Emergence of behaviour in a selforganized living matter network
Abstract
What is the origin of behaviour? Although typically associated with a nervous system, simple organisms also show complex behaviours. Among them, the slime mold Physarum polycephalum, a giant single cell, is ideally suited to study emergence of behaviour. Here, we show how locomotion and morphological adaptation behaviour emerge from selforganized patterns of rhythmic contractions of the actomyosin lining of the tubes making up the networkshaped organism. We quantify the spatiotemporal contraction dynamics by decomposing experimentally recorded contraction patterns into spatial contraction modes. Notably, we find a continuous spectrum of modes, as opposed to a few dominant modes. Our data suggests that the continuous spectrum of modes allows for dynamic transitions between a plethora of specific behaviours with transitions marked by highly irregular contraction states. By mapping specific behaviours to states of active contractions, we provide the basis to understand behaviour’s complexity as a function of biomechanical dynamics.
Editor's evaluation
We have judged that the response to the referee's residual comments are sufficient to allow this paper to proceed to publication. In particular, the detailed analysis of the mode spectrum and its relationship to behavior is novel and possibly of general use in this field. Also, the experimental data per se should be interesting to a wide spectrum of readers.
https://doi.org/10.7554/eLife.62863.sa0Introduction
Survival in changing environments requires from organisms the ability to transition between diverse behaviours (Angilletta and Sears, 2011; Wong and Candolin, 2014). In higher organisms, a plethora of neural dynamics enable this capacity, ranging from almost random to strongly correlated firing patterns of neurons (Mochizuki et al., 2016). Decoding the origin of behaviour from neuronal activity has been called the ‘holy grail of neuroscience’ (Bando et al., 2019), a task especially challenging given the vastly complex networks of neurons (Berman, 2018). Significant progress has been made by simultaneous tracking of neuronal activity and behaviour – defined as trajectories through spaces of postural dynamics – in the fruit fly Drosophila melanogaster (Honegger et al., 2020) and the nematode Caenorhabditis elegans (Nguyen et al., 2016). Behaviours of these systems have been identified as lowdimensional (Stephens et al., 2008) and hierarchical (Berman et al., 2016).
While these discoveries have advanced our understanding of the origin of behaviour, the complexity and size of biological neural networks make the acquisition and interpretation of experimental data especially challenging. Curiously, organisms without a nervous system may offer an ideal intermediate step towards understanding behaviour. Certain nonneural organisms readily transition between a multitude of behaviors similar in dynamic variability to that of organisms with a nervous system (Berg and Brown, 1972; Otto and Kessin, 2001; McMains et al., 2008; BenJacob et al., 1994; BenJacob et al., 2000; Wan and Goldstein, 2014; Wan, 2018) and thus provide the opportunity to study the link between the underlying biophysical process and behaviour.
A nonneural organism with an exceptionally versatile behavioural repertoire is the slime mould Physarum polycephalum  a unicellular, networkshaped organism (Sauer, 1982) of macroscopic dimensions, typically ranging from a millimeter to tens of centimeters. P. polycephalum’s complex behaviour is most impressively demonstrated by its ability to solve spatial optimisation and decisionmaking problems (Nakagaki et al., 2000; Tero et al., 2010; Nakagaki and Guy, 2007; Dussutour et al., 2010; Reid et al., 2016), exhibit habituation to temporal stimuli (Boisseau et al., 2016), and use exploration versus exploitation strategy (Aono et al., 2014). Recently, P. polycephalum was found capable of encoding memory about food source locations in the hierarchy of its body plan (Kramar and Alim, 2021) in a process much reminding of synaptic facilitation– the brain’s way of creating memories (Jackman and Regehr, 2017). The generation of such rich behaviour requires a mechanism allowing not only for longrange spatial coordination but also the flexibility to enable switching between different specific behavioural states.
The behaviour generating mechanism in P. polycephalum are the active, rhythmic, crosssectional contractions of the actomyosin cortex lining the tube walls (Yoshimoto and Kamiya, 1984; Ueda et al., 1986; Kamiya et al., 1988). The contractions drive cytoplasmic flows throughout the organism’s network (Iima and Nakagaki, 2012; Alim et al., 2013), transporting nutrients and signalling molecules (Alim et al., 2017). Cytoplasmic flow is responsible for mass transport across the organism and thereby contractions directly control locomotion behaviour (Rieu et al., 2015; Lewis et al., 2015; Zhang et al., 2017; Bäuerle et al., 2020; Rodiek et al., 2015).
So far, only one type of networkspanning peristaltic contraction pattern has been described experimentally (Alim et al., 2013; Oettmeier et al., 2017). However, for small P. polycephalum plasmodial fragments various other shortrange contraction patterns have been observed (Lewis et al., 2015; Zhang et al., 2017) and predicted by theory of active contractions (Bois et al., 2011; Radszuweit et al., 2013; Radszuweit et al., 2014; Julien and Alim, 2018; Kulawiak et al., 2019). Similarly, up to now unknown complex, largescale contraction patterns might play a role in generating the behaviour of large P. polycephalum networks. Furthermore, transitions between such largescale patterns are needed to allow for switching between specific behaviours, for example taking sharp turns during migration in the absence of stimuli (Rodiek and Hauser, 2015).
