mechanism of fading memory [306]. As stated in [2], these experiments show
that “ (1) the information about a particular stimulus persists in the
activity of the network for up to a second after the end of the stimulus
(fading memory). (2) Two subsequent stimuli and the order of their
presentation can be correctly classified with a linear classifier sometime
after the end of the second stimulus, suggesting that the network is capable
of performing non-linear XOR operations and (3) Stimulus identity is
distributed across many neurons ($>$30) and encoded both in the rate vector
and the temporal correlation structure of the responses”. The above evidence
may explain the fading memory from the perspective of neuronal oscillations,
suggesting that short-term memory is not only a property of the networks, but
also a consequence of the oscillations and reverberations.

In addition, long-term memory is also important, and it is a more complex one.
Early experiments showed that the “default” state of the unperturbed, sleeping
brain is a complex system of numerous self-governed oscillations, particularly
in the thalamocortical system [301, 305]. The content of these oscillations
reflects spike sequence patterns created by prior waking experience. Moreover,
these oscillations are spontaneously replayed (e.g., during sleeping), leading
to an “off-line” synaptic modification. Such replays might be the way to the
formation of long-term memory. Overall, a reason why short-term memory rather
than long-term memory is one of the necessary requirements for building a RC
model might be that we usually keep the random connections in the reservoir
fixed without modifications. As a result, RC models generally struggle to form
long-term memory, since replays in terms of synaptic modifications are
required (i.e., investigating long-term memory is a more challenging task). In
fact, some learning algorithms, such as STDP [307] and FORCE learning [35],
are trying to modify the synaptic connections, and therefore they are likely
to be able to possess long-term memory.

####  6.2.5 Examples of Synchronization in RC

Multiple reservoirs. Deep reservoir computing was proposed in [49], in which
multiple reservoirs are concatenated together to form a hierarchical network
structure. Detailed model descriptions are discussed in Section 4. It has been
proved that the deep RC structure can achieve (1) multiple timescale
representation, ordered along the network’s hierarchy; (2) multiple frequency
representation, where progressively higher layers focus on progressively lower
frequencies. According to the Arnold tongue regime shown in Fig. 17, if we
keep the coupling strengths at a low level (week synaptic links, or even zero
weight connections), neurons with similar preferred frequencies can be
synchronized, at different frequency bands. Therefore, this may explain the
reason why multiple reservoirs can achieve these while the conventional ESN
cannot, from the perspective of neuronal oscillations.

GACTRNN. Gating Adaptive Continuous Time Recurrent Neural Network (GACTRNN),
is another research taking the inspirations from neuronal oscillation proposed
in machine learning by [63] in 2020. It extended the classic RNN to adaptive
timescales RNN, which shares some similarities to reservoir computing models.
GACTRNN is claimed to be able to learn to gate its timescale characteristic
during activation and thus dynamically change the timescales in processing
sequences; in other words, by changing their timescales during processing,
neurons can learn to simultaneously represent temporally different primitives
(Fig. 18A).

Small-world topology. An ESN based on the topology of small-world (SW) wiring
was proposed by [96] (Fig. 18B). The model incorporated SW structure with RC
and further investigated echo state property. It was found that the SW
topology plays the roles of both efficient signal propagation and enhancement
of the ESP in neural computation. In fact, this idea partially originated from
the cortical anatomical connectivity of the human brain. According to [301],
“complex brains have developed specialized mechanisms for the grouping of
principal cells into temporal coalitions”. In order to reduce the complexity
of the connections without excessive wiring, the number of long-range
connections between neurons decreases in growing brains; in other words, the
synaptic path lengths between distant cell assemblies are reduced, keeping the
path lengths short and maintaining fundamental functions.

