A fractal revival.
Self-similarity in time.
Thanks to the interest of my colleague Miguelina Guirao, a former collaborator of S. S. Stevens at Harvard and director of the Laboratorio de Investigaciones Sensoriales LIS in Buenos Aires, I dug in my old protocols and notes of 1979 to unearth my fractal model of eye movements. With her help we recalculated some experimental samples and obtained a statistical expression of saccadic eye movements during a free search (geometric mean of 5 subjects) and the negative slope of the log/log transform of the data, with a fractal dimension D = 1.27 (fig. 30).
Figure 30. Log/log graph of number (frequency) and amplitude (degrees) of saccades D = 1.27, with negative slope.
But as usual these new and rigourous calculations led me to new insights in the subject matter, some 15 years later! I invited my young colleague Juan P. Garrahan, a graduate student at Oxford and now a Ph.D.candidate in theoretical physics at the University of Buenos Aires, to perform some computer simulations on the model. As told before my model is based in two "machines" or programs: a) the hyperbolic distribution of the amplitude of saccades and b) the isotropic distribution of the angle of saccades in all the directions of the visual space. At every fixation point a new saccade of some amplitude and direction is generated by these two programs. We can play now with the saccadic path, changing only the fractal dimension D. As a result we obtain a family of different simulated trajectories that can be compared with the real ones. Unfortunately there is not the equivalent of a "holter" equipment for eye movements tracking during long periods of time but the computer simulation may run for hours reproducing a visual free search for a determinate fractal dimension D (two short runs are shown in fig. 15). We can now simulate any kind of ocular movements for a fixed temperature of sight 1/D.
A new concept has appeared to me after so many years of latency. Instead of searching for a hidden self-similarity in the "saccadic tree", I discovered that a collection of saccades could be self-similar in time. In fact, the complex path of saccades in short or long ranges of time is self-similar. Thus scaling remains intrinsic to the fractal nature of eye movements. Of course this is only supported by a computer simulation of saccadic movements and implies the same D in short or long records of saccades. But this hypothesis can now be experimentally tested. Therefore the fractal model has a simple scaling property in time, that could be of some predictive importance. This idea will be submitted to publication as the 7th version of this fractal model. It is interesting to note that the concept of self-similarity for different scales of time is a re-interpretation of my first reading of physical Brownian movements, that triggered this whole research. Perrin, quoted by Mandelbrot (1982) suggested self-similarity as following : "if the particle positions were marked down 100 times more frequently, each interval would be replaced by a polygon smaller than the whole drawing but as complicated, and so on" (my emphasis). In our case each computer graph is "as complicated" as the other, for different running times of the simulation!
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