1 Introduction

The use of model organisms is key for the understanding of life in a broad sense, helping to unveil a large number of fundamental principles governing biological processes (Alfred and Baldwin, 2015; Ankeny and Leonelli, 2020). However, this also poses many challenges (Rine, 2014; Russell et al., 2017; Bertile et al., 2023; Myslivecek et al., 2023). Indeed, choosing a particular species as a “generalist case” might raise relevant questions such as its representativeness of holobionts, its reaching as a genetic prototype, and the universality of its translational physiology, to list just a few issues. Moreover, to adequately serve as a model, one needs a proper, quoting (Leonelli, 2013): “standardization of the organism in question and the accumulation of knowledge and resources on the organism on a large scale.”

Although general methods have been proposed to deal with non-model species (Bertile et al., 2023; Williams et al., 2011), to work with certain particular organisms is still the main trend, whose usual choices tend to be bacteria, yeast, worms, mice and flies (Perillo, 2017). In particular, Drosophila melanogaster, commonly known as the fruit fly, has long been central in developmental biology studies (Yamaguchi and Yoshida, 2018; Lewis, 2004). For example, its relatively simple genetic makeup and short generation time makes the Drosophila ideal for the investigation of a wide range of processes, e.g., related to genetics, ontogeny, learning, development, aging, etc. (Stearns et al., 2000; Phillips et al., 2022). Furthermore, Drosophila nervous system is complex enough to share many structural similarities with more intricate nervous systems, including human, yet simple enough to be feasibly modeled.

Among the various life stages of the Drosophila, the larval one represents a critical period of growth and transformation, punctuated by changes in physiology, morphology and behavior (Bate and Arias, 1993). This is precisely why such stage has been proposed as an important instance to survey the relationship between brain progressive development and the associated behaviors (Gerber and Stocker, 2007). Thus, substantial efforts have been made toward mapping the Drosophila melanogaster neural system (Pfeiffer et al., 2008; Zheng et al., 2018; Naddaf, 2023). Characterizing the neuromuscular network has been particularly fruitful, as motor activity translates directly into observable and measurable actions. Techniques such as calcium (Ca) imaging have been instrumental in visually identifying neural clusters of motor neuron activity associated with muscular contractions in the body segments (reviewed by Kohsaka et al., 2017 and Kohsaka, 2023). Also, analyses have indicated potential similarities between motifs of neural activity and segmental contractions in D. melanogaster larvae (Pulver et al., 2015).

Nonetheless, the depiction of the resulting data are usually aimed to disclose the activity propagation patterns in the ventral nerve cord (VNC), e.g., forward and backward wave fronts, not addressing a proper quantification of the underlying dynamics. Actually, despite all the progress, it is still not totally clear how these nervous system patterns control alternations and duration of the crawls and turns (the elementary movement steps given rise to the locomotion trajectories). In fact, such control—conceivably functioning at different hierarchical (and time-scale) levels (Anteneodo and Da Luz, 2010) of the individual body structural organization—needs to be mediated by feedbacks with the environment. But this kind of trade-off interaction is also not fully known (Koyama et al., 2020; Davies et al., 2021). The bottom line is that contrary to the rapid advances in experimental techniques (Davis, 2023), the aforementioned knowledge gaps are, at least partially, due to an insufficiency of more appropriate mathematical frameworks. It is conceivable that more robust analysis and simulation methods could describe the brain patterns involved, allowing to link them to emergent motor responses and actions. As recently demonstrated in a breakthrough work (Shiu et al., 2024), theoretical tools are paramount to comprehend the sensorial processes in the Drosophila brain.

Given the above, as a novel protocol to delineate and typify the neural-motor activity patterns of Drosophila larvae, we propose a graph theory approach to characterize the structural patterns of neuronal activity waves. It allows to quantitatively assess asymmetry and propagation patterns across the VNC. This is achieved by processing Ca imaging data such that each burst of neuronal signal corresponds to an “event,” represented by a graph node ascribed to the VNC activated region. As activity propagates across different VNC regions, the associated activation flow is represented by directed graph edges. This ensures that the geometrical structure of the resulting graph mirrors the physical arrangement of the triggered VNC regions.

