Brains are usually described as input/output systems: they transform sensory input into motor output. However, the motor output of brains (behavior) is notoriously variable, even under identical sensory conditions. The question of whether this behavioral variability merely reflects residual deviations due to extrinsic random noise in such otherwise deterministic systems or an intrinsic, adaptive indeterminacy trait is central for the basic understanding of brain function. Instead of random noise, we find a fractal order (resembling Lévy flights) in the temporal structure of spontaneous flight maneuvers in tethered Drosophila fruit flies. Lévy-like probabilistic behavior patterns are evolutionarily conserved, suggesting a general neural mechanism underlying spontaneous behavior. Drosophila can produce these patterns endogenously, without any external cues. The fly's behavior is controlled by brain circuits which operate as a nonlinear system with unstable dynamics far from equilibrium. These findings suggest that both general models of brain function and autonomous agents ought to include biologically relevant nonlinear, endogenous behavior-initiating mechanisms if they strive to realistically simulate biological brains or out-compete other agents.

Analyzing the structure of behavioral variability may provide evidence for understanding whether the variability is the result of cumulated errors in an imperfectly wired brain (system noise) or whether the variability is under neural control. In this study, we take advantage of turning behavior in tethered Drosophila; this system provides superb control over the perceived environment for a true assessment of the spontaneity of the behavior, while at the same time offering easily quantifiable behavioral dynamics ( Fig. 2 ). Most importantly, we eliminate any potential nonlinear effects which could arise from a closed reafferent feedback loop between the animal's behavior and its environment by opening this loop to study intrinsically generated behavior, without any environmental feedback. Thus, the environment is kept so constant (both between and within experiments), that any remaining minute variation in it must be infinitely smaller than any of the stimuli known to trigger turning behavior [30] . Moreover, the temporal distribution of any such remaining environmental fluctuations can be assumed to be Gaussian. We know of no other intact preparation affording such minute control. We chose the temporal sequence of highly stereotyped flight maneuvers producing short bursts of yaw-torque (‘torque spikes’; corresponding to body-saccades in free flight [31] ) for our analysis, because they have been repeatedly both classified as single units of behavior and used for quantitative behavioral analysis. Tethered Drosophila produce these spikes in a probabilistic manner not only in response to visual stimulation [14] , but also if the stimulus situation is constant [30] (see also Figs. S1 and S2 ). Freely flying flies do not offer this distinction, as one cannot discern spontaneous body-saccades from elicited body-saccades [32] .

A–According to the robot-hypothesis, there is an unambiguous mapping of sensory input to behavioral output. If the behavioral output is not constant in a constant environment, there are a number of possible sources of noise, which would be responsible for the varying output. B–In a competing hypothesis, non-constant output is generated intrinsically by an initiator of behavioral activity. Note that the sources of noise have been omitted in B merely because their contribution may be small, compared to that of the initiator, not because they are thought to be non-existent.

However, even the best-understood behavioral systems display a residual of variability, which has so far prevented exact predictability of individual behavior. There are a number of systems from single neurons and synapses [15] , [16] to invertebrate [17] , [18] and vertebrate animals including humans [19] – [21] , which even generate variable output despite no variations in input at all, leading to difficulties reproducing even tightly controlled experiments [22] . This variability is often classified as random noise, a by-product of a complex brain [23] , [24] . Documented sources of noise range from genetic and historical variations [23] to neural noise [24] , [25] or stochastic fluctuations in macromolecule number [26] . This noise requires compensatory homeostatic mechanisms to ensure stable neuronal and network function over extended periods of time [27] . Because of the obvious analogy, we term the hypothesis that brains are deterministic input/output systems with added noise the ‘robot-hypothesis’ ( Fig. 1a ). A less prominent alternative explanation contends that some of the variability is adaptive and irreducible [19] , [20] , [28] . According to this latter view, individual behavior is fundamentally indeterministic (not fundamentally deterministic but noisy) and precise prediction principally (not only technically) impossible ( Fig. 1b ). It is critical to emphasize at this point that the processes leading to behavioral indeterminacy may very well be deterministic: indeterministic output of deterministic systems is a well-known phenomenon [29] .

