Smoothness analyses of trace fossils: expanding quantitative toolkits to unravel bilaterian movement ecology during the Ediacaran–Cambrian transition

Abstract

Trace fossils provide unique insights into early animal evolution during the Ediacaran–Cambrian transition. Throughout this critical interval, the smoothness (continuity of derivatives) of combined locomotory and feeding trace fossils (i.e. grazing) significantly increased, potentially indicating diverse evolutionary innovations. We propose several quantitative metrics, including smoothness criteria, minimum normalized radius of curvature and power spectral density, to explore these trace fossils, benchmarked by diverse modern analogues. We classify the trace fossils into unsmooth, regional-smooth and smooth trajectories. A noticeable appearance of regional-smooth and smooth trace fossils among different localities took place around the terminal Ediacaran (since ~545 Ma). Through comparison with extant animals, this period is consistent with the rise to dominance of slender mobile bilaterian eumetazoans with a hydrostatic body, coordinated musculature and long-range sensation to maintain course. The increase of body-length-to-width ratio, and locomotory and navigational capacities of the trace-makers are also quantitatively assessed during metazoan evolutionary radiation. Our results pinpoint the time window for the emergence of ecologically dominant benthic mobile bilaterians in an exponential fashion. We also expand quantitative ichnology, offering new criteria to explore palaeobiological and evolutionary implications of animal–sediment interactions at macroevolutionary scale.

1. Introduction

Combined locomotory and feeding (i.e. grazing) trace fossils are a unique source of data in palaeobiology. They provide valuable insights not only on the mode of locomotion and feeding strategies, but also on the evolution of sensory–nervous systems of trace-makers. In the case of the Ediacaran–Cambrian transition, the ichnological record has long played a substantial role in revealing groundbreaking palaeobiological and palaeoecological changes at community and ecosystem levels [1–6]. The trace fossil record helps to calibrate the earliest appearance of body plans (e.g. presence of articulated appendages [7]) and diverse behaviours including feeding (e.g. sophisticated grazing strategies [8]). Ichnological evidence has been instrumental in filling gaps in the body fossil record, particularly helping to reconstruct the cryptic history preceding the Cambrian explosion [9].

The striking differences between Ediacaran and early Cambrian trace fossils are well known, with most Ediacaran trace fossils fitting within the architectural category of ‘simple horizontal trails and burrows’ [10]; these trace fossils tend to be millimetric and lack well-defined morphological features. In contrast, early Cambrian trace fossils display remarkable varieties of architectures, cross-scale sizes (from micrometric to centimetric) and high fine-scale morphological complexity that reflects novel animal–sediment interactions [11]. A transition between these two scenarios is revealed by a detailed exploration of the terminal Ediacaran that has documented the emergence of larger bioturbators and more complex and penetrative trace fossils, signalling a prelude to the Cambrian explosion [12–17]. Although an expanded ichnological toolkit has been recently developed to explore this complex scenario [5,9,11,18], there are few established metrics to quantify trace-fossil morphology, resulting in limited quantitative studies in analytical palaeobiology. In the 1990s, the pioneering work of Hoffmann [19] numerically generated horizontal trajectories using computer programs and found that the trace-makers with larger deviation angles per step could create more random and tortuous paths like Helminthoidichnites. Plotnick [20,21] used individual-based models upon Lattice maps to simulate traces, recording multiple behavioural responses to changes in environmental heterogeneity. Combined with experiments, Koy & Plotnick [22] also found an increase in tortuosity when the trace-maker foraged upon isolated food patches, and identified such behaviour as early as the Ediacaran–Cambrian transition. Mitchell et al. [23] studied the spatial correlation of trace fossils from the Shibantan member of the Dengying Formation and identified the positive aggregation of Ediacaran trace-makers around food resources, indicating the ability of early animals to detect food in their proximity. More recently, the frequency spectrum and autocorrelation function were proposed to calculate the major frequency and scale of turnings, which largely reveals the putative body shape and some aspects of the locomotion mechanism used by potential producers of some well-known ichnotaxa [24]. Also, autocorrelation and a method to discretize horizontal movement paths as reflected by trace fossils were used to assess early evolutionary innovations in bilaterians within the framework of movement ecology [25]. Fractal indexes, which can be related to both behaviour and density of food, have been occasionally used as well to explain changes in trajectory [26,27].

All the above studies focused on the overall pattern of the trace fossils using statistical metrics like fractal index, tortuosity, sinuosity or undulation periods, which can be significantly influenced by behaviours related to external conditions like food distribution, type of substrate and consistency, flow currents or predation pressure. A metric that can reflect organismal body profile and reveals the inherent nature of the trace-makers’ locomotion has not yet been developed, and it is worth exploring.

In mathematics, smoothness refers to whether a function (at any dimension, including both curves and surfaces) has continuous derivatives along its profile, measuring the fluency of its trend. For surfaces, as extensively done in previous ichnological studies, surficial smoothness describes (lack of) surficial ornamentation (i.e. bioglyphs, tension microfractures related to concentric infill). For curves, trajectory smoothness describes the existence of abrupt turns. These two smoothness definitions reflect derivative continuity of the (surface or curve) function. Conceptually, trajectory smoothness (hereafter ‘smoothness’) measures how fluent trace-maker’s turning behaviour/locomotion is, even within a single turn, as opposed to jagged turns (figure 1D). It also mathematically reflects the differentiability of a curve, which is independent of the overall trajectory profiles like sharpness (i.e. the magnitude of turning, figure 1D) or tortuosity (i.e. both the frequency and magnitude of turning). This smoothness has significant potential in ichnology, while it has not been quantitatively discussed or defined before from an ichnological standpoint.