##  7 Perspectives and Future Research

Reservoir computing is becoming increasingly popular due to its simple network
structure, hardware-friendly features, low computational cost, and fast
training process. These benefits enable RC to extend far beyond machine
learning into a wide range of research fields. In this paper, we provide a
thorough overview of RC’s history, strengths and weaknesses from the
perspectives of machine learning, dynamical systems, physics, biology, and
neuroscience. We also summarize recent advanced approaches and architectures
for RC optimizations and implementations. Besides, applications of RC are
reviewed, from which we have seen how this interdisciplinary idea can be
applied in various research areas.

While RC still remains an unconventional computational framework compared to
other machine learning techniques like deep learning, its impact can be
enhanced by addressing various challenges. Recent developments have unveiled
new directions and perspectives for RC, indicating its untapped potential and
promising prospects that may even surpass those of mainstream methods. In this
section, we present perspectives and discuss the open problems that motivate
further research in this field.

###  7.1 Reservoir Design and Optimization

A consensus view of conventional RC is that initializing a random RNN as a
reservoir is not the optimal solution, and that connecting a linear readout
with the reservoir limits the generation of the downstream responses. It is
also known that neurons in cortical networks in the brain are not randomly
connected, while their structures and synapses exploit an evolutionary and
developmental process [308]. Recent research, especially on ESNs and LSM,
mainly focuses on network structure designs (e.g., deep reservoir), parameter
optimizations (e.g., particle swarm optimization) and training rule
determinations (e.g., STDP and Hebbian learning). Even if the optimal synaptic
weights were discovered, the performance of various concrete tasks would still
vary. As pointed out by Jaeger [45], “currently available insights are mostly
distilled from experimental studies of timescale profiles or frequency spectra
in input data and provide no comprehensive guides for optimizing reservoir
designs”. In other words, one should find a way to analyze and abstract both
the characteristics of input/output and task specifications, which can be used
to design the reservoir dynamics. One possible solution is called reservoir
Learning-to-learn (L2L) [308], in which a set of (hyper)parameters of the
reservoir are optimized by BPTT for a whole family of learning tasks; note
that this shares some similarity to meta learning in machine learning and
neuroscience. This L2L method was investigated on LSM models and it can also
be implemented by other RC architectures. Moreover, to have better and faster
learning, it is possible to train the reservoir by the L2L method even without
changing synaptic weights to readout neurons. Nevertheless, whether the
performance takes advantage of other reservoirs is still an open question.

###  7.2 Easy-access Tools, Coding frameworks and Recipes

One pressing open problem in RC is the relative lack of user-friendly coding
environments, libraries, and computation frameworks. Unlike the well-developed
infrastructure supporting deep learning and other scientific computational
paradigms, the coding ecosystem for RC remains relatively underdeveloped and
fragmented. As shown in this paper that although numerous models and
architectures have been proposed for RC, there is a notable dearth of unified
frameworks that researchers can leverage with ease. This poses a significant
challenge as it lowers the speed and efficiency of research, requiring
additional time and effort to navigate through a variety of individual tools
and frameworks, and often necessitates the development of custom code for each
research project.

