The generation of rhythmic patterns of movement is a key feature of behaviours that are conserved across the animal kingdom (Ijspeert, 2008). Central pattern generators (CPGs) are motor neural circuits capable of generating rhythmic activity when activated that require no sensory or descending inputs of phasic timing information (Ijspeert, 2008; Marder et al., 2005; Berni et al., 2012; Berni, 2015). However, complex sequences of behaviour entirely organised within the nervous system without the need for sensory inputs from environmental influences or sensory feedback during its course are seemingly rare in nature (Marder et al., 2005; Slater, 1999). For instance, finding resources requires efficient movement through complex environments and is an ability key to animal survival (Ijspeert, 2008; Sims et al., 2008). Foraging, in which an animal searches widely to obtain food resources, is one such complex sequence of behaviour, comprising physiological, sensory and cognitive processes modified in response to environmental fluctuations. Apparently complex patterns of behaviour can potentially arise from simple movement ‘rules’ (Sims et al., 2008; Maye et al., 2007; Wearmouth et al., 2014), however it remains unknown whether autonomous movements resulting in spatial exploration could result in optimal searches.

Autonomous generation of movement patterns that are optimal for finding resources should be advantageous in naïve and inexperienced animals. It has been proposed (Maye et al., 2007; Wearmouth et al., 2014; Kölzsch et al., 2015; Humphries et al., 2012; de Jager et al., 2014; Sims, 2015; Reynolds, 2015) that foraging patterns described by apparent Lévy random walks in freely moving animals may represent intrinsically generated behaviour that is exhibited when a searcher has incomplete information about resource location, such as when there is an absence of external clues or cues, or no information about the environment provided by memory (Sims et al., 2008). Therefore, where there is a lack of knowledge about resource location, autonomously generated search programs based on Lévy walks may represent efficient patterns to adopt until clues and cues are encountered by moving.

A Lévy random-walk search pattern can be generated by drawing displacements (move steps) from a probability distribution with a heavy power-law tail, such that P(l)~l ^-µ, with 1 < µ ≤ 3 where l is the move step length between turns and µ the power law exponent (Figure 1A,B). A Lévy walk iterated over many time intervals will be distributed much further from its starting position than a Brownian walk of the same length because small-step ‘walk clusters’ are interspersed by long ‘steps’ to new locations, with this pattern repeating across all scales resulting in a fractal pattern of walk clusters with no characteristic scale (Sims et al., 2014; Reynolds, 2018). Theory predicts that Lévy walk search strategies are optimal where resources are sparse and distributed unpredictably (Viswanathan et al., 1999; Viswanathan et al., 2011; Humphries and Sims, 2014; Wosniack et al., 2015) – such as in uncertain or dynamic environments where the spatial scaling patterns of search behaviour cannot be tuned to target distributions (Reynolds, 2018) – whereas Brownian (exponential) walks are sufficiently efficient when resources are highly abundant (Viswanathan et al., 1999; Viswanathan et al., 2011; Humphries and Sims, 2014; Wosniack et al., 2015). However, a Lévy walk with an inverse square power-law distribution of move steps (µ ~2) is optimal across a very broad range of resource distributions and abundances (Humphries and Sims, 2014; Wosniack et al., 2015). Because Lévy walks can optimise search efficiencies in this way, it has been proposed that natural selection should have led to adaptations for Lévy walk foraging [the Lévy foraging (LF) hypothesis] (Sims et al., 2008; Viswanathan et al., 1999; Viswanathan et al., 2011; Viswanathan et al., 2008). Movement patterns resembling Lévy walks have been observed in diverse organisms (Sims et al., 2008; Maye et al., 2007; Kölzsch et al., 2015; Humphries et al., 2012; Cole, 1995; Humphries et al., 2010; de Jager et al., 2011; Bazazi et al., 2012; Hays et al., 2012; Raichlen et al., 2014), such as jellyfish, molluscs, insects, fish, birds and mammals including human hunter-gatherers, and even in fossil trails of extinct invertebrates (Sims et al., 2014), leading to the idea that search strategies naturally evolved to exploit optimal Lévy walks (Sims et al., 2008; Viswanathan et al., 2008). However, the origin of apparent Lévy-like behaviour is unclear (Sims, 2015; Sims et al., 2014) and remains contentious (Reynolds, 2015; Reynolds, 2018; Viswanathan et al., 2011; Benhamou, 2007), not least because a candidate generative mechanism for Lévy movement has not been identified unequivocally (Kölzsch et al., 2015; de Jager et al., 2014; Reynolds, 2015; Sims et al., 2014).

Figure 1

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Although Lévy-like patterns appear prevalent in organism movement patterns it has been argued that Lévy walks lack a biological basis or are too simplistic to explain the complexity and diversity of animal movements (Reynolds, 2015; Reynolds, 2018; Benhamou, 2007). While it is not expected that Lévy-like movement will be temporally ubiquitous in animal behaviour, since it is only advantageous when searching with incomplete information (Sims et al., 2008; Viswanathan et al., 1999; Humphries and Sims, 2014), recent studies have hypothesised two principal mechanisms to account for Lévy movement patterns (Sims, 2015): it arises from (i) endogenous neurophysiological processes (Maye et al., 2007; Wearmouth et al., 2014; Kölzsch et al., 2015; Humphries et al., 2012; de Jager et al., 2014; Sims et al., 2014) resulting in behavioural adaptations to different resource distributions (Sims et al., 2008; Humphries et al., 2010) – the intrinsic hypothesis; or that (ii) sensory interactions of animals moving along straighter, ballistic type paths in fractal environments, for example, power-law distributions of resource patches, give rise to Lévy patterns as an emergent phenomena (Reynolds, 2015; Reynolds, 2018; Boyer et al., 2006) – the extrinsic hypothesis. Nevertheless, there have been no studies to date that have unequivocally tested both hypotheses in the same model organism.