Figure 1.

Concept of trajectory smoothness. (A) Sketch of turning angle at sampling point A. (B) Sketch of the definition of first-order smoothness, where left trace is regionally smooth around (A) while the right is not. (C) Radius of curvature at (A). Dashed line is the imaginary circle of $R_i$. (D) Sharpness measures the magnitude of turning, while smoothness measures the differentiability.

Informed by modern analogues, the smoothness of the curve of a horizontal trace is directly related to the body shape, body length and behavioural fluency (smooth transition between behaviours/movements without abrupt changes), and can be linked to the basic locomotory mode of the trace-maker. This metric can also be influenced by the sensory range of the organism to a certain degree. By calibrating the smoothness of trace fossils with the smoothness of modern animal traces, we can gain crucial insights on the palaeobiology of vagile, benthic animals including their body shape, basic locomotion mechanism and navigational capacities (i.e. the organism’s ability to sense and respond to signals from the environment, sensu [28]). Accordingly, smoothness offers a novel quantitative tool to expand our understanding of evolutionary breakthroughs that took place during the Ediacaran–Cambrian transition.

Previous evolutionary studies of trajectories have been descriptive, with very limited quantitative criteria, or comparison with modern animals (see Laing [25] for an exception). In this article, we mathematically apply smoothness measures to trace fossils using definitions for limits [29] and classify their smoothness into three categories, smooth, regional-smooth and unsmooth, which we propose relates to distinct basic locomotion modes, such as slow crawling/creeping, muscular-foot gliding/short-arthropod walking and peristalsis/undulation, respectively. We also present two other metrics that may assist ichnologists to better understand the nature of smoothness, including the minimum normalized radius of curvature, which is related to the size of the locomotory unit (and which informs on the locomotory mode), and the power spectral density (PSD), which reflects contribution of each turning frequency to the total variance of turning signals.

Using these metrics and trails left by extant animals, we attempt to calibrate the smoothness of different horizontal trails to best assess changes in body shape/length, basic mode of locomotion and sensory range across the Ediacaran–Cambrian transition from a variety of perspectives. We have quantified selected ichnotaxa from the most extensive database available [5].

Our core hypothesis states that this transition signals significant changes in body shape accompanied by anatomical and locomotion developments indicating phenotypic evolution. We potentially link these changes to the evolution of animal sensory–nervous systems that resulted in an increased smoothness index in the context of more complex, heterogenous terminal Ediacaran–early Cambrian environments. This quantitative approach offers a novel perspective to reveal the bilaterian radiation during the Ediacaran–Cambrian transition.

2. Material and methods

(a). Materials

We included in our analysis 172 trace-fossil occurrences from 38 stratigraphic units, and trails left by modern foraminiferans, arthropods (and vermiform larvae), gastropods, annelids, nematodes and acorn worms collated in comprehensive databases [4]. For trace fossils, ichnotaxa include simple horizontal grazing traces Helminthoidichnites (some referred to as ‘Planolites’ in previous studies), Helminthopsis, Gordia, Cochlichnus and Archaeonassa. These simple horizontal trails are considered a representative sample for this period, as they compose 59% of all documented Ediacaran trace fossils worldwide [4], being present in diverse depositional environments to an extent unknown by any other assemblages in Phanerozoic rocks [4]. Other ichnotaxa included are Psammichnites (originally described as Helminthoida or Taphrhelminthopsis in some original sources), Parapsammichnites and Spirodesmos. These ichnotaxa represent morphologically more complex biogenic structures that have either worldwide distribution or belong to architectural designs that cross the Ediacaran–Cambrian transition [4,30]. We have also included in our analysis the Shibantan grazing trails left in open nomenclature [30,31]. Other trace fossils not included in our study, such as Lamonte [31,32] and putative resting traces Yilingia [33] and trackways [34] warrant further analyses [11,35]. Morphologically more complex, Cambrian-type trace fossils, such as Cruziana, Didymaulichnus or Treptichnus, record other architectural designs and behaviours [11] and are not the focus of the present analysis.

Analysed ichnotaxa come from the Ediacaran White Sea assemblage to Cambrian Stage 2. Based on U-Pb radiometric dating (when available) and the relative stratigraphic position of the ichnofossil-bearing strata within a particular lithostratigraphic unit, the age of the specimens studied is estimated.

