Kohei Nakajima

Yansong Li, Kai Hu, Kohei Nakajima et al.

Echo state network (ESN), a kind of recurrent neural networks, consists of a fixed reservoir in which neurons are connected randomly and recursively and obtains the desired output only by training output connection weights. First-order reduced and controlled error (FORCE) learning is an online supervised training approach that can change the chaotic activity of ESNs into specified activity patterns. This paper proposes a composite FORCE learning method based on recursive least squares to train ESNs whose initial activity is spontaneously chaotic, where a composite learning technique featured by dynamic regressor extension and memory data exploitation is applied to enhance parameter convergence. The proposed method is applied to a benchmark problem about predicting chaotic time series generated by the Mackey-Glass system, and numerical results have shown that it significantly improves learning and prediction performances compared with existing methods.

Quoc Hoan Tran, Sanjib Ghosh, Kohei Nakajima

Current technologies in quantum-based communications bring a new integration of quantum data with classical data for hybrid processing. However, the frameworks of these technologies are restricted to a single classical or quantum task, which limits their flexibility in near-term applications. We propose a quantum reservoir processor to harness quantum dynamics in computational tasks requiring both classical and quantum inputs. This analog processor comprises a network of quantum dots in which quantum data is incident to the network and classical data is encoded via a coherent field exciting the network. We perform a multitasking application of quantum tomography and nonlinear equalization of classical channels. Interestingly, the tomography can be performed in a closed-loop manner via the feedback control of classical data. Therefore, if the classical input comes from a dynamical system, embedding this system in a closed loop enables hybrid processing even if access to the external classical input is interrupted. Finally, we demonstrate preparing quantum depolarizing channels as a novel quantum machine learning technique for quantum data processing.

Tomoyuki Kubota, Yudai Suzuki, Shumpei Kobayashi et al.

Quantum computing has been moving from a theoretical phase to practical one, presenting daunting challenges in implementing physical qubits, which are subjected to noises from the surrounding environment. These quantum noises are ubiquitous in quantum devices and generate adverse effects in the quantum computational model, leading to extensive research on their correction and mitigation techniques. But do these quantum noises always provide disadvantages? We tackle this issue by proposing a framework called quantum noise-induced reservoir computing and show that some abstract quantum noise models can induce useful information processing capabilities for temporal input data. We demonstrate this ability in several typical benchmarks and investigate the information processing capacity to clarify the framework's processing mechanism and memory profile. We verified our perspective by implementing the framework in a number of IBM quantum processors and obtained similar characteristic memory profiles with model analyses. As a surprising result, information processing capacity increased with quantum devices' higher noise levels and error rates. Our study opens up a novel path for diverting useful information from quantum computer noises into a more sophisticated information processor.

Mitsumasa Nakajima, Katsuma Inoue, Kenji Tanaka et al.

The ever-growing demand for further advances in artificial intelligence motivated research on unconventional computation based on analog physical devices. While such computation devices mimic brain-inspired analog information processing, learning procedures still relies on methods optimized for digital processing such as backpropagation. Here, we present physical deep learning by extending a biologically plausible training algorithm called direct feedback alignment. As the proposed method is based on random projection with arbitrary nonlinear activation, we can train a physical neural network without knowledge about the physical system. In addition, we can emulate and accelerate the computation for this training on a simple and scalable physical system. We demonstrate the proof-of-concept using a hierarchically connected optoelectronic recurrent neural network called deep reservoir computer. By constructing an FPGA-assisted optoelectronic benchtop, we confirmed the potential for accelerated computation with competitive performance on benchmarks. Our results provide practical solutions for the training and acceleration of neuromorphic computation.

Taichi Haruna, Kohei Nakajima

We present an inequality that bounds the short-term memory capability of dynamical systems from below. It can be interpreted as an uncertainty relation between a measure of short-term memory and that of the size of state fluctuations induced by input signals. The lower bound can be achieved by a readout weight and thus represents a suboptimal memory called harmonic memory. We examine analytically and numerically the inequality in a number of reservoir systems subject to input noise. We illustrate cases in which equality is achieved exactly, equality holds asymptotically, and the inequality is strict. We also study the effect of a state-space regularization to elucidate the inequality in terms of the fluctuation structure of the state-space. We find that a certain strength of input noise induces extra memory under the regularization, and we refer to this phenomenon as noise-induced memory. We observe that the memory uncertainty relation does not hold in general for the regularized memory and harmonic memory. This fact is explained in terms of the mechanism of noise-induced memory.