According to Laplace, randomness is only a measure of our “ignorance of the different causes involved in the production of events.” [1] Probably the most fundamental feature of modern scientific inquiry is the ability to find these causes and predict future events [1] , [2] . Reflecting this view, animals are thought to operate according to laws firmly tying behavioral ‘responses’ to environmental variables: “[N]euroscience, over the last 30 years, […] each year brings a greater understanding of the mechanical way with which we perceive, we remember, we speak, we feel.” [3] Once these laws are known, the behavior of any animal at any time can be predicted from the current environmental situation [4] : “We cannot prove […] that human behavior […] is fully determined, but the position becomes more plausible as facts accumulate.” [5] This does not necessarily imply that the same stimulus always elicits the same behavior, but that each behavior is a response to a stimulus: “Indeed, so pervasive is the basic assumption of this model that it is common to refer to any behaviour as a ‘response’ and thus by implication […] assume that there must be an eliciting stimulus.” [6] This basic tenet not only guides basic neurobiological and psychological research but has been the foundation for a great many robotics applications [7] – [9] as well as for speculations on the future societal impact of neuroscience [3] , [10] , [11] . Basically, the brain is seen an input/output device: “brain function is ultimately best understood in terms of input/output transformations and how they are produced” [12] . Contending that less complex brains would be more amenable to this research, the study of invertebrate and in particular fly behavior developed into a prominent focus of attention [7] , [8] , [13] , [14] .

Results

Spontaneous behavior is not simply random Naively, if the production of torque spikes in our featureless or uniform environment were due to random noise in the Drosophila brain or from any uncontrollable input, the time intervals between spikes (inter-spike interval, ISI) should reflect this stochasticity, much like the hiss of static from a radio between stations. Given a certain mean spike rate, the most straightforward assumption is to expect a stochastic procedure to behave according to a Poisson process [24], [25], [33]. In other words, this situation should represent a natural system for generating random numbers. Therefore, we adapted a recently developed computational method, Geometric Random Inner Products (GRIP) [34], to quantify the randomness of the ISI sequences of three groups of flies. The first group (‘openloop’) flew in a completely featureless white panorama (i.e., without any feedback from the uniform environment–open loop). The ISI sequence in these flies must be generated entirely spontaneously. The second group (‘onestripe’) flew in an environment that contained a single black stripe as a visual landmark (pattern) in a flight simulator situation that allowed for straight flight in optomotor balance (i.e. the fly could use its yaw torque to control the angular position of the stripe–closed loop). Flies from this group not only received reafferent feedback from the effects their maneuvers had on the angular position of the stripe, but it is also known that such stripes elicit optomotor and fixation responses [35] (see also Fig. S2), providing for an input/output control group. The third group (‘uniform’) flew in a uniformly textured environment that was otherwise free of any singularities (i.e., closed loop, the fly could use its yaw torque to control the angular position of the evenly dashed environment). This arrangement also allows for straight flight in optomotor balance but it does not elicit any fixation or directional preferences as the onestripe situation. Therefore the uniform group constitutes an intermediate case. A significant deviation from ideal randomness in any of these groups would contradict the ‘robot-hypothesis’. GRIP results show that fly behavior deviates from perfect randomness (Fig. 3a). In all our groups, this deviation even exceeds the values from a computer-generated Poisson process (Kruskal-Wallis ANOVA: H(3, N = 52) = 17.2; p<0.0007. In post-hoc tests, all fly values were significantly higher than the poisson control values, p<0.03 in all cases). Plotting the number of ISIs as a function of ISI duration reveals an overrepresentation of long ISIs with respect to an exponential distribution (so-called heavy-tailed distributions; see Fig. S3). Thus, the simplest hypothesis that first-order noise underlies variable spike generation in a constant environment has to be rejected. PPT PowerPoint slide