(b). Smoothness criteria

Smoothness criteria ensure no abrupt change in turning angles (i.e. continuous first-order derivatives) along a trajectory. The turning angle at the i-th sampling point A$(x_i,y_i)$ is defined as the change between its directional vector $n_i$ and that of the previous point A’ $(x_{i-1},y_{i-1})$, $n_{i-1}$ (figure 1A), which is computed according to Wang & Imran [24]:

$$\text{d}{\theta }_i=\{\begin{array}{cc}\arccos (n_{i-1}\cdot n_i)& \begin{array}{ccc}\begin{array}{cc}& \end{array}{n}_{i-1}\cdot n_i<0& \text{and}& {n_{i-1}\times n_i|}_z>0\end{array}\\ -\arccos (n_{i-1}\cdot n_i)& \begin{array}{ccc}\begin{array}{cc}& \end{array}{n}_{i-1}\cdot n_i<0& \text{and}& {n_{i-1}\times n_i|}_z<0\end{array}\\ \arcsin ({n_{i-1}\times n_i|}_z)& \begin{array}{cc}& \end{array}{n}_{i-1}\cdot n_i\ge \mathrm{0,}\end{array}$$ (2.1)

where ${|}_z$ denotes the z-component of the vector. Anti-clockwise turn has a positive deviation angle, according to Right-hand Corkscrew Rule of cross product between radial vector (from turning centre) and movement vector in Cartesian coordinates.

Based on the obtained $\text{d}{\theta }_i$ computed with the normalized trace length ($L^{*}=L/W$, where L is the real trace length and W is the width), the definition of regional (first-order) smoothness around point A$(x_i,y_i)$ reads:

For all extremely small values $\epsilon >0$, there always exists a tiny length $\delta$, such that whenever the distance between the two points $A^{-}$ and A is smaller than $\delta$, we can always have their deviation angle $d\theta <\epsilon$.

Based on the $\epsilon –\delta$ definition, the left trace in figure 1B is regionally smooth around point A: when ${\text{A}}^{-}$ and A are close enough (i.e. $\le \text{δ}$), the deviation angle between tangent line ${\text{A}}^{-}-\text{A}$ and tangent line $\text{A}-{\text{A}}^{+}$eventually approaches 0 (i.e. $\le \text{ε}$). However, in the unsmooth trajectory with an abrupt turn at A, the tangent line ${\text{A}}^{-}-\text{A}$ approached a fixed direction (α degrees above horizon) no matter how close they are, while the tangent line $\text{A}-{\text{A}}^{+}$ keeps horizontal. Therefore, there is a fixed lower limit to the deviation angle in this case, which is α. One can never find a δ (no matter how small it is) so that the deviation angle at A is smaller than α, not say an even smaller random value ε. In this case, the point that causes disruption in the continuity of turning angle series is termed as disrupted point or knot. Hence, the criterion for (first-order) trace smoothness reads:

For each and every point that could be found along a trace, if none of them are disrupted points, this trace is a smooth trace. If there only exists limited number of disrupted points (e.g., 10 knots per 100 normalized units’ length) along a trace, like a spline [36], this trace is a regional smooth trace. If there are more knots, or even each and every point is a knot, it is an unsmooth trace.

It should be noted that smoothness is a theoretical and intrinsic nature (differentiability) of a curve indicating the fluency of behaviour/locomotion, therefore is unrelated with the statistical metrics measuring overall patterns/complexity like sinuosity, tortuosity, fractal index or sharpness. For example, sharpness indicates the magnitude of the turning angle, as illustrated in figure 1D. However, the definition of the limit involves arbitrarily small values, making its implementation in vectorized trajectories challenging. As a result, the existence of a knot (i.e. a sudden change in the tangent line) can only be visually assessed by decreasing δ. To address this, we propose the following quantifiable metrics.

(c). Minimum normalized radius of curvature

Imagine there is a circle that can closely and perfectly be attached to the curvature of the trace-fossil trajectory, the radius of circle at this attaching point is its radius of curvature. As shown in figure 1C, from geometrical relationships, we know that:

$${\displaystyle \mathrm{\angle }3=2\mathrm{\angle }2=\mathrm{\angle }1=\text{d}{\theta }_i.}$$ (2.2)

Therefore, the normalized radius of curvature $R_i^{*}$ at point A is:

$${\displaystyle R_i^{\ast }=\frac{R_i}{W}=\frac{\text{d}x}{W|\text{d}{\theta }_i|}=\frac{W}{WF|\text{d}{\theta }_i|}=\frac{1}{|F\text{d}{\theta }_i|}=\frac{1}{|{\kappa }_i^{\ast }|},}$$ (2.3)

where dx is the sampling distance between two points along the trace and F is the sampling frequency: $W=F\cdot \text{d}x$. ${\kappa }_i^{*}$ is the normalized curvature at i-th sample point A. Hence, the minimum normalized radius of curvature ($R_{\min }^{*}$) along a trace with N sampling points reads:

$${\displaystyle R_{\mathrm{m}\mathrm{i}\mathrm{n}}^{\ast }=\underset{i\le N}{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{\frac{1}{F|\text{d}{\theta }_i|}\right\}.}$$ (2.4)

Theoretically speaking, an abrupt turn could give rise to an infinitely small $R_{\min }^{*}$ along the trajectory.

(d). Power spectral density

A common approach to analyse signals is to place them into the frequency domain where amplitudes of outputs represent the contribution from each frequency to the total variance. For trajectories, the frequency spectrum of the turning angle series informs on the frequencies/scales of inherent self-repeating patterns, therefore serving as a proxy to indicate small-scale zigzag locomotion. In this article, we will compute the PSD (mathematically denoted as S) using a fast Fourier transform [24]. PSD informs the distribution of signal power across different (normalized) frequency components $f^{*}=\frac{kF}{N}$ (k is the frequency index).

