Computational capability of ecological dynamics

Ecological dynamics is driven by complex ecological networks. Computational capabilities of artificial networks have been exploited for machine learning purposes, yet whether an ecological network possesses a computational capability and whether/how we can use it remain unclear. Here, we developed two new computational/empirical frameworks based on reservoir computing and show that ecological dynamics can be used as a computational resource. In silico ecological reservoir computing (ERC) reconstructs ecological dynamics from empirical time series and uses simulated system responses for information processing, which can predict near future of chaotic dynamics and emulate nonlinear dynamics. The real-time ERC uses real population dynamics of a unicellular organism, Tetrahymena thermophila. The temperature of the medium is an input signal and population dynamics is used as a computational resource. Intriguingly, the real-time ecological reservoir has necessary conditions for computing (e.g. synchronized dynamics in response to the same input sequences) and can make near-future predictions of empirical time series, showing the first empirical evidence that population-level phenomenon is capable of real-time computations. Our finding that ecological dynamics possess computational capability poses new research questions for computational science and ecology: how can we efficiently use it and how is it actually used, evolved and maintained in an ecosystem?

MU, 0000-0003-4831-7181; KN, 0000-0001-5589-4054 Ecological dynamics is driven by complex ecological networks. Computational capabilities of artificial networks have been exploited for machine learning purposes, yet whether an ecological network possesses a computational capability and whether/how we can use it remain unclear. Here, we developed two new computational/empirical frameworks based on reservoir computing and show that ecological dynamics can be used as a computational resource. In silico ecological reservoir computing (ERC) reconstructs ecological dynamics from empirical time series and uses simulated system responses for information processing, which can predict near future of chaotic dynamics and emulate nonlinear dynamics. The real-time ERC uses real population dynamics of a unicellular organism, Tetrahymena thermophila. The temperature of the medium is an input signal and population dynamics is used as a computational resource. Intriguingly, the real-time ecological reservoir has necessary conditions for computing (e.g. synchronized dynamics in response to the same input sequences) and can make near-future predictions of empirical time series, showing the first empirical evidence that populationlevel phenomenon is capable of real-time computations. Our finding that ecological dynamics possess computational capability poses new research questions for computational science and ecology: how can we efficiently use it and how is it actually used, evolved and maintained in an ecosystem? theoretical studies have shown that prey-predator, mutualistic, competitive and biotic-abiotic interactions are prevalent, and that they play a pivotal role in ecological community dynamics [1][2][3][4]. In nature, the interactions shape an ecological network. Information of a node, for example, species abundance or the state of an abiotic variable, can be processed through interactions and transferred to another node in a complex way that is often difficult to be accurately represented by equations. Population or community dynamics includes temporal fluctuations in species abundance and is a consequence of the 'information processing'. Ecologists have tried to discern rules that govern the ecological dynamics or the information processing.
The terminologies 'network' and 'information processing' also appear in computational science, but they have been studied from a different viewpoint. In computational science, information processing capability of artificial networks is exploited as a computational resource. Artificial neural networks are represented by a network of neuron-like processing units (nodes) interconnected via synapse-like weighted links (interactions), which are typically classified into feedforward neural networks [5] and recurrent neural networks (RNNs) [6]. A machine learning approach called reservoir computing (RC) is a special type of RNN that is suitable for temporal information processing such as time series analysis [7,8]. In RC, input data are nonlinearly transformed into patterns in a high-dimensional space by an RNN called a 'reservoir'. Then, a pattern analysis from the transformed patterns is performed in the readout. The main characteristic of RC is that the input weights (W in ) and the weights of the recurrent connections within the reservoir (W) are not trained, whereas only the readout weights (W out ) are trained with a simple learning algorithm such as a linear regression. This simple and fast training process makes it possible to drastically reduce the computational cost of learning compared with standard RNNs, which is the major advantage of RC [7,8].
Recently, RC implementation using a physical material has been gaining growing attention in machine learning and engineering fields ( physical RC) [9]. A nonlinear, complex information processing capability is embedded in a physical material (i.e. embodiment) [10], and thus one can replace a reservoir in RC with a physical material. For example, a soft robotic, tentacle-like arm can process an input signal from a motor that initiates a movement of the robotic arm, and then the signal transmits through the arm in a way that depends on physical characteristics of the robotic arm such as length, material and shape. Nakajima et al. [11,12] have shown that such a soft robotic arm has a shortterm memory and can be used to solve several computational tasks in real time. In addition, a recent study has shown that even biological entities such as plants may be used as a physical reservoir [13].
Several successful examples of physical RC [8,9,14] imply that we may be able to use information processing capability of other types of networks as a computational resource. Here, we show that ecological dynamics can be used as a computational resource. We call this approach 'ecological reservoir computing (ERC)' and implement two types of ERC in this study (figure 1). The first type of ERC is in silico ERC; it reconstructs ecological dynamics from empirical time series using a time-delay embedding [15] (i.e. an equation-free ecosystem model) and simulates the system dynamics in response to hypothetical input signals. In silico ERC uses the reconstructed dynamics and the simulated responses as a reservoir and reservoir states, respectively, which successfully predicts the near future of chaotic dynamics and emulates nonlinear dynamics. The second type of ERC is real-time ERC; we set up an experimental system that enables continuous monitoring of population dynamics of a unicellular eukaryotic organism, Tetrahymena thermophila, in a small chamber. We manipulated the temperature of the medium in the chamber as input signal, and monitored changes in population abundance as reservoir states (i.e. a model-free empirical system). Surprisingly, the real-time ERC has the necessary conditions for RC and is able to make near-future predictions of model and empirical time series. The computational performance of the real-time ERC is currently lower than that of other types of RC, but the finding that untrained ecological dynamics may possess necessary conditions for RC would open up new research directions in computational science and ecology.

