Point three is a very valuable comment in this respect, I believe. It is very common to attribute behavioral variability to random noise and therefore it is straightforward to analyze animal behavior using stochastic methods. However, the variability in animal behavior has a decided evolutionary advantage. It has been selected for and is under control of the brain. Using stochastics to analyze animal behavior with the goal of a neurobiological explanation is therefore doomed to fail. We also tried to analyze fly behavior in this straightforward way and also failed. Specifically, we also found a Lévy distribution in our data and later found that the Lévy-like behavior of the flies was not due to randomness at all:
<a href="http://www.plosone.org/do...">Maye et al. 2007, PLoS One</a>

1. The intent of this paper was to introduce a set of metrics defined on the trajectory of a fly to quantify locomotor behavior. An application of these metrics to a larger set of flies to answer specific hypotheses about fly behavior, will be published elsewhere. This was not the intent here, and we feel that data from a single fly was sufficient to introduce the trjectory measures. This should not be taken to mean that we have only looked at data from a single fly, rather that this was the data set we used in the paper to illustrate the measures.

As for how many flies will be needed to gain control over individual variability, this is an interesting question, and the answer is not a priori obvious. While it is customary in T-maze experiments to use N=100, this is not the case in studies involving individual flies; for example, in flight simulator studies of flight behavior of individual flies, N=10 has been used (see for example the paper by Frye and Dickinson in Journal of Experimental Biology 207, 123-131 (2004)).

2. The fact that we use constructs from stochastic process theory to quantify the locomotor trajectory of a fly, does not have any bearing on the validity, or the lack thereof, of psychological constructs such as intentionality. Observations of human behavior could equally well be quantified using stochastic process theory without any implications, about the presence or absence of intentionality in the agent. Stochastic process theory in describing time series, and psychological explanations in terms of motivations and intentions, are both epistemological constructs having their limited domains of validity when describing behavior. Conflating their domains of application is to produce epistemological confusion. In particular, if the goal is to develop bona fide statistical measures (as was our goal in this paper), invocation of internal psychological states of the fly has no relevance.

We are well prepared to believe that if one had a well developed theory of "fly psychology", then *within* that theory, intentionality may well play an important role. In the human, it certainly does, and given biological and evolutionary continuity, it would not be a great surprise if such a construct were to be useful in the fly as well. However, this has little bearing, we feel, on what are good statistical measures, to be used in practical experimental settings (such as phenotyping), where the more important questions are to get robust and parsimonious statistical measures, for which sampling properties can be determined, etc.

3. We are well aware that the Wiener process is a Levy process. We wished to avoid being pedantic (while it is true that the Levy distributions include the Gaussian, the Gaussian process is sufficiently familiar that it perhaps does not require this extra characterization in a paper written for a biology audience). By Levy flights we were referring to processes corresponding to long tailed Levy laws. If credentials are necessary here, perhaps we can point to a recently published book from one of the co-authors of the paper, including a tutorial on stochastic process theory (Observed Brain Dynamics, OUP, by Mitra and Bokil, see http://www.amazon.com/Obs...).

Our intent in the brief discussion where this reference is made, which is indeed not central to the paper, was to point to the inappropriateness of either Gaussian processes or long tailed Levy processes for describing the trajectories at hand. The Gaussian processes are simply ruled out (the marginal distribution of the velocity is strongly non-Gaussian). Levy flights (heavy tailed) are also similarly easy to rule out for our data set. Such Levy laws have proved to be a wild goose chase in the past in describing animal behavior: see
Nature 449, 1044-1048 (25 October 2007)
Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer
http://www.nature.com/nat...
and associated media coverage
http://news.independent.c...

Heavy tailed distributions appear frequently in biology due to underlying heterogeneity, but fitting these heavy tailed distributions to power laws may have less grounding in underlying phenomena than in the case of phase transitions and critical phenomena. We for ourselves have not found such fits of very much use.