Since the trajectory is smooth when normalized frequency components approach 0 (i.e. infinitely large manoeuvres) and unsmooth when frequency components approach 1 (i.e. the zigzag period equals the trace/body width), it is reasonable to set 0.5 as a divide. On this basis, we can calculate the signal power representing small-scale unsmooth locomotion ($P_{\text{unsmooth}}$) by integrating the PSD from f = 0.5 to 1. For discrete signals, we have:

$${\displaystyle P_{\text{unsmooth}}=\frac{F}{N}\underset{0.5}{\overset{1}{\int }}S\left(f^{\ast }\right)\text{d}{f}^{\ast }.}$$ (2.5)

It should be noted that the word ‘power’ here is not equivalent to that in classic physics (rate of work). For clearer comparison, we further define the smoothness index with one minus the ratio between the signal power representing unsmooth small-scale wriggles and the total signal power ($P_{\text{total}}$), indicating the fluency of behaviour/locomotion during foraging:

$${\displaystyle \gamma =1-\frac{P_{\text{unsmooth}}}{P_{\text{total}}}=1-\frac{\underset{0.5}{\overset{1}{\int }}S\left(f^{\ast }\right)\text{d}{f}^{\ast }}{\underset{0}{\overset{1}{\int }}S\left(f^{\ast }\right)\text{d}{f}^{\ast }}.}$$ (2.6)

Tests of convergence of smoothness index over increasing trajectory length and sample number of trajectories obtained by individual-based foraging modelling [20]. This confirms the robustness of the metrics employed.

To study how perceptual range (within which stimuli are sensed and recognized) could influence trajectory smoothness, the individual-based foraging model (electronic supplementary material, SC) is also used to produce trajectories under diverse perceptual range.

3. Results

In this section, we present the quantitative results for the minimum normalized curvature, power spectral density and smoothness criteria of seven types of modern traces and 187 trace fossils across the Ediacaran–Cambrian transition. Sampling frequency of all the specimens is 2.

(a). Radius of curvature under different trajectory smoothness

We present the results of the calculations of the minimum normalized radius of curvature ($R_{\min }^{*}$) of the horizontal trails/burrows (figure 2). Those trace fossils classified as ‘unsmooth’ have extremely small, normalized radii of curvature (e.g. around 0.5). This is a reasonable outcome of the frequent abrupt changes of directions along unsmooth traces: the radius of curvature should theoretically be infinitely small around the disrupted point. Unsmooth trace fossils with many abrupt turns are clearly dominant in the Ediacaran. This type of trace fossils persisted in the Phanerozoic (e.g. extant foraminiferan trails, $R_{\min }^{*}\approx 0.3$) but statistically occupied a smaller proportion of the trace-fossil repertoire. The second group of trace fossils, regional smooth ones, display a limited number of disrupted points and proliferated after approximately 545 Ma during the terminal Ediacaran, with only three specimens (two from the Shibantan Member and one from the Ediacara Member of the Rawnsley Quartzite) being present in older Ediacaran rocks. Regional smooth traces reveal slightly larger minimum normalized radii of curvature (up to 2) compared with unsmoothed ones, resembling structures produced by short-bodied modern arthropods (horseshoe crab) and gastropods ($0.5\le R_{\min }^{*}\le 1$). This indicates that the trace-makers can make abrupt, even sharp turns, but normally their trajectory is locally smooth. The earliest evidence of smooth trace fossils is recorded approximately 545 Ma. In contrast to regional smooth trace fossils, no older representatives are known. The minimum radii of curvature of these trace fossils can be as large as 6. Modern insect larvae, nematodes, annelids and acorn worms produce smooth traces with comparable $R_{\min }^{*}$. Statistically, $R_{\min }^{*}$ increases with the body length-to-width ratio of the studied extant animals.

Figure 2.

Temporal distribution of minimum radius of curvature, with modern analogues plotted on y-axis, accompanied by silhouettes. Smoothness increases with minimum normalized radius of curvature. The bars indicate the estimated errors from dating. Colour of modern analogue corresponds to smoothness category: blue: unsmooth; red: regional unsmooth; black: smooth.

(b). Power spectral density and smoothness index

Through Fourier Transform of the turning angle series, we can compute the PSD of trails/burrows across Ediacaran and Cambrian. As an example, the PSD of a modern smooth acorn worm trail and an unsmooth foraminiferan horizontal trail are computed [37] (figure 3). The spectrum of acorn worm trail has major peaks at extremely low frequency components (figure 3E), indicating large-scale self-repeating patterns. For the rest of the spectrum with higher frequency components ($f^{*}\to 1$), the magnitude of normalized PSD is extremely low (approaching 0). This means for the same trace-maker (fixed I and v), it did not allocate much power/energy into small-scale high-frequency unsmooth locomotion. To be precise, after integrating the spectrum over $f^{*}\in [0.5,1]$, we obtain ${\displaystyle P_{\text{unsmooth}}\text{=0}\text{.00135}}$, only occupying 9.4% of the total signal power (γ = 0.906). This turning angle series is a pink noise signal where spectral density is basically inversely proportional to frequency (i.e. $S\propto 1/f^{*}{}^{\alpha }$, where $\alpha \approx 1$), and this indicates the trace-maker’s well-organized large-scale locomotion pattern and low (physical) power/energy-allocation for small-scale unsmooth movement [38].

