Adaptive behaviour and learning in slime moulds: the role of oscillations

(a) Physarum polycephalum: a unique model

The slime mould P. polycephalum is a remarkable organism belonging to the Amoebozoa. It is often used as a model system to study problem-solving in aneural biological systems (figure 1) [24–27]. Its vegetative state, the plasmodium, is a giant mobile cell, whose basic structure consists of a syncytium of nuclei and an intracellular cytoskeleton that forms a complex cytoplasmic network of veins (figure 1). The plasmodium can extend up to hundreds of square centimetres and can be severed into viable and structurally similar yet smaller subunits. Upon contact, plasmodia can fuse with each other, giving rise to the (re-)formation of a giant plasmodium [28]. Physarum polycephalum can be reared and maintained in a liquid-shaking culture as micrometre-sized fragments (microplasmodia) or as macroplasmodia on a solid substrate. Lastly, a starving plasmodium can encapsulate and enter a dormant stage called sclerotium (figure 1). Physarum polycephalum can turn back from this resting stage to a plasmodium within 24 h after transfer to fresh food medium.

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Both P. polycephalum's motion and its behaviour rely on its internal architecture. Its cytoplasm consists of a visco-elastic (gel, also named ectoplasm) and a liquid (sol, also named endoplasm) phase [28]. This compartmentation is due to a sharp concentration change of the fibrous proteins forming the cytoskeleton [29]. Ecto- and endoplasm are interconvertible. The cell body of P. polycephalum is a transport network consisting of interconnected tubes (figure 1). Ectoplasm, which contains actin and myosin, forms the contractile walls of the tubes (figure 1). Within these veins flows the endoplasm, which contains organelles such as nuclei and mitochondria (electronic supplementary material, movies S1 and S2). The veins contract and relax periodically, causing the endoplasm to flow back and forth, a phenomenon termed ‘shuttle streaming’ [30]. The vein network of P. polycephalum bears two analogies with the vertebrates' circulatory systems. First, like the vasculature, the veins transport and distribute the endoplasm throughout the cell body. Second, the veins are pore-like capillaries, allowing respiratory gases, molecules and organelles to be exchanged with the surrounding cytoplasm [31]. This open network allows homeostasis to be maintained in cells ranging from 10 µm² to 10 m². The spatial structuration of P. polycephalum's vein network and its dynamics allow an efficient regulation of the cytoplasmic flow within the entire cell [32,33]. This vein network and the interactions between the ectoplasm and the endoplasm have been studied extensively by numerous biophysicists [34,35].

The vein network is also responsible for cellular motility. Physarum polycephalum can migrate at a speed of up to 4 cm h⁻¹ [32] through the interplay of intracellular flow, adhesion and rhythmic contractions of the intracellular actomyosin cytoskeleton [30,32]. These contractions produce a pressure gradient that pushes the endoplasm towards the cell periphery, where the veins vanish and the endoplasm can flow freely. Local cytoskeletal reorganization and local alteration of the actin–myosin cortex lead to the formation of pseudopods or fan-shaped leading fronts which extend and retract in synchrony with the shuttle streaming of the endoplasm [30,36,37]. Based on internal and external cues, P. polycephalum can adapt and alter its shape, size and motion. The frequency and the amplitude of the cytoskeleton contractions depend on the quality of the environment [38–40]. For instance, higher frequencies are observed in response to attractive, high-quality resources [41], whereas lower frequencies are recorded when P. polycephalum encounters repulsive stimuli such as chemical repellents [40]. As a result, slime moulds migrate towards or away from a variety of external stimuli such as chemicals [38], light [42], temperature [43], humidity [44], gravity [45] or substrate distortion [46].

(b) Physarum polycephalum's learning abilities

Although slime moulds lack the complex dedicated neural hardware of neurozoans, the environment they evolved in is no less complex and they face the same challenges: they must search for and choose between resources, adapt to changing conditions, avoid dangers and find suitable microclimates to inhabit. Physarum polycephalum demonstrates key aspects of complex decision-making [24–27,47,48] (figure 1). It can find its way in a maze [49], construct efficient transport networks [50], distinguish how different masses distort the substrate [46], avoid obstacles [51] and risky environments [52], optimize its nutrient intake [53,54], and use conspecifics' cues to choose what substrate to exploit [55,56]. All these feats rely on the abilities of P. polycephalum to sense and respond simultaneously to a wide range of biotic and abiotic stimuli [38–45].

