Amoebae Anticipate Periodic Events

When plasmodia of the true slime mold Physarum were exposed to unfavorable conditions presented as three consecutive pulses at constant intervals, they reduced their locomotive speed in response to each episode. When the plasmodia were subsequently subjected to favorable conditions, they spontaneously reduced their locomotive speed at the time when the next unfavorable episode would have occurred. This implied the anticipation of impending environmental change. We explored the mechanisms underlying these types of behavior from a dynamical systems perspective.

Figure 1

SPS and SPSD responses. Typical time course of locomotion for an organism (wet $\mathrm{\text{weight}}=15\mathrm{mg}$) before and after three periodic applications ($S1$, $S2$, and $S3$) and single application ($S4$) of dry stimulation. (a) Temperature is denoted by the upper line (TM) and humidity by the lower line (HU). (b) Locomotion speed calculated over successive 10 min intervals. (c) Acceleration ($\Delta$ speed) defined as the difference in speed between successive intervals in (b). $S1$, $S2$, $S3$, and $S4$ indicate the time points of the real stimulations, whereas $C1$, $C2$, …, $C7$ indicate the time points of the virtual stimulations. The SPS response was induced at time point $C1$, $C2$, and $C3$ by periodic stimulation (closed arrows), and at time point $C6$ and $C7$ by trigger stimulation at $S4$, after disappearing once (open arrows).

Figure 2

Statistical analysis of the SPS and SPSD responses in terms of mean speed with standard error (a) and statistical occurrence of spontaneous slowdown (b). The mean speed was calculated from 43 examples for the tested group (open circles), and from 39 samples for the control group with no in-advance periodic stimulation (closed circles). Points $S1$–$S4$ indicate the times of the real stimulations, whereas points $C1$–$C7$ indicate the times of the virtual stimulations. The SPS response was induced at $C1$ by periodic stimulation (closed arrow), and at $C6$ by trigger stimulation at $S4$, after disappearing once (open arrow). The organisms had weights ranging from 10 to 19 mg.

Figure 3

SPS responses over a range of stimulation periods ($\tau$). The gray levels of the bars indicate the number of occurrences of SPS: black denotes three, dark gray denotes two, light gray denotes one, and white denotes none. The twofold criteria for SPS were that there should be a unimodal evolution of locomotion speed between the last stimulation and the time of the subsequent virtual stimulation, and that the minimum locomotion speed should occur at the time of this virtual stimulation. Bimodal time courses and fluctuations of locomotion speed were often observed, and were regarded as an absence of SPS, as were instances when the minimum speed occurred a little earlier or later than the virtual stimulation. *: Dry stimulation at $26°\mathrm{C}$ and 70% humidity, which is a weaker stimulus than in all other cases. Number of repeats ($\tau \min $): 36 (30), 46 (40), 24 (50), 121 (60), 20 (70), 20 (80), and 19 (90).

Figure 4

Simple dynamics model and simulation of SPS and SPSD responses. (a) Simulation of the variation of locomotion speed $S$ with time (black line), and the function form of $H(t)$ (dashed line). (b) Schematic illustration of the model behaviors. Many oscillators with different frequencies (the same symbols denote the same frequencies) rotate counterclockwise on the ring in the state space spanned by two kinds of chemical species that are not specified and might be any chemicals. In the initial conditions the phase $\theta$ of the oscillator is distributed at random; no coherent motion is observed, so that locomotion speed $S$ is nearly one (b1). After periodic perturbations of frequency ${\omega }_{\mathrm{ext}}$ are applied to all the oscillators, the multiple oscillators with ${\omega }_j={\omega }_{\mathrm{ext}}$ (open stars) tend to synchronize because they are pushed leftward on the ring during the perturbation (b2). This synchronization is observed in oscillators with each frequency of ${\omega }_j\simeq {\omega }_{\mathrm{ext}}$ (open squares and open diamonds). At this point, $S$ shows clear periodic variations with the same frequency as the external perturbations, and this state is maintained until the coherent oscillators desynchronize in phase, due to differences between their intrinsic frequencies ${\omega }_j$ (b3). Note that groups of oscillators of the same frequency remain synchronized even though $S$ does not show any clear periodic behavior. When the perturbation is applied again, the oscillator groups of the same frequency tend to synchronize in phase, so that $S$ again shows periodic behavior (b2). ${\omega }_j=0.1$, 0.11, 0.12, 0.13, …, 4.49, 4.50; $N=1000$; $\alpha =0.5$; $-0.001<\xi <0.001$.