User:Lakinekaki/Bios theory

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Bios theory, little quoted theory developed by H. Sabelli and colleagues, attempts to characterise the behavior of certain nonlinear dynamical systems that are sensitive to initial conditions and generate novelty (as defined by Sabelli and colleagues). Biotic patterns are found in physical, biological and psychological processes [1], including heartbeat intervals and other physiological processes, meteorological time series, economic data. According to authors, bios seems to be a generic, widespread pattern of natural processes, extending from physics to psychobiology. Bios is generated mathematically by different equations[2].

Contents

[edit] History

RRI (heart rate intervals), bios and chaos (generated with equation written above)
RRI (heart rate intervals), bios and chaos (generated with equation written above)

Biotic pattern was first defined in the study of the heart rate variation, when it was reproduced with mathematical recursions [3]. Although this pattern and equation have been studied before, Louis Kauffman and Hector Sabelli thought that this pattern deserves a distinct name and they called it bios. Much of the mathematics of bios theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand.

The term bios comes from the greek: bio-, comb. form of bios "life, course or way of living" (as opposed to zoe "animal life, organic life"). It was adopted by physician Hector Sabelli for this theory.

[edit] Mathematical bios

Mathematical bios was first defined as a distinct phase in the time series generated with the recursion

A(t+1) = A(t) + g*sin(A(t)) \,.

When g is kept constant, then, depending on its value, this recursion generates either a steady state, periodicity, chaos, bios or infinitation (output increases in size toward infinity). When g = k*t, where k is a small constant, this recursions generates all above patterns on different values of g. Without a conserved term, A(t), this recursion cannot produce bios. This recurrence relation is related to the circle map.

Steady state, bifurcations, chaotic phase, and bios in development of recursive equation. Note that y-axis does not show expanding series, so that bifurcations and chaos could be seen.
Steady state, bifurcations, chaotic phase, and bios in development of recursive equation. Note that y-axis does not show expanding series, so that bifurcations and chaos could be seen.
Transition from chaos to bios (detail from the image above with y-axis showing the full range of the series and x-axis focusing on the transition/expansion of the series).
Transition from chaos to bios (detail from the image above with y-axis showing the full range of the series and x-axis focusing on the transition/expansion of the series).


[edit] Biotic motion

In order to classify the behavior of a system as biotic, the system must exhibit the following properties:

  • it must be sensitive to initial conditions
  • it must generate temporal patterning
  • it must generate complexity
  • it must generate diversity
  • it must generate asymmetry
  • it must generate novelty

For the understanding of novelty, we have to understand recurrence plots first.

Recurrence plot enables us to investigate the m-dimensional phase space trajectory through a two-dimensional representation of its recurrences. Such recurrence of a state at time i at a different time j is pictured within a two-dimensional squared matrix with black and white dots, where black dots mark a recurrence, and both axes are time axes. This representation is called recurrence plot.

Sabelli made a distinction between two types of recurrences: isometry recurrence (or shorter isometry) and similarity recurrence. For the definition of novelty, definition of isometry recurrence is necessary.

[edit] Isometry recurrence

Recurrence (isometry) plot can be mathematically expressed as

\mathbf{R}(i,j) = \Theta(\varepsilon - |(||\vec{x}(i)|| - ||\vec{x}(j)||)|), \quad \vec{x}(i) \in \Bbb{R}^m, \quad i, j=1, \dots, N,

where N is the number of considered states \vec{x}(i), \varepsilon is a threshold distance, || \cdot || a norm (Euclidean norm), \Theta( \cdot ) the Heaviside step function, and

\vec{x}(i) = (u(i), u(i+1), \ldots, u(i+m-1)),

where u(i) is the time series, m the embedding dimension.

The difference between isometry and similarity recurrences can be expressed as follows:

in similarity recurrence \vec{x}(i)\approx \vec{x}(j),\,

in isometry recurrence ||\vec{x}(i)||\approx ||\vec{x}(j)||,\,

which means that for the similarity recurrence both the direction and length of vectors are being calculated, while for isometry recurrence only length of vectors is being calculated.

[edit] Novelty

Novelty can be detected in the embedded series (m > 1). The time series x where total number of isometric points is smaller than the total number of isometric points in the series x after shuffling, has a feature called novelty.