Yongbo Zhang, Katsuma Inoue, Mitsumasa Nakajima et al.

Spiking neural networks (SNNs), the models inspired by the mechanisms of real neurons in the brain, transmit and represent information by employing discrete action potentials or spikes. The sparse, asynchronous properties of information processing make SNNs highly energy efficient, leading to SNNs being promising solutions for implementing neural networks in neuromorphic devices. However, the nondifferentiable nature of SNN neurons makes it a challenge to train them. The current training methods of SNNs that are based on error backpropagation (BP) and precisely designing surrogate gradient are difficult to implement and biologically implausible, hindering the implementation of SNNs on neuromorphic devices. Thus, it is important to train SNNs with a method that is both physically implementatable and biologically plausible. In this paper, we propose using augmented direct feedback alignment (aDFA), a gradient-free approach based on random projection, to train SNNs. This method requires only partial information of the forward process during training, so it is easy to implement and biologically plausible. We systematically demonstrate the feasibility of the proposed aDFA-SNNs scheme, propose its effective working range, and analyze its well-performing settings by employing genetic algorithm. We also analyze the impact of crucial features of SNNs on the scheme, thus demonstrating its superiority and stability over BP and conventional direct feedback alignment. Our scheme can achieve competitive performance without accurate prior knowledge about the utilized system, thus providing a valuable reference for physically training SNNs.

JingChuan Guan, Tomoyuki Kubota, Yasuo Kuniyoshi et al.

The effects of noise on memory in a linear recurrent network are theoretically investigated. Memory is characterized by its ability to store previous inputs in its instantaneous state of network, which receives a correlated or uncorrelated noise. Two major properties are revealed: First, the memory reduced by noise is uniquely determined by the noise's power spectral density (PSD). Second, the memory will not decrease regardless of noise intensity if the PSD is in a certain class of distribution (including power law). The results are verified using the human brain signals, showing good agreement.

Alexandre Pitti, Kohei Nakajima, Yasuo Kuniyoshi

The body morphology plays an important role in the way information is perceived and processed by an agent. We address an information theory (IT) account on how the precision of sensors, the accuracy of motors, their placement, the body geometry, shape the information structure in robots and computational codes. As an original idea, we envision the robot's body as a physical communication channel through which information is conveyed, in and out, despite intrinsic noise and material limitations. Following this, entropy, a measure of information and uncertainty, can be used to maximize the efficiency of robot design and of algorithmic codes per se. This is known as the principle of Entropy Maximization (PEM) introduced in biology by Barlow in 1969. The Shannon's source coding theorem provides then a framework to compare different types of bodies in terms of sensorimotor information. In line with PME, we introduce a special class of efficient codes used in IT that reached the Shannon limits in terms of information capacity for error correction and robustness against noise, and parsimony. These efficient codes, which exploit insightfully quantization and randomness, permit to deal with uncertainty, redundancy and compacity. These features can be used for perception and control in intelligent systems. In various examples and closing discussions, we reflect on the broader implications of our framework that we called Informational Embodiment to motor theory and bio-inspired robotics, touching upon concepts like motor synergies, reservoir computing, and morphological computation. These insights can contribute to a deeper understanding of how information theory intersects with the embodiment of intelligence in both natural and artificial systems.

Lianxiang Cui, Kohei Nakajima, Kazuyuki Aihara

Physical reservoir computing exploits the intrinsic dynamics of physical systems for information processing, while keeping the internal dynamics fixed and training only linear readouts; yet the role of input encoding remains poorly understood. We show that optimal input encoding is a geometric problem governed by the system's fluctuation-response structure. By measuring steady-state fluctuations and linear response, we derive an analytical criterion for the input direction that maximizes task-specific linear memory under a fixed power constraint, termed Response-based Optimal Memory Encoding (ROME). Backpropagation-based encoder optimization is shown to be equivalent to ROME, revealing a trade-off between task-dependent feature mixing and intrinsic noise. We apply ROME to various reservoir platforms, including spin-wave waveguides and spiking neural networks, demonstrating effective encoder design across physical and neuromorphic reservoirs, even in non-differentiable systems.