Based on such mathematical construction, among other aspects one should be able to compare activity behavior in larvae across different developmental stages, thereby describing the evolution of motifs of neuronal activity underlying movement. To demonstrate that this is indeed the case, we have performed many controlled measurements (details in the following) explicitly considering the larvae's specific developmental phases. After emerging from the egg, the Drosophila undergoes three growth stages, the instars (see Figure 1). Each stage is separated from the previous one by a cuticle molt to accommodate their remarkable size increase. Throughout the first, second, and the beginning of the third instars, the larvae display foraging behavior characterized by continuous feeding and a permanent search for food. This feeding frenzy is vital to sustain the mentioned rapid growth and to guarantee enough energy reserves for surviving metamorphosis (Tennessen and Thummel, 2011). So, the significance of larval movement extends beyond mere locomotion. In consequence, the regulation of foraging patterns has far-reaching implications for ecological interactions, sensory perception, and ultimately, the fitness of the organism. Naturally, all these processes are driven by the patterns of neuromotor activity in the animal's VNC.

Figure 1. Drosophila full life cycle. In particular, the three larval phases, the instars, are explicitly shown. The characteristics times of the transitions between different developmental stages are also indicated.

In the experiments, we recorded detailed neuromotor activity of the first (L1) and third (L3) larvae instars (with an one stage lag chosen so to warranty a clear developmental maturation between phases) and applied the proposed graph-based scheme to investigate the structural characteristics of neuronal activity underlying specific behavioral outputs, such as turning and crawling. Additionally, we tracked larval trajectories in a custom-designed measurement arena to examine dispersal behavior (Almeida-Carvalho et al., 2017) and evaluated how differences in neuromotor coordination influence movement dynamics. For this type of locomotion data, besides L1 and L3, we also considered the second (L2) instar. By comparing the neural structures across the stages, our treated data clearly established that activity propagation and generation in L1 larvae were less coordinated compared to L3, reflecting the immaturity of the former neural networks. More importantly, these differences could be concretely quantified by relating distinct “motifs” to the graphs representing the neural firing circuits. Indeed, their types, frequency of occurrence, correlation and temporal sequence indicated the basic motion steps being executed and how efficiently and synchronized the neural signaling induced movement and spatial dispersion.

The structure of our works is as follows. In section 2 we first describe the experimental procedures, including the rearing of Drosophila larvae, the recording of their locomotion trajectories and the Ca imaging techniques employed to monitor neuromotor activity. Second, we introduce our mathematical modeling approach based on graph theory and motifs identification, which is used to analyse and quantify the propagation patterns of neuronal activity within the VNC. Section 3 presents our key findings, highlighting the differences in spontaneous activity generation, activity propagation efficiency, and symmetry patterns between first instar (L1) and early third instar (L3) larvae. Finally, section 4 summarizes the implications of our study, emphasizing the developmental progression of neuromotor coordination and suggesting potential directions for future research.

2 Methods 2.1 Drosophila larvae rearing

Larvae neuronal activity was assessed at two distinct stages of their development, first instar (L1) and early third instar (L3). For the recording of locomotion trajectories we also assessed second instar (L2). To establish the experimental sample, 30 female and 20 male adult flies of the wild type OregonR strain (OrR) were placed in a laying pot and provided with a standard corn media enriched with yeast paste Petri dish to lay eggs. The eggs were collected for 2 h and incubated either for 24 h (first instar; L1), 48 h (second instar; L2) or 74 h (early third instar; L3) at 25°C under a 12-h light-dark cycle. Three sets of larvae were produced for each developmental stage. This controlled setup ensures synchronized development and consistent conditions for subsequent analyses of locomotion and trajectory patterns.

2.2 Recording locomotion trajectories

We conducted observations of larval exploratory behavior in controlled environments under minimal external stimuli. The recordings were conducted in the dark apart from infrared light that the larvae can not see, maintaining a consistent temperature of 25°C. Each experimental trial spanned 60 min, during which the larvae were tracked within a 240 × 240 mm2 arena. The arena surface was prepared with a 0.4% agar-based coating, 2 mm in thickness.