Despite their increasing recognition as a model for the study of complex behaviours in unicellular organisms, surprisingly little attention has been paid to the ability of slime moulds to learn and to modify their behaviour accordingly. Four studies have shown that P. polycephalum displays the capacity to learn [23,57–59]. In the first study, Saigusa et al. [57] showed primitive anticipatory and recall behaviours. Individuals exposed to periodically occurring negative stimuli were found to reduce their locomotive speed in response to each stimulus. When the negative stimuli were withdrawn, the slime moulds spontaneously reduced their locomotive speed at the point in time when the next negative stimulus would have occurred, i.e. they learned the periodicity. Once the spontaneous slowdowns stopped, they could be induced again by a single negative stimulus, which implies recall of the memorized periodicity. This study has sparked several models considering the involvement of oscillations in aneural learning, as will be further discussed in §4. In the second study, Boisseau et al. [23] demonstrated habituation, the simplest form of learning, in P. polycephalum. Habituation is a progressive decrease in the magnitude of a behavioural response to an iterative stimulus. Using locomotion as the behavioural output and diverse chemicals as repellent stimuli, the authors showed that P. polycephalum learned to ignore a chemical when it was encountered repeatedly for 5 days (habituation), but responded again when the chemical was withheld for a certain time (spontaneous recovery). They also demonstrated stimulus specificity and ruled out sensory and motor fatigue, often mistaken for habituation. In the third study, Vogel & Dussutour [58] showed that non-habituated slime moulds were able to directly acquire a learned behaviour from a habituated slime mould via cell fusion. They revealed that the transfer of learned behaviour was effective only when the vein networks merged and the cytoplasm streamed back and forth between the fused individuals. In the last study, Boussard et al. [59] showed that the memory of habituation can be stored for a month during the dormant stage and retrieved once the slime mould is revived. Chemical analyses indicated an uptake of repellent during the process of habituation and its retention throughout the dormant stage. The authors showed a strong correlation between the level of intracellular repellent and the habituation performance, and revealed that constrained absorption of the repellent induced habituation in 2 h. This study proposed that P. polycephalum absorbs the repellent and uses it as a ‘circulating memory’.

(c) Inferring learning from motion analysis

Motion is straightforward to capture in almost any living system. In this regard, time scale is the only difference between the wing flapping of hummingbirds and the pulsatile movement of P. polycephalum. While it is easy to detect motion, given the right sampling frequency, it is much harder to infer intent from movement [60]. This highlights the difficulty of designing experiments addressing cognitive abilities in aneural organisms. In other words, when do we interpret a moving system as cognitive, as opposed to a purely biochemical or biomechanical process? Regardless of the organism, at the microscopic scale, motion results from simple biological processes. In most cells, diverse organization of the actin cytoskeleton can produce pushing, pulling and resistance forces responsible for the whole cell shape and migration [61]. Studies investigating learning or decision-making in P. polycephalum interpret motion as behaviours. For instance, Boisseau et al. [23] measured the migration speed of P. polycephalum when facing a repellent and concluded that habituation occurs when the speed ceased to be affected by the repellent. Thus, they implied that motion can reflect cellular ‘memory’ to previous stimuli. One can question the relevance of interpreting motion as a cognitive output as opposed to simple biochemical processes.

Contrary to motion, demonstrating a cognitive ability in any living organism requires testable criteria. Cognition can be defined as the individual capacity to perceive, process and use information. Despite much evidence showing that cognitive abilities such as sensing, communication, memory, learning and decision-making are found in prokaryotes, single-celled eukaryotes, plants and fungi [17], some authors restrict the existence of cognition to neural organisms. For instance, Dukas [62, p. 347] defines cognition as ‘the neuronal processes concerned with the acquisition, retention, and use of information’. Yet, a cognitive ability observed in several organisms could either rely on similar mechanisms and share a common evolutionary origin or result from an evolutionary convergence [63]. For instance, in neurozoans, habituation results from the decreased release of chemical neurotransmitters at synaptic terminals which modulates the weights of certain neural connections. However, in P. polycephalum and other single-celled organisms, habituation cannot be encoded in a neural network. Yet, a detailed analysis of P. polycephalum's behaviour [23] suggests that this organism is capable of habituation as it fulfils all the criteria that define a habituation phenomenon [64]. Thus, should we restrict a cognitive ability to its underlying mechanisms, and by doing so, preclude its existence in many systems, or should we consider together its biological function and its implementation [4]? As suggested by Marr [65], to study a cognitive system we need to answer three questions: What problem is the system trying to solve? How does the system solve the problem at the behavioural level? and What are the mechanisms underlying the behavioural response? The accumulating evidence supporting the existence of cognitive abilities in slime moulds challenges us to understand what kind of physical systems underlie such abilities. One of the hypotheses we make is that learning in slime moulds might result from intracellular oscillators, as oscillations can encode information in their frequency, amplitude or duration. We especially focus on how the coordination of relatively simple oscillatory processes at the cell and molecular levels results in the emergence of learning in P. polycephalum.