Our ability to specifically and acutely manipulate neuronal activity in Drosophila larvae during a spatial exploration sequence makes it a suitable model for testing the Lévy foraging hypothesis and to explore its mechanisms. Here, we used two transgenic lines of flies: BL (for brain lobes) in which the brain and part of the SOG function can be acutely and reversibly impaired in free-moving larvae, and BLsens (for BL and sensory), which allows us to block the brain lobes, the SOG and all somatosensation (Berni et al., 2012; Berni, 2015). Movements in these larvae comprise sequences of forward peristaltic contractions (crawls) interspersed with pauses and reorientations (turns) in which the brain and part of the SOG are not required for normal performance of an exploratory routine; patterns are produced by autonomous activity of the thoracic and abdominal nervous system (Berni et al., 2012; Berni, 2015). Because the BL line may not be expressed in all neurons, we performed electrophysiology experiments to evaluate the strength of neuronal activity blockade. We also tested a transgenic line specific for the mushroom body, MB247 (Pauls et al., 2010), to determine the potential contribution of cholinergic neurons from this structure of the brain in free-moving larvae. Finally, to confirm the results obtained affecting activity in the brain lobes, SOG and sensory system, we induced neuronal death expressing the proapoptotic genes (rpr and hid) with the BLsens line.

We used this model system to test whether extended substrate exploration of Drosophila larvae is an intrinsic, spontaneous pattern resembling a Lévy search in the absence of exteroception and proprioception.

We recorded movement paths of individual larva using high-resolution video tracking extending for 1 hr (Figure 1C–E) (Materials and methods). A Kalman filter used for track smoothing to reduce effects of video tracking errors combined with computation of track step lengths reconstructed the larva movement paths reliably (Figure 2A; see Materials and methods). Across all controls and treatments in this study 658 larva tracks were recorded; 187 (28.4 %) did not meet the criteria required for reliable track analysis to support testing of the Lévy foraging hypothesis (Humphries et al., 2013) and were discarded prior to track modelling analysis (Supplementary file 1). The criteria for discarding individual larva tracks were: (i) tracks with fewer than 200 move steps (126 tracks), or (ii) tracks where the larva interacted with the arena edge (e.g. travelled along it; 52 tracks), or (iii) where no computed steps were longer than the 0.44 mm minimum resolution for a step length (nine tracks) (see Materials and methods) (Humphries et al., 2010). Figure 2A and Figure 2—figure supplement 1 provides illustrations of the track processing for an example track with the detailed analysis of the turning behaviour. Turns were identified as the reversal in direction in a 1D projection of the 2D movement patterns. This method is unbiased and preserves the distribution properties of the original 2D trajectory (Humphries et al., 2013) (see Materials and methods). We undertook sensitivity analyses to examine how variations in Kalman filter parameters and in the minimum step resolution value affected truncated power-law fitting and exponents (µ). We found no significant effects of different Kalman filter parameters or minimum resolution step values on mean µ values (Supplementary file 2), demonstrating that the µ values were robust to relatively broad alterations in video tracking resolution and track smoothing (Figure 2—figure supplement 2). Similarly, there were no significant differences between movement patterns as measured by µ values of larva before and after collisions with other larva or the arena edge within BL/+ and shi^ts/+ treatments (Figure 2—figure supplement 3) – indicating stationarity in movement data statistics (for additional stationarity tests see ‘Exploration strategy in an environment with minimal external cues’ section) – thus, track sections occurring after brief collisions were retained for analysis.

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For the remaining 471 tracks across all controls and treatments we tested for Lévy walks by fitting truncated power law (Pareto-Lévy) and exponential distributions to empirical move-step length frequency distributions using Maximum Likelihood Estimation (MLE) for parameter fitting and Akaike Information Criteria weights (wAIC) for model selection (illustrated in Figure 1E,F; Materials and methods) (Humphries et al., 2013). We tested for truncated power laws rather than pure power laws because animal movements are naturally bounded, even within the relatively large experimental arena used (Figure 1C). As well as truncated power laws and exponential models we also tested power-law, truncated exponential, log-normal (a heavy-tailed distribution), and gamma distributions in addition to composite Brownian walk distributions comprising proportions of two, three, and four exponential distributions (Sims et al., 2014).

Exploration strategy in an environment with minimal external cues

At the control temperature of 22°C where normal movements and feeding responses were expected (Hoffmann, 2010), we found for control larva experiencing minimal external stimuli, e.g. visual, olfactory and gustatory, that 30 trajectories (70 %) of BL/+ (elav-Gal4, tsh-Gal80/+,+; cha3.3-Gal80/+) larvae with intact nervous systems, and 34 (79 %) of shi^ts/+ (UAS-shi^ts/+) larvae with intact nervous systems and inactivated shibire^ts, were best fitted by truncated Lévy power-law distributions (Supplementary file 1; Figure 5). Only 14 tracks at 22°C (16%) were best fit by an exponential distribution, while eight tracks (9%) were best fit by neither model (Supplementary file 1).

For the BL/+ larva used, despite 33°C being above the described preferred temperature of the larvae, we found that it was still within a range where feeding behaviour was not affected (Figure 3B). This suggests that movements of BL/+ at the restrictive temperature reflected normal searching movements to locate resources. Similarly, in control larva at 33°C to determine the effects on movement paths of the restrictive temperature (33°C) at which UAS-shi^ts blocks neuronal transmission (Figures 3 and 4), we found 57% of BL/+ larvae and 77% of shi^ts/+ larvae move-step distributions were best fitted by truncated Lévy power-law distributions (Figure 5), with 14 tracks (13%) best fit by the exponential model; 23 tracks (21%) were best fit by neither model) (Supplementary file 1 and 3). There were no fits to any other model distributions tested, except for seven tracks that were best fit by pure power law distributions (Supplementary file 4 and 6).