Here, we decompose experimentally recorded contractions of a large P. polycephalum network of stable morphology into a set of physically interpretable contraction modes using Principal Component Analysis. We find a continuous spectrum of modes and high variability in the activation of modes along this spectrum. By perturbing the network with an attractive stimulus, we show that the resulting locomotion response is coupled to a selective activation of regular contraction patterns. Guided by these observations, we design an experiment on a P. polycephalum specimen reduced in morphological complexity to a single tube. This allows us to quantify the causal relation between locomotion behaviour, cytoplasmic flow rate and varying types of contraction patterns, thus revealing the central role of dynamical variability in generating different behaviours.
Results
Continuous spectrum of contraction modes reveals large variability in organism’s contraction dynamics
To characterize the contraction dynamics of a P. polycephalum network, we record contractions using brightfield microscopy (Video 1) and decompose this data into a set of modes using Principal Component Analysis (PCA). At first, networks in brightfield images are skeletonized, with every single skeleton pixel representing the local tube intensity as a measure of the local contraction state (Bäuerle et al., 2017). Thus, any network state at a time t_{i} is represented by a list of pixels, ${\overrightarrow{I}}^{{t}_{i}}$, along the skeleton, see Figure 1A and ‘Data processing’ (Appendix 1). Performing PCA on this data results in a linear decomposition of the intensity vectors ${\overrightarrow{I}}^{{t}_{i}}$ into a basis of modes ${\overrightarrow{\varphi}}_{i}$:
See ‘Principal Component Analysis (PCA) (Appendix 2)’ for details. The modes, ${\overrightarrow{\varphi}}_{\mu}$, are orthonormal eigenvectors of the covariance matrix of the data and represent linearly uncorrelated contraction patterns of the network, and ${a}_{\mu}^{{t}_{i}}$ denotes the timedependent coefficients of the modes.
We rank modes according to the magnitude of their eigenvalues. Contrary to the small number of large eigenvalues found in a number of biological systems (Stephens et al., 2008; Jordan et al., 2013; Gilpin et al., 2016), here the spectrum of relative eigenvalues, see ‘Principal component analysis (PCA)’ (Appendix 2) for technical details, is continuous with no clear cutoff (Figure 1B) and as a result the contraction dynamics is highdimensional. Notably, this is even the case when we disregard eigenvalues which lie below the upper noise bound (black line), computed from randomised data. Therefore, PCA does not directly lead to a dimensionality reduction of the data. Instead, we here investigate the characteristics of mode dynamics that result from a continuous spectrum and how these shape the organism’s behaviour.
The highestranking modes shown in Figure 1C(i) have a smooth spatial structure that varies on the scale of network size. As we will discuss below, such largescale modes are associated with the long wavelength peristalsis observed in Iima and Nakagaki, 2012; Alim et al., 2013. Interestingly, we also find modes highlighting specific morphological characteristics of the network. For example, the structure of mode ${\overrightarrow{\varphi}}_{4}$, Figure 1C(i), corresponds to the thickest tubes of the network Figure 1A, which suggests a special role of these tubes in the functioning of the network. Finally, as we go to lower ranked modes, the spatial structure of the modes becomes increasingly finer. Yet, despite lacking an obvious interpretation for their structures, like for mode ${\overrightarrow{\varphi}}_{30}$, Figure 1—figure supplement 1, it is not possible to ignore their contribution relative to highranking modes.
Next, we turn to the timedependent coefficients of modes shown in Figure 1C(ii). In accordance with the known rhythmic contractions (Kamiya, 1960) the coefficient $a}_{1$ of the highest ranked mode ${\overrightarrow{\varphi}}_{1}$ oscillates with a typical period of $T\sim 100\phantom{\rule{thinmathspace}{0ex}}\mathrm{sec}$ . Most strikingly, amplitudes of mode coefficients vary significantly over time  even on orders of magnitude, as shown in Figure 1—figure supplement 2.
To map out the complexity of contractions over time, we define a set of significant modes for every time point. We quantify the activity of a mode by its relative amplitude
where $\stackrel{~}{{a}_{\mu}^{2}}$ denotes the amplitude of the square of the mode’s coefficient. By definition the sum over the relative amplitudes of all modes is normalized to one at any given time, ${\sum}_{\mu}{p}_{\mu}^{{t}_{i}}=1$. For any given time point, we order the modes by their relative amplitude from largest to smallest and take the cumulative sum of their values until a chosen cutoff percentage is reached, see Figure 2A. We find that the percentage of modes required to reach a specified cutoff value varies considerably over time. For a 90% cumulative amplitude cutoff, we find that on average 6.06% ($\approx 70$ modes) of the 1500 modes are significant. As discussed in more detail in ‘Choice of the cutoff of mode coefficient amplitudes’ (Appendix 6), defining a cutoff for the cumulative sum of mode amplitudes is related to the problem of defining a cutoff for a continuous spectrum of eigenvalues. One common method is to define the cutoff with respect to the largest eigenvalue of the spectrum computed from a randomised version of the original data (Berman et al., 2014). In ‘Choice of the cutoff of mode coefficient amplitudes’ (Appendix 6), we find that the 90% cumulative amplitude cutoff considered above is consistent with this definition of cutoff for eigenvalues. As an important feature, we observe that there is large variation in the number of significant modes over time, with a standard deviation of 36.96% from the mean value. This is an indicator for the complexity of the contractions in the network.