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larger image TIFF original image Download: Figure 3. Spontaneous behavior is not simply random. A–GRIP analysis of ISIs. Plotted are the mean standard deviations from the theoretically expected random value for fly ISI series and the random series generated by a Poisson process. The fly deviations are all significantly larger than the values for the computer-generated series. B–Log-log plots of ISIs. The Lévy exponent μ is calculated from the inclination of the linear fit. A Lévy distribution is defined as 1<μ<3. Smaller values indicate a larger proportion of long ISIs. A Cox Process (cox) reveals a similar power-law structure as the flies. Error bars are S.E.M.s throughout. See Methods for details and statistics. https://doi.org/10.1371/journal.pone.0000443.g003 One may argue that the assumption of a constant spike rate is arbitrary, overly simplistic and that more complex stochastic processes are likely to be at work, even in flies. A well-known example of such stochastic processes is a doubly stochastic Poisson process (or Cox Process) [36], [37]. A Cox process is essentially a Poisson process in which the rate is not constant, but fluctuates randomly. In our example, a fly's spike rate may change in response to uncontrolled, random events in the fly's environment or to random events within the fly. Cox processes can generate heavy-tailed distributions, sometimes also called power-law distributions. Power laws are among the most frequent scaling laws that describe the scale invariance found in many natural phenomena and can be seen as a straight line on a log-log graph of the data. Therefore, we plotted the number of ISIs as a function of ISI duration on a double logarithmic scale. To simulate a Cox process, we used the instantaneous spike rates from the flies in the openloop group to drive the rate of a Poisson process (cox; see Methods for details). A very similar process has previously been used to successfully model the spike trains of neurons such as those in the cat visual cortex [38]. We found inverse power-law distributions both in the timing of fly ISIs and in the cox group (Fig. 3b). For the two fly groups without a singularity in the environment (openloop and uniform) and for the Cox process, the duration of ISIs decayed according to a non-Gaussian Lévy distribution (with the Lévy exponent 1<μ<3). Conspicuously, the Cox process is also Lévy distributed. Do such results provide any leads for investigating the potential mechanisms underlying spontaneous turning behavior? Lévy flights, a special class of Markov processes, are scale invariant and often associated with power-laws described in many other systems [39]–[41]. A Lévy flight can be conceptualized as a process which first chooses a direction at random and then keeps flying for a distance drawn at random from a Lévy distribution [42]. The Cox process, although not working in this way, still yields a Lévy distribution. It has also been proposed that systems with a large number of nonlinearly coupled subsystems also may exhibit Lévy distributions [43], [44]. Clearly, “the presence of such distributions tells us nothing about the mechanisms that give rise to them” [45]. Notwithstanding, all the more common stochastic processes which can give rise to Lévy distributions imply second-order (or conditional) stochastics. These processes share the property that the conditional probability distribution of the next step depends only on their current state and not on the steps in the past (i.e., no memory). The Cox process is a classic representative of this class of conditional stochastic processes.

Spontaneous behavior reveals a fractal order A standard method of testing for renewal processes without memory (i.e., Markov, Lévy or Cox processes) is to compare the original sequence to randomly shuffled (“surrogate”) sequences. This surrogate data set maintains the same relative frequency of ISI durations as the original data, but destroys the ordering of the intervals. A significant difference between surrogate data and original data indicates that conditional probabilities are not involved in the generation of the series. For this comparison, we first computed the correlation dimension [46] for the original ISI series which yields a sequence-dependent measure for each fly. The correlation dimension is a measure of the dimensionality of the space occupied by a particular ISI sequence (similar to the less reliable fractal dimension). If the correlation dimension converges on a fractional value, the ISI sequence is termed ‘fractal’. This first step of computing individual correlation dimensions already hints at a difference between the stochastic ISI series and the fly series: all four traces appear very similar, but the fly data each converge on a specific dimension while the cox series diverges with increasing embedding dimensionality (Fig. 4a). The convergence of the correlation dimensions for fly data suggests a fractal order in the fly ISI series and not in the cox series. However, these differences are rather subtle and somewhat subjective. In the decisive second step, we calculated the probability that any randomly shuffled sequence of ISIs could have produced the same outcome. The results show that most likely the recorded sequence of ISIs–and not any random shuffling thereof–is responsible for the computed correlation dimensions, rejecting the hypothesis of second-order stochastics dominating the generation of spontaneous turning behavior in Drosophila (Fig. 4b). Similar to sequences of ISIs recorded in the monkey basal ganglia [47], sequences of fly ISIs are not entirely defined by their probability distribution. In contrast, we can not reject the hypothesis that any sequence could generate the computed correlation dimension for the cox series, at the .05 criterion. A Kruskal-Wallis ANOVA was significant for the shuffled correlation dimension probabilities: H(3, N = 52) = 24.7; p<0.0001. All fly probabilities were significantly lower than the cox probability (p<0.02 in all cases). This outcome rules out renewal processes as the main mechanism generating spontaneous turns in Drosophila. Specifically, this excludes Cox processes or other superpositions of random processes, which one could assume if several separate processes in the brain lead to torque spike production or for the superposition of environmentally and endogenously triggered torque spikes. PPT PowerPoint slide

