Scale, self-similarity and the
saccadic tree
"Chaque portion de la
matière peut être conçue comme un jardin plein de
plantes et comme un étang plein de poissons. Mais chaque
rameau de la plante, chaque membre de l'animal, chaque goutte de ses
humeurs est encore un tel jardin ou un tel étang". Leibniz,
"Monadologie".
The
first step of my research was to understand the geometrical nature of
the saccadic paths and I verified, as I expected, its hyperbolic
distribution. Below the surface image of a messy trail of saccades of
different amplitudes and orientations, I could recognize the deep
structure of a simple underlying power function This was an essential
step, of the "empirical" kind, like Zipf's or Pareto's law for words
or salaries. But I also knew that this finding didn't suffice. The
better I understood Mandelbrot's theory the more I became aware of
the importance of scaling and self-similarity to complete the fractal
model of eye movements, but it was difficult, for me at least, to
find a ground for this hypothesis. It is obvious that scaling plays a
quite different role in mathematics, physics and in eye physiology.
Take the Brownian motion of very
fine particles (less than 1 micron) as an example. When its motion is
examined in the microscope (see Perrin's Atoms , 1909, quoted in
Mandelbrot 1977), the succesive positions can be marked at very small
time intervals and joined by segments. The (constructed) prodigious
entangled path left behind is a curve of topological dimension DT=
2)! The disparity between these two values DT< D marks the
"fractal nature" of Brownian motion. Take another case: the small and
large details of coastlines are geometrically identical except for
scale. Coast lengths also increase without bound under close
scrutiny!
The trajectory of saccadic eye
movements is also a monstruous entanglement (see pictures above) but
it cannot be compared to a Brownian path nor to a coastline, because
of its physiological nature. A saccade is a ballistic eye movement
that you cannot control or dissect into smaller and smaller
independent micromovements, although a linear function from
micro-saccades to macro-saccades has been established. Therefore
saccades are certainly not "geometrically" self-similar in the sense
of the other two examples. The search of scaling structures in nature
or society is more difficult than in pure mathematics. Below some
lower limit the concept of coastline ceases to belong to geography,
(Mandelbrot, p. 38) and Pareto also said that his law "non vale per
gli angioli". The same for saccadic movements, I understood that the
scaling problem should be tackled from another point of view. In
order to find some proof I changed from geometry, my first research
stage, to Mandelbrot's lexicographic trees. This was a subtle shift
indeed, but I was guided by the master's hand.
In fact, Mandelbrot, who made the
necessary modifications to the Zipf Law in the fifties, gave also
some new insights about D as a similarity dimension in the field of
linguistics in his 1977 book on fractals. Since the Zipf law of word
frequency is near perfectly hyperbolic, I quote from his 1982
version, "it is eminently sensible to try and relate it to some
underlying scaling property. As suggested by the notation, the
exponent plays the usual role of dimension. An object that could be
scaling does indeed exist in the present case: it is a
lexicographical tree. A lexicographical tree has N+1 trunks, numbered
from 0 to N. The first trunk corresponds to the "word" constituted by
the improper letter "space" taken by itself,and each of the other
trunks corresponds to one of N proper letters.The "space" trunk is
barren but each of the other trunk carries N+1 leaders corresponding
to the "space" and to N proper letters. The next generation space
leader is barren and others branch out into N+1 as before. Hence the
barren tip of each space leader corresponds to a word made of proper
letters followed by a space. And the construction continues ad
infinitum. Each barren tip is inscribed with the corresponding word's
probability. A tree can be termed scaling if each branch taken by
itself is in some way a reduced-scale version of the whole tree" (p.
345).
|
Fig. 19. Graph of the
saccadic tree of a movement of amplitude A =
2°
|
This idea proved enough
for me. Instead of a word written between two "spaces" or #
marks, like #fractal# (word's length = 7 letters) I tried to
represent a saccade of amplitude A as a movement between two
fixation points #, for instance #aaa# (saccadic amplitude =
3 degrees). I represented this tree as the most simple
dichotomic branching. I pictured the amplitude of a saccadic
movement as the sum of (abstract) elements of unit
amplitude, from one stopover or fixation point # to the
other#.
|
I went a step further. I gave an a
priori probability for every elementary unit. The probability of a
saccade of amplitude A could be interpreted as the product of the
probabilities of every elementary unit a. For instance, in the graph
above if each unit has probability 1/2, the final product for a
saccade of amplitude 3° should be 1/8. If we continue the
calculation for larger saccades the resulting graph is, of course, an
hiperbolic function of decreasing probabilities as in the function
obtained for the empirical distribution of amplitude and frecuency of
saccades!
This was certainly not a
coincidence. Perhaps it was a deep intuition. I made the first
drawings of this self-similar probabilistic tree in the "Café
Le Rubi", Place de la Petite Fusterie, a place very dear in my
memory, by the river Rhône. Some sixteen years before, newly
married, my wife and I decided in this same café to rent an
apartment in Fribourg and to spend the whole summer in Switzerland.
Fig.20. Place de la Petite
Fusterie
In the afternoon I returned again
to the laboratory and I discussed this probabilistic saccadic tree
with Bullinger and Kauffman, the two experts who provided me with the
most fascinating computerized eye movements graphs. They told me that
some observations suggested a linear continuum from "micro-saccades"
(of minute amplitude) to "macro-saccades" of the kind I was examining
(larger than 1°). Later (1994) I discovered a striking
similarity between Yarbus' records of eye (micro) movements during
fixation (fig. 21) and Mandelbrot's fractional Brown trails (fig.
22).
|
Fig. 21. Records of eye movements during fixation
on a stationary point by a subject. a) Fixation for 10
sec;b) fixation for 30 sec; c) fixation for 1 min. (Yarbus,
op. cit. fig. 54)
|
Fig. 22. Fractional Brown trails, D~1.1111,
D~1.4285 (Mandelbrot, op cit. plate 255)
|
I imagined a myokinetic tree for eye
movements to replicate Mandelbrot's lexicographic tree in the search
of a deep self-similar fractal structure. I wrote the first English
version (4 pages) of my paper at their Lab. That night I visited
Ariane Etienne's laboratory at the Palais Wilson to see her hamsters
at work. Ariane studied and worked with both Jean Piaget and Konrad
Lorenz, a disciple of two great masters. I walked back by the
splendid lake to my hotel quite late in the night, but I found "Les
Armures" still opened. I entered and asked for a fondue. I was hungry
and tired. The following day I managed to visit the Nestlé
Foundation at Lausanne and a remarkable laboratory of architecture at
the University where clients could play with 1-1 maquettes of their
future homes. Again the problem of scale and my personal story (III)
with urban and open spaces.
The next days I continued my
analysis of the power function, log /log transforms and the like for
a while. An incredible amount of events follow a power function law,
but not of all them have a fractal interpretation. Zipf based this
most general behavior on the "principle of the least effort". In fact
his 1949 book is called Human behavior and the principle of least
effort. I wonder at that time wether this "principle" could explain
saccadic eye movements too. Some months later I received a charming
letter from Mandelbrot telling me that he appreciated my findings but
he wasn't very sure about Zipf's interpretation of the law of least
effort. In the last version of my paper I eliminated it.