Apart from the number of significant modes, the dynamics of the network depend on the temporal correlation of modes. While the modes form a spatially uncorrelated basis, the temporal correlation of mode activation is nontrivial. In Figure 2B, we show the distribution of temporal correlations between mode coefficients as a function of the number of significant modes, see ‘Distribution of temporal correlations’ (Appendix 3) for technical details. For a small number of significant modes, the coefficients are strongly (anti)correlated in time, while for a large number of significant modes, correlations values between coefficients are more uniformly distributed. Here, correlated coefficients result in coordinated pumping behaviour/contractions, while least correlated coefficients coincide with irregular networkwide contractions. The above analysis shows that the dynamics of network contractions covers a wide range in complexity, from superposition of few largescale modes strongly correlated in time, to superpositions of many modes of varying spatial scale and temporal correlations. This gives rise to strong variability in the regularity of the contraction dynamics over time. Up to now, we investigated an ‘idle’ network not performing a specific task, so we next stimulate the network to provoke a specific behaviour and scrutinize how the continuous spectrum of modes contributes to it.
Stimulus response behaviour is paired with activation of regular, largescale contraction patterns interspersed by manymode states
To probe the connection between a specific behaviour and network contraction dynamics, we next apply a food stimulus to the same network, see Figure 3A. Food acts as an attractant and causes locomotion of the organism toward the stimulus in the long term. The stimulus immediately triggers the tubes in the network to grow in a concentric region around the stimulus site. Also, the thick transport tubes oriented toward stimulus location increase their volume, see Figure 3A. Altogether these morphological changes are typical for the specific behaviour induced here, namely the generation of a new locomotion front.
In Figure 3B, we quantify this stimulus response behaviour by tracking the growth of the most active regions of the network, defined by the boxes shown in the 81 min in Figure 3A. The tracked regions are located on opposing sides of the network. Starting approximately at 85 min, the part of the network next to the stimulus site grows rapidly (burgundy curve in Figure 3B), at the expense of the fanshaped locomotion front in the lower left corner of the network (green curve in Figure 3B). In Figure 3—figure supplement 2, we additionally show that prior to the stimulus, the network grows the fanlike shaped locomotion front in the lower left corner. Taken together, the application of the stimulus leads to a reversal of the network’s growth direction.
To identify potential changes in the contraction dynamics due to stimulus application, we perform PCA on a 700 frames long subset of the data subsequent to the ‘idle’ data of the previous section. First, we rediscover a continuous spectrum of modes, see Figure 3—figure supplement 1, resembling that of the ‘idle’ dynamic state. However, now the highestranked contraction modes, see Figure 3C, show spatial patterns which can be directly related to the network’s growth behaviour. This includes activation of the upper region of the network close to the stimulus, as well as activation of the thick tubes extending from top to bottom of the network. In fact, for more than 500 frames after the stimulus has been applied, the rhythmic contraction dynamics of the network are dominated by the three highestranked modes, see Figure 3D and Figure 3—figure supplement 3 for the oscillatory dynamics of mode coefficients. During this period, every time a single mode is the most active one for a duration of $\S gt;30$ frames, its amplitude exceeds that of any other mode by 20–30%.
Next we link the stimulusinduced reversal in growth direction to the changes in the contraction pattern. Specifically, we observe that the time interval of growth reversal Figure 3B coincides with the activation of the thirdranked mode ${\overrightarrow{\varphi}}_{3}$, (orange curve in Figure 3D), as indicated by the pink shaded box extending across Figure 3D and B. The structure of this mode shows a clear distinction of the growth area close to the stimulus and an activation of the two thick tubes stretching from bottom to the top of the network. This mode is followed by an activation of mode ${\overrightarrow{\varphi}}_{2}$ (blue curve), clearly marking the growth region within its spatial structure.
Finally, over time the growth of the stimulus response region tapers off and we find reactivation of mode ${\overrightarrow{\varphi}}_{1}$ (red curve) which was the dominant mode before stimulus application. We note that the spatial structure of mode ${\overrightarrow{\varphi}}_{1}$ is remarkably similar to mode ${\overrightarrow{\varphi}}_{1}$, the topranked mode that we find for PCA on the prestimulus ‘idle’ data Figure 3—figure supplement 2B. The reactivation of this mode indicates that this contraction pattern is intrinsic to the network and is not simply erased by the stimulus.
Strikingly, the regular contraction dynamics shown in Figure 3D are interspersed with manymode states where the number of significant modes increases considerably, see Figure 3E. The number of significant modes oscillates after the stimulus. The oscillation maxima coincide with times at which the organism switches from one dominant contraction pattern to another, as indicated by the blueshaded boxes extending across Figure 3D and E. Our results suggest that prolonged regular dynamics dominated by a few or even a single mode are associated with specific behaviour like locomotion and growth, while the manymode states seem to serve as transition states between them.