In each trial, a group of 10 larvae of similar size was put to roam at the arena, and each developmental stage (L1 and L3) was tested with three repetitions (summing up 30 larvae per stage). Their movements were recorded through a frustrated total internal reflection (FTIR)-based imaging technique (Risse et al., 2013) using a Basler acA2040-180 km CMOS camera set at a resolution of 2,048 × 2,048 pixels. The recording was performed at two frames per second, optimized for accurate representation of forward displacements and actual pause-turn events, while minimizing the inclusion of “flickering” movements (often associated with peristaltic contractions). To enhance the observations precision, an advanced imaging setup was employed, featuring a 16 mm KOWA IJM3sHC.SW VIS-NIR lens and a 825 nm high-performance longpass filter (Schneider, IF-093). For the recording of the first instar larvae L1, an additional 2 × amplification lens was used.

The images were then processed with the FIM-track software (Risse et al., 2017), yielding positional time series for each of the larva's trajectories, see the schematics in Figure 2A. To make the trajectories as free from the other organisms as possible, a tracking was always interrupted upon any collision between two distinct larvae. So, two new trajectories were assumed as the interacting larvae would get enough away apart. Similarly procedure was considered when the larvae reached the arena's edge. Figure 2A illustrates the experimental setup.

Figure 2. Schematics of the movement experiment and type of data gathered. (A) Arena, image processing setup, locomotion trajectory. (B) Positional measurement points along the larvae's body. (C) Displacement aggregation method to generate random walk steps.

2.3 Recording locomotion wavelength

To evaluate the number of body segments along which a peristaltic wave of muscle contraction progresses (here operationally defined as “wavelength”), we recorded movies in L1 (15 individuals) and L3 (10 individuals). Larvae were transferred to a 5 cm petri dish coated with 0.6 ml of 0.9% agarose. The plate was inverted to view the denticle bands and 2 min movies were captured at 30 fps with a ximea MQ013CG-ON camera mounted on a Leica M420 microscope at 25 × (L1) and 6 × (L3) magnification. Larvae only executed forward peristaltic wave and their progression was determined by the number of segments contracting as revealed by denticle belts movements that are located on the boundary of each segment and clearly visible from A8 to A1. The movement of the thoracic segments T3, T2 and T1 was evaluated by the movement of A1 and the front of the animal, thus representing three segments. The events were quantified with the open source software VCode 1.2.1 (http://social.cs.uiuc.edu/projects/vcode.html).

2.4 Descriptive statistics of larval trajectory

Positional data, i.e., (x, y) coordinates of an individual larvae trajectory i in each video frame j (so a characterization of time), is represented by the vector

Ri⃗(tj)=(xi(tj),yi(tj)),    (1)

where (xi(tj), yi(tj)) represents the centroid coordinates of the larvae at time tj = jΔt. Here, Δt = 0.5 s is the temporal resolution, j = 1, 2, …, N the frame index and N = 6000 the number of frames per experiment. Measurements include the positions of the head, tail, and three spine points for each larva in every frame (Figure 2B). The body length Li(tj) at tj is computed as the sum of distances between consecutive body points; an average 〈Li〉 is calculated for each trajectory.

Trajectories are segmented into quasi-linear portions, defined as stretches of more or less straight displacements, delimited by the organism directional changes. The movement in a sequence of frames are aggregated into a single step if the intermediate frame positions all fall within a distance ϵ from the line connecting the first and last frame positions (Turchin, 1998)—refer to in Figure 2C. The distance ϵ is defined as half the average body length at each trajectory, plus half the standard deviation, namely,

ϵ=〈Li〉/2+〈Li2〉/2.    (2)

The aggregation of frame-movements into steps is performed with a Ramer-Douglas-Peucker algorithm so to find the combination that best represents the trajectory (Wosniack et al., 2022; Ramer, 1972). The velocity Vk is defined in terms of the average velocity across each linear step

where ℓk is the linear displacement of the step k and ΔTk its duration. Vk is then normalized in relation to body-size by dividing it by 〈Li〉.