Figure 3

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Figure 4

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Figure 5 with 1 supplement see all

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Movements comprised spatially intensive clusters of shorter move steps with frequent turns interspersed with more extensive, straighter steps of varying length (Figure 2B and 5). This intensive/extensive movement pattern was evident at increasing scales within the test arena (Figure 2B) and is a characteristic of scale-free Lévy patterns (Kölzsch et al., 2015; Viswanathan et al., 1999). We confirmed this observation using tortuosity analysis of the control tracks at both temperatures (Table 1; Materials and methods). MLE analysis of the frequency distributions of straightness index ‘jumps’, describing the transition from spatially intensive to extensive movement clusters across different scales (cluster sizes) within each individual time-series, demonstrated good fits to truncated power-laws (127 of 136 tracks; 93%) averaging 1.98 orders of magnitude of the data (Table 1), which are characteristics that are expected for a Lévy pattern (Kölzsch et al., 2015).

Table 1

T (°C)	Treatment	No. ofTracks	Mean µ	SD	Orders of Magnitude	SD
22	BL/+	29	1.70	0.34	1.98	0.41
33	BL/+	31	1.54	0.38	1.99	0.62
22	shits /+	33	1.85	0.44	1.89	0.39
33	shits /+	34	1.61	0.40	2.04	0.40
33	MB247/+	21	1.39	0.27	1.87	0.45
33	BLsens > shits	6	1.75	0.65	1.37	0.51
33	BL > shits	23	1.67	0.40	1.61	0.61
33	MB247 > shits	22	1.35	0.18	1.93	0.34
22	New Blsens > rpr,hid_control	61	1.74	0.33	2.11	0.37
32	New BLsens > rpr, hid	22	1.41	0.29	1.55	0.26

At the environmentally relevant control temperature (22°C), model fitting of move step length frequency distributions showed that all 31 BL/+ and 34 shi^ts/+ tracks fitting truncated power-laws showed characteristics that are consistent with Lévy random walks. Firstly, we found BL/+ and shi^ts/+ move-step distributions to have power-law exponents (µ) within the Lévy range (1 < µ ≤3) (BL/+: mean µ, 1.47 ± 0.36 S.D., n = 30; shi^ts/+: mean µ, 1.49 ± 0.35 S.D., n = 34) (Table 2; Supplementary file 3). Cumulative probability distribution (CPD) analysis confirmed that for the sample exponent parameters it was predicted that 93.2% and 98.4% of BL/+ and shi^ts/+ µ exponents within populations, respectively, were estimated to fall within the Lévy range (Table 3). In addition, we confirmed stationarity in the movement pattern data by finding no differences in mean µ values from the first and second halves of larva tracks each fitted by a truncated power-law (Supplementary file 5 for results for all treatments). Second, the mean squared displacement (MSD) of a 2D trajectory describes the spatial extent of random motion where the scaling behaviour indicates short (α = 1) and long-range (α >1) correlations in the random walks. Hence, α = 1 describes normal Brownian random walks whereas α >1 characterises scale-free Lévy random walks (Sims et al., 2008; Viswanathan et al., 2011; Bartumeus et al., 2005). Therefore, we computed the MSD for each track and found that the means of α for individual BL/+ and shi^ts/+ larvae were 1.34 and 1.26, respectively, which supports the presence of long-range correlations in the control larvae paths (Table 4). Lastly, the best fit truncated power-law model for each individual larva move-step distribution was greater than 1.5 orders of magnitude of the data (BL/+ mean, 1.63 ± 0.41 S.D., n = 30; shi^ts/+ mean, 1.52 ± 0.32 S.D., n = 34) (Table 2; Figure 6), with some individual tracks spanning two orders of magnitude which is considered a rule of thumb for a candidate power law (Stumpf and Porter, 2012). These results demonstrate that when moving in environments with ambient temperatures similar to those found in Drosophila natural habitats, control BL/+ and shi^ts/+ larvae exhibit movement patterns with Lévy-like characteristics.

Table 2

T (°C)	Treatment	N tracks	Mu	SD	Orders of magnitude of the data	SD
22	BL/+	30	1.47	0.36	1.63	0.42
33	BL/+	33	1.35	0.32	1.48	0.49
22	shits/+	34	1.49	0.35	1.52	0.32
33	shits /+	39	1.54	0.43	1.52	0.38
33	MB247/+	23	1.74	0.48	1.33	0.44
33	BL > shits	26	1.66	0.50	1.15	0.40
33	BLsens > shits	11	1.96	0.39	1.20	0.23
33	MB247 > shits	23	1.56	0.35	1.39	0.46
22	BLsens > rpr, hid_control	62	1.51	0.38	1.71	0.40
32	BLsens > rpr, hid	23	2.14	0.47	0.93	0.52

Table 3

		Cumulative Probability Distribution (%)
T (°C)	Treatment	μ exponent range
		1–3	1.25–2.75	1.5–2.5	1.75–2.25
22	BL/+	93.2	71.2	41.3	17.1
33	BL/+	90.4	59.6	26.4	8.1
22	shits /+	98.4	76.8	33.6	8.0
33	MB247/+	96.2	83.9	60.2	30.6
33	BL > shits	92.5	76.0	51.6	25.3
33	BLsens > shits	98.7	95.1	80.6	47.3
33	MB247 > shits	96.6	79.9	50.6	22.4
22	BLsens > rpr, hid_control	94.8	75.0	45.2	19.3
32	BLsens > rpr, hid	92.9	85.1	67.9	38.4