(a) Short-period oscillations in Physarum polycephalum: the contraction cycle

When facing a repellent in its environment, P. polycephalum reduces its migration speed. However, as mentioned earlier, constrained absorption of the repellent quickly attenuates this behaviour, and the slime mould behaves almost as if the repellent was absent (habituation) [23]. Thus, the entry of repellents into the cell might interact with the shuttle streaming responsible for the migration of the organism [59]. In P. polycephalum, several well-defined chemical oscillators control regular relaxation–contraction patterns, which locally drive the intracellular flow and are at the root of further large-scale patterns such as migration and cell shape. In turn, the shape and motion of the cell influence these local oscillations [73–75]. Relaxation–contraction patterns are generated through intrinsic nonlinear oscillatory phenomena in the plasmodium supported by complicated mechano-chemical reactions among intracellular chemicals such as ATP, free intracellular calcium (Ca²⁺) and actin, but also among organelles. These oscillators are interconnected, often coupled and partially dependent on each other [76]. Most of the observed chemical oscillators are associated with energy metabolism and respiration, with mitochondria being most prominently involved at the organelle level. In the following, we will present the chemical oscillators that have been described in P. polycephalum.

Calcium, as the universal signalling ion in cells, is involved at almost every level of intracellular signalling and bridges the gap between locomotion and energy metabolism. Oscillations in cytosolic [Ca²⁺] represent one of the most important examples of cellular rhythm in most organisms. In P. polycephalum, free intracellular [Ca²⁺] is directly linked to actomyosin contraction [77–79]. [Ca²⁺] and actomyosin contractions oscillate with a similar period but in approximate anti-phase [80,81], because the actin–myosin interaction is inhibited by Ca²⁺ [82]. In the resting state, the cytoplasmic concentration of Ca²⁺ is kept as low as possible. Excess Ca²⁺ is either extruded via the cell membrane or sequestered in mitochondria and cytoplasmic vesicles. When the cell is stimulated, [Ca²⁺] increases owing to its release from storage or the entry of extracellular Ca²⁺. This transient increase is used by the cell as a secondary messenger to regulate the interaction between actin and myosin. However, in contrast to animal cells, in P. polycephalum intracellular Ca²⁺ is an inhibitor [83]. High [Ca²⁺] coincides with relaxation, whereas low [Ca²⁺] induces contraction. The [Ca²⁺] oscillator was first modelled by Smith & Saldana [84], using a phosphorylation–dephosphorylation cycle of myosin light chain kinase. [Ca²⁺] waves travel at approximately 5 µm s⁻¹, suggesting that they are not propagated by shuttle streaming [85], which is much faster (up to several mm s⁻¹). Furthermore, in the absence of external stimulations, Ca²⁺ is not homogeneously distributed within the cell but forms a gradient throughout the plasmodium [86]; the mean concentration at the front is higher than that at the rear, establishing cellular polarization during migration. Both repellents and attractants change the spatio-temporal pattern of Ca²⁺. A stimulation with an attractant at the rear increases the frequency of [Ca²⁺] oscillations, raising [Ca²⁺] at the stimulation site, whereas a repellent has the opposite effect.

Another oscillatory compound is ATP, which is distributed in patterns throughout the plasmodium [87] and exhibits oscillations in phase with the relaxation–contraction cycle [76,80]. ATP efflux increases during the contracting phase, whereas it decreases during the relaxing phase. In a migrating plasmodium, ATP concentrations are approximately twice as high at the front [88] than at the rear. ATP provides energy for the contraction of the actomyosin cytoskeleton, which in turn generates cytoplasmic streaming. The ATP-producing pathways, e.g. glycolysis in the cytoplasm and biological oxidation in mitochondria, influence directly the contractile activity of P. polycephalum. Hence, there is a strong link between energy metabolism and locomotion. When the plasmodium is starved, ATP synthesis is impacted and the slime mould is constrained to switch to different energy-generating mechanisms [89] or initiate more drastic measures (e.g. sclerotization or sporulation). Inhibition of both ATP synthesis and respiration in mitochondria stops locomotion, but inhibition of ATP synthesis alone does not. Thus, it was suggested that another mitochondrial function, such as [Ca²⁺] regulation, controls the rhythmic contractions [76,90].