Reconstructed ecological dynamics as a reservoir
A seminal RC framework, echo state network (ESN), was proposed by Jaeger [7,16]. This model uses an RNN-based reservoir consisting of discrete-time artificial neurons. When the feedback from the output to the reservoir is absent, the time evolution of the neuronal states in the reservoir is described as where t denotes discrete time, X t is the state vector of the reservoir units, u t is the input vector, W in is the weight matrix for the input-reservoir connections and W is the weight matrix for the recurrent connections in the reservoir. The function f represents an element-wise activation function of the reservoir units, which is typically a sigmoid-type activation function. In the context of a population or community dynamics model in ecology, the weight matrix, W, represents a rule such as prey-predator interactions that govern the dynamics. Therefore, if equations governing the ecological dynamics are known, we can use the system for RC. This is clearly demonstrated in Methods and electronic supplementary material, figure S1 using a simple two-species model system, i.e. Lotka-Volterra equations, in the subsection 'Demonstration of the concept of ecological reservoir computing' in Methods (the computational performance of the system is low due to a small reservoir size). Unfortunately, however, we usually do not know equations that govern real, complex ecological dynamics. In other words, although previous studies demonstrated universal rules on 'model' ecological communities with explicit equations [17,18], whether and how these formulations accurately represent real ecological dynamics are usually unknown. In this circumstance, an 'equation-free modelling' approach enables extracting a potential mechanism that drives the dynamics. Among such approaches, a nonlinear time series analysis called 'empirical dynamic modelling' [19][20][21] may provide a promising way to use an empirical ecological time series as a reservoir. According to a delay embedding theorem [15,22], multivariate system dynamics may be reconstructed from a single time series using a time-delay embedding (i.e. reconstruction of system dynamics by plotting timelagged coordinates in a multi-dimensional space; figure 2a and electronic supplementary material, appendix I) even when equations governing system dynamics are unknown, which is known as state space reconstruction (SSR). One may add one or more variables (ordinates) in the reconstructed state space, allowing simulations of ecosystem response to external forces, which is a forecasting method of empirical dynamic modelling known as 'scenario exploration' [21]. Scenario exploration predicts ecosystem response to hypothetical changes in external forces by averaging near-future behaviours of nearest neighbours of a target state in the reconstructed state space [21,24]. Deyle et al. [21] developed scenario exploration and predicted how changes in sea surface temperature influence population abundance of Pacific sardine. In the context of RC, changes in sea surface temperature and predicted population abundance of Pacific sardine may be regarded as 'input' and 'reservoir state', respectively.
We demonstrate in silico ERC, as a scenario-exploration-based approach, using empirical ecological time series: (i) fish-catch time series collected from pelagic regions in Japan and (ii) DNA-based quantitative prokaryote time series taken from experimental rice plots (figure 2a) [23]. State spaces of the system were first reconstructed using an optimal embedding dimension (E) determined following a previous study [25]. Then, reservoir states were calculated as follows: where f simp indicates a 'simplex projection' [25], a nonlinear forecasting method that predicts a behaviour of a target state based on behaviours of nearest neighbours in the reconstructed state space. Briefly, if a target state is a state at time t Ã (= X t Ã ), then f simp looks for nearest neighbours around the target state X t Ã , and predicts the future state X t Ã +1 by calculating the weighted average of the future states of the nearest neighbours ('1. Overview of in silico ERC' in electronic supplementary material, methods). The behaviour conceptual illustrations of 'ecological reservoir computing (ERC)' (model or reconstructed system) (real ecological system) y t + 1 = y t (r yr y y t -β yx x t ) Figure 1. Conceptual illustrations of ecological reservoir computing (ERC) and how in silico ERC is implemented. (a) Conceptual illustrations of in silico ERC and real-time ERC. In silico ERC uses either equations or empirical dynamics reconstructed by empirical time series as a reservoir. In the present study, we mainly investigated the properties and performance of in silico ERC using reconstructed dynamics. Real-time ERC uses an empirical ecological interaction network as a reservoir. A node in an ecological reservoir may represent an individual, species or abiotic variable in this study.
royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 10: 221614 of X t is predicted by f simp so that X t follows the rule of the empirical ecological dynamics. In other words, f simp reflects the rules or behaviours of nearest neighbours, and thus may implicitly include the consequences of interspecific interactions such as prey-predator interactions and competitions. Then, a hypothetical input, u t , is added to the state after transformation by W in . We choose an identity function as f so that equation (2.2) can be interpreted as the population dynamics. Alternatively, one may apply f simp after adding a hypothetical input, u t , to X t (figure 2 and supplementary material, methods). In addition, one may easily multiplex reservoir states generated by different species (species multiplexing; figure 2b), which improves the performance of in silico ERC. This type of in silico ecological reservoir is an equation-free ecosystem model and it possesses a specific memory capacity and shows echo state property (ESP), which are necessary for successful RC scenario exploration based on simplex projection