Figure 3.

(A,C) Smooth acorn worm trace (photograph by Olmo Miguez-Salas). (B,D) Unsmooth foraminiferan horizontal traces [37]. (E) Power spectral density of modern traces in a dimensionless system. The numerous small-scale abrupt turns in the foraminiferan trail led to higher normalized spectral density at higher frequency components (close to 1).

Nevertheless, the unsmooth foraminiferan trace has multiple fluctuations in the spectrum (figure 3E), indicating that no regular self-repeating locomotion pattern exists, which can be inferred the extracted coordinates from the trace in the photo (figure 3B,D). Therefore, this trajectory has cross-scale radii of curvatures, and the trace-maker needs to allocate a certain amount of physical power/energy for generating small-scale abrupt turns. After calculation, we have $P_{\text{unsmooth}}=0.0687$, much larger than that of acorn worm trail, and γ = 0.523. This is a typical white noise signal with comparable spectral density over different frequencies, indicating stochastic patterns/behaviours [38].

Therefore, unsmooth traces with profuse small-scale abrupt turns would have higher PSD at higher frequency components compared to smooth ones. For the same trace-maker with the fixed inertia and velocity, an unsmooth trace inevitably requires more power/energy (both signal and physical) allocation for small-scale orientations. However, signal power per se does not reflect physical power (or energy consumption) among different trace-makers.

To test how perceptual range influences trajectory smoothness, we performed foraging simulations where trajectory was left by crawling organisms following maximum food gradient [20,21]. Five different normalized perceptual ranges ($L_{\text{P}}^{*}$) were used: 0.5, 1.5, 3, 4.5, 6 under the same food density (100 particles per unit area). From figure 4, the produced traces are becoming smoother parallel to an increase of perceptual range. This trend is also confirmed by the exponentially decreasing signal power from unsmooth locomotion (0.09735, 0.032755, 0.009155, 0.003017 and 0.0018295 for the five traces in figure 4), and less high-frequency components in the spectra.

Figure 4.

Signal power representing simulated small-scale unsmooth locomotion decreases with perceptual range. The different curves are the simulated trajectories and the corresponding spectral density. Black-lined subfigures: simulated trace; blue-lined subfigures: PSD of trace.

Based on the above understandings, we computed the $P_{\text{unsmooth}}$ and $\gamma$ for all the studied traces. All the unsmooth traces, along with foraminiferan traces, have large $P_{\text{unsmooth}}$ (up to 0.07) and low smoothness index (a minimum value around 0.5) (figure 5A,B), while low γ indicates the existence of kinesis (undirected responses to the intensity variation of the stimulus [39]). These trace fossils are dominant in the Ediacaran Period, as recorded by Helminthoidichnites, Gordia, Archaeonassa and Helminthopsis. Like the PSD of the Cambrian Archaeonassa (figure 3E), Ediacaran trace fossils with turns of hierarchical scales tend to have irregular patterns.

Figure 5.

Signal power representing small-scale abrupt turns (A) and smoothness index (B) of the studied trace fossils, with modern analogues on the y-axis.

Regional-smooth, extant arthropod and gastropod traces have moderate $P_{\text{unsmooth}}$ and smoothness index. However, in trajectories left by four types of extant slender worms and entirely smooth trace fossils that appeared after around 545 Ma according to our data, $P_{\text{unsmooth}}$ is predominantly below 0.005 and smoothness indices are generally larger than 0.8. The $P_{\text{unsmooth}}$ of trace fossils decreases exponentially through time.

4. Discussion

Our quantification revealed three features that can explain the transition in trace fossils across the Ediacaran–Cambrian transition. Trajectory smoothness can be related to one or all the following: (i) body shape, mostly related to aspect ratio; (ii) locomotory capacities, mostly related to the velocity and coordination of movement; and (iii) sensory capacities, mostly related to the distance at which sensory inputs can be detected and acted upon.

(a). Body shape and locomotory capacities

Unsmooth trajectories with a small radius of curvature characterize most trace fossils before approximately 545 Ma, although they still exist in tiny organisms like extant foraminiferans. Their apparent rarity in the Phanerozoic is not necessarily a loss, but indicative of the increased diversity and abundance of trace fossils resulting from the Cambrian Explosion. Out of the 382 early Cambrian stratigraphic units analysed in [5], only 3 of the trace-fossil associations summarized consist of just grazing trails.

The small minimum radius of curvature of Ediacaran trajectories may indicate that the producers of these trace fossils (figure 6A) had a round/short body, which might lack clear geometrical anterior–posterior differentiation. This morphotype facilitates sudden turns, even spot turns, in any direction. This could also be a result of small-sized locomotory units, such as amoeboid movement by extending parts of the body, or using crawling cilia [43–46], which would create turns at a (much) smaller scale compared to the body size. These features observed in linear traces correspond with the crawling/creeping behaviours of single-cell organisms (e.g. modern shelled Foraminifera or potentially Platysolenites [47–49]), but could also characterize the slow mode of movement of early bilaterians with an aspect ratio close to 1 (e.g. modern grazing horseshoe crabs occasionally have $R_{\min }^{*}\approx 0.5$). A similar locomotory mode can also be inferred for some macro-scale trace-makers from the White Sea Assemblage, which are now regarded as animals [50], as exemplified by the reconstructed trajectory of Dickinsonia, composed of a series of body imprints (Epibaion) with axes oriented in diverse directions [51,52]. This high manoeuvreability likely resulted from very slow, possibly ciliary movement, facilitated by the organism’s nearly circular body shape with an aspect ratio close to 1.