Thomas Geert de Jong, Hirofumi Notsu, Kohei Nakajima

Nature is pervaded with oscillatory dynamics. In networks of coupled oscillators patterns can arise when the system synchronizes to an external input. Hence, these networks provide processing and memory of input. We present a universal framework for harnessing oscillator networks as computational resource. This computing framework is introduced by the ubiquitous model for phase-locking, the Kuramoto model. We force the Kuramoto model by a nonlinear target-system, then after substituting the target-system with a trained feedback-loop it emulates the target-system. Our results are two-fold. Firstly, the trained network inherits performance properties of the Kuramoto model, where all-to-all coupling is performed in linear time with respect to the number of nodes and parameters for synchronization are abundant. The latter implies that the network is generically successful since the system learns via sychronization. Secondly, the learning capabilities of the oscillator network, which describe a type of collective intelligence, can be explained using Kuramoto model's order parameter. In summary, this work provides the foundation for utilizing nature's oscillator networks as a new class of information processing systems.

Yuichiro Terasaki, Kohei Nakajima

A continuous one-dimensional map with period three includes all periods. This raises the following question: Can we obtain any types of periodic orbits solely by learning three data points? In this paper, we report the answer to be yes. Considering a random neural network in its thermodynamic limit, we first show that almost all learned periods are unstable, and each network has its own characteristic attractors (which can even be untrained ones). The latently acquired dynamics, which are unstable within the trained network, serve as a foundation for the diversity of characteristic attractors and may even lead to the emergence of attractors of all periods after learning. When the neural network interpolation is quadratic, a universal post-learning bifurcation scenario appears, which is consistent with a topological conjugacy between the trained network and the classical logistic map. In addition to universality, we explore specific properties of certain networks, including the singular behavior of the scale of weight at the infinite limit, the finite-size effects, and the symmetry in learning period three.

Ryo Terajima, Katsuma Inoue, Kohei Nakajima et al.

Recent studies have demonstrated that the dynamics of physical systems can be utilized for the desired information processing under the framework of physical reservoir computing (PRC). Robots with soft bodies are examples of such physical systems, and their nonlinear body-environment dynamics can be used to compute and generate the motor signals necessary for the control of their own behavior. In this simulation study, we extend this approach to control and embed not only one but also multiple behaviors into a type of soft robot called a tensegrity robot. The resulting system, consisting of the robot and the environment, is a multistable dynamical system that converges to different attractors from varying initial conditions. Furthermore, attractor analysis reveals that there exist "untrained attractors" in the state space of the system outside the training data. These untrained attractors reflect the intrinsic properties and structures of the tensegrity robot and its interactions with the environment. The impacts of these recent findings in PRC remain unexplored in embodied AI research. We here illustrate their potential to understand various features of embodied cognition that have not been fully addressed to date.

JingChuan Guan, Tomoyuki Kubota, Yasuo Kuniyoshi et al.

State space models (SSMs) have gained attention by showing potential to outperform Transformers. However, previous studies have not sufficiently addressed the mechanisms underlying their high performance owing to a lack of theoretical explanation of SSMs' learning dynamics. In this study, we provide such an explanation and propose an improved training strategy. The memory capacity of SSMs can be evaluated by examining how input time series are stored in their current state. Such an examination reveals a tradeoff between memory accuracy and length, as well as the theoretical equivalence between the structured state space sequence model (S4) and a simplified S4 with diagonal recurrent weights. This theoretical foundation allows us to elucidate the learning dynamics, proving the importance of initial parameters. Our analytical results suggest that successful learning requires the initial memory structure to be the longest possible even if memory accuracy may deteriorate or the gradient lose the teacher information. Experiments on tasks requiring long memory confirmed that extending memory is difficult, emphasizing the importance of initialization. Furthermore, we found that fixing recurrent weights can be more advantageous than adapting them because it achieves comparable or even higher performance with faster convergence. Our results provide a new theoretical foundation for SSMs and potentially offer a novel optimization strategy.

Tempei Kabayama, Motomasa Komuro, Yasuo Kuniyoshi et al.

Reservoir computing can embed attractors into random neural networks (RNNs), generating a ``mirror'' of a target attractor because of its inherent symmetrical constraints. In these RNNs, we report that an attractor-merging crisis accompanied by intermittency emerges simply by adjusting the global parameter. We further reveal its underlying mechanism through a detailed analysis of the phase-space structure and demonstrate that this bifurcation scenario is intrinsic to a general class of RNNs, independent of training data.

Tomoyuki Kubota, Yusuke Imai, Sumito Tsunegi et al.