Table 4

T (oC)	Treatment	No. of tracks	Mean α	SD	Mean R2	SD
22	BL/+	30	1.34	0.51	0.88	0.10
33	BL/+	33	1.41	0.53	0.90	0.10
22	shits /+	34	1.26	0.45	0.88	0.16
33	shits /+	39	1.37	0.73	0.83	0.21
33	MB247/+	23	1.40	0.81	0.84	0.12
33	BL> shits	26	1.21	0.55	0.81	0.24
33	BLsens> shits	11	0.94	0.78	0.67	0.25
33	MB247> shits	23	1.03	0.42	0.80	0.18
22	New BLsens> rpr,hid_control	62	1.17	0.49	0.81	0.20
32	New BLsens> rpr, hid	23	1.30	0.53	0.80	0.14

For control treatments BL/+, shi^ts/+ and MB247/+ (MB247-Gal4/+) larvae at the restrictive temperature (33°C) similar results were found. In summary, frequency distributions of straightness index cluster size and move step length distributions followed truncated power-law distributions over several orders (Table 1; Supplementary file 3), with mean µ exponents between 1.35 and 1.74 (Table 2; Figure 5) and cumulative probability distributions showing >90% of larvae trajectories were within the Lévy range (Table 3). Finally, MSD α values were between 1.37 and 1.41 confirming long-range correlations (Table 4). Therefore, BL/+, shi^ts/+ and MB247/+ larvae (without synaptic blockade) moving for up to 1 hr time periods in larger environments with minimal exposure to external cues at the higher (restrictive) temperature of 33°C, exhibited movement patterns with Lévy-like characteristics that were similar to those of larvae at 22°C.

Furthermore, we also found that the turn angle distribution for all individual larva paths that were best fitted by truncated power-laws showed a near uniform distribution of large turn angles (50 and 175^o, left or right) (Figure 5—figure supplement 1), a characteristic consistent with the uniform random turn-angle distribution (i.e. where all turn angles have equal probability) expected in a pure, idealised Lévy random walk (Viswanathan et al., 1999; Bartumeus et al., 2005). The presence of some directional persistence in our larva tracks (higher frequency of turn angles around 0^o) indicates some short-range angle correlations. However, it has been widely demonstrated that the efficiencies of random Lévy walk searches remain robust to the presence of short-range correlations in directional persistence (Bartumeus et al., 2005; Viswanathan et al., 2001; da Luz et al., 2001; Raposo et al., 2003). This, taken together with the presence of truncated power-law (Lévy) fits to move step-length and walk-cluster size distributions, and the presence of long-range correlations in movement paths confirms the appropriateness of applying the Lévy random walk model to Drosophila larva tracks.

Lévy-like movement patterns are generated when brain processing is impaired

The demonstration of Lévy-like movement patterns in control larvae in an environment with minimal external stimuli is not conclusive support for an intrinsic mechanism underlying Lévy-like behaviour. Figure 2A shows the effect on movement of larva with panneural expression of a dominant negative temperature-sensitive form of Shibire^ts that blocks synaptic transmission (but does not affect electrical synapses) by depleting rapidly and almost completely the synaptic vesicles at the restrictive temperature of 33°C (Koenig et al., 1983; Ohyama et al., 2013). To test whether Lévy-like exploration movements are intrinsic to the neural circuitry of larvae in the absence of environmental sensory processing by the brain, we tracked individual BL >shi^ts (>stands for Gal4/UAS) larva with reduced synaptic activity in the brain and part of the suboesophageal ganglion (SOG) (Berni et al., 2012; Berni, 2015) (Materials and methods). Extracellular recordings of neuronal activity confirm that above 33°C, BL >shi^ts brain activity is impaired in a manner indistinguishable from the positive control elav >shi^ts, such that the proportion of activity left at 32°C compared to activity at 23°C was 0.30 ± 0.48 (S.D.) and 0.19 ± 0.53 (S.D.), respectively (Tukey post hoc test: BL >shi^ts vs elav >shi^ts: p=0.91) (Figure 4).

In 1 hr long trials of 77 tracks, 30% were discarded from further analysis (see criterion above; Supplementary file 1). However, 54 BL >shi^ts larva paths were sufficiently extensive to support reliable parameter estimation (Humphries et al., 2013). The BL >shi^ts larvae explored the arena as expected (Berni et al., 2012; Berni, 2015) (Figure 5), confirming here that movement paths were produced from peristaltic contractions and reorientations (turns) generated within the thoracic and abdominal circuitry alone. It was interesting that the movement patterns showed spatially intensive walk clusters interspersed with straighter, longer steps that, overall, were of similar complex form to Lévy-like movements in controls (Figure 2B and 5). Overall, 26 paths were best fit by truncated power-laws (Table 2; Supplementary file 3; Figure 5). The exponential was the best fit for 13 tracks, while a further 15 were best fit by neither model (Supplementary file 1). No other model distribution tested provided better fits (Supplementary file 4 and 6).

Analysis showed that the 26 BL >shi^ts larvae movement paths best fit by truncated power-law (Lévy) distributions had a mean exponent of 1.66 (± 0.50 S.D.; (Figures 5 and 6; Table 2), with CPD estimates for these exponent parameters predicting that 93% of the population would fall within the Lévy range, and 52% between μ exponent values of 1.5 and 2.5 (Table 3). The mean exponent was higher than BL/+ and shi^ts/+ larvae power-law exponents at both 22°C and 33°C (Table 2), but not significantly (Figure 6). Furthermore, computation of MSD confirmed similar long-range correlations (Table 4). Collectively, these results indicate that the movement path patterns of BL >shi^ts larvae with blocked brain and SOG were also Lévy-like and statistically indistinguishable from those of BL/+ and shi^ts/+ larvae without brain and SOG block at 33°C, and at an environmentally preferred temperature (22°C).