Novelty can be measured with recurrence and embedding plots.

[edit] Isometry Recurrence plots

Different series show different patterns when presented in recurrence plots. Random and chaotic series produce uniform plots. Periodic series produce periodic plots. Biotic series produce plots with distinct complexes that are lost after shuffling the data.

Recurrence plot (embedding 10) of longitude of a part of a British coastline
Recurrence plot (embedding 10) of longitude of a part of a British coastline
Recurrence plot (embedding 10) of mathematically generated bios.
Recurrence plot (embedding 10) of mathematically generated bios.
Recurrence plot (embedding 10) of mathematically generated chaos.
Recurrence plot (embedding 10) of mathematically generated chaos.


Total number of calculated isometries as percent of the number of total possible isometries is a useful quantification that is used in embedding plots.

[edit] Embedding plots

Embedding plots display quantifications of recurrence plots[4] of the original series (bold line) and its shuffled copy (thin line) on many embedding dimensions. Each point in embedding plot represents the number of isometries (as percent of all possible isometries) calculated in the recurrence plot for the corresponding embedding dimension. When this is done for many embeddings, and also for the shuffled series, few patterns become evident. In general, number of isometries can: decrease with shuffling as observed with chaotic series; increase with shuffling (novelty) as observed with biotic series; periodically increase and decrease in original series reaching the maximum when the number of embedding corresponds with the period of the series, as observed with periodic series.

Embedding plot for time series generated with the process equation with g=4.43 (chaos) r=10% of the range. Shuffling the series decreases the number of isometries.
Embedding plot for time series generated with the process equation with g=4.43 (chaos) r=10% of the range. Shuffling the series decreases the number of isometries.
Embedding plot for time series generated with the process equation with g=4.65 (bios) r=10% of the range. Shuffling the series increases the number of isometries (novelty).
Embedding plot for time series generated with the process equation with g=4.65 (bios) r=10% of the range. Shuffling the series increases the number of isometries (novelty).
Embedding plot of the sine wave. Shuffling the series eliminates periodicity.
Embedding plot of the sine wave. Shuffling the series eliminates periodicity.


[edit] Criticisms of bios theory

One criticism of bios theory is that it represents simply a diffusion of a chaotic system. This criticism is disputed by proponents of the theory, on the ground that bios can also be bounded (as is demonstrated with some recursions) and with finding properties that define a bounded bios in the heart-rate interval series (RRI).

What characterizes bios is not diffusion but the generation of novelty, which is absent in non-diffusive chaotic attractors but is present in non-diffusive bios (it is independent from diffusion). Novelty can be easily observed in embedding plots.

[edit] See also

[edit] References

[edit] Academic papers

  1. ↑  Kauffman, L. and Sabelli, H. (2003) Mathematical Bios. Kybernetes 31: 1418-1428.
  2. ↑  Kauffman, L. and Sabelli, H. (1998) The Process equation. Cybernetics and Systems 29: 345-362
  3. ↑  Zbilut, J.P. and C.L. Webber. (1992). Embeddings and delays as derived from quantification of recurrence plots. Physics letters A 171: 199-203
  4. ↑  Sabelli, H. (2001). Novelty, a Measure of Creative Organization in Natural and Mathematical Time Series. Nonlinear Dynamics, Psychology, and Life Sciences. 5: 89-113.
  5. ↑  A. Sugerman and H. Sabelli. Novelty, Diversification And Nonrandom Complexity Define Creative Processes. Kybernetes 32: 829-836, 2003
  6. ↑  Hector C. Sabelli, Linnea Carlson-Sabelli, Minu K. Patel, Joseph P. Zbilut, Joseph V. Messer, and Karen Walthall, Psychocardiological Portraits: A Clinical Application of Process Theory. In Chaos theory in Psychology (1995), F. D. Abraham and A. R. Gilgen (Eds). Greenwood Publishing Group, Inc., Westport, CT. pp 107-125.

[edit] Semitechnical and popular works

  • Sabelli, Hector (2005). Bios: a Study of Creation (with Bios Data Analyzer on CD-Rom). World Scientific. ISBN 9812561676. 

[edit] External links

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