A physical neural network (PNN) has both the strong potential to solve machine learning tasks and intrinsic physical properties, such as high-speed computation and energy efficiency. Reservoir computing (RC) is an excellent framework for implementing an information processing system with a dynamical system by attaching a trained readout, thus accelerating the wide use of unconventional materials for a PNN. However, RC requires the dynamics to reproducibly respond to input sequence, which limits the type of substance available for building information processors. Here we propose a novel framework called generalized reservoir computing (GRC) by turning this requirement on its head, making conventional RC a special case. Using substances that do not respond the same to identical inputs (e.g., a real spin-torque oscillator), we propose mechanisms aimed at obtaining a reliable output and show that processed inputs in the unconventional substance are retrievable. Finally, we demonstrate that, based on our framework, spatiotemporal chaos, which is thought to be unusable as a computational resource, can be used to emulate complex nonlinear dynamics, including large scale spatiotemporal chaos. Overall, our framework removes the limitation to building an information processing device and opens a path to constructing a computational system using a wider variety of physical dynamics.

Tempei Kabayama, Yasuo Kuniyoshi, Kazuyuki Aihara et al.

Chaotic dynamics are ubiquitous in nature and useful in engineering, but their geometric design can be challenging. Here, we propose a method using reservoir computing to generate chaos with a desired shape by providing a periodic orbit as a template, called a skeleton. We exploit a bifurcation of the reservoir to intentionally induce unsuccessful training of the skeleton, revealing inherent chaos. The emergence of this untrained attractor, resulting from the interaction between the skeleton and the reservoir's intrinsic dynamics, offers a novel semi-supervised framework for designing chaos.

Katsuma Inoue, Soh Ohara, Yasuo Kuniyoshi et al.

Language is an outcome of our complex and dynamic human-interactions and the technique of natural language processing (NLP) is hence built on human linguistic activities. Bidirectional Encoder Representations from Transformers (BERT) has recently gained its popularity by establishing the state-of-the-art scores in several NLP benchmarks. A Lite BERT (ALBERT) is literally characterized as a lightweight version of BERT, in which the number of BERT parameters is reduced by repeatedly applying the same neural network called Transformer's encoder layer. By pre-training the parameters with a massive amount of natural language data, ALBERT can convert input sentences into versatile high-dimensional vectors potentially capable of solving multiple NLP tasks. In that sense, ALBERT can be regarded as a well-designed high-dimensional dynamical system whose operator is the Transformer's encoder, and essential structures of human language are thus expected to be encapsulated in its dynamics. In this study, we investigated the embedded properties of ALBERT to reveal how NLP tasks are effectively solved by exploiting its dynamics. We thereby aimed to explore the nature of human language from the dynamical expressions of the NLP model. Our short-term analysis clarified that the pre-trained model stably yields trajectories with higher dimensionality, which would enhance the expressive capacity required for NLP tasks. Also, our long-term analysis revealed that ALBERT intrinsically shows transient chaos, a typical nonlinear phenomenon showing chaotic dynamics only in its transient, and the pre-trained ALBERT model tends to produce the chaotic trajectory for a significantly longer time period compared to a randomly-initialized one. Our results imply that local chaoticity would contribute to improving NLP performance, uncovering a novel aspect in the role of chaotic dynamics in human language behaviors.

Quoc Hoan Tran, Kohei Nakajima

Quantifying and verifying the control level in preparing a quantum state are central challenges in building quantum devices. The quantum state is characterized from experimental measurements, using a procedure known as tomography, which requires a vast number of resources. Furthermore, the tomography for a quantum device with temporal processing, which is fundamentally different from the standard tomography, has not been formulated. We develop a practical and approximate tomography method using a recurrent machine learning framework for this intriguing situation. The method is based on repeated quantum interactions between a system called quantum reservoir with a stream of quantum states. Measurement data from the reservoir are connected to a linear readout to train a recurrent relation between quantum channels applied to the input stream. We demonstrate our algorithms for quantum learning tasks followed by the proposal of a quantum short-term memory capacity to evaluate the temporal processing ability of near-term quantum devices.

