A Poincaré plot, named after

Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

, is a type of

recurrence plot In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment i in time, the times at which the state of a dynamical system returns to the previous state at i, i.e., when the phase space trajectory visits rou ...

used to quantify

self-similarity __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...

in processes, usually

periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...

s. It is also known as a return map. Poincaré plots can be used to distinguish chaos from randomness by embedding a data set in a higher-dimensional

state space A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the toy ...

. Given a time series of the form :

x_t, x_, x_, \ldots,

a return map in its simplest form first plots (''x''_''t'', ''x''_''t''+1), then plots (''x''_''t''+1, ''x''_''t''+2), then (''x''_''t''+2, ''x''_''t''+3), and so on.

Applications in electrocardiography

An electrocardiogram (ECG) is a tracing of the voltage changes in the chest generated by the heart, whose contraction in a normal person is triggered by an electrical impulse that originates in the

sinoatrial node The sinoatrial node (also known as the sinuatrial node, SA node or sinus node) is an oval shaped region of special cardiac muscle in the upper back wall of the right atrium made up of cells known as pacemaker cells. The sinus node is approximat ...

. The ECG normally consists of a series of waves, labeled the P, Q, R, S and T waves. The P wave represents

depolarization In biology, depolarization or hypopolarization is a change within a cell, during which the cell undergoes a shift in electric charge distribution, resulting in less negative charge inside the cell compared to the outside. Depolarization is esse ...

of the atria, the Q-R-S series of waves depolarization of the ventricles and the T wave

repolarization In neuroscience, repolarization refers to the change in membrane potential that returns it to a negative value just after the depolarization phase of an action potential which has changed the membrane potential to a positive value. The repolarizat ...

of the ventricles. The interval between two successive R waves (the RR interval) is a measure of the heart rate. The heart rate normally varies slightly: during a deep breath, it speeds up and during a deep exhalation, it slows down. (The RR interval will shorten when the heart speeds up, and lengthen when it slows.) An RR tachograph is a graph of the numerical value of the RR-interval versus time. In the context of RR tachography, a Poincaré plot is a graph of RR(''n'') on the ''x''-axis versus RR(''n'' + 1) (the succeeding RR interval) on the ''y''-axis, i.e. one takes a sequence of intervals and plots each interval against the following interval. The recurrence plot is used as a standard visualizing technique to detect the presence of oscillations in non-linear dynamic systems. In the context of electrocardiography, the rate of the healthy heart is normally tightly controlled by the body's regulatory mechanisms (specifically, by the

autonomic nervous system The autonomic nervous system (ANS), formerly referred to as the vegetative nervous system, is a division of the peripheral nervous system that supplies viscera, internal organs, smooth muscle and glands. The autonomic nervous system is a control ...

). Several research papers demonstrate the potential of ECG signal-based Poincaré plots in detecting heart-related diseases or abnormalities. PoincarePlot

References

Scaling symmetries Dynamical systems Chaos theory Statistical charts and diagrams

Plot Plot or Plotting may refer to: Art, media and entertainment * Plot (narrative), the story of a piece of fiction Music * ''The Plot'' (album), a 1976 album by jazz trumpeter Enrico Rava * The Plot (band), a band formed in 2003 Other * ''Plot' ...

{{chaos-stub

Applications in electrocardiography

See also

References