To quantify the degree of dispersion of the trajectory i (starting at ta and ending at tb), we consider the Mean Square Displacement (MSD) over a time window τ, MSDi(τ), computed by averaging all possible displacements of duration τ along i

MSDi(τ)=1n∑t=tat=tb-τ[(xi(t+τ)-xi(t))2+(yi(t+τ)-yi(t))2].    (4)

Here, n is the number of displacements counted in each i. The overall MSD(τ) for the entire set of trajectories follows by averaging the MSDi(τ)'s, normalized by their respective mean body length (so to yield values in units of individuals sizes).

2.5 Calcium imaging

For calcium imaging of the central nervous system (CNS), whose location in the individuals body is indicated in Figure 3A, the GAL4-UAS system was employed to activate the calcium indicator GCaMP3 (Tian et al., 2009). Specifically, the OK371-GAL4 driver (Mahr and Aberle, 2006) was used for expression in motor neurons and R36G02-GAL4 for A27h (Fushiki et al., 2016).

Figure 3. Calcium imaging technique and representative results. (A) Drosophila's body with indicated morphological structures. (B) The VNC and corresponding ROIs located in each segment. Correlation between bodily movements (C) and VNC activity (D).

Thus, individual first and third instar larvae were dissected using hypodermic needles. The central nervous system (CNS), including the brain, subesophageal ganglion (SOG), and ventral nerve cord (VNC), was delicately separated from the larval body wall (Figures 3A, B). It was then mounted dorsal side up on a cover slide coated with 0.1% poly-L-Lysine (Sigma P8920). To ensure stability, recordings commenced at least 5 min post-dissection. Throughout the dissection and subsequent Ca imaging experiments, the CNS was bathed in a physiological saline solution containing (in mM) 135 NaCl, 5 KCl, 2 CaCl2, 4 MgCl2, 5 TES and 36 sucrose.

For live imaging of the isolated CNS, wide-field epifluorescence microscopy was employed. We used a cool LED simply better control PE-300white at 1% intensity for uniform illumination at 488 nm, with imaging conducted using an Olympus BX50WI compound microscope (Olympus, Center Valley, PA). The emitted light was filtered through GFP emission filters before being captured by an Hamamatsu Orca Flash 4.0 (Hamamatsu Photonics K.K). Image capture rates were set at 5 Hz using HCImage software, maintaining constant gain settings. We analyzed fluorescence values from regions of interest (ROIs) in thoracic (T1–T3) and abdominal (A1–A8) ganglia (Figure 3B). The optical intensity Ii = Δf/f were smoothed with a moving average in the range of 3 s, the asymmetric least squares method (Peng et al., 2010) was used to correct the signal baseline, and the data normalized to put the signals within a quantitative range from 0 to 1, where 0 shows minimum activity and 1 maximum activity. Typical results are presented in Figures 3C, D.

2.6 Mathematical modeling and a novel graph characterization

Calcium imaging measurements lead to curves as shown in Figure 4A, each representing neural activity within one neuronal ROI and ranging from A1 to T3 on both the left and right sides (Figure 4B). Given that crawling behavior emerges from a coordinated pattern of peristaltic segment contractions (Heckscher et al., 2012; Gjorgjieva et al., 2013), we should expect the ganglia neuromotor activity to correlate with muscular contractions in the corresponding body segments, as shown in Figures 3C, D. In this way, calcium imaging of the larval VNC reveals a pulse propagation along ganglia, similar to peristaltic contractions (Figure 5).

Figure 4. Typical calcium imaging signals. (A) A set of neural activity pulses for left and right A1–A8 and T3 (T1 and T2 are very similar, so not shown). (B) Peak patterns along a pulse of a given ROI (here A8). Each peak corresponds to a neuromotor activity event in a ROI. For the later graph construction, vertices will be associated to left (red) and right (blue) peaks as the present ones.

Figure 5. Illustration of neural and muscular activities correlation. (A) Activity pulses traces taking place in distinct ROIs, as measured by calcium imaging. As indicated, the emerging pattern corresponds to a forward wave (from A8 onward). (B) The dynamics of VNC—equivalent to vertebrates spinal cord—wave behavior and the resulting segment contractions during a forward crawling wave.