Figure 6.

Typical Ediacaran–Cambrian trace fossils. (A) Unsmooth Helminthoidichnites tenuis from the lower member of the Blueflower Formation, western Canada [40], with a large number of abrupt turns. (B) Regional-smooth Gordia isp. from the uppermost Brioverian, France [41], with few abrupt turns (e.g. in the red box). (C) Smooth Cochlichnus luguanensis from the Chapel Island Formation, eastern Canada [42]. (D) A sketch of a curving trace with a normalized radius of curvature of 2 (two times the trace width). (E) Smooth Parapsammichnites pretzeliformis from the Feldschuhhorn Member of the Urusis Formation, Namibia [14,16], with the minimum radius of curvature (R) shown in the red box. (F) Smooth Psammichnites isp. from the Shiyantou Formation, China, photo by Ziwei Zhao. Scale bars are 1 cm.

Regional smooth traces could be left by short-bodied animals, resulting from locomotion mechanisms involving the use of coordinated musculatures either controlled by internal hydrostatic pressure (analogous to slug-like molluscs) or by an antagonistically muscle system attached to an exoskeleton (analogous to arthropods) [53,54]. Transferring forces and displacements in predictable and stable ways [55] (e.g. muscular foot or aligned appendages) resulted in high coordination and the ability to maintain a smooth course. Albeit with a few earlier representatives, the flourishing of regional-smooth trace fossils happened during the terminal Ediacaran. Although paleobiologic affinities of the producers generating regional smooth trace fossils are unknown, their kinetic capacities indicate a major evolutionary breakthrough linked to new anatomies and the use of musculature. This provides strong evidence for a bilaterian equipped with well-coordinated muscles in a locomotory organ (e.g. a molluscan foot or arthropod limb). However, a relatively short body (i.e. aspect ratio comparable to 1−2) would be necessary for the trace-makers to make occasional abrupt or spot turns, likely having similar body plans, as those of short-bodied molluscs (e.g. Mono-/Polyplacophora, Helcionelloida) or arthropods (e.g. isopod-like). Smooth trails can only be produced by slender animals (e.g. extant annelids, nematomorphs, acorn worms vermiform larvae). These trajectories result from biomechanical constraints related to the body rigidity (due to existence of consistent longitudinal connective tissues or use of longitudinal muscles), i.e. the Euler–Bernoulli equation of a twisting worm’s body ensures a continuously differentiable (smooth) deflection curve where the body will bend alike [56]. Trajectory smoothness is maintained even with abrupt changes in body diameter, mechanical properties or external forces like organism–substrate interactions [57]. The appearance of smooth trace fossils by the terminal Ediacaran indicates larger, elongated bodies like those of palaeoscolicids, nematodes, polychaetes or acorn worms. A classic Fortunian example is Cochlichnus from the Chapel Island Formation at Fortune Head, Canada (figure 6B). It displays a large minimal radius of curvature along its trajectory (figure 6C). Even for a curve with a minimum radius of curvature of 2 (P. pretzeliformis; figure 6E) or 1.14 (Psammichnites isp.; figure 6F), the trajectory still shows the signature of a slender body incapable of abrupt bends, but equipped with complex longitudinal and circular muscles to support peristaltic movement (and likely responsible for the packed sediment pads in Parapsammichnites backfill). Slender body profile also helps directional sensation, movement stability and resistance/drag reduction. Therefore, the putative trace-maker of the looping P. pretzeliformis may have had a morphology like that of palaeoscolecids [58[58]]. Alternatively, the inferred link with the younger ichnogenus Psammichnites, which has an axial ridge indicating a snorkel device, might also suggest affinities with relatively elongated, vermiform molluscs [59].

The increase in body size/length versus grain size during the Ediacaran–Cambrian transition [60] also reduced the irregularities in the trace-fossil trajectory by enhancing the geometrical resolution and distributing pressure more evenly at organism–substrate contact surfaces, resulting in a smoother trend. It should also be noted that for an organism comparable in size or smaller than the surrounded grains, smoothness or many other quantitative metrics may not be supported due to the background noise/perturbations in the trajectory signal.

The low $P_{\text{unsmooth}}$ of smooth traces (both modern and fossil; figure 5) also indicates a fluent locomotion mode without small-scale shaky movements. These traces tend to have high continuity of curvature distribution, as a result of high kinetic coordination and flexibility of a trace-maker with smooth musculature action [53,61]: the organism was capable of making turns in a wide and continuous range of curvatures, with more predictable/organized/purposed locomotion. This is backed up by the trace-makers’ capacity to accurately control the direction of locomotion (kinematic coordination) in response to external stimuli based on a well-developed sensory–neuromuscular system [61].