M M M M M M M M M M M M t t t t t t
input vector 1) used for the learning process one-step future of is predicted by the weighted average of future states of (i.e., a simulated response of the reconstructed system) = eqn. S1 in electronic supplementary material.

Algorithm
Step 1. Prepare time series.
Step 3. Identify nearest neighbours of a target point.  , and blue point indicates the predicted response of the system by simplex projection (denoted by, e.g. f simp (W in u t + M t )). As in the main text, the predicted response may be calculated by W in u t + f simp (M t ).
royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 10: 221614 (figure 3 and electronic supplementary, methods). Even if initial conditions of an ecological reservoir are different, the difference in the reservoir states converges to zero when the same input sequence is used (figure 3a,b; uniform random values are used as the input sequence), suggesting that the in silico ecological reservoir possesses ESP. In terms of memory capacity, the in silico ecological reservoir remembers the information of input data about 10-20 time steps ago (figure 3c; species-multiplexed reservoir shows a higher memory capacity). We also measured information processing capacities, which can evaluate the expressiveness of the reservoir in terms of memory capacity and nonlinear processing of inputs, using both species systematically (electronic supplementary material, figure S2 and methods). Interestingly, the analysis on information processing capacity suggests that the in silico ecological reservoir has relatively higher nonlinear processing capacity than the linear one compared with the profile of the conventional ESNs (electronic supplementary material, figure S2c). These results together suggest that in silico ERC satisfies necessary conditions for RC. We tested the performance of in silico ERC by several standard tasks: prediction of chaotic dynamics, emulation of nonlinear autoregression moving average (NARMA) time series and generation of an autonomous system (Mackey-Glass equation) (for detailed parameters, see electronic supplementary material, tables S1 and S2 and methods). First, in silico ERC with species multiplexing accurately predicts Lorenz attractor (figure 4a; time series of 500 prokaryotic species in the same experimental system were multiplexed [23]; total reservoir size = 3271; electronic supplementary material, table S1), which outperforms predictions made by the simple two-species model reservoir (electronic supplementary material, figure S1e). Interestingly, the prediction accuracy measured by a correlation coefficient increases with the number of species-multiplexed (figure 4b), suggesting that species diversity of a community might be related to the computational capability of an ecological community. Second, NARMA2 (eqn. 8 in electronic supplementary material, methods) can be royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 10: 221614 accurately emulated with species-multiplexed in silico ERC (figure 4c) though the performance is still lower than that of ESN, a typical RC method (figure 4d for NARMA2, 3, 4, 5 and 10; total reservoir size of ESN = 2000; electronic supplementary material, table S1). Third, the Mackey-Glass equation cannot be embedded in a closed loop with in silico ERC in our current numerical experiments, but in silico ERC generates different attractor dynamics (figure 4e,f ). Altogether, though the performance is still lower than that of ESN, these results show that the method based on scenario exploration can be used as RC and solves several standard tasks. More importantly, in silico ERC implies that real ecological dynamics may also be used as a computational resource.