While the network morphology is characteristic for P. polycephalum, reducing network complexity may help to conclude on the role of regular dynamics in driving specific behaviours, and the role of manymode states and the therefrom arising continuous distribution of modes.
Number of significant modes determines maximum cytoplasmic flow rate in the minimal morphological representation of the network
We next perform exactly the same course of experiments as before but on a P. polycephalum specimen reduced in complexity to a single tube with a locomotion front at either end, see inset in Figure 4 and Video 2. Strikingly, when performing PCA on this specimen of simple morphology we again find a continuous spectrum of modes (Figure 5—figure supplement 1) and large variability, including spikes of manymode states, in the number of significant modes (Figure 5A). This observation finally underlines that the continuous spectrum of modes and its variability in activation is intrinsic to the organism’s behaviour, ruling out that the complexity of contraction modes only mirrors morphological complexity. Foremost, this minimal constituent of a network allows us now to directly map the effect of variations in the contraction dynamics onto behaviour.
From the experimentally quantified tube contractions, we calculate the maximal flow rate at any point along the tube (Li and Brasseur, 1993) and over time correlate the strength of the flow rates, driving locomotion behaviour at the tube ends, with the number of significant modes, see ‘Flow rate calculation in a P. polycephalum cell with singletube morphology’ (Appendix 4). For both the flow rate at the left and right end of the tube, shown in Figure 4, and Figure 4—figure supplement 1, respectively, we find that large flow rates are only achieved when the number of significant modes is small. We had previously found that few significant modes are highly (anti)correlated, whereas states with many significant modes are not, see Figure 2B. This observation now confirms our physical intuition that the irregularity of states consisting of many modes goes hand in hand with reduced pumping efficiency and thus unspecific behaviour. Since a small number of significant modes not necessarily always implies a large flow rate, we next turn to analyze their exact spatial structure and instantaneous temporal correlation to determine how cytoplasmic flow rates impact behaviour.
Instantaneous coupling and selective activation of modes determine locomotion behaviour
We now demonstrate the impact of changes in the dynamics of a small number of modes on the organism’s behaviour. For this, we quantify the locomotion behaviour of the single tube by tracking the area of the locomotion fronts protruding from each end of the tube over time, see Figure 5A. The growth curves of the tube ends are shown in Figure 5B. While initially the right end is protruding faster at the expense of the left end, a food stimulus applied to the left end of the tube reverses the direction of locomotion.
As for the network, we use PCA to analyse the contraction dynamics of the single tube and link it to behaviour. We apply PCA to contraction data along the tube which we parameterize by a longitudinal coordinate. The spatial shapes of the two topranked modes ${\overrightarrow{\varphi}}_{1}$ and ${\overrightarrow{\varphi}}_{2}$ approximate Fourier modes, see Figure 5C and Figure 5—figure supplement 2. Examining the activation of modes, we find that over long time intervals, and in particular after the stimulus, the two topranked modes dominate the tube’s contraction dynamics, see Figure 5D. To illustrate the connection between the nature of tube contraction dynamics and locomotion behaviour, we pick two representative time intervals after the stimulus where either only mode ${\overrightarrow{\varphi}}_{1}$, or modes ${\overrightarrow{\varphi}}_{1}$ and ${\overrightarrow{\varphi}}_{2}$ equally, dominate overall, see vertical pink bars in Figure 5D. During the first interval when mode ${\overrightarrow{\varphi}}_{1}$ alone is dominating, the tube is driven by a standing wave contraction pattern  yielding only a low cytoplasmic flow rate. Correspondingly, the size of the locomotion front at either end shows no significant change in area during this interval. In contrast, during the interval when both modes ${\overrightarrow{\varphi}}_{1}$ and ${\overrightarrow{\varphi}}_{2}$ are equally active, the resulting superposition is a lefttraveling wave producing a large cytoplasmic flow rate in that direction. The lefttraveling wave is in accordance with the growth of the left and retraction of the right locomotion front as quantified in Figure 5B. See ‘Mode superpositions in a P. polycephalum cell with singletube morphology’ (Appendix 4) for more details. In Figure 5E, we highlight the most pronounced manymode states during changes of dominant contraction dynamics.
These two examples solve the conundrum of Figure 4, which shows that a small number of significant modes does not necessarily lead to high cytoplasmic flow rates. Yet, the direct mapping of contraction dynamics onto ensuing cytoplasmic flows confirms that a small number of significant modes is associated with specific behaviour. High cytoplasmic flow rates at the tube ends drive locomotion, while lower flow rates likely lead to other behaviours such as mixing. Furthermore, manymode states seem necessary for transitions in a multibehavioural space.
Our explanation of behaviour – from contractions via flows to locomotion behaviour – in the single tube is a template for an analogous explanation in the network morphology. The analogy is justified by the strong resemblance of the continuous mode spectrum, dynamics of significant modes, activation of regular contraction patterns and the nature of growth behaviour in both the network and single tube. Therefore, while it is beyond the scope of this study, we expect a detailed analysis of the link between contractions and flows in the network morphology to yield qualitatively similar results to those of the single tube, thus completing the mechanism of behaviour generation.