These results clearly indicate that clusters of neural activation—whose local intensities variation can be monitored from the images—are essential for producing functional locomotor behavior. Nevertheless, the study of neuronal propagation across ganglia tends to be mostly descriptive. Indeed, it expresses bursts as “waves,” but typified just in terms of either forward (toward the anterior neural system part, i.e., A8 → T1) or backward (toward the posterior neural system part, i.e., T1 → A8) propagation. Further classifying the stimulations as either symmetrical, on both the left and right sides, or asymmetrical, on one side only.

It is well known that the proper characterization of an evolving network structure representing a system dynamics is able to uncover many of its properties (Albert and Barabási, 2000; Wang et al., 2015; Perra et al., 2012; Zhang et al., 2021). Hence, based on the type of data we typically acquire from calcium imaging experiments, we have developed a mathematical framework, a graph-based model, to quantitatively describe the mentioned neural processes. The primary objective of such an approach is to prevail over the limitations of the previously mentioned qualitative analyses. Concretely, to identify different kinds of VNC activities and even ascertain how the movement patterns may be regulated by left/right ganglia sides switches. Hence, rather than just classifying a signal as forward or backward, the generated graphs should also single out the local traits of the propagating waves (in each ROI). Additionally, they should unveil symmetrical features, such as intensity and phase differences, something not possible by dividing the images into two halves and treating them as simplified binary data.

So, the key presumption is to be able to map each set of neural activity, represented by a certain collection of signals as in Figure 4A, to a graph. Naturally, the stemmed graphs are only a partial representation of the full process, but as we demonstrate in the following sections, they already yield significant information about the actual phenomenology. Consecutive activities result in distinct graph architectures, generating successive motifs. The investigation of the sequential engendering of these motifs and their correlation and frequency should give relevant quantifiable information about the neurons signals propagation and how they might related to motor behavior.

The protocol to construct a graph j is implemented as the following (refer to Figure 6 as a schematic guide). We assume a specific collection (labeled j) of signals, as those displayed in Figure 4A (for how to specify the different j's, see below). Then, we first identify the different peaks i (i = 1, 2, 3, …) (Figure 4B) for each activity curve from every ROI in j. Notice the peaks have two fundamental properties: (a) intensity Ii = Δf/f (more details in Section 2.5), which can be normalized between 0 and 1, where 0 and 1 indicate, respectively, the baseline and the overall maximum along the whole activity curve; and (b) occurrence time ti, i.e., the instant the peak i reaches the intensity Ii. In this way, each i corresponds to an activity event ai(j)(si,li,Ii,ti) of j, where si is the segment (A1–A8, T1–T3), li is the side (left or right), Ii is the normalized intensity and ti is the peak time instant. Second, vertices (or nodes) vi are assigned to all these peaks i in the collection j, forming the set j (Figure 6A). Last third, we represent the j sequential firing patterns across ROIs—essentially an activity chain—by means of a directed graph Gj(v,e→). For so, we consider the vertices set j, connecting them through a set of directed edges j, classified into two kinds, symmetry and propagation (Figures 6A–C). The construction of j obeys the straightforward rule: if two adjacent ROIs are on the same segment (side), but correspond to different sides (segments), and their activity peaks, represented by vertices vi″ and vi′ taking place at ti″ and ti′, are such that 0<Δt=ti″-ti′≤τ, a symmetry (propagation) directed edge e→i′i″ from vi′ to vi″ is established, refer to (Figures 6B, C).

Figure 6. Neural activity signals and graph representation. (A) Basic elements (vertices and edges) used to portray neural events as a graph. A directed edge linking two vertices in adjacent ROIs (refer also to Figure 4) is established when both events, firing peaks, are within a time range Δt ≤ τ = 3s —determined from data, as discussed at the end of Section 2.6. (B) The creation of a symmetry edge, joining vertices in different sides. (C) The creation of a propagation edge, joining vertices in different segments. (D) Activity signals in each neural cluster with the constructed motifs (sub-graphs of the full G) representing the associated pulses.