Figure 6

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The path tortuosity analysis of BL/+, shi^ts/+ and BL >shi^ts larvae showed that there was complex spatial clustering of intensive and extensive movements (Figure 5) and that in both treatments the frequency distributions of straightness-index cluster sizes across scales were best fitted by truncated power-law models (Table 1). Previous studies (Sims et al., 2014; de Jager et al., 2012) have identified that spatially nested composite Brownian (CB) walks may have parameters finely tuned to Lévy walks (Reynolds, 2015). To investigate this further, we compared larvae move-step distributions with three composite Brownian (CB) walk distributions comprising proportions of two, three, and four exponential distributions (CB2, CB3 and CB4; Materials and methods). Comparing truncated power-law, power, exponential and the three CB walks within the model set demonstrates that 13% of BL/+ larvae move-step distributions (8 of 63 distributions) were best fit by second or third-order CB walks, with two tracks (3.2%) best fit by pure power laws. In contrast, in shi^ts/+ larvae 92% of move step distributions (36 of 39 distributions) were best fit by a pure or truncated power law, with three best fit by second or third or CB walks. Similarly, 85% of BL >shi^ts distributions (22 of 26 distributions) were best fit by truncated power laws over exponential and second, third or fourth order CB walks, with 4 (15%) best fit by a CB2 distribution (Supplementary file 5). This suggests that BL/+ movements were typified by Lévy-like patterns resembling multi-scale and hierarchically nested Brownian walk clusters, confirming the results found from the tortuosity analysis (Table 1), but less extensive in terms of a repeating pattern across increasing scales shown by the shi^ts/+ and BL >shi^ts larvae. The strong support within the model set for truncated power laws in BL >shi^ts larvae combined with a higher power law exponent may indicate a fundamental difference whereby the lack of brain and SOG inputs may alter movements in such a way as to more closely approximate optimal Lévy-like walks.

The mushroom body is the main associative centre in insects for processing sensory information of several modalities ultimately contributing to the selection of appropriate behavioural outputs (Erber et al., 1987; Schürmann, 1987). It has been shown that the cholinergic driver Cha3.3-Gal80, present in the BL line, blocks Gal4 activity in the anterior dorsolateral cells (eight neurons) of the mushroom body (Tastekin et al., 2015). Therefore, to eliminate the possibility of a potential contribution of neurons of the mushroom body, we used the specific MB247 driver line and impaired mushroom body activity using UAS-shi^ts (Thum et al., 2006). In controls, we found 74% of MB247/+ larvae tracks, with intact mushroom bodies in the brain, were best fit by truncated power laws with a mean exponent of 1.74 (± 0.48 S.D.) over 1.33 orders of magnitude of the data (Figure 6; Table 2; Supplementary file 3, 4 and 6), and a CPD predicting 96% within the Levy range and 60% between exponent values of 1.5 and 2.5 (Table 3). A truncated power law best fit to straightness index cluster sizes (Table 1) and an MSD α >1 also supported power law behaviour in the movement trajectories (Table 4). For larvae with blocked mushroom bodies (MB247 >shi^ts), we found very similar results: 74% of track move-step length distributions were best fit to truncated power laws with a mean exponent of 1.56 (S.D. 0.35) over 1.39 orders of magnitude of data (Figure 5; Table 2; Supplementary file 1, 3, 4 and 6) and power-law distributed cluster sizes and long-range correlations in move step length time series (Tables 1 and 4). The mean exponent of MB247 >shi^ts was not significantly different from its hemizygote control shi^ts/+, therefore the results indicate that active or inactive mushroom bodies in free-moving larvae had no major effect on the Lévy-like patterns of movement observed.

In conclusion, when brain processing of sensory information is impaired, extended substrate exploration emerges as multi-scale movement paths resembling truncated Lévy walks, strongly supporting the intrinsic hypothesis.

Optimal searching is generated autonomously by thoracic and abdominal circuits

To further test the intrinsic hypothesis, we evaluated whether the Lévy-like move patterns were generated by the thoracic and abdominal CPG by tracking BLsens > shi^ts (elav-Gal4, tsh-Gal80/+,+; UAS-shi^ts/+) larvae where synaptic transmission in the brain, SOG, and all sensory neurons is affected at the restrictive temperature (proportion of activity left at 33°C compared to activity at 23°C was −0.01 ± 0.22 S.D.; Figure 4) (Berni et al., 2012; Berni, 2015). We tracked 30 BLsens > shi^ts larvae and, as expected (Berni et al., 2012; Berni, 2015), we found that these larvae moved more slowly than the other treatments. It was evident that tracks had fewer long move steps than observed for the other larvae (lower straightness index values; Figure 5), with trajectories instead being characterised by having a high frequency of highly spatially intensive movements occurring intermittently along the path (Figure 5). In total, 21 tracks had sufficient computed move step lengths to support robust analysis (Supplementary file 1).