Most simple horizontal trace fossils in Ediacaran White Sea deposits are unsmooth, having non-differentiable curves dominated by many abrupt turns, revealing a short-bodied (including spherical and flat circular), slow-moving organism, as informed by the traces produced by modern analogues. Whether the producers were early primitive bilaterians or protozoans cannot be determined in the absence of additional evidence (see below). The rise to dominance of regional-smooth traces coincides with the appearance of smooth traces produced by bilaterians with undulating/peristaltic locomotion by 545 Ma. Especially for smooth traces, the stiffness of the long body and the natural undulatory frequency precluded the trace-maker from making abrupt turns, as illustrated by undulatory locomotion recorded by Cochlichnus (figure 6C) in the Fortunian. By the terminal Ediacaran, Cambrian-type bilaterian locomotion trajectories emerged, recording larger body size and bodies with high manoeuvreability well-designed for heading to or maintaining a specific movement direction.

(b). Sensory and perceptual capacities

As demonstrated in numerical simulations, smoothness could also be an indication of perceptual distance or the existence of kinesis. In the presence of a simple sensory–nervous system, the trace-makers can only capture limited information from their environment during navigation. This limited perceptual range causes the information received from stimuli to be spatially discrete, creating short-range shifting from one direction to another over time for each nearby stimulus. Therefore, the stimuli lattice is the smallest spatial scale at which an organism recognizes spatial heterogeneity [62]. This uncertainty forces the animal to run into unknowns with lower speed and to move in stochastic directions. In turn, slow-moving short-range sensory organisms could experience multiple sensorial inputs (e.g. changing flow field, local heterogeneity) within a short distance compared with their body size, causing noise in the trajectory. This type of unsmooth poorly navigated kinesis-dominant movement was typical during the Ediacaran (e.g. Helminthoidichnites (figure 6A) from the Nasep–Huns transition around 540 Ma [63] has a$P_{\text{unsmooth}}$ as high as 0.025 with $\gamma =0.662$). These metrics indicate the stochasticity of the turning directions of these trails, suggesting limited sensorial distance and slow, poorly navigated grazing behaviour on substrates with randomly distributed food resources. In Ediacaran benthic settings, there was actually no need of long-range perception as short-range may have worked efficiently to identify and maintain feeding within a regionally food-rich area [64]. This is in accordance with the Seilacherian scenario of matground seascapes with homogeneously abundant food resources mostly close to the sediment–water interface and in the virtual absence of predators. It has been proposed that incipient patchiness, slowly generated by the presence of Ediacaran animals on microbial mats through time, may have acted as points of carbon concentration playing a decisive role at enabling the evolution of bilaterian bioturbators (see ‘the savannah hypothesis’ [65]) despite the paucity of trace fossils associated with Ediacaran body fossils [66]. In turn, bilaterians would also have enhanced the heterogeneity of substrates.

For animals with a longer sensory range and slender body (that might have sensors along the body), stimuli (including close to 0 intensity) come from all distant directions, changing continuously but gradually: the larger perceptual range coarse-grains the landscape for the organism, averaging out local small-scale variations [39,62]. These reduced sensorial perturbations would have enabled more fluent (including straight) manoeuvres. The trace-maker can gradually make decisions and maintain a direction for a relatively long distance. This increased sensory understanding provides the conditions to allow sticking to the course and even acceleration, generating smoother trajectories. This long-range signal detection and correction in the path most likely involved a well-developed nervous and sensorial system able to maintain monitoring stimuli. Interestingly, animals moving at higher speed to specific destinations would result in a strong inertia, which might also lead to a smooth trajectory. Intentionally keeping a straight/smooth trajectory moving at low speed requires additional attention and nerve–muscle coordination, which might not have been a requirement or possible for Ediacaran grazers. On the other hand, mathematically speaking, smoothness in a curve indicating continuously differentiability, that is, the observer can compute the derivatives and predict the race-maker’s locomotion of the next step. This regional predictability also supports extended perceptual range, enabling a fixed route when a stimulus is still far away: locomotion is, in principle, predictable to the observer during this process.

For example, smooth coiling traces show strong thigmotaxis (the inherent habit of keeping in touch with its body or following a previous trail) and have very low $P_{\text{unsmooth}}$ (0.00183 ~ 0.00319) and high $\gamma$ (around 0.885). Examples include: acorn worm traces (figure 3) [27,67], polychaete traces [68], as well as the Spirodesmos archimedeus from the Ediacaran–Fortunian transition at the uppermost Brioverian beds [69]. This guided and predictable behaviour suggests nerve and sensorial systems that support efficient navigation to form a space-filling curve (following its own trail in the case of acorn worms). The simple trail H. tenuis from Member 3 of the Chapel Island Formation in Grand Bank Head (Figure 7(8) in [70]), also has $P_{\text{unsmooth}}=0.00177$ and $\gamma =0.112$, exhibiting an extremely smooth profile with fluent transition between large and small curvatures, suggesting a relatively long-range perceptual and route-planning capacity.

Therefore, increased smoothness in the trace-fossil record could reflect a linear increase in perceptual range and improved navigational capacity. In addition to tactile chemoreception, distance chemoreception offered more efficient ways to exploit mixgrounds [4] in an increasingly more heterogeneous seascape [20,71,72] with partitioned niches [73] and well-established prey–predator relations [74]. In this scenario, animals need efficient, long-range navigational capacities for food searching (e.g. therefore switching between intermittent search and Lévy flights [75,76]), escaping predation, migration, reproduction or avoiding environmental stress [57,77]. Increased smoothness in emerging lineages was a pre-requisite to support the Cambrian Information Revolution and the emergent Phanerozoic-style ecology [66,72,78]. This more efficient movement evolved in the context of significant environmental change during the Ediacaran–Cambrian transition, most notably an increase in oxygen levels that among other changes may have supported an increase in body size [79–82].