Real-time ecological reservoir computing
In this section, we show that ERC is possible even with real ecological dynamics. For real-time ERC, we set up an experimental system to use the population dynamics of a eukaryotic unicellular organism,    royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 10: 221614 concentration, cell-to-cell interactions and individual behaviours interact to generate complex, nonlinear dynamics [27][28][29] (figure 5d,e), which we expected would be suitable for real-time ERC. Thus, the number of cells captured in an image is an outcome of the complex system response to an external force (see Methods for the formulation of the system response in the context of RC), and this system is a model-free empirical system for RC.
To obtain reservoir states of the system and improve its computational capability, we adopted several strategies: data preprocessing, time-multiplexing and space-multiplexing. First, the Tetrahymena population dynamics was preprocessed to be stationary and unbiased (electronic supplementary material, figure S5b-d), which represents responses of the Tetrahymena population to the inputs and was used as reservoir states. Second, five reservoir states were multiplexed for one input (timemultiplexing; i.e. a reservoir state was taken every minute and an input signal was manipulated every 5 min; electronic supplementary material, methods and appendices II and III). Third, reservoir states taken using three medium concentrations (1.6%, 4% and 10% modified Neff medium) were multiplexed to further increase the reservoir size (space-multiplexing; electronic supplementary material, methods and appendices II and III).
We first tested whether the Tetrahymena reservoir has a memory capacity and ESP by inputting uniform random values as medium temperature using the three strategies. Examples of the monitoring results with the same input sequence and the same medium concentration for different trials are shown in figure 5f and electronic supplementary material, movie S1 (https://www.youtube. com/watch?v=z_QeEka4W3w), which shows a clear common-signal-induced synchronization that is a signature of ESP [30,31]. A different medium concentration generated different population dynamics (figure 5f ), suggesting that the population dynamics under a different medium concentration may be used as a reservoir with different hyperparameters. We further tested the correspondences between the two runs for each medium concentration, and found that the state differences become smaller when the same input sequence is inputted to the system (figure 6a-f ). On the other hand, with a different input sequence in the 4% medium system, the population dynamics show a different pattern and the state difference does not converge ( figure 6g,h). These results suggest that the system has ESP [30,31]. In addition, the population dynamics have a specific memory capacity; the dynamics recover the input values at 5-15 min ago (= 1-3 steps ago) (figure 7a-c). These characteristics enable the Tetrahymena reservoir to measure the medium temperature (figure 7d,e), showing that, by using the short-term memory of the community dynamics (not the medium), the Tetrahymena population dynamics can work as a 'thermometer' of the system. Together, these results suggest that the ecological reservoir may be used as a computational resource.
To explicitly show that the Tetrahymena reservoir can solve computational tasks, we predicted three time series: Lorenz attractor (model time series) and two fish-catch time series (empirical time series). As with the uniform random inputs, the same inputs generate similar population dynamics under the same medium concentration, showing ESP of the system (electronic supplementary material, figure S6). By time-and space-multiplexing those reservoir states, the Tetrahymena reservoir reasonably predicts the near future of the three time series (figure 8a-c; see electronic supplementary material, movie S2 for how the Tetrahymena reservoir predicts the near future in real time; https://www.youtube.com/watch?v=SUmkYAnfjFk). The predictions made by the Tetrahymena reservoir are more accurate than those made by linear readout at certain time points, suggesting that the computational capability of Tetrahymena population dynamics has been successfully extracted and used by the experimental system. The Tetrahymena reservoir predicts 15 time-step future of Lorenz attractor, 19 time-step future of flatfish time series and 30 time-step future of Japanese jack mackerel time series (figure 8d-f ).

