Open Access Highly Accessed Technical advance

Dynamical density delay maps: simple, new method for visualising the behaviour of complex systems

Anton Burykin1, Madalena D Costa12, Luca Citi13 and Ary L Goldberger12*

Author Affiliations

1 Wyss Institute for Biologically Inspired Engineering at Harvard University, Boston, MA 02115, USA

2 Margret and H.A. Rey Institute of Nonlinear Dynamics in Physiology and Medicine, Division of Interdisciplinary Medicine and Biotechnology, Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, MA 02215, USA

3 School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park, Colchester CO4-3SQ, UK

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BMC Medical Informatics and Decision Making 2014, 14:6  doi:10.1186/1472-6947-14-6

Published: 18 January 2014



Physiologic signals, such as cardiac interbeat intervals, exhibit complex fluctuations. However, capturing important dynamical properties, including nonstationarities may not be feasible from conventional time series graphical representations.


We introduce a simple-to-implement visualisation method, termed dynamical density delay mapping (“D3-Map” technique) that provides an animated representation of a system’s dynamics. The method is based on a generalization of conventional two-dimensional (2D) Poincaré plots, which are scatter plots where each data point, x(n), in a time series is plotted against the adjacent one, x(n + 1). First, we divide the original time series, x(n) (n = 1,…, N), into a sequence of segments (windows). Next, for each segment, a three-dimensional (3D) Poincaré surface plot of x(n), x(n + 1), h[x(n),x(n + 1)] is generated, in which the third dimension, h, represents the relative frequency of occurrence of each (x(n),x(n + 1)) point. This 3D Poincaré surface is then chromatised by mapping the relative frequency h values onto a colour scheme. We also generate a colourised 2D contour plot from each time series segment using the same colourmap scheme as for the 3D Poincaré surface. Finally, the original time series graph, the colourised 3D Poincaré surface plot, and its projection as a colourised 2D contour map for each segment, are animated to create the full “D3-Map.”


We first exemplify the D3-Map method using the cardiac interbeat interval time series from a healthy subject during sleeping hours. The animations uncover complex dynamical changes, such as transitions between states, and the relative amount of time the system spends in each state. We also illustrate the utility of the method in detecting hidden temporal patterns in the heart rate dynamics of a patient with atrial fibrillation. The videos, as well as the source code, are made publicly available.


Animations based on density delay maps provide a new way of visualising dynamical properties of complex systems not apparent in time series graphs or standard Poincaré plot representations. Trainees in a variety of fields may find the animations useful as illustrations of fundamental but challenging concepts, such as nonstationarity and multistability. For investigators, the method may facilitate data exploration.

Atrial fibrillation; Delay map; Heart rate variability; Nonlinear dynamics; Poincaré plot; Sleep; Time series; Visualisation