(c). Ediacaran trace-fossil affinities

The most common trace fossils recorded in Ediacaran strata are those assigned to Helminthoidichnites [83,84], although some of them may best fit within Archaeonassa or Helminthopsis [85,86]. These ichnotaxa appeared as early as the Nilpena Sandstone Member of Australia. Ediacaran simple trails have been attributed to bilaterians since the pioneer work by Seilacher [87]. However, it has been subsequently noted that similar structures can have other origins, including compressed carbonaceous filaments [8,88], mechanical transport of microbial aggregates [89,90], locomotion by large single-celled eukaryotes like Gromia sphaerica [91], a series of gas domes blocked by slender obstacles [92] or some simple multicellular organism comparable to slime moulds [93]. Some benthic Foraminifera can produce simple surficial trails/burrows and even more complex, maze-like or one-ended tube-like penetrative structures, especially on fine sediments [37,46,94–96], whose existence can be traced to the early Cambrian [97] and Ediacaran (e.g. Platysolenites) [47–49]. Placozoans, with their external digestion and random walk locomotion, are another group devoid of muscles that could have potentially produced simple unsmooth trails [98–100]. Therefore, in addition to tiny short-bodied primitive bilaterian animals, organisms such as unilocular naked and agglutinated foraminiferans and placozoans also need to be thoughtfully considered as potential producers of some of the Ediacaran surficial horizontal trails [101]. This more nuanced scenario has prompted the approach of analysing these occurrences critically and on a case-by-case basis. This set of criteria can now be complemented by a whole new battery of metrics for ichnological analysis.

5. Conclusions

The studied metrics provide valuable insights into the nature and timing of evolutionary innovations in animal locomotion that took place during the Ediacaran–Cambrian transition. Smoothness may become an indispensable part of quantitative ichnology. The quantitative criterion for smoothness using $\epsilon –\delta$ definition allows us to classify trace fossil trajectories into three categories: smooth, regional-smooth and unsmooth. Auxiliary quantitative indices like minimum normalized radius of curvature and PSD of locomotion trajectories, along with comparisons to modern analogues, may help illuminate the rise to dominance of early animals at the Ediacaran–Cambrian transition as the prelude to modern Phanerozoic ecology.

Unsmooth trails with unlimited number of disrupted points have small minimal radii of curvature and high $P_{\text{unsmooth}}$. They appeared during the Ediacaran White Sea timespan and were the most representative trace fossils at that time. Their trace-makers tend to have a body with a small aspect ratio, without a distinctive anterior–posterior axis, and moved on the substrates slowly in any direction. These trace fossils could have been produced by movement under short-range kinesis responses of early short-bodied benthic bilaterians, placozoans or foraminiferans grazing on homogeneous Ediacaran microbial mats with limited sensory range and thus low smoothness index.

Regional-smooth traces with limited disrupted points tend to have moderate values in all metrics. These trace fossils rose to dominance at around 545 Ma. Regional-smooth traces were likely produced by short-bodied eumetazoan animals with a defined anterior–posterior axis, musculature enabling well-coordinated locomotion and probably equipped with a long-range sensing. These trace fossils composed of smooth curves connected by abrupt turns are of uncontroversial bilaterian origin.

Smooth traces have large minimal radius of curvature, likely due to long-range sensory capacities, a slender body or higher locomotory inertia. The trace-makers have extremely low energy allocation for unsmooth parts in their locomotion, therefore a high smoothness index. The smooth parts indicate a well-organized pattern (lower randomness) and fluent behaviour/locomotion, implying more advanced locomotory, perceptual and navigational capacities typical of bilaterians characterized by well-tuned kinetic coordination. This efficient locomotion can be extensively inferred from the trace-fossil record after approximately 545 Ma.

All the above metrics suggest that the critical transition to smoothness in locomotion trajectories took place by the terminal Ediacaran, coinciding with the roots of the Cambrian explosion and most likely as a pre-requisite to the Cambrian Information Revolution and the emergent Phanerozoic-style ecology. This evolutionary breakthrough in locomotion and sensory–nerve systems experienced an exponential trend that is typical of emergent phenomena, as evidenced by careful quantification of trace fossils.

Ethics

This work did not require ethical approval from a human subject or animal welfare committee.

Data accessibility

All calculations are conducted on Matlab whose codes can be accessed through [102], along with codes converted to R.

Supplementary material is available online [103].

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Authors’ contributions

Z.W.: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, software, supervision, validation, visualization, writing—original draft; M.G.M.: data curation, investigation, resources, supervision, validation, writing—review and editing; L.A.B.: data curation, investigation, resources, validation, writing—review and editing; N.H.: data curation, software.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

Funds were provided by the Royal Society (NIF R1 221871) and UK Research and Innovation (UKRI) guarantee funding for MSCA (EP/Z000920/1) to Z.W. and Natural Sciences and Engineering Research Council (NSERC) Discovery Grants 311727-20 to M.G.M. and 422931-20 to L.A.B., respectively. M.G.M. acknowledges support by the George J. McLeod Enhancement Chair in Geology.