Discussion
To uncover the origin of behaviour in P. polycephalum, we quantified the dynamics of this living matter network and linked it to its emerging behaviour. The simple build of this nonneural organism allows us to trace contractions of the actomyosinlined tubes, compute cytoplasmic flows from the contractions and finally link these dynamics to the emerging mass redistribution and wholeorganism locomotion behavior. Decomposing the contractions across the network into individual modes, we discover a large intrinsic variability in the number of significant modes over time along a continuous spectrum of modes. By triggering locomotion through application of a stimulus, we identify that states with few significant modes and regular contraction patterns correspond to specific behaviors, in this case locomotion. Yet, irregular contraction patterns consisting of a large number of significant modes are also present, particularly marking the transitions between different regular contraction states. The use of an organism with a singletube morphology allows us to obtain quantitative insights into the mechanism connecting contraction dynamics and locomotion behavior and in first approximation serves as an analogue system for the large P. polycephalum with network morphology. Our findings suggest that a continuous spectrum of contraction modes allows the living matter network P. polycephalum to quickly transition between a multitude of behaviours using the superposition of multiple contraction patterns.
Networks are ubiquitous in biology, including examples such as ecological networks (García Martín and Goldenfeld, 2006) and biomolecular interaction networks (Albert, 2005). Measurable quantities of these networks, for instance the degree distribution of the network, typically follow continuous distributions and are oftentimes powerlaws. The spectrum of eigenvalues Figure 1B that we find for the contraction dynamics in P. polycephalum may similarly suggest a powerlaw. However, the presence of a powerlaw is generally difficult to prove and interpret. Instead, our sole focus is on the continuous nature of the spectrum. It is important to emphasise that the continuity of the eigenvalue spectrum is not simply the result of the organism’s complex network morphology. This is demonstrated by the fact that we find a similar spectrum also for the singletube morphology Figure 5—figure supplement 1. Therefore, here the continuous spectrum of eigenvalues is distinctively a property of the dynamic state of the organism.
Our observation of interlaced regular and irregular contraction patterns in P. polycephalum reminds of the strongly correlated or random firing patterns of neurons in higher organisms (Mochizuki et al., 2016). In neural organisms, stereotyped behaviours are associated with controlled neural activity, as for example for locomotion in C. elegans (Liu et al., 2018) or the behavioural states of the fruit fly Drosophila melanogaster (Berman et al., 2014; Berman et al., 2016). Variability in the dynamics of behaviour is also widely observed in these neuronal organisms (Grobstein, 1994; Renart and Machens, 2014; Werkhoven et al., 2021; Honegger et al., 2020; Ahamed et al., 2020). It is thus likely that the transition role of irregular states consisting of many significant modes observed here for P. polycephalum parallels the mechanisms of generating behaviour in the more complex forms of life.
P. polycephalum is renowned for its ability to make informed decisions and navigate a complex environment (Nakagaki et al., 2000; Tero et al., 2010; Nakagaki and Guy, 2007; Dussutour et al., 2010; Reid et al., 2016; Boisseau et al., 2016; Aono et al., 2014; Ueda et al., 1976; Miyake et al., 1991). It would be fascinating to next follow the variability of contraction dynamics during more complex decisionmaking processes. Furthermore, it would be interesting to observe ‘idle’ networks during foraging over tens of hours. It is likely that the contraction states with many significant modes here act as noisy triggers that can spontaneously cause the organism to reorient its direction of locomotion.
In the context of P. polycephalum’s foraging behaviour, another exciting line of research opened by our results is the link between contraction modes and the organism’s metabolic changes. The foraging networks displays a plethora of morphological patterns which are linked to the underlying metabolic states (Takamatsu et al., 2017; Lee et al., 2018). It has recently been shown that in the neural organism Drosophila melanogaster, behaviour stemming from neural activity causes largescale changes in metabolic activity (Mann et al., 2021). Exploring the relationship between behaviour emergence and metabolism in P. polycephalum will bring key insight about the interplay between the mechanical and the biochemical machinery of the organism.
P. polycephalum’s bodyplan as a fluidfilled living network with emerging behaviour finds its theoretical counterpart in theories for active flow networks developed recently (Woodhouse et al., 2016; Forrow et al., 2017). Strikingly, these theories predict selective activation of thick tubes which we observe in the living network as well, prominently appearing among the top ranking modes, see ${\overrightarrow{\varphi}}_{4}$ in Figure 1C(i) or ${\overrightarrow{\varphi}}_{3}$ in Figure 3C. This is a first hint that dynamics states arising from first principles in active flow networks could map onto behavioural and transition states observed here.
