Chart of the variations of the "aster" according to ratio I/E:

     Each respiratory cycle is represented by an "aster" (fresnel diagram) which includes the complex components from 1 to 4 of the FFT of the signal.
Each aster is represented by 8 points placed on the same vertical line and the whole of the points representative of each component is a scatter plot.
We chose the representation in wire of iron to improve legibility.
The cycles were classified under Excel according to their ratio I/E. This ratio varies from ~0.7 to ~1 in the spontaneous breathing of the recorded subject.
In the recordings of studied at rest it does not appear frank discontinuity in the evolution of the harmonics.

     In front of this "regular" evolution of the components we resumed the study of the variations of the form of the respiratory cycle according to ratio I/E with a model worked out in collaboration with André Eberhard.
The model suggested aims to reproduce the form of the respiratory cycle and to approximate the real respiratory cycle as well as possible when I/E varies.
The respected physiological conditions are:
       - a variation compatible with the breathing at rest
       - equality of inspired and expired volumes.
The working hypothesis is that the subject tries to preserve its "pattern" respiratory which would correspond for him to an optimal breathing.
Independently of the validity of this assumption the model can offer a reference we called M.A.C (Mean Adjusted Cycle) for a study of the variations of the form of the cycle under different experimental conditions.

Model:

      The inspiration and expirations are modelled using Bernstein polynomials which one can vary the parameters (number and amplitude of the points of the polygon of control). The desired curves are modelled by the displacement of points of control observing the conditions of equality of inspiratory and expiratory volumes.
When the parameters are acquired, one can vary I/E by moving the point of passage from inspiration toexpiration using the mouse or the keys < - - and --> of the keyboard.

     The aster is visualized and the variations of its components according to I/E are recorded in a file allowing a later study under Excel.
The respiratory signal reconstituted starting from the components of the aster is superimposed (blue) on the signal of origin (red)
For our study we introduced the possibility of providing to the program a preset form. In the event the "average cycle" of a subject (with 3 points of control for inspiration and 4 points for expiration), corresponding to its "respiratory identity card" defined by the former studies of the PRETA laboratory.

Results of simulation :

     Below we superimposed the two graphs after a correction of scale. We observe an encouraging correlation between the two layouts which is found in the few recordings we carried out and this in spite of the imperfections of the model.

    After reconstitution of the signals from Asters we can visualize the real signal, the M.A.C and the difference between them on Excel (partial graphics)

Matlab programs of the model:
File compressed Stuffit
Autoextractible compressed file (BernsteinCycle.exe)