However, this does not imply that no progress is being made towards building a
more unified and accessible coding infrastructure for RC. Indeed, we have seen
some promising developments over the years. Back in the early 2010s, [33]
provided a comprehensive guide on how to implement an ESN, which widely
impacts future studies. The author also released a demonstration of coding ESN
from scratch in Julia, Matlab, Octave, Python, and R language. In 2012, a
toolbox was developed for RC called Oger (OrGanic Environment for Reservoir
computing) to train and evaluate recurrent neural networks, particularly ESNs
and LSMs [309, 310]. From 2017 on, several tools were released in terms of
different RC models, such as ESNs, LSMs, and FORCE-based algorithms. The
easyesn library was released [311, 312], providing a more easy-to-use API for
automatic gradient based hyperparameter tuning (of ridge regression penalty,
spectral radius, leaking rate and feedback scaling), as well as transient time
estimation. Meanwhile, a hands-on LSM implementation using NEST simulator in
Python was proposed [97, 98], which is considered to be the starting point for
new researchers who are interested in spiking-based RC models. Later, another
open-source spiking model framework, Nengo, was developed for FORCE learning
and its variation implementations [81]. In 2018, EchoTorch was proposed, and
perhaps it is the first Python package to simplify the evaluation and
implementation of ESNs and RC [313]. In 2019, a Matlab toolbox for DeepESNs
[49] was released that extends the RC paradigm towards deep networks. One of
the most common deep learning frameworks, TensorFlow, also supports the ESN
layer in 2020 (see TF Addons). More recently, in 2022, [82] present tension,
an object-oriented, open-source Python package that implements a TensorFlow /
Keras API for FORCE learning. Another Julia package for RC is
ReservoirComputing.jl [314]. It aims to provide a simple and flexible
framework to work with ESNs and other models. Additionally, [315] present a
Python library that facilitates the creation of RC architectures, from ESNs
and FORCE learning, to complex networks such as DeepESNs and other advanced
architectures with complex connectivity between multiple reservoirs with
feedback loops.

Despite progress, the full potential of RC is yet to be realized, and the goal
of a unified and accessible environment for RC still eludes us. There is a
need for more work in this area to ensure that the full potential of RC can be
explored and utilized effectively.

Figure 19:  Self-oscillatory echo state network (SO-ESN) generating
oscillations to reproduce desired waveforms without any inputs (figure from
[316], with permission). The figure shows the ratio (%) of reservoir that
triggers oscillations with regard to leaking rate (x-axis) and spectral radius
(y-axis). A stable synchronization by oscillation can be seen in (E), in which
neurons 0 and 1 were randomly selected from the population.

###  7.3 Physical RC and Extremely Efficient Hardware

As mentioned earlier, RC has reawakened and gained attention because of the
fast development of PRC designs. Unlike conventional RCs that suffer from the
information processing speed limit, PRCs can overcome this limit and process
massive amounts of data in real-time. We review PRC models that use different
physical materials from different areas such as electronics, optics,
chemistry, and quantum. As more and more PRC approaches are being proposed,
there are several open problems to be solved. For example, some PRCs get rid
of the massive recurrent connections in RNNs, yet they are hard to design and
tune (e.g., implementing a delayed feedback loop of the single node reservoir
is quite a challenging task). Similarly, setting hyperparameters for PRC is
not straightforward. In 2021, [317] proposed a framework to evaluate what
makes a good computing substrate, providing a new perspective on how to build
and compare physical reservoir computers.

Another challenge lies in harnessing the full potential of RC for machine
learning applications and achieving highly efficient hardware implementations.
Very recently, this issue has been explored and discussed in [318], where the
future directions of inquiry are segmented into three categories. Simply put,
the first category focuses on the theoretical aspects of RC, aiming to drive
efficient model design and ensure reliable RC applications. This includes
works like reservoir memory machines [319], consistency capacity [320] and
curve fitting abilities [321] analysis. The second category delves into the
exploration of novel model designs and applications of RC, with an aim to
enhance computational performance and efficiency in tasks related to pattern
recognition. Possible solutions include (1) industrial applications such as
adaptive practical nonlinear model predictive control [322] and digital twins
[323]; (2) integrating RC with deep learning methods such as convolutional and
graph neural networks [324, 325, 326, 262, 263]. Lastly, the third category is
to keep investigating new architectures and mechanisms in physical hardware
that are suitable for RC implementations. Latest research includes a new FPGA-
based RC for low-power pattern recognition [327], networks based on the
Schrödinger equation [328], and a new cellular automata implementing rule
(CA90) [329]. Overall, these studies use a wide range of new hardware,
showcasing efficient RC-based methods and stimulating further growth in this
research domain.