Analysis showed 11 (of 21 tracks) had best fits to truncated power-law distributions (Table 2; Supplementary file 3; Figure 5), with two tracks best fit by the exponential distribution and eight tracks best fit by neither distribution (Supplementary file 1). Despite the tracks showing exponential fits having sufficient computed move step lengths, these larvae did not undertake extensive movements instead spending major portions of the 1 hr trial within a small area. The 11 truncated power law fits had a mean exponent of 1.96 (± 0.39 S.D.) (Table 2), higher than the mean exponent of the BL >shi^ts larvae (Figure 6), and closer to the theoretical optimum, µ_opt ≅ 2 (Viswanathan et al., 1999; Viswanathan et al., 2011; Humphries and Sims, 2014; Wosniack et al., 2015) and over a broad range of move step lengths (mean orders of magnitude, 1.20 ± 0.23 S.D., n = 11) (Table 2; Figure 5). The CPD predicts 81% of randomly chosen BLsens > shi^ts exponent values would fall between 1.5 and 2.5, and 47% between 1.75 and 2.25 (Table 3), indicating the majority of exponents were close to μ_opt = 2. No other model distributions provided better fits to the data (Supplementary file 4 and 6). Long-range correlations in MSD of BLsens > shi^ts larvae trajectories were present with α ~1 (Table 4). As with the BL/+ and BL >shi^ts larvae, we also found the frequency distributions of straightness index cluster sizes were best fit by truncated power laws over a mean of 1.4 orders of magnitude of the data (Table 1). Furthermore, 55% of tracks were best fit to truncated or pure power laws rather than exponential or composite Brownian walks (Supplementary file 5), similar to the results found for BL >shi^ts larvae.

To validate our results we used a second approach; instead of blocking synaptic transmission with UAS-shi^ts we induced specific neuronal death early during larval development in the brain lobe, SOG and sensory system (Figure 7A–D). We generated a new and stronger BLsens driver line (BLsens^R57C10) by replacing elav-gal4 with R57C10-gal4 and crossed it to the proapoptotic genes activators (UAS-rpr and UAS-hid) using tubulin-gal80^ts to restrict the moment of expression (UAS-rpr, UAS-hid/+,+; elav-Gal4, tsh-Gal80/+, tub-Gal80^ts/+; Materials and methods; Figure 7A–D). After determining a set of temperatures that kill the neurons and allow the larvae to reach the third instar, we performed behavioural experiments in the same way as the other lines tested. At the control temperature of 22°C with no induced apoptosis, we found 80% of BLsens > rpr, hid larvae had move step length distributions best fit by truncated power laws with an exponent of 1.51 over 1.71 orders of magnitude of the data (Figure 8; Table 2; Table 3; Supplementary file 3), and cluster and MSD analyses confirming power law behaviour (Tables 1 and 4). Of the 77 tracks analysed, five were best fit by the exponential and 10 were best fit by neither distribution. For BLsens > rpr, hid larvae tracked at the restrictive temperature of 32°C, inducing apoptosis of brain plus sensory neurons, the trajectories were generally more spatially restricted for the majority of individuals. Although this resulted in more exponential fits to step-length frequency distributions (10 tracks from 62 retained), it was striking that 37% of distributions were best fit by truncated power laws (Table 2; Supplementary file 1; Figure 7 and 8). Moreover, the mean exponent for BLsens > rpr, hid was 2.14, with CPD analysis demonstrating 68% of population exponents fall within the μ value range of 1.5 to 2.5, and 38% between 1.75 and 2.25 (Table 3). All other analyses supported the presence of Lévy-like walks in movement patterns best fit by truncated power laws (Tables 1 and 4). Interestingly, the power law exponent of BLsens > rpr, hid larvae movement patterns at 32°C, killing most brain and sensory neurons, was significantly higher and closer to the theoretical optimum than controls at 22°C (Figure 7B, D, E). This was similar to the result found for power law exponents of BLsens > shi^ts compared to shi^ts/+ control larvae (Figure 6). Therefore, the results obtained with proapoptotic driver lines were similar to those seen in the experiments with UAS-shi^ts, which reinforces our finding that even without brain function larvae explore their environment with movement patterns well approximated by Lévy walks.

Figure 7

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Figure 8

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In summary, it is striking that larvae with dramatically disrupted brain, SOG or sensory neural activity were still able to undertake complex, multi-scale movement paths that approximate a Lévy-like search pattern, a theoretically optimal strategy for ‘blind’ foragers searching for sparse target resources in unknown locations. Our result is the strongest evidence yet for an intrinsic, autonomous mechanism underlying Lévy-like patterns of movement in an animal.

Intrinsic Lévy walks in animal searches have been hypothesised based on experiments showing Lévy-like movements of insects (Maye et al., 2007; Reynolds, 2015; Cole, 1995; Bazazi et al., 2012) and molluscs (Kölzsch et al., 2015; de Jager et al., 2014) exposed to minimal external stimuli under controlled conditions. Similarly, we have demonstrated that freely moving Drosophila BL/+, shi^ts/+ and MB247/+ control larvae tracked over large relative time and space scales with minimal sensory inputs (visual, olfactory, gustatory) and homogeneous temperatures at 22°C and 33°C display exploration (search) patterns well approximated by Lévy-like searches. All previous observations, even in favour of the extrinsic hypothesis (Censi et al., 2013) were demonstrated in animals with intact nervous systems, thus the influence of the sensory inputs from the environment could not unambiguously be asserted.