Likely our most broadly relevant finding in this work is that irregular dynamics, here arising in states with many significant modes, play an important role in switching between behaviours. This should inspire theoretical investigations to embrace irregularities rather than focusing solely on regular dynamic states. The most powerful aspect of P. polycephalum as a model organism of behaviour lies in the direct link between actomyosin contractions, resulting in cytoplasmic flows and emerging behaviours. The broad understanding of the theory of active contractions (Bois et al., 2011; Radszuweit et al., 2013; Radszuweit et al., 2014; Julien and Alim, 2018; Kulawiak et al., 2019) might therefore well be the foundation to formulate the physics of behaviour not only in P. polycephalum but also in other simple organisms. This would not only open up an new perspective on life but also guide the design of bioinspired soft robots with a behavioural repertoire comparable to higher organisms.
Materials and methods
Experiments
View detailed protocolThe specimen was prepared from fused microplasmodia grown in a liquid culture (Daniel et al., 1962) and plated on 1.5%agar. The network was trimmed and imaged in the bright field setting in Zeiss ZEN two imaging software with a Zeiss Axio Zoom V.16 microscope equipped with a Hamamatsu ORCAFlash 4.0 digital camera and Zeiss PlanNeoFluar 1 x/0.25 objective. The acquisition frame rate was 3 sec. The stimulus was applied in a form of a heatkilled HB101 bacterial pellet in close network proximity.
Appendix 1
Data
The typical thickness of tubes in a P. polycephalum network is $\sim 50100\mu \mathrm{m}$ and the contraction amplitude about ~10% of the tube’s typical thickness (Alim et al., 2013). This change in tube thickness can be detected from a brightfield microscopy recording. We record one brightfield frame every three seconds. Since the periodic contractions of the tubes take place on the time scale of 100 sec, they are thus well resolved by the selected frame rate. Typically an idle network keeps a stable morphology and does not move significantly over a period of 1.5 h to 2.5 h which we use for recording its contraction dynamics. Since no two P. polycephalum specimens ever have the same network morphology, we are naturally constrained to one biological and one technical replicate in our experiments.
Data processing
Our data is a stack of brightfield images recorded from the P. polycephalum network with a rate of one frame every three seconds (Video 1; Video 2). Each brightfield frame has a time label t_{i} and the total number of frames is given by $T$. We process this data in the following steps. First, we mask the network in the brightfield images through thresholding. It is important to note that we use the same mask for all the images in the stack. This is possible since we consider a network that does not significantly move or change its morphology. This is true even when we apply a stimulus to the network, since we only consider the initial stages of stimulus response, before the network starts to display strong movement. From the masked regions of the brightfield frames, we extract pixel intensity values which we convert to 8bit format. Since we are here primarily interested in the contraction dynamics of the organism and not in the the actual base thickness of tubes or its longterm growth dynamics, we detrend the data using a movingaverage filter (rational transfer function) with a window size of two contraction periods (~ 200 sec) (Bäuerle et al., 2017). This leaves us only with the desired information about contractions taking place in the time scale of several minutes. We store the intensity values of each frame in a vector ${\overrightarrow{I}}^{{t}_{i}}$ of dimension $M$ equal to the number of pixels in the network, and $i$ indexes the frames in the range $i=1,\mathrm{\dots},T$. From the postprocessed data, we define the following data matrix
where $t$ denotes the matrix transpose.
Appendix 2
Principal component analysis (PCA)
The contraction modes are computed from the covariance matrix of the data. We compute the covariance matrix from the data matrix $\mathbf{X}$ after subtracting the mean from each column. The covariance matrix is given by
The sought after contraction modes ${\overrightarrow{\varphi}}_{\mu}$ are the eigenvectors of the covariance matrix
and ${\lambda}_{\mu}$ is the eigenvalue. The number of nonzero eigenvalues is equal to the rank of the covariance matrix. The eigenvalue captures the variance of the data along the direction of mode ${\overrightarrow{\varphi}}_{\mu}$. We also define the relative eigenvalue as
The mode coefficient ${a}_{\mu}$ is obtained by projecting the data onto mode ${\overrightarrow{\varphi}}_{\mu}$.
We note that we perform PCA on data segments with at least 700 frames (=35 min). Since it is well known from the literature (Kamiya, 1960) that the period of the contraction dynamics in P. polycephalum is on the order of 100 sec the analysed data contains on the order of 20 contraction periods at minimum. We can therefore be sure that we use enough data to resolve the characteristic dynamical features investigated here. As a further reassuring result, we recover the typical contraction period of 100 sec in our analysis, see Figure 1C(ii).
We add a brief comment on Fourier analysis, as alternative decomposition method to PCA. First, in one dimension, PCA is equivalent to Fourier decomposition. This is indeed apparent in our PCA analysis of the singletube data set where the principal components shown in Figure 5C correspond precisely to half a period of a sine and cosine Fourier mode. In two dimensions, the situation is more complicated. While we could in principle apply 2D Fourier decomposition we would need to apply Fourier analysis separately to every frame in our data set. However, this would mean that we have no information about the temporal evolution of mode activation. The Fourier modes would be different from one frame to the next and the activation of largescale patterns over time would be obscured.
Appendix 3
Distribution of temporal correlations