Takahiro Goto, Quoc Hoan Tran, Kohei Nakajima

Encoding classical data into quantum states is considered a quantum feature map to map classical data into a quantum Hilbert space. This feature map provides opportunities to incorporate quantum advantages into machine learning algorithms to be performed on near-term intermediate-scale quantum computers. The crucial idea is using the quantum Hilbert space as a quantum-enhanced feature space in machine learning models. While the quantum feature map has demonstrated its capability when combined with linear classification models in some specific applications, its expressive power from the theoretical perspective remains unknown. We prove that the machine learning models induced from the quantum-enhanced feature space are universal approximators of continuous functions under typical quantum feature maps. We also study the capability of quantum feature maps in the classification of disjoint regions. Our work enables an important theoretical analysis to ensure that machine learning algorithms based on quantum feature maps can handle a broad class of machine learning tasks. In light of this, one can design a quantum machine learning model with more powerful expressivity.

Quoc Hoan Tran, Kohei Nakajima

Quantum reservoir computing (QRC) is an emerging paradigm for harnessing the natural dynamics of quantum systems as computational resources that can be used for temporal machine learning tasks. In the current setup, QRC is difficult to deal with high-dimensional data and has a major drawback of scalability in physical implementations. We propose higher-order QRC, a hybrid quantum-classical framework consisting of multiple but small quantum systems that are mutually communicated via classical connections like linear feedback. By utilizing the advantages of both classical and quantum techniques, our framework enables an efficient implementation to boost the scalability and performance of QRC. Furthermore, higher-order settings allow us to implement a FORCE learning or an innate training scheme, which provides flexibility and high operability to harness high-dimensional quantum dynamics and significantly extends the application domain of QRC. We demonstrate the effectiveness of our framework in emulating large-scale nonlinear dynamical systems, including complex spatiotemporal chaos, which outperforms many of the existing machine learning techniques in certain situations.

Kohei Nakajima

Understanding the fundamental relationships between physics and its information-processing capability has been an active research topic for many years. Physical reservoir computing is a recently introduced framework that allows one to exploit the complex dynamics of physical systems as information-processing devices. This framework is particularly suited for edge computing devices, in which information processing is incorporated at the edge (e.g., into sensors) in a decentralized manner to reduce the adaptation delay caused by data transmission overhead. This paper aims to illustrate the potentials of the framework using examples from soft robotics and to provide a concise overview focusing on the basic motivations for introducing it, which stem from a number of fields, including machine learning, nonlinear dynamical systems, biological science, materials science, and physics.

Katsuma Inoue, Kohei Nakajima, Yasuo Kuniyoshi

Chaotic itinerancy is a frequently observed phenomenon in high-dimensional and nonlinear dynamical systems, and it is characterized by the random transitions among multiple quasi-attractors. Several studies have revealed that chaotic itinerancy has been observed in brain activity, and it is considered to play a critical role in the spontaneous, stable behavior generation of animals. Thus, chaotic itinerancy is a topic of great interest, particularly for neurorobotics researchers who wish to understand and implement autonomous behavioral controls for agents. However, it is generally difficult to gain control over high-dimensional nonlinear dynamical systems. Hence, the implementation of chaotic itinerancy has mainly been accomplished heuristically. In this study, we propose a novel way of implementing chaotic itinerancy reproducibly and at will in a generic high-dimensional chaotic system. In particular, we demonstrate that our method enables us to easily design both the trajectories of quasi-attractors and the transition rules among them simply by adjusting the limited number of system parameters and by utilizing the intrinsic high-dimensional chaos. Finally, we quantitatively discuss the validity and scope of application through the results of several numerical experiments.

Taichi Haruna, Kohei Nakajima

The ability of discrete-time nonlinear recurrent neural networks to store time-varying small input signals is investigated by mean-field theory. The combination of a small input strength and mean-field assumptions makes it possible to derive an approximate expression for the conditional probability density of the state of a neuron given a past input signal. From this conditional probability density, we can analytically calculate short-term memory measures, such as memory capacity, mutual information, and Fisher information, and determine the relationships among these measures, which have not been clarified to date to the best of our knowledge. We show that the network contribution of these short-term memory measures peaks before the edge of chaos, where the dynamics of input-driven networks is stable but corresponding systems without input signals are unstable.