Our finding that BL >shi^ts larvae with strongly impaired brain processing and part of SOG function explore with multi-scale, Lévy-like movement patterns provides the first strong evidence that a Lévy-like pattern was not an emergent property of interactions with fractally distributed resources or sensory landscapes (rejecting the extrinsic hypothesis). In contrast, we show that substrate exploration resembling a Lévy walk pattern is an intrinsic routine for exploration (support for the intrinsic hypothesis). That BLsens >shi^ts larvae exhibit multi-scale, Lévy-like patterns and heavy-tailed distributions – via forward crawls interspersed by pauses then turns – indicates that multi-scale movement patterns are generated spontaneously by the central networks of the thoracic and abdominal system that operate independently of input from either the brain or the sensory system. This was confirmed in larvae BLsens >rpr, hid larvae expressing proapoptotic gene activators that kill brain and sensory neurons, which were also free-moving with patterns from long trajectories well approximated by Lévy-like walks. Therefore, we conclude that multi-scale, Lévy-like search patterns arise intrinsically and autonomously in Drosophila larvae.

We also found evidence for intrinsic Lévy-like patterns that are theoretically optimal for sparse resource target encounter in patchy landscapes. Model fits to movement patterns gave power law exponent values of µ = 1.66 for BL >shi^ts and µ = 1.96 for BLsens > shi^ts larvae, values close to the theoretically optimal Lévy search pattern of µ ~2. Recent simulations demonstrate that Lévy walks with µ ~2 are optimal for a much broader range of resource densities and distributions than generally realised (Humphries and Sims, 2014; Wosniack et al., 2015). In addition, numerous studies show that Lévy walks with power law exponents < 2 yield more ballistic searches which are optimal for finding target resources more distantly located (Viswanathan et al., 2011; Humphries and Sims, 2014). It is also apparent that Lévy searches with µ ~1.5, and similar to exponents shown by BL/+ and shi^ts/+ larvae (1.35 and 1.54 respectively) at the restrictive temperature of 33°C, have in simulations been shown to have the lowest variance in foraging efficiency compared to theoretically optimal Lévy foragers, in addition to having low ‘famine’ durations that are only slightly greater than for µ ~2 Lévy searches (Humphries and Sims, 2014). Taken together, this suggests Drosophila larval exploration patterns were similar to theoretically optimal Lévy searches for efficient location of sparse and distant resources.

The restrictive temperature of 33°C did not affect patterns of movements in larvae because there was no significant difference in power law exponents for BL/+ or shi^ts/+ at 33°C or 22°C, a preferred temperature for these larvae. The difference between exponents of BL/+ and shi^ts/+ larvae with BL >shi^ts larvae may be as a result of sensory responses of BL/+ and shi^ts/+ search patterns to minimal external cues likely to have been present in our experimental set-up. For these larvae, the lack of food cues may have resulted in a higher proportion of longer steps (more extensive searching) giving rise to the lower exponent. This implies that the theoretically near-optimal search pattern of BL >shi^ts larvae emerges from central pattern generation in the absence of external sensory inputs.

Precisely how intrinsic, autonomous Lévy-like search patterns are generated in the nervous system of Drosophila larvae was not investigated in this study. Recent studies however, have localised the networks controlling the elements of the behaviour sequence (Berni et al., 2012; Berni, 2015) which provides some insight. Exploratory behaviour is already present in newly hatched Drosophila larvae consisting of straight-line forward crawls driven by peristaltic contractions that are interrupted by pauses, turns and redirected forward crawls (Berni et al., 2012). Genetic dissection of neural circuitry of this exploratory behaviour shows that larval straight-line crawls rely on symmetrical output of the thoracic and abdominal network, whereas thoracic segments generate the asymmetric activity resulting in a turn (Berni, 2015). An intermittent interruption to crawling determined the time-dependent alternation between crawls and turns, but it was not known which part of the system was responsible or whether it was dependent on a fluctuating property of the whole network such as excitation level (Berni, 2015).

A Lévy movement pattern will emerge when straight-line, ballistic movement is interrupted by pauses with intervals timed stochastically according to a heavy-tailed (e.g. power-law) distribution of pause times. It is possible that Lévy movement we observed in Drosophila larvae could arise from the temporal structure of pauses arising from a ‘Lévy timing mechanism’ (LTM) within the thoracic and abdominal network. We propose that the LTM is generated by temporally scale-free changes in neuronal activity of the circuit controlling the transition between straight-line crawls and turns. Scale-free temporal structuring of activity has been observed from single neurons to circuits and is a widespread feature of nervous systems (Mazzoni et al., 2007), suggesting that these patterns of activity could underlie the generation of a Lévy movement. Furthermore, an LTM arising within the thoracic and abdominal neural network of free-moving Drosophila larvae could be analogous to hypothetical ‘fractal reorientation clocks’ in intermittent locomotion (Bartumeus and Levin, 2008). It was proposed in a simulation study that a scale-free (Lévy) distributed pattern of times between pauses (stops or strong changes in speed) punctuating the movement path of a model organism results in Lévy intermittence that may come from reorientation mechanisms capable of organising directional persistence on time (Bartumeus and Levin, 2008). The neural coding processes underpinning a potential mechanism such as this remains to be elaborated precisely, but a link between fractal timing dynamics and propagation patterns in neural networks has been proposed from experimental observations (Beggs and Plenz, 2003). Patterns of spontaneous activity in cortical networks can occur as cascades or ‘avalanches’, where it is proposed that the activity of an individual neuron causes activities in other units in turn, thereby initiating a neuronal avalanche through the network (Beggs and Plenz, 2003). The spatial and temporal distributions of neuronal avalanches appear well described by power laws (Beggs and Plenz, 2003), which indicates the network to be in a critical state such that its system dynamics occur at many different spatio-temporal scales (scale-free) (Beggs and Plenz, 2003). Furthermore, a neural coding scheme based on bursty, multifractal time-series patterns of spike potentials recorded from rat hippocampus during foraging has been proposed to generate neurologically based superdiffusive (Lévy) search patterns (Gutiérrez and Cabrera, 2015). In this context, our empirical study provides some support for the idea of an LTM or ‘fractal reorientation clock’ organising movement paths. Indeed, scale-free or fractal initiators of signalling pathways underlying movement patterns in the absence of prior information about resource location may be common in nature across diverse taxa, from individual cells to animals (Maye et al., 2007; Korobkova et al., 2004; Proekt et al., 2012; Namboodiri et al., 2016).