For a given time point t_{i}, the significant modes are determined based on the 70% criterion curve from Figure 2A. Next, the temporal correlations among the coefficients are computed in a time interval of ±15 frames around the time point t_{i}. The correlations are then counted in bins of the appropriate row of Figure 2A. Repeating this processing for all time points and normalising each row by the total number of correlations in that row, we obtain the final distribution shown.
Appendix 4
Flow rate calculation in a P. polycephalum cell with singletube morphology
To compute the flow rate of the cytoplasm in a P. polycephalum specimen with singletube morphology we use the theory developed in Shapiro et al., 1969; Li and Brasseur, 1993. In that work the flow of an incompressible Newtonian fluid inside an axisymmetric tube of fixed length is considered and the equations for the flow velocity field are written in the lubrication theory approximation. Furthermore, a timedependent thickness profile of longitudinal waves is imposed in the tube. Assuming noslip boundary conditions, the flow field can be fully determined at every point along the tube as a function of the timedependent tube profile. For the case when the tube profile is a periodic train of waves, we compute the volume flow rate averaged over an oscillation period by evaluating equation (13) of Li and Brasseur, 1993. We express the flow rate in units of volume of the entire tube divided by the oscillation period. This serves to characterize the performance in pumping of the significantly contracting P. polycephalum cell. We determine the time period over which to average the volume flow rate directly from the flowrate curve. Furthermore, the thickness profile of the tube is given by the measured pixel intensity profile.
Appendix 5
Mode superpositions in a P. polycephalum cell with singletube morphology
We are interested in how the contraction dynamics of the cell controls the cell’s locomotion behavior. In our analysis we therefore focus on the tube segment connecting the locomotion fronts at either end of the tube and perform Principal Component Analysis only on this part of the cell. Since the tube is effectively onedimensional, we find that the modes we obtain closely approximate Fourier modes. This means that superpositions of these modes afford a clear interpretation in terms of different contractionwave patterns. Such an interpretation is even further facilitated by the fact that we find that over large time intervals after the stimulus, the number of significant modes is very small. Indeed, over such time intervals it is sufficient to approximate the contraction dynamics with only one or two modes, as can be seen from Figure 5D. Hence we are essentially studying a superposition of modes ${\overrightarrow{\varphi}}_{1}$ and ${\overrightarrow{\varphi}}_{2}$ shown in Figure 5C and Figure 5—figure supplement 2 of the main text with their oscillating mode coefficients shown in Figure 5—figure supplement 3. To develop intuitive understanding of the nature of the superposition, we note that the modes ${\overrightarrow{\varphi}}_{1}$ and ${\overrightarrow{\varphi}}_{2}$ approximate sine and cosine functions over the length of the tube. Given a sine and cosine spatial contraction profile, different types of superpositions can be formed depending on the nature of their timedependent coefficients. To illustrate further, let us assume the idealised case where both coefficients are sine functions that can have different phases and amplitudes. Then, if the coefficient of one contraction profile is very small compared to the other, the resulting superposition is a standing wave. In case the coefficients have equal amplitudes, but are phase shifted by $\pi /2$, the superposition is a traveling waves. Finally, if the coefficient amplitudes are not equal and the phase shift lies somewhere between zero and $\pi /2$, the nature of the superposition is a mix of standing and traveling wave. Extrapolating this idealised picture allows us to infer the contraction dynamics resulting from our twomode approximation. We see that the coefficients of the two modes ${\overrightarrow{\varphi}}_{1}$ and ${\overrightarrow{\varphi}}_{2}$ shown in Figure 5—figure supplement 3 change in amplitude and phase relative to each other. It is easy to identify from this plot together with the plot of relative amplitudes in Figure 5D time intervals which approximate one of the contraction dynamics that we have described for the idealised system. Therefore we conclude that the superposition of the two top modes changes in its nature over time, ranging form a pure standing wave to a pure traveling wave.
Appendix 6
Choice of the cutoff of mode coefficient amplitudes
Our analysis of contraction dynamics requires us to place a cutoff on the amplitude of mode coefficients. Our chosen cutoff of 90% is supported in two ways:
First, the problem of choosing a cutoff for the coefficients is related to the problem of choosing a cutoff for the eigenvalue spectrum, since an eigenvalue is the variance of the mode coefficient. Given the continuous nature of the eigenvalue spectrum, there is no unique way to choose a cutoff. In (Berman et al., 2014), it is proposed to define the cutoff by the largest eigenvalue of the spectrum of the randomised data, see the black line in Figure 1B. We tested their criterion on our data and find a 93% cutoff, equivalent to roughly 70 modes. This is consistent with our choice of a 90% cutoff for the amplitudes.
Second, our main qualitative observation  considerable variation in the number of significant modes over time  is robust to different choices of cutoff values. In Figure 2A and similarly in Figure 3E, we show the number of significant modes for two different values of the cutoff, namely 70% and 90%.
Data availability
The two datasets from which Figures 1,2 and 3 and Figures 4 and 5 were generated are included as videos of raw brightfield time series in the article.
References

Scalefree networks in cell biologyJournal of Cell Science 118:4947–4957.https://doi.org/10.1242/jcs.02714

Coordinating theoretical and empirical efforts to understand the linkages between organisms and environmentsIntegrative and Comparative Biology 51:653–661.https://doi.org/10.1093/icb/icr091

Cooperative selforganization of microorganismsAdvances in Physics 49:395–554.https://doi.org/10.1080/000187300405228

Mapping the stereotyped behaviour of freely moving fruit fliesJournal of the Royal Society, Interface 11:20140672.https://doi.org/10.1098/rsif.2014.0672

Pattern formation in active fluidsPhysical Review Letters 106:028103.https://doi.org/10.1103/PhysRevLett.106.028103

Habituation in nonneural organisms: evidence from slime mouldsProceedings. Biological Sciences 283:1829.https://doi.org/10.1098/rspb.2016.0446

Hematinrequiring plasmodial myxomyceteJournal of Bacteriology 84:1104–1110.https://doi.org/10.1128/jb.84.5.11041110.1962

Mode Selection in Compressible Active Flow NetworksPhysical Review Letters 119:028102.https://doi.org/10.1103/PhysRevLett.119.028102

Variability in behavior and the nervous systemEncyclopedia of Human Behavior 4:447–458.

Peristaltic transport and mixing of cytosol through the whole body of Physarum plasmodiumMathematical Medicine and Biology 29:263–281.https://doi.org/10.1093/imammb/dqr010

Physics and Chemistry of Protoplasmic StreamingAnnual Review of Plant Physiology 11:323–340.https://doi.org/10.1146/annurev.pp.11.060160.001543

Dynamic organization of Physarum plasmodiumCell Motility and the Cytoskeleton 10:107–116.https://doi.org/10.1002/cm.970100115

A novel growth mode of Physarum polycephalum during starvationJournal of Physics D 51:244002.https://doi.org/10.1088/13616463/aac2b0

Coordination of contractility, adhesion and flow in migrating Physarum amoebaeJournal of the Royal Society, Interface 12:20141359.https://doi.org/10.1098/rsif.2014.1359

Nonsteady peristaltic transport in finitelength tubesJournal of Fluid Mechanics 248:129–151.https://doi.org/10.1017/S0022112093000710

Similarity in Neuronal Firing Regimes across Mammalian SpeciesThe Journal of Neuroscience 36:5736–5747.https://doi.org/10.1523/JNEUROSCI.023016.2016

Physarum polycephalum —a new take on a classic model systemJournal of Physics D 50:413001.https://doi.org/10.1088/13616463/aa8699

Intracellular mechanochemical waves in an active poroelastic modelPhysical Review Letters 110:138102.https://doi.org/10.1103/PhysRevLett.110.138102

Decisionmaking without a brain: how an amoeboid organism solves the twoarmed banditJournal of The Royal Society Interface 13:20160030.https://doi.org/10.1098/rsif.2016.0030

Variability in neural activity and behaviorCurrent Opinion in Neurobiology 25:211–220.https://doi.org/10.1016/j.conb.2014.02.013

Periodic traction in migrating large amoeba of Physarum polycephalumJournal of The Royal Society Interface 12:20150099.https://doi.org/10.1098/rsif.2015.0099

Migratory behaviour of Physarum polycephalum microplasmodiaThe European Physical Journal Special Topics 224:1199–1214.https://doi.org/10.1140/epjst/e2015024552

Patterns of cell thickness oscillations during directional migration of Physarum polycephalumEuropean Biophysics Journal 44:349–358.https://doi.org/10.1007/s0024901510287

BookDevelopmental Biology of PhysarumCambridge, United Kingdom: Cambridge University Press.

Peristaltic pumping with long wavelengths at low Reynolds numberJournal of Fluid Mechanics 37:799–825.https://doi.org/10.1017/S0022112069000899

Dimensionality and dynamics in the behavior of C. elegansPLOS Computational Biology 4:e1000028.https://doi.org/10.1371/journal.pcbi.1000028

Energysaving with low dimensional network in Physarum plasmodiumJournal of Physics D 50:154003.https://doi.org/10.1088/13616463/aa635a

Rhythmicity, recurrence, and recovery of flagellar beatingPhysical Review Letters 113:238103.https://doi.org/10.1103/PhysRevLett.113.238103

Coordination of eukaryotic cilia and flagellaEssays in Biochemistry 62:829–838.https://doi.org/10.1042/EBC20180029

Behavioral responses to changing environmentsBehavioral Ecology 26:665–673.https://doi.org/10.1093/beheco/aru183

ATP and calciumcontrolled contraction in a saponin model of Physarum polycephalumCell Structure and Function 9:135–141.https://doi.org/10.1247/csf.9.135
Article and author information
Author details
Funding
Simons Foundation (400425)
 Philipp Fleig
IMPRS for Physics of Biological and Complex Systems
 Mirna Kramar
Max Planck Society
 Philipp Fleig
 Mirna Kramar
 Michael Wilczek
 Karen Alim
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Version history
 Received: September 7, 2020
 Preprint posted: September 8, 2020 (view preprint)
 Accepted: December 20, 2021
 Version of Record published: January 21, 2022 (version 1)
Copyright
© 2022, Fleig et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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