Significance, current limitations and future perspectives
Recently, machines built completely from biological tissues, called living machines or biological robots, are gaining attention due to the extreme adaptability and flexibility provided by the protean nature of their donor organisms [32]. Our work shares the same line of motivation and serves as one of the examples of living machines or biological robots. However, ecological dynamics are often more complex and less understood than individual-level phenomena that have previously inspired living machines or biological robots. Perhaps partly due to this complexity, the computational capability of ecological dynamics has not been explored so far. This study illustrates for the first time that ecological dynamics has computational capability that could be harnessed. In this section, we will discuss the significance of the finding, its current limitations and future perspectives.
royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 10: 221614 In the present study, we show the first empirical evidence that ecological dynamics can be used as a computational resource in two ways: in silico ERC and real-time ERC. The former provides a numerical framework to quantify the 'potential' computational capability of the ecological dynamics and to use the reconstructed dynamics as a computational resource. In the context of computational science, while in silico ERC currently has poorer computational performance than traditional RC (figure 4), the  royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 10: 221614 First, it provides an efficient way to search for ecological dynamics with potentially high information processing capability in the real world. Second, comparing the computational capabilities with ecological factors (such as species identity, phylogeny and environmental factors) may provide insights into how and why the computational capability of ecological dynamics evolved. For example, communities in a rapidly changing environment might possess higher information processing capacity than those in a stable environment. The latter, i.e. the fact that untrained ecological dynamics possess necessary conditions for RC and can solve several machine learning tasks, is more intriguing, and its significance is multi-fold. In the context of computational science, real-time ERC is a novel computational framework. Though the computational royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 10: 221614 performance of ERC is still lower than that of the typical RC, other ecological dynamics (e.g. high-diversity community dynamics) with different experimental settings (e.g. different input signals such as light) will possess different reservoir properties (e.g. with/without ESP and different memory capacity) and such ecological reservoirs might outperform the typical RC. To achieve this, the development of methods for efficient (eco)system monitoring is key. In the context of ecology, the real-time ERC enables quantifications of the computational capability of empirical ecological populations or communities. The computational capability may be regarded as an 'extended' functional trait of organisms, which should evolve by interacting with biotic and abiotic factors in a natural habitat. As we stated in the previous paragraph, comparing the computational capabilities with ecological factors may provide insights into how and why the computational capability evolved. In addition, identifying responsible genes for the computational capability and designing organisms with a high computation capability would be a fascinating direction. Also, a community with high diversity may potentially have a high reservoir size, which could beget a high computational capability (as shown in figure 3b). The high computational capability, high community diversity and stable ecosystem functions might be interdependent, and the potential positive relationship between community diversity and computational capability may add a new value to biodiversity. Lastly, if the 'closed-loop' approach as shown in the Mackey-Glass equation in in silico ERC (figure 3e,f ) is successful in real-time ERC, it would imply that we may be able to design specific dynamics in real-time ecological dynamics. Developing a method for efficient ecosystem management is a pressing but challenging task and an ERC-based method might be a basis of a novel approach to manipulate ecological dynamics.

Conclusion
The present study provides the first empirical evidence that ecological dynamics may possess computational capabilities, and we demonstrate how it can be used in terms of RC. In the context of computational science, ERC does provide a novel framework for computing, and its potential is worth exploring. In the context of ecology, introducing the concept of 'computational capability' into ecological studies may open up new research directions. While how and why the computational capability of ecological dynamics evolved cannot be understood in the present study, answering this question may deepen our understanding of how ecological dynamics is driven and how ecosystem functions are maintained, which may contribute to better management and forecasting of ecological dynamics.

A classic reservoir computing framework: echo state network
In the early 2000s, echo state networks (ESNs) as well as liquid state machines (LSMs) were proposed as a seminal reservoir computing (RC) approach [7,34]. ESNs (and LSMs) are different from conventional recurrent neural networks (RNNs) in that weights on the recurrent connections in the reservoir are not trained, but only the weights in the readout are trained [7]. To apply a simple machine learning method to the readout, the reservoir should be appropriately designed in advance. The characteristics of ESNs are briefly described in equation (2.1). Equation (2.1) represents a non-autonomous dynamical system forced by the external input u t . The output is often given by a linear combination of the neuronal states as follows: z t = W out X t , where z t is the output vector and W out is the weight matrix in the readout. In supervised learning, this weight matrix is trained to minimize the difference between the network output and the desired output for a certain time period.