Thus, Gj gives a general picture of the full neural activity instance j, with the vertices of Gj associated to the different events ai(j) and the directed edges of Gj indicating their temporal connection. In fact, j is a direct portrayal of the neuronal activity propagation, as illustrated in Figure 6D. Consequently, typical neural activity patterns (for example, the forward wave in Figure 6D) are represented as motifs, i.e., simple sub-graphs of Gj, see Figure 7. By investigating successive graphs, …, Gj−1, Gj, Gj+1, …, it is possible to infer statistical properties of activity propagation by comparing ordering, correlation and relative frequencies of such motifs.

Figure 7. Tree of possible propagation motifs resulting from the graph construction and their nomenclature (depending on the directional flow). The motifs, basic small sub-graphs, have a central reference vertex, red dot, which then may or may not be connected to other vertices, yellow dots, by oriented edges.

We finally mention how to identify different Gj's. Typically, the span of a full calcium imaging experimental run is of about Trun = 10 min, along which one clearly observes intense neural firing activity in the brain. But such long series is composed by many bursts, namely, chains of activation separated by short temporal intervals. Each one of them—e.g., the collection of signals in Figure 4A—constitutes thus a specific j and accordingly a graph Gj. Taking the whole runs and computing the average interval between successive (in time) peaks, we found Δt¯=0.618 s and σΔt = 0.427 s for L1 and Δt¯=0.755 s and σΔt = 0.234 s for L3, hence in both cases with a fair effective Δteff=Δt¯+σΔt≈1 s. In this way, for the above threshold we set τ = 3Δteff, establishing the maximum delay between two connected peaks of τ = 3 s. Importantly, in most cases 3 s fits with the normal times separating empirically clear distinct bursts (identified by direct visual inspection), despite the fact that the above factor of 3 was chosen somewhat arbitrarily. We define as J the total number of graphs constructed from an experimental series. Among all the J graphs, we define J1 as the number of trivial ones, i.e., those having just a single vertex.

3 Results

Based on the novel protocol of analysis proposed (relying on a graph construction), next we demonstrate its usefulness in characterizing and interpreting the experimental data obtained.

3.1 Spontaneous activity

We shall name “spontaneous” activity events those whose associated vertices have no incoming edges, refer to Figure 7. So, they are not triggered (at least within a time interval Δt) by spikes in neighboring ROIs. On the contrary, these events are the ones that eventually can initiate waves of neural activity, meaning they are the starting vertices of any graph Gj (see the discussion at the end of Section 2.6).

The L1 and L3 larvae exhibit very similar main frequencies of spontaneous activity (Table 1), calculated as the average of J/Trun (Section 2.6). However, spontaneous events in L1 larvae are slightly more prone to not triggering a chain of neural activity (27.50%) than in L3 (20.00%), generating trivial graphs with a unique vertex. Statistical analysis supports this observation, as the metric nodes_no_incoming_no_outgoing_edges was significantly higher in L1 than in L3 (t = 3.13, P = 0.012), indicating a greater occurrence of trivial graphs in L1. Meanwhile, the metric nodes_no_incoming_edges showed no statistically significant difference between groups (t = 1.67, P = 0.129), suggesting that while L1 larvae exhibit a higher mean, the difference is not substantial between developmental stages. This higher failure rate results in a larger value of J1/J for L1. On the other hand, among the spontaneous activity events that successfully initiate an activity propagation (AP); for L1 we find a fairly equal distribution between forward and backward waves (~35%) and just a little lower percentage of both directions waves (~30%); but for L3 close to half (~45%) of the waves are forward, whereas 35% are backward and only 20% are both directions waves. But despite the mentioned contrasts, the standard deviations of each one of these proportions are similar for the L1 and L3 larvae, details in Table 1.

Table 1. Frequency of the spontaneous events (SE) among all events (J/Trun) and relative percentage of their associated activities, averaged over each larvae population.