Our results indicate that multi-scale movement paths that appear complex are generated intrinsically and we show that a relatively simple neural network generated within the thoracic and abdominal nervous system is the principal driver. That free-moving Drosophila larvae without brain and sensory function crawl and explore their environment with apparently complex search patterns that approximate optimal Lévy walks should perhaps not be considered so surprising since it is well established that higher animals can have spared function following decerebration. In vertebrates such as rats for example, much behavioural capacity is spared following decerebration at the mesencephalic and precollicular levels with preservation of a number of complex and well-coordinated behaviour patterns (Woods, 1964; Lovick, 1972). These include locomotion and components of feeding, grooming and defensive behaviour (Woods, 1964; Lovick, 1972). These results for vertebrates clearly suggest that the neural circuitry for many elements of species-typical behaviours can be organised in lower regions of the neuraxis and that these can be integrated to form complex patterns of behaviour. Thus, the intrinsic Lévy-like search patterns we identify in Drosophila with no brain or sensory functions can be viewed as an underlying ‘free running’ optimal search pattern that is modified by environmental stimuli. We propose that when sensory inputs are received to the brain, such as olfactory stimuli, descending signals (Tastekin et al., 2015) to the thoracic and/or abdominal circuits modulate neural activity from Lévy-like patterns to deterministic patterns of movement driven by environmental information, e.g. chemotaxis (Gomez-Marin et al., 2011).

Multi-scale movement patterns exhibited by brain and sensory blocked larvae are the intrinsic output of the thoracic and abdominal nervous system and appear finely tuned to theoretically optimal Lévy walks, which suggests selection pressure for Lévy walk characteristics (Reynolds, 2015; Sims et al., 2014). Indeed, comprehensive computer simulations have confirmed that Lévy walks arise as the optimal search strategy in a wide range of different environments, providing the density of resources is sparse and the searcher’s information about resource location is limited to its close vicinity (Wosniack et al., 2017). This has been suggested to provide the strongest theoretical evidence underpinning the evolutionary origins of empirically observed Lévy walks (Wosniack et al., 2017). Thus, because searching behaviour is crucial to mobile animals’ success (e.g. for food and mate location), the simple neural circuitry giving rise to complex, intrinsic Lévy-like searches that are theoretically optimal in diverse patchy landscapes (Humphries and Sims, 2014; Wosniack et al., 2015; Wosniack et al., 2017) may have adapted early in the evolution of nervous systems. Centralised nervous systems comprised of cerebral ganglia were likely present in the Ediacaran (Northcutt, 2012) and brains and trunk ganglia are evident in early Cambrian arthropods (Tanaka et al., 2013). Early presence of Lévy-like behavioural outputs from simple neural circuitry, such as that observed for Drosophila larvae, could help explain the widespread occurrence of multi-scale, Lévy-like behaviour among widely separated modern taxa, such as fish and birds (Sims et al., 2008; Humphries et al., 2012; Humphries et al., 2010), that have a last common ancestor over 450 million years ago. We cannot exclude, however, the possibility for convergent evolution of similar movement patterns across diverse taxa. Nevertheless, an advantage as beneficial for resource acquisition by ‘blind foragers’ as Lévy search patterns (Viswanathan et al., 2011; Humphries and Sims, 2014; Wosniack et al., 2015), together with our finding of intrinsic neuronal generation of multi-scale, Lévy-like behaviour for substrate exploration in an animal, presents the intriguing possibility that search strategies evolved to exploit Lévy patterns.

All data generated and analysed in this study are available in Dryad (http://doi.org/10.5061/dryad.7m0cfxpq0). Results from analysis are also included in the manuscript and supporting files.

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Author details

1. David W Sims

   1. The Marine Biological Association of the United Kingdom, Plymouth, United Kingdom
   2. Ocean and Earth Science, National Oceanography Centre Southampton, University of Southampton, Southampton, United Kingdom
   3. Centre for Biological Sciences, University of Southampton, Southampton, United Kingdom

   Contribution

   Conceptualization, Resources, Formal analysis, Supervision, Funding acquisition, Visualization, Methodology, Writing—original draft, Project administration

   Contributed equally with

   Nicolas E Humphries

   For correspondence

   dws@mba.ac.uk

   Competing interests

   No competing interests declared

2. Nicolas E Humphries

   The Marine Biological Association of the United Kingdom, Plymouth, United Kingdom

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Visualization, Methodology, Writing—review and editing

   Contributed equally with

   David W Sims

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-3741-1594

3. Nan Hu

   Department of Zoology, University of Cambridge, Cambridge, United Kingdom

   Contribution

   Investigation

   Competing interests

   No competing interests declared

4. Violeta Medan

   1. Departamento de Fisiología, Biología Molecular y Celular, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Buenos Aires, Argentina
   2. Instituto de Fisiología, Biología Molecular y Neurociencias (IFIBYNE-UBA-CONICET), Buenos Aires, Argentina

   Contribution

   Investigation, Methodology

   Competing interests

   No competing interests declared

5. Jimena Berni

   Department of Zoology, University of Cambridge, Cambridge, United Kingdom

   Contribution

   Conceptualization, Resources, Formal analysis, Supervision, Funding acquisition, Investigation, Visualization, Methodology, Project administration, Writing—review and editing

   For correspondence

   jb672@cam.ac.uk

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-5068-1372

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