Demonstration of the concept of ecological reservoir computing
We demonstrate the concept of ERC using a toy model that is frequently used in ecology. Equation (4.1) shows two coupled difference equations that can be interpreted as a model of two-species royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 10: 221614 population dynamics where x t and y t indicate a population density of species x and y at discrete time t, respectively. r i indicates the population growth rate of species i, β ij indicates influences from species j to species i (i.e. interspecific interactions), and the second term in the parenthesis indicates density-dependent effects. We used the same r i for the linear term and squared term following the previous study [19]. The simple nonlinear model can be used as a small reservoir (i.e. reservoir size = 2) in the context of RC. First, any inputs, u t , can be converted using the weight matrix for the input-reservoir connections, W in , and then reservoir dynamics follow equation (4.2). This information processing can be described as follows: where t denotes discrete time, X t (= {x t , y t }) is the state vector of the reservoir and g is the two-species population dynamics model (equation (4.1)). f represents an element-wise activation function. While hyperbolic tangent is often used as f for ESN, here we choose an identity function as f so that equation (4.2) can be interpreted as the population dynamics of two species in response to the addition or removal of individuals of species x and y (thus, equation (4.2) is not ESN, but used here to explain the population dynamics may be regarded as an analogue of ESN). Then, the reservoir states, X t , are used to train readout weights by a ridge regression. We used the following parameter values that were determined by a grid search of r x , r y , β xy and β yx : r x = 3.0, r y = 2.7, β xy = -0.2, β xy = 0.2. The sparsity (i.e. the proportion of 0 in the matrix elements) of W in (= a matrix that transforms the input value) was set to 0. Matrix elements of the input weight were chosen from a uniform random distribution [-1,1], and they were multiplied by 0.3 to adjust the influence of the input vectors. Readout was trained using a ridge regression (λ for the regularization = 0.05). For detailed information of the parameters, see electronic supplementary material, tables S1 and S2. Also, see electronic supplementary material, figure S1a for an example of the dynamics. This framework enables transforming a traditional ecological population model into an RC system. We used Lorenz attractor as an input, u t , and this simple reservoir predicts one-time-step future of the chaotic time series better than a simple ridge regression (electronic supplementary material, figure S1b-e).

In silico ERC: empirical ecological time series and scenario exploration
In in silico ERC, an empirical ecological time series is used to simulate an ecological system's responses. For this purpose, we developed a framework based on SSR, a method to reconstruct an original dynamics from a single time series [15,22] (figure 2a). In Deyle et al. [21], a numerical method to predict an ecosystem's responses to external forces called 'scenario exploration' was proposed based on SSR (figure 2a). We used the scenario exploration to simulate an ecosystem's response to hypothetical inputs and used the simulated responses as reservoir states. In other words, rules/mechanisms governing the reconstructed ecosystem dynamics were used as a computational resource. The scenario exploration-based in silico ERC is fully explained in electronic supplementary material, methods and visually explained in figure 2a and electronic supplementary material, appendix I. The in silico ecological reservoir shows ESP and possesses a specific memory capacity (figure 3 and electronic supplementary material, methods).
The performance of in silico ERC was evaluated using three tasks: prediction of chaotic dynamics, emulation of nonlinear autoregression moving average (named NARMA2, 3, 4, 5 and 10; eqns. S3-S5 in electronic supplementary material were used to generate NARMA2, 3-5 and 10, respectively) time series [35], and generation of an autonomous system (Mackey-Glass equation). All parameters used in the tasks are described in electronic supplementary material, methods and tables S1 and S2.
The performance of in silico ERC was compared with that of ESN. ESN was implemented following equation (2.1) and Jaeger [7,16], and detailed parameters are described in electronic supplementary material, tables S1 and S2. The reservoir size (N = 2000) was chosen because it showed the highest performance based on our preliminary analysis that tested the effects of reservoir size on the performance (i.e. reservoir sizes from 20 to 2000 were tested). The implementation of in silico ERC can be found in the '01_ERCinsilico' folder at the Github repository.
royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 10: 221614 13 4.4. Real-time ERC: a target unicellular microbe In real-time ERC, real-time ecological dynamics is used as a reservoir. In the present study, the population dynamics of Tetrahymena thermophila was used as a reservoir. Tetrahymena thermophila (hereafter, Tetrahymena) is a unicellular, eukaryotic organism that belongs to the ciliates [26]. Tetrahymena is commonly found in a freshwater ecosystem, and is widely used as a model organism in molecular biology studies. Tetrahymena can easily be cultured using a wide variety of media, chambers and conditions, and its doubling time is ca 2 h under optimal conditions [26]. More detailed physiological characteristics are shown in electronic supplementary material, figures S3 and S4. Experimental conditions to incubate and maintain Tetrahymena are described in the subsection 'A target unicellular organism for real-time ERC: T. thermophila' in electronic supplementary material, methods.