###  7.4 RC with Cognitive Science and Neuroscience

In terms of modelling RC in cognitive science and neuroscience, we have
reviewed several mechanisms of cognitive brains and tried to bridge the
concept of neuron population to RC, such as mixed selectivity [24] which
reveals that the prefrontal cortex can generate high-dimensional mixed
selective dynamics to assure the separability in the downstream readout units.
Future research directions include finding new analogies for biological
characteristics in RC, allowing for a deeper understanding of brain’s and
bodies’ mechanisms.

The domain of oscillation with synchronization, as previously discussed, is
one area warranting further investigation. In fact, progress has been made in
using ESNs to produce oscillatory outputs without any inputs, by mimicking the
central pattern generators (CPGs) shown to be involved in rhythmic human
movement [190, 191, 192, 193]. CPGs are important circuits present in the
neural system of live beings. In fact, vertebrates have a spinal cord composed
of many of CPG circuits, and it was shown that the spinal cord and mostly CPGs
are sufficient for complex locomotion (e.g., walking and running in cats)
[330, 331]. In 2023, Tham and Vargas [316] show that even the most basic ESN
can be trained to reproduce the trajectory of dynamical systems from simple
sinusoidal and square waves to complex Lorenz chaotic time series with high
precision, without any external excitation.  Fig. 19 depicts the different
probabilities of having a reservoir that triggers oscillations in terms of
both leaking rate and spectral radius (i.e., the echo state property, ESP),
where pure yellow represents the highest probability. Here, it is important to
note that the ESP, by design, typically restricts spontaneous oscillation, as
it ensures that the reservoir’s internal state should eventually lose memory
of its initial conditions, hence creating a fading “echo” of past inputs. This
can be seen in the cases of Fig. 19C-D, that the reservoirs’ states converge
to 0 and remain stable afterward (refer to damped oscillation in [316]). In
contrast, as shown in Fig. 19E, oscillations occur when the ESP is most likely
not to be guaranteed. Therefore, this study clearly indicates the necessity
for additional research, particularly regarding the outcomes and implications
when the ESP is not strictly adhered to.

On the other hand, high-dimensionality is another feature shared by the
encoding of neuron populations and RC models. Recent studies have also
proposed other approaches to understanding the role of high-dimensionality in
biological and artificial neural networks. Neural population geometry, for
example, is an approach that provides a useful population-level mechanistic
descriptor underlying task implementation. Here, the geometry of
representation can be represented by high-dimensional neural activity, and it
is further observed that the neural activity lies on lower-dimensional
subspaces, i.e., the so-called intrinsic dimensionality or neural manifolds
[332]. Similar to the reservoir responses, these lower-dimensional subspaces
can then be well-separated by using simple classifiers.

Although all of these remain in an early stage of development, the
investigations look promising for getting deeper insights into both artificial
and natural intelligence.

###  7.5 RC from an Evolutionary Perspective

In addition to artificial intelligence and neuroscience, some researchers are
exploring RC through an evolutionary lens, which brings a fresh angle to the
discussion. Natural systems exhibiting reservoir-like behaviors are prevalent
in nature (e.g., nonlinearities in liquids [11], soft robotic in muscles
[188], electric and chemical dynamics in neural networks [333, 116], and brain
mechanisms [13, 28, 24, 2]). This suggests that such reservoirs might have
evolved as advantageous structures for processing complex information,
similarly to other computational approaches such as feed-forward networks [1],
attractor networks [29], and self-organized maps [334] that have been
influenced by and have influenced biology. As stated by [335], however, the
current evidence is not as compelling as the well-established similarities
between, for example, the structure of the human visual system and deep
convolutional neural networks. From an evolutionary perspective, [335] argue
that although reservoir-like systems may initially emerge due to their
simplicity, their long-term persistence could be hindered by evolutionary
trends towards specialization and scaling. In other word, as the reservoir
evolves to specialize, integrate diverse sensory information, or scale up, the
generalizing properties of the reservoir may become less advantageous compared
to highly specialized circuits. This leads to another question: under what
conditions might reservoirs maintain their original architecture with
redundant dynamics, and when might they evolve towