathematical experts confirmed that ManyLands helps them to visualize and analyze complex 4D dynamics, while facilitating mathematical experiments and simulations. CCS Concepts • Human-centered computing → Scientific visualization; Visual analytics; Web-based interaction; c © 2019 The Author(s) Computer Graphics Forum c © 2019 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. A. Amirkhanov et al. / ManyLands: A Journey Across 4D Phase Space of Trajectories

[1]  David H. Laidlaw,et al.  Exploring 3D DTI Fiber Tracts with Linked 2D Representations , 2009, IEEE Transactions on Visualization and Computer Graphics.

[2]  Torsten Möller,et al.  Sliceplorer: 1D slices for multi‐dimensional continuous functions , 2017, Comput. Graph. Forum.

[3]  Orlando Merino,et al.  Discrete Dynamical Systems and Difference Equations with Mathematica , 2002 .

[4]  Peter L. Brooks,et al.  Visualizing data , 1997 .

[5]  Geoffrey E. Hinton,et al.  Visualizing Data using t-SNE , 2008 .

[6]  Peter Szmolyan,et al.  Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle , 2016, Journal of mathematical biology.

[7]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[8]  Daniel Weiskopf,et al.  Continuous Scatterplots , 2008, IEEE Transactions on Visualization and Computer Graphics.

[9]  Chris North,et al.  A radial focus+context visualization for multi-dimensional functions , 2002, IEEE Visualization, 2002. VIS 2002..

[10]  Rüdiger Westermann,et al.  Visual Analysis of Spatial Variability and Global Correlations in Ensembles of Iso‐Contours , 2016, Comput. Graph. Forum.

[11]  Wolfgang Berger,et al.  Eurographics/ Ieee-vgtc Symposium on Visualization 2010 Hypermoval: Interactive Visual Validation of Regression Models for Real-time Simulation , 2022 .

[12]  Axel Kowald,et al.  Systems Biology - a Textbook , 2016 .

[13]  Ivan Viola,et al.  Metamorphers: storytelling templates for illustrative animated transitions in molecular visualization , 2017, SCCG.

[14]  John T. Stasko,et al.  Effectiveness of Animation in Trend Visualization , 2008, IEEE Transactions on Visualization and Computer Graphics.

[15]  Jeffrey Heer,et al.  Animated Transitions in Statistical Data Graphics , 2007, IEEE Transactions on Visualization and Computer Graphics.

[16]  Ben Shneiderman,et al.  Using vision to think , 1999 .

[17]  Daniel Asimov,et al.  The grand tour: a tool for viewing multidimensional data , 1985 .

[18]  Jock D. Mackinlay,et al.  Cone Trees: animated 3D visualizations of hierarchical information , 1991, CHI.

[19]  Pierre Dragicevic,et al.  Rolling the Dice: Multidimensional Visual Exploration using Scatterplot Matrix Navigation , 2008, IEEE Transactions on Visualization and Computer Graphics.

[20]  Rüdiger Westermann,et al.  Time-Hierarchical Clustering and Visualization of Weather Forecast Ensembles , 2017, IEEE Transactions on Visualization and Computer Graphics.

[21]  E. Abbott,et al.  Flatland: a Romance of Many Dimensions , 1884, Nature.

[22]  Richard A. Becker,et al.  Brushing scatterplots , 1987 .

[23]  Jack Snoeyink,et al.  Computing contour trees in all dimensions , 2000, SODA '00.

[24]  Benjamin B. Bederson,et al.  Does animation help users build mental maps of spatial information? , 1999, Proceedings 1999 IEEE Symposium on Information Visualization (InfoVis'99).

[25]  Ivan Viola,et al.  DimSUM: Dimension and Scale Unifying Map for Visual Abstraction of DNA Origami Structures , 2018, Comput. Graph. Forum.

[26]  Helwig Löffelmann,et al.  Visualizing the behaviour of higher dimensional dynamical systems , 1997 .

[27]  Alfred Inselberg,et al.  The plane with parallel coordinates , 1985, The Visual Computer.

[28]  G. Feichtinger,et al.  The geometry of Wonderland , 1996 .

[29]  Sebastian Grottel,et al.  Visual Analysis of Trajectories in Multi‐Dimensional State Spaces , 2014, Comput. Graph. Forum.

[30]  Gerik Scheuermann,et al.  LineAO - Improved Three-Dimensional Line Rendering , 2013, IEEE Trans. Vis. Comput. Graph..

[31]  Ross T. Whitaker,et al.  Curve Boxplot: Generalization of Boxplot for Ensembles of Curves , 2014, IEEE Transactions on Visualization and Computer Graphics.