Real-time ERC: Tetrahymena population dynamics as a reservoir
Although there is only a single species in the system, the population dynamics at the bottom of the aluminium chamber is a result of complex interactions among biotic and abiotic factors such as medium temperature, cell physiological states, cell-cell interactions and behaviours. Indeed, previous studies demonstrated that Tetrahymena population dynamics and behaviour may be influenced by temperature and medium concentrations, and complex nonlinear interactions seem to govern the dynamics and behaviour [27][28][29]. These studies imply that complex, but deterministic, nonlinear interactions drive the population dynamics of Tetrahymena. Specifically, the cell dynamics can be formulated as ð4:3Þ where W in determines how the effects of temperature (a scalar value), u t , propagate to the population dynamics and f tetra determines how temperature influence (W in u t ) and population density captured at the bottom of the chamber (X t ) interact in the chamber. Importantly, we do not know exact formulations of f tetra and W in , but we can still use this system for RC if f tetra and W in are time-invariant.

Real-time ERC: experimental system to monitor Tetrahymena population dynamics
The computational capability of RC positively correlates with the reservoir size, and to increase the reservoir size, we used three concentrations of modified Neff medium (see electronic supplementary material, methods) in the experiments ('low nutrient,' 'med. nutrient' and 'high nutrient'; see electronic supplementary material, methods). The Tetrahymena population in the medium was incubated in an aluminium chamber (figure 5a-d), and the temperature inside the aluminium chamber was automatically regulated using a custom temperature regulator system (E5CC; OMRON, Kyoto, Japan). A user can set a maximum of 256 consecutive temperature values at flexible time intervals. Medium temperatures were changed every 5 min because it took some time to change the medium temperature, and thus the total incubation time for each experiment was 256 time steps × 5 min = 1280 min. The medium temperature was also monitored every minute using a temperature logger/sensor (Ondotori TR-52i; T&D, Matsumoto, Japan). During the incubation, images of the Tetrahymena population at the bottom of the aluminium chamber were taken every minute, resulting in 1280 images for each run. Experimental conditions during the monitoring are described in the subsection 'A target unicellular organism for real-time ERC: T. thermophila' in electronic supplementary material, methods.

4.7.
Real-time ERC: monitoring Tetrahymena population dynamics, preprocessing the cell count data and reservoir state multiplexing In the experiment, we used four time series as inputs, u t : (i) uniform random, (ii) Lorenz attractor, (iii) empirical fish-catch time series (flatfish; Paralichthys olivaceus), and (iv) empirical fish-catch time series (Japanese jack mackerel; Trachurus japonicus). The first one was used to quantify the memory capacity of the ecological reservoir ( figure 7), and the other three were used to test the predictive capability of the Tetrahymena reservoir ( figure 8).
Responses of the Tetrahymena population to changing medium temperatures were monitored semiautomatically using a time-lapse camera (figure 5) and the number of Tetrahymena cells was counted using a standard particle analysis (electronic supplementary material, figure S5). We preprocessed royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 10: 221614 the raw data so that the time series is stationary and unbiased (electronic supplementary material, figure  S5). Then, time-and space-multiplexing techniques were applied to increase the reservoir size (see electronic supplementary material, appendices II and III for visualized explanation of the time-and space-multiplexing). For explanation purposes, we name the reservoir state S i,j t , where i, j and t indicate a nutrient condition ('l', 'm' and 'h' denote low, medium and high, respectively), a replicate of the experiment (1 or 2), and time step, respectively. For example, S m,1 t indicates the reservoir state taken from the first run of the medium nutrient concentration (4% modified Neff ). For the quantification of memory capacity (uniform random value inputs), {S l,1