Interestingly, the distributions of spontaneous activity among segments display certain distinctions for L1 and L3. Indeed, as the forward propagation, the L3 larvae present a clear dominating peak at A8, absent for L1 (Figure 8A). However, for backward propagation, although the highest peak for L3 is at T1, we still have other important peaks, notably at T2. Such trend for backward propagation, in the sense there is not an unique very pronounced peak, is the same for L1 (Figure 8B). But notice that for L1, the second most frequent activity is not at T2, but instead at A2. The prominence of A8 and T1 corresponds to the expected initiation points of behavioral contraction waves (tail for forward, head for backward). To better characterize these dissimilarities between L1 and L3, we used the Kolmogorov–Smirnov (KS) test (Dodge, 2008), which evaluates whether or not two distributions differ significantly by comparing their cumulative distribution functions (CDF). For the forward propagations per segment of L1 and L3, the KS statistic resulted in 0.545, with a P-value of 0.07. This near-threshold P-value suggests that L1 and L3 distributions are likely governed by distinct processes, albeit with a limited statistical certainty. Conversely, for backward propagations, the KS statistic was 0.273, with a P-value of 0.83, indicating minimal distinction between the distributions, supporting the hypothesis of akin drives for L1 and L3 in the case of backward propagations.

Figure 8. Spontaneous activity per segment of L1 and L3 larvae. (A, B) The distributions for segments initiating forward and backward waves (the elementary motifs in the inset). (C–F) Heat maps in which the horizontal axes marks the origin of the activity chains, thus corresponding to the locations of the spontaneous segments (or vertices). The propagation occurs along each row (as shown by arrows). The color intensity indicates how frequently a given segment contributes to the associated propagation chain.

The relevant distinctions and similarities between L1 and L3 spontaneous activity generation are more clearly evidenced through heat-maps (Figures 8C–F). Such type of plots provides relevant insight into the directional flow and distribution of spontaneous activity across segments. As previously mentioned, forward wave propagation reveals striking differences between L1 and L3 (Figures 8C, E). In L3, forward activity is highly concentrated at A8 and evolves consistently through most of the larva's body. In contrast, L1 displays a concentration of forward activity generation predominantly at A1 and T3, highlighting an important difference regarding the initiation vertices of contraction waves for the L1 and L3 individuals. Nonetheless, backward propagating waves for L1 and L3 exhibit fairly comparable patterns (Figures 8D, E).

3.2 Propagation of activity

Differently from spontaneous, “excited” events relate to activity triggered by preceding—but not later than Δτ—spikes in neighboring segment ROIs. In our construction, these events are represented by vertices attached to oriented edges. When these edges income from the posterior (anterior) region, inset in (Figures 9A, B), we have a forward (backward) propagating wave. Further, if these vi's are also tied to outgoing oriented edges e→ii′, thus directed toward other vertices vi′'s, they form the interior links of a propagation chain, or equivalent, “inner” vertices of a graph Gj. Obviously, the first vertex of any Gj is spontaneous, whereas the last is excited, but not an inner vertex. We denote as VinnerL(Gj) (VinnerR(Gj)) the number of inner vertices of Gj related to the left (right) segments. The length of Gj is then assumed as the largest between VinnerL(Gj) and VinnerR(Gj). Such definition is useful if the interest is to characterize the neural activity just in terms of type of segments, regardless their locations.

Figure 9. Inner excited activity per segment (so A8 and T1 are not included) of L1 and L3 larvae. The probability for a segment to continue the (A) forward and the (B) backward propagation once excited. The insets show a typical link (i.e., a very basic motif in the graph representation) of the propagation chain. The graph length (main text) distribution for Gj's representing (C) forward and (D) backward propagation.

Given a graph we can count the number of its vertices corresponding to a certain segment (A1 to A8 and T1 to T3). So, considering all the non-trivial J-J1 graphs obtained from the data, we can determine the relative contribution of any segment to the neural activity processes along a full experimental run. For instance, in Figures 9A, B we depict the probability of each inner segment to participate in, respectively, forward and backward waves. As one can see, with the exception of T2, all other subsequent segments for the L3 larvae are more likely to continue a propagating wave, either forward or backward.

From the comparatively lower propagation probability in the L1 larvae, one can infer that their neural signals present a higher tendency to fade away while traveling along the VNC, resulting in shorter activity chains, i.e., smaller Gj's. This can be quantified by computing the distribution of the previously defined graph length for Gj