[32]  J Jaap Molenaar,et al.  Continuum modeling in the physical sciences , 2007, Mathematical modeling and computation.

[33]  Roberto Tamassia,et al.  Algorithms for Plane Representations of Acyclic Digraphs , 1988, Theor. Comput. Sci..

[34]  Valerio Pascucci,et al.  Visual Exploration of High Dimensional Scalar Functions , 2010, IEEE Transactions on Visualization and Computer Graphics.

[35]  Michael Gleicher,et al.  A Framework for Considering Comprehensibility in Modeling , 2016, Big Data.

[36]  Helwig Löffelmann,et al.  Visualizing the behaviour of higher dimensional dynamical systems , 1997, Proceedings. Visualization '97 (Cat. No. 97CB36155).

[37]  D. M. Y. Sommerville,et al.  An Introduction to The Geometry of N Dimensions , 2022 .

[38]  Allison Woodruff,et al.  Guidelines for using multiple views in information visualization , 2000, AVI '00.

[39]  Tobias Isenberg,et al.  A Systematic Review on the Practice of Evaluating Visualization , 2013, IEEE Transactions on Visualization and Computer Graphics.

[40]  A. Hanson,et al.  Meshview : Visualizing the Fourth Dimension , 1999 .

[41]  Torsten Möller,et al.  Hypersliceplorer: Interactive visualization of shapes in multiple dimensions , 2018, Comput. Graph. Forum.

[42]  Cynthia A. Brewer,et al.  ColorBrewer in Print: A Catalog of Color Schemes for Maps , 2003 .

[43]  Peter Lindstrom,et al.  Topological Spines: A Structure-preserving Visual Representation of Scalar Fields , 2011, IEEE Transactions on Visualization and Computer Graphics.

[44]  Wilmot Li,et al.  Exploded View Diagrams of Mathematical Surfaces , 2010, IEEE Transactions on Visualization and Computer Graphics.

[45]  Albert Goldbeter,et al.  A model for the dynamics of bipolar disorders. , 2011, Progress in biophysics and molecular biology.

[46]  C. Chicone.  Ordinary Differential Equations with Applications , 1999, Texts in Applied Mathematics.

[48]  Rafael Ballester-Ripoll,et al.  Visualization of High-dimensional Scalar Functions Using Principal Parameterizations , 2018, ArXiv.

[49]  Peter Szmolyan,et al.  Scaling in Singular Perturbation Problems: Blowing Up a Relaxation Oscillator , 2011, SIAM J. Appl. Dyn. Syst..

[50]  Ross T. Whitaker,et al.  Contour Boxplots: A Method for Characterizing Uncertainty in Feature Sets from Simulation Ensembles , 2013, IEEE Transactions on Visualization and Computer Graphics.

[51]  Ken Brodlie,et al.  Visualizing and Investigating Multidimensional Functions , 2002, VisSym.

[52]  Tobias Isenberg,et al.  Depth-Dependent Halos: Illustrative Rendering of Dense Line Data , 2009, IEEE Transactions on Visualization and Computer Graphics.

[53]  Barbara Tversky,et al.  Animation: can it facilitate? , 2002, Int. J. Hum. Comput. Stud..

[54]  R. Abraham,et al.  Dynamics--the geometry of behavior , 1983 .

[55]  Andreas Buja,et al.  Interactive data visualization using focusing and linking , 1991, Proceeding Visualization '91.

[56]  Graham J. Wills,et al.  Linked Data Views , 2008 .

[57]  Christian Kuehn,et al.  Multiscale Geometry of the Olsen Model and Non-classical Relaxation Oscillations , 2014, J. Nonlinear Sci..

[58]  Nancy Argüelles,et al.  Author ' s , 2008 .

[59]  Daniel Weiskopf,et al.  Continuous Parallel Coordinates , 2009, IEEE Transactions on Visualization and Computer Graphics.

[60]  Freddy Dumortier,et al.  Canard Cycles and Center Manifolds , 1996 .

[61]  Sang-Cheol Seok,et al.  Using projection and 2D plots to visually reveal genetic mechanisms of complex human disorders , 2009, 2009 IEEE Symposium on Visual Analytics Science and Technology.

[62]  Martin Wattenberg,et al.  ManyEyes: a Site for Visualization at Internet Scale , 2007, IEEE Transactions on Visualization and Computer Graphics.

[63]  Steven K. Feiner,et al.  Worlds within worlds: metaphors for exploring n-dimensional virtual worlds , 1990, UIST '90.

[64]  K. Sneppen,et al.  Minimal model of spiky oscillations in NF-κB signaling , 2006 .