Tomoyuki Kubota, Hirokazu Takahashi, Kohei Nakajima

A dynamical system can be regarded as an information processing apparatus that encodes input streams from the external environment to its state and processes them through state transitions. The information processing capacity (IPC) is an excellent tool that comprehensively evaluates these processed inputs, providing details of unknown information processing in black box systems; however, this measure can be applied to only time-invariant systems. This paper extends the applicable range to time-variant systems and further reveals that the IPC is equivalent to coefficients of polynomial chaos (PC) expansion in more general dynamical systems. To achieve this objective, we tackle three issues. First, we establish a connection between the IPC for time-invariant systems and PC expansion, which is a type of polynomial expansion using orthogonal functions of input history as bases. We prove that the IPC corresponds to the squared norm of the coefficient vector of the basis in the PC expansion. Second, we show that an input following an arbitrary distribution can be used for the IPC, removing previous restrictions to specific input distributions. Third, we extend the conventional orthogonal bases to functions of both time and input history and propose the IPC for time-variant systems. To show the significance of our approach, we demonstrate that our measure can reveal information representations in not only machine learning networks but also a real, cultured neural network. Our generalized measure paves the way for unveiling the information processing capabilities of a wide variety of physical dynamics which has been left behind in nature.

Estefany A. Torres, Kohei Nakajima, Isuru S. Godage

Soft Continuum arms, such as trunk and tentacle robots, can be considered as the "dual" of traditional rigid-bodied robots in terms of manipulability, degrees of freedom, and compliance. Introduced two decades ago, continuum arms have not yet realized their full potential, and largely remain as laboratory curiosities. The reasons for this lag rest upon their inherent physical features such as high compliance which contribute to their complex control problems that no research has yet managed to surmount. Recently, reservoir computing has been suggested as a way to employ the body dynamics as a computational resource toward implementing compliant body control. In this paper, as a first step, we investigate the information processing capability of soft continuum arms. We apply input signals of varying amplitude and bandwidth to a soft continuum arm and generate the dynamic response for a large number of trials. These data is aggregated and used to train the readout weights to implement a reservoir computing scheme. Results demonstrate that the information processing capability varies across input signal bandwidth and amplitude. These preliminary results demonstrate that soft continuum arms have optimal bandwidth and amplitude where one can implement reservoir computing.

Hisashi Iwade, Kohei Nakajima, Takuma Tanaka et al.

The inherent transient dynamics of recurrent neural networks (RNNs) have been exploited as a computational resource in input-driven RNNs. However, the information processing capability varies from RNN to RNN, depending on their properties. Many authors have investigated the dynamics of RNNs and their relevance to the information processing capability. In this study, we present a detailed analysis of the information processing capability of an RNN optimized by recurrent infomax (RI), which is an unsupervised learning scheme that maximizes the mutual information of RNNs by adjusting the connection strengths of the network. Thus, we observe that a delay-line structure emerges from the RI and the network optimized by the RI possesses superior short-term memory, which is the ability to store the temporal information of the input stream in its transient dynamics.

Keisuke Fujii, Kohei Nakajima

Quantum computer has an amazing potential of fast information processing. However, realisation of a digital quantum computer is still a challenging problem requiring highly accurate controls and key application strategies. Here we propose a novel platform, quantum reservoir computing, to solve these issues successfully by exploiting natural quantum dynamics, which is ubiquitous in laboratories nowadays, for machine learning. In this framework, nonlinear dynamics including classical chaos can be universally emulated in quantum systems. A number of numerical experiments show that quantum systems consisting of at most seven qubits possess computational capabilities comparable to conventional recurrent neural networks of 500 nodes. This discovery opens up a new paradigm for information processing with artificial intelligence powered by quantum physics.

Kohei Nakajima, Nico Schmidt, Rolf Pfeifer

Soft robots can exhibit diverse behaviors with simple types of actuation by partially outsourcing control to the morphological and material properties of their soft bodies, which is made possible by the tight coupling between control, body, and environment. In this paper, we present a method that will quantitatively characterize these diverse spatiotemporal dynamics of a soft body based on the information-theoretic approach. In particular, soft bodies have the ability to propagate the effect of actuation through the entire body, with a certain time delay, due to their elasticity. Our goal is to capture this delayed interaction in a quantitative manner based on a measure called momentary information transfer. We extend this measure to soft robotic applications and demonstrate its power using a physical soft robotic platform inspired by the octopus. Our approach is illustrated in two ways. First, we statistically characterize the delayed actuation propagation through the body as a strength of information transfer. Second, we capture this information propagation directly as local information dynamics. As a result, we show that our approach can successfully characterize the spatiotemporal dynamics of the soft robotic platform, explicitly visualizing how information transfers through the entire body with delays. Further extension scenarios of our approach are discussed for soft robotic applications in general.