Pezard L., Martinerie J., Müller J., Varela F. J. & Renault B. (1996) Multichannel measures of average and localized entropy. Physica D 96: 344–354.

Pezard L., Martinerie J., Müller J., Varela F. J. & Renault B.
(

1996)

Multichannel measures of average and localized entropy.
Physica D 96: 344–354.
We present a procedure to quantify spatio-temporal dynamics applied here to brain surface recordings during three distinct cognitive tasks. The method uses 19 sites of EEG recording as spatial embedding for the reconstruction of trajectories, global and local linear indices, and non-linear forecasting methods to quantify the global and local information loss of the dynamics (K-entropy). We show that K-entropy can differentiate between raw and multivariate phase random surrogate data in a significant percentage of EEG segments, and that relevant non-linear indices are best studied in time segments not longer than 4 s. We also find a certain complementarity between local non-linear and linear indices for the discrimination between the three cognitive tasks. Moreover, localized projections onto electrode site of K-entropy provide a new kind of brain mapping with functional significance.

Schwegler H. & Tarumi K. (1986) The “protocell”: A mathematical model of self-maintenance. Biosystems 19(4): 307–315.

Schwegler H. & Tarumi K.
(

1986)

The “protocell”: A mathematical model of self-maintenance.
Biosystems 19(4): 307–315.
The concepts of self-generation, autonomous boundary and self-maintenance are explained briefly. The “protocell” is presented as a model of self-maintenance which is based on simple physical mechanisms of diffusion and reaction. The time evolution of the surface of the protocell is taken into account explicitly in the form of a Stefan condition giving rise to a non-linear feedback of the surface motion to the reaction and diffusion processes inside the protocell. The spatio-temporal dynamics are investigated, particularly in the neighbourhood of the stationary states, showing a self-maintaining behaviour under a certain range of nutritional conditions. Under another set of conditions we find an instability leading to a division process so that the population of protocells becomes self-maintaining instead of the single individual. The presented formulation of the protocell model is crucially improved compared with a previous version which required boundary conditions at infinity. The previous version was not strictly self-maintaining since dynamics outside the cell were essential for its behaviour.