Oscillations of [H⁺] and cAMP also accompany the oscillatory contractility of P. polycephalum [91]. Cytosolic [H⁺] oscillations are a by-product of oxidative phosphorylation and are in phase with ATP oscillations; they are related to the energy metabolism [91]. cAMP oscillates approximately 60° ahead of the [Ca²⁺] oscillations [85,88] and is also involved in cell polarization. It forms a gradient from front to back, with the back showing several times higher concentrations than the front. Knowing that ATP and H⁺ trigger contraction and oscillate in phase, whereas cAMP and Ca²⁺ are responsible for the relaxation, Ueda et al. [88] proposed that a feedback loop consisting of ATP, H⁺, cAMP and Ca²⁺ controls the contraction cycle. They postulated that a negative feedback occurs from Ca²⁺ to ATP and H⁺ in the mitochondria, where ATP is produced owing to H⁺ influx and Ca²⁺ entrance in exchange for H⁺. Ueda et al. [88] suggested that these interactions between Ca²⁺, ATP and H⁺ are involved in the time-keeping of the rhythmic oscillations.

Experimental and theoretical evidence indicates that mitochondrial energetic variables oscillate autonomously as part of a network of coupled oscillators [92]. The concentration of intracellular NAD(P)H in plasmodia of P. polycephalum was found to oscillate with the same period as the rhythmic contractions, but a third of a period ahead [93]. When irradiated with UV light, the contractions can be stopped without affecting the rhythmic variation of NAD(P)H. Thus, the NAD(P)H oscillator works independently of the rhythmic contractile system, but seems to be entrained with it. Continuous energy production in the form of ATP appears to be necessary for oscillatory contractions. Both glycolysis, an oscillatory process in itself, and mitochondrial respiration influence the regulation of the contraction oscillations.

In addition to the chemical oscillators, it should be mentioned that spatio-temporal oscillations can also originate from mechano-chemical properties of an active material [94–96]. Spatial coordination can be achieved by mechanical interactions between different regions (see e.g. [97]), and local positive feedback between deformation and contraction of the contractile apparatus induces the observed characteristic contractile behaviour by hydrodynamic interaction alone via the streaming of endoplasm. However, more realistic models combine the interaction of mechanical and chemical oscillations. It was found that the temporal frequency is set by the chemical oscillator, whereas wavelength is determined by a mechano-chemical instability [98]. In addition, only a combined model such as [94,98] allows for migration of microplasmodia. Radszuweit et al. [96] proposed a model for the oscillations that is based on both chemical (i.e. exchange of calcium between the cytosol and storage vessels) and mechanical factors (i.e. flow and deformation).

We can assume that the contraction cycle in P. polycephalum relies on a feedback system between the contractile apparatus (i.e. the actomyosin system), the calcium regulatory system, and the energy metabolism [99,100]. It stands to reason that, since all of these oscillators are highly interconnected, they must be synchronized in some manner to produce coordinated behaviours as a function of environmental conditions (see e.g. [101]). These oscillators might be involved in learning and memory as any encounter with a stimulus could be encoded and stored as changes in the contraction cycle and, by extension, as modifications of the vein network architecture.

(b) Long-period oscillations in Physarum polycephalum: the cell cycle

In P. polycephalum, a lower level of intracellular repellent is sufficient to trigger the memory of habituation after the dormant stage [59]. This suggests that the interaction between the repellent and the relaxation–contraction cycle, which could support short-term habituation, is not the only mechanism supporting long-term habituation. In the brain, memory consolidation—the process by which a short-term memory is transformed into long-term memory—requires gene transcription and de novo protein synthesis to strengthen synaptic connections, e.g. facilitation and maintenance of synaptic plasticity, growth of new functional synapses or physical insertion of new receptors [102]. Gene transcription and protein synthesis could be responsible for memory consolidation in P. polycephalum also. In all eukaryotic cells, gene transcription requires the chromatin to be ‘open’, allowing easy access of transcription factors to target genes. In a non-dividing cell, such as a neuron, chromatin can be readily opened and gene transcription can occur at any time. By contrast, in a dividing cell such as P. polycephalum, chromatin undergoes dramatic compaction during mitosis, which precludes gene transcription [103]. Hence, in P. polycephalum, gene transcription is temporally restricted, setting up critical temporal windows during which this process must take place. In addition, as the cell-division cycle globally disrupts and reorganizes the molecular content of the cell, it can profoundly influence learning and memory performance. In the following, we will briefly describe the mitotic cycles of P. polycephalum.