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larger image TIFF original image Download: Figure 4. Correlation dimension. A–While the correlation dimension converges on a group-specific value with increasing embedding dimension for fly-generated ISIs (openloop, onestripe, uniform), a number sequence generated randomly by a Cox Process (cox) diverges. B–Probability to obtain the computed correlation dimensions in A by random shuffling of the original data. While the cox group exceeds an alpha value of .05, the three fly groups stay well below that threshold. https://doi.org/10.1371/journal.pone.0000443.g004

Long-range correlations in the behavior imply nonlinearity However, there are yet more complex composite stochastic models which, like the fly data, can exhibit a fractal structure [15], [48]. These models combine a multitude of stochastic processes by deterministic rules. For instance, the so-called “branched Poisson process” (BPP, see Fig. S4a) consists of a cascade of Poisson processes each driving the rate of the next via a filter function [48]. The combined output of all these processes constitutes the output of the entire BPP. Such processes can produce ISI series which do show fractal characteristics and their probability of shuffled data to yield the same correlation dimension comes to lie in-between standard stochastics and fly data, such that they cannot easily be distinguished from either of the two (data not shown). The results from surrogate data imply a form of memory in both spontaneous flight behavior and to a certain degree also in BPPs that lasts beyond the current time point. Specific ISI durations are determined in part by the timing of other spike(s), and ISI durations fluctuate over time rather than relaxing to a homeostatic steady state. Such a memory can lead to long-range correlations in the data which may be the reason why the shuffled data fail to reproduce the original correlation dimension. A sensitive method to detect these correlations is to calculate the root mean square (r.m.s.) fluctuations in the ISI series (see Methods). For uncorrelated time series r.m.s. fluctuations decay according to a power-law with an exponent α of ½. If the exponent deviates from ½, long-range correlations exist in the time series [32], [49]. This computation shows significant deviations from ½ for all the fly series (Fig. 5; t-test against single value: p<0.001 for all three groups). Besides the fly data, we tested two forms of BPP, one with a linear filter function and one with a nonlinear filter. We found that the presence of long-range correlations was dependent on the nonlinearity of the filter function (Fig. 5; t-test against single value: p<0.3 for BPP with linear filter and p<0.04 for BPP with nonlinear filter). However, the value for the BPP with the nonlinear filter function is still significantly smaller than the value for the openloop group, to which it was fitted (Mann-Whitney U-Test, p<0.005), ruling out even BPPs with nonlinear filters as an appropriate model for spontaneous flight behavior in Drosophila. PPT PowerPoint slide

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larger image TIFF original image Download: Figure 5. Long-range correlations in fly ISIs. If the slope of the log-log plots of the r.m.s. fluctuation (exponent α, see Methods) deviates significantly from ½, long-range correlations exist in the time series. All three fly groups show a significant deviation from 0.5. The deviation of branched Poisson processes (BPP), however, depends on the nonlinearity of the filter function used to drive the Poisson processes and is significantly smaller than that of fly ISI series. *-significant difference from 0.5. https://doi.org/10.1371/journal.pone.0000443.g005 The dependence of the α-values on the nonlinearity contained in the BPPs entices to hypothesize that what is needed to achieve long-term correlations such as those observed in flies (this study and [32]) and other animals such as albatrosses [49] are not essentially random processes connected by nonlinear mechanisms, but rather essentially nonlinear processes containing random noise. We thus employed a recently developed method which distinguishes essentially stochastic from essentially nonlinear time series.