The plasmodium of P. polycephalum is a polynucleated cell that undergoes nuclear division every 8–12 h [28]. In contrast to most cells, mitosis is not followed by cytokinesis and it occurs within the nuclear membrane (closed mitosis), which is common in protists and fungi. In P. polycephalum, the phase G1 is absent, the S-phase lasts for 2–3 h, followed by a long G2 phase and 30 min mitosis (the nuclear division itself) [28]. In P. polycephalum, gene transcription, and in particular mRNA transcription, goes from zero during mitosis to a maximum in early S-phase [103]. Remarkably, the plasmodium exhibits synchronous mitotic division of its entire population of nuclei, and therefore the number of nuclei doubles every cell cycle [28]. Pieces cut off from a plasmodium maintain this synchrony for a certain time and two asynchronous plasmodia can synchronize their mitotic rhythm after fusion [104,105]. Experiments on the cell cycle, particularly those employing fusion techniques in P. polycephalum [104,105], gave rise to pioneering theoretical models [106,107] that presented the mitotic oscillator as a dynamical system and suggested that synchronous mitotic division emerges from dynamic interactions of regulatory molecules. In the 1970s Physarum polycephalum was considered to be the ideal organism to study the cell cycle. However, owing to the lack of tools to disrupt gene function in P. polycephalum, in the 1980s most researchers turned to other model systems in which they could test their hypotheses.

In the last decades, numerous independent genetic and biochemical studies using different organisms such as Xenopus and yeast identified a major evolutionarily conserved regulatory network for cell-division cycle (i.e. mitosis) regulation [108]. This network includes a cyclin that is periodically synthesized and degraded at specific cell cycle phases and kinases (cyclin-dependent kinases, CDKs) that is regulated by phosphorylation and interaction with activators and inhibitors. The oscillations in CDKs activity, in complex with regulatory cyclin subunits, drive the cell-division cycle. In parallel to experimental measurements, mathematical modelling of the cell cycle has proved helpful in unravelling how interacting proteins such as cyclin and their partner CDKs control the dynamics of cell division. These models contributed to the modern understanding of the cell as a complex dynamical system [109–111].

Cyclins and CDKs are present in all eukaryotes, including P. polycephalum [112]. In P. polycephalum, the vigorous actomyosin-driven cytoplasmic streaming described in §3a ensures that local accumulation of cyclin/CDKs complexes diffuses [112] and triggers the activation of cyclin/CDKs complexes in neighbouring nuclei, maintaining mitotic synchrony [105]. Until now, yeasts have historically been a major model system to understand the control of the cell-division cycle in animals, taking advantage of their powerful genetics and rapid duplication times. However, otherwise conserved cell cycle regulators have been lost in this organism. By contrast, the complete set of P. polycephalum cell cycle regulatory molecules has been identified [112]. Hence, with new options for experimental manipulation and in conjunction with mathematical modelling, P. polycephalum could become once more an attractive model organism to study cycle regulation at the regulatory complexity of a mammalian cell.

Mitotic and genetic oscillators are relevant to understanding learning and memory in P. polycephalum not only because of their oscillatory nature, but also because they provide a way to encode past experiences as epigenetic changes. The concept of epigenetic memory was defined by D'Urso & Brickner [113, p. 230] as ‘a stable propagation of a change in gene expression induced by developmental or environmental stimuli’. Epigenetic memory impacts gene expression and the associated feedback loops over both short and long time scales. Several pieces of evidence suggest that epigenetic mechanisms can regulate the ability to store long-term memories in neurozoans; it might also be true for aneural organisms [114].

(a) Synchronization, a prerequisite

Synchronization, a phenomenon arising from the interaction of two or more oscillators, can be observed in many biological systems. The position of an oscillator in its cycle can be described in terms of its phase, which advances at a pace set by the oscillator's natural frequency. Interacting oscillators influence each other's phase progression (figure 2). A paradigm for synchronization is a population of globally coupled oscillators (i.e. each oscillator in the population is affected by the dynamics of the rest), in which the natural frequencies are initially heterogeneously distributed. As time progresses, the oscillators minimize their pairwise phase differences, resulting in a pattern known as full synchronization, where all oscillators operate at the same instantaneous frequency and phase. This is described mathematically by the canonical Kuramoto model [115], which predicts a phase transition between desynchronized and synchronized states, based on the global coupling strength (figure 2d,f; electronic supplementary material, movie S3). Synchronous mitosis in P. polycephalum illustrates such global coupling leading to a state of full synchronization. Here, nuclear division is robustly controlled across the entire organism and occurs when the ratio of DNA to cytoplasm in the plasmodium drops below a certain value owing to an increase of the cellular volume [116].

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Modifications of the Kuramoto model give access to a much richer spectrum of synchronization patterns, as observed in many biological systems, such as cardiac cells, populations of fireflies or assemblies of neurons. Strikingly, employing local instead of global coupling or combinations thereof can lead to the emergence of complex spatio-temporal patterns. These patterns range from full synchronization or desynchronization to spiral waves, travelling waves (figure 2e), and even to curious chimera states [117], where part of the population is synchronized and part is not. Notably, the Kuramoto model can be modified to account for local coupling and is also well suited for describing populations of oscillators coupled through a network topology [115], as shown in figure 2d,e and electronic supplementary material, movie S4 for a random planar network. Indeed, synchrony in a population of oscillators is crucially affected by the topology of their network (regular lattice, random network and small-world network). This lies at the foundation of diverse dynamics observed in biological networks, such as the central nervous system. Synchrony in the brain supports neural communication and neural plasticity and is relevant for many cognitive processes [118]. Similarly, synchronization phenomena observed in the plasmodium's intrinsic cellular oscillations have been suggested to support complex behaviours in P. polycephalum [119].

As described previously (§3a), transport of cytoplasm in P. polycephalum is actively driven by rhythmic contractions associated with biochemical oscillations. The rhythmic contractions typically organize as travelling waves [32,120], but other patterns such as rotating spirals are observed [121]. Although a variety of active materials are capable of generating such patterns, they can also be indicative of locally coupled oscillator systems. Unlike neural networks or cardiac cells, the plasmodium of P. polycephalum is not composed of discrete, interacting units but instead forms a continuous yet inhomogeneous material (§2a). It has been shown that P. polycephalum reproduces the dynamics of coupled oscillators at large spatial scales when introducing an artificial discretization by shaping the plasmodium into extended patches connected by thin veins [73,122]. In this setting, owing to the imposed asymmetry, the connections were largely passive. By contrast, in naturally grown plasmodia, the connecting veins contribute actively to the oscillatory dynamics. Here, it appears necessary to move to smaller spatial scales by regarding the plasmodium as a continuum of oscillators coupled locally via diffusion, via advection of chemicals through active transport, and via mechanical interaction. This approach has been pioneered by Kobayashi et al. [123].

(b) Integration of environmental information

Both in nerves and in P. polycephalum's veins, chemical oscillations are responsible for conveying macro-oscillations as well as transport or motion at the organism scale. What we call macro-oscillations might be the rhythmic movement of the human legs, the wings of a bird, or the centre of gravity of a P. polycephalum plasmodium [30]. Why are these motions rhythmic? An enticing argument arises from a neuroscience perspective, stating that rhythmicity provides a powerful tool for organisms to infer easily from their environment. Buzsaki [70] sees leg movements as an oscillating system in which perturbations indicate an environmental change. Considering leg movement, a change of the floor topology will disturb the movement rhythmicity and therefore will become a piece of integrated information allowing motion continuation.

Hence, an environmentally driven behavioural change can be seen as integration of environmental information through change in the oscillatory pattern. A prominent example is found in rats, in which active movement of the whiskers generates the sense of touch. The perturbation of the whisker movements following a contact with an object elicits a sensory signal that carries information about the object. In rats, whisker motion over a texture produces a specific temporal profile of velocity, which is encoded in the temporal pattern of neural activity, for instance rougher surfaces evoke higher firing rates [124]. Hence, the information given by mechanoreceptors is directly encoded into a new oscillatory pattern in the brain. This kind of functioning pervades the neural sensing of external stimuli, and changes in oscillatory patterns are likely to be a common mechanism by which neurozoans integrate and store information.

Physarum polycephalum's biochemical and cytoplasmic oscillations may have a similar purpose and might offer a possible explanation for the organism's ability to process, integrate and store information. Physarum polycephalum exhibits few sustained large-scale oscillations and a continuous spectrum of short-scale oscillations of varying spatio-temporal scale when not faced with any stimuli (i.e. resting state) [125]. Adding a stimulus in the environment has been suggested to activate regular large-scale oscillation patterns within this continuous spectrum [125].

Upon encountering an external stimulus, P. polycephalum responds first by adjusting its oscillation frequency and amplitude locally. This change then spreads over the entire network [126] and finally the morphology of the slime mould adapts to the environment [127]. Positive stimuli such as chemoattractants lead to increased frequencies, whereas negative stimuli such as chemical repellents lead to a decrease (§3a). Consequently, it appears reasonable to assume that the stimulated regions increase or decrease their importance for the global oscillation pattern: positively stimulated fronts act as pacemakers, whereas negatively stimulated regions concede. Indeed, it is observed in small fragments [120] and in macroplasmodia (A Fessel, H-G Döbereiner 2020, personal observation), that travelling waves responsible for mass displacement are directed towards those fronts where the slime mould extends, underlining the plausibility of this concept.

Hence, an encounter with an external stimulus could be memorized as a change in the spatio-temporal dynamics of the system. Returning to the results of Boisseau et al. [23], we might wonder: how does the global oscillation pattern change in response to repetitive stimulations to allow habituation? We do not have the answer yet, but we believe that the process of habituation in P. polycephalum relies on both an attentive resting state and mechanisms to adapt the oscillation patterns to changing environments.

(c) Maintaining excitability: resting-state activity and self-organized criticality

As outlined in the previous section, the disturbance of an oscillatory resting state via an input is an efficient mechanism for integrating environmental information. This mechanism is extensively used in neurozoans' central nervous systems. Here, we describe how the resting state affects neural functioning and introduce a number of advanced mechanisms that can maintain or modify the resting state of an oscillatory system. These mechanisms have proved to be relevant in the context of self-organization or adaptive control, in particular in neural systems. Their involvement in P. polycephalum is unclear yet appears to be entirely possible in the light of the analogy proposed in this article.

Resting-state activity in the brain is not immediately related to any motor task or sensory input. It represents spontaneous activity that is closely related to the functional organization of the neural network and encompasses both sub-threshold membrane potential oscillations and patterns of neural spiking activity. By assisting spike-timing-dependent plasticity (see §4d), synchronization of the sub-threshold activity facilitates reliable and precise adaptation of synaptic strengths [128]. Therefore, the resting state is involved in organization of the neural activity, optimization of the learning process, and network development in the brain, and is believed to be implicated in higher functions, such as different states of consciousness and focus [129,130]. Finally, it interacts with external inputs (see §4b).

Recent results suggest that the observed resting-state patterns indicate a state at which the brain is stable yet responsive. This is summarized as the critical brain hypothesis [131], stating that the brain operates, controlled by a self-organizing process, in a subcritical regime, yet close to the critical point [132], where it is most excitable. Criticality is an umbrella term that denotes the behaviour of a system perched between order and disorder. In the brain, the subcritical regime is considered as unexcitable, whereas the critical regime displays cascades of activity unrelated to any input. Neither regime is favourable for normal brain function and excitations lead to a controlled response only slightly below the critical point.

Evidence suggests that the phase transition associated with the critical point in question is, in fact, the transition between desynchronization and synchronization [133] (see also §4a), highlighting once more that synchronization, a very basic feature of coupled oscillatory systems, is at the very heart of coordination of complex functions. The mechanism that appears to be responsible for restoring the system to the working point after it has shifted too far into the critical or subcritical regimes is sleep [134], during which resting-state activity is pronounced. Curiously, during sleep, the resting-state activity couples to the flow of blood and cerebrospinal fluid [135], the latter being driven by the synchronized motion of cilia [136]. This mechanism is hypothesized to facilitate, for instance, waste disposal and hints at another analogy: throughout this article, spatio-temporal synchronization of relaxation–contraction oscillations has been pointed out as the driving force behind fluid flow in the slime mould. Here, a quite similar mechanism that is rooted in the spatio-temporal synchronization of neural oscillations drives fluid flow in the brain.

It is unclear whether self-organized criticality is involved in the dynamics of P. polycephalum. Coggin & Pazun [137] indicate that the spectrum of frequencies observed in P. polycephalum may constitute a signature of criticality. The authors further support their hypothesis by showing that the oscillating system operates at the edge of chaos (at the boundary between order and disorder), another indication for criticality that has been studied in the context of self-organization in general, and in particular in neural networks [138]. The peculiar power spectrum of P. polycephalum has been noted not only in the context of criticality but also with regard to the anticipatory learning behaviour of the slime mould described in §3b. Here, it serves as an underlying assumption to the model of Saigusa et al. [57]. Despite its implications, the mechanisms generating the power spectrum are as of now unclear and thus the hypothesis has not been further verified.

How can an oscillatory system be maintained at the edge of chaos? Mechanisms facilitating this include coherence resonance and stochastic resonance [139], which can stabilize a spectrum by noise-induced entrainment and synchronization, which lead to frequency selection [140,141]. Alternatively, modification of the Kuramoto model to include adaptively controlled coupling leads to a similar effect. The scheme facilitating this adaptive control in dynamical systems that exhibit chaos builds on the self-organized control of unstable periodic orbits through inclusion of a delayed intrinsic feedback [142]. Intriguingly, by application of this scheme to coupled Kuramoto oscillators, Kaluza & Mikhailov [143] were able to generate a system that robustly adapts desired synchronization levels through feedback control of coupling strengths. This corresponds to the adaptation of synaptic weights in a neural network (see §4d), highlighting that the combination of these rather general mechanisms can facilitate learning in addition to maintaining criticality. Recently, these principles of adaptation have been experimentally verified in an artificial system of optically coupled chemical oscillators [144] and have been extended to complex, modular synchronization patterns [145].

Albeit operating on a slow time scale, the adaptation of coupling strengths is a motive in the structure formation of P. polycephalum [146] (§4d). It is driven by the oscillatory shuttle streaming and used by the organism to adapt to its environment by eventually modifying the topology of its network. Notably, such topological adaptation is another form of maintaining criticality if it stabilizes the system in the vicinity of the percolation phase transition [147]. In networks, percolation is the transition between a low-connectivity, fragmented state and a state with high connectivity, indicated by the existence of a giant component [146]. Owing to the slow time scale, percolation appears as a rather unlikely candidate in P. polycephalum or neurozoans, unless the highly dynamic network of flow directions (in the slime mould) or neural activations (in neurozoans) is considered instead of topologies of veins or neurons [147].

(d) Adaptation and computation

Although topological adaptation is a weak candidate for explaining criticality, it is clearly involved in the formation of memory in neurozoans. Here, it manifests as the adaptation of weights (synaptic strengths) between neurons and is controlled by a mechanism referred to as spike-timing-dependent plasticity: connections are promoted if neurons fire in synchrony and demoted otherwise. This feedback mechanism for the adaptation of the neural network topology is reminiscent of the current reinforcement model brought forward by Tero et al. [148], which states that veins in the tubular network of P. polycephalum are promoted if there is a sufficient volume flow, and demoted otherwise. This oscillatory flow is driven by relaxation–contraction oscillations that organize as a wave [32]. For efficient transport, phase and frequency relationships along a single vein need to be intricately maintained [149,150], indicating that precise control of the spatio-temporal oscillation pattern is required in the slime mould as well.

Intriguingly, the analogy applies not only to synaptic plasticity and current reinforcement, but also to memristors, electrical components with a resistance that is based on the history of currents flowing through them. Memristor-based computational models have been successful at modelling learning in P. polycephalum in the form of anticipation of periodic events [151] (see also §§3b and 4c), as well as the computation of shortest paths [152]. Indeed, the ‘electric circuit analogy’ is more general and encompasses hydraulic counterparts for all basic electrical components. In consequence, microfluidic devices can be manufactured that mimic the function of electrical circuits [153], including the generation and control of oscillatory flows [154], thus extending the analogy to alternating currents. This has been widely applied, for instance, for the description of oscillatory blood flow in the field of haemodynamics [155]. There it has led to the formulation of the Windkessel model, a motif of electrical networks consisting elementarily of a resistor and a capacitor in parallel. The Windkessel model accounts for the influence of vessel elasticity on pulsatile flows and has been successfully employed to model frequency selection, migration and symmetry breaking in P. polycephalum [156]. In neuroscience, the approach of modelling biological transport processes by a corresponding electrical circuit has seen great success, pioneered for instance by the Hodgkin–Huxley model [157]. Intriguingly, the Windkessel model has also been applied by Cabral et al. [158] for modelling the organization of resting-state activity (see §4c) in the brain in conjunction with the Kuramoto model for synchronization [158]. Notably, the model of Cabral et al. [158] also requires the interaction between oscillators to be time delayed, an effect that was first shown to exist in biological systems in P. polycephalum [73].

The emergent computational principles of neural systems have broad implications beyond fundamental neuroscientific research. The development of silicon-based multi-purpose computing architecture is facing severe limitations, making bioinspired innovations indispensable. Neuromorphic computing [159] mimics basic functions of neurons and synapses, paving the way for efficient hardware implementation of deep learning, potentially using memristors [160]. Furthermore, oscillator-based computing [161] seeks to harness the complexity of oscillatory systems for computation. Both can make use of reservoir computing [162], the concept of adapting to the inherent computational powers of naturally or randomly generated systems instead of tailoring them to the task. Intriguingly, systems have been generated in this fashion that unite oscillatory properties, signatures of criticality, memristivity and learning [163–165]. Irrespective of the details of the implementation, understanding the rules governing how natural oscillatory systems such as P. polycephalum process information and learn is paramount to the advance of computing, and any advances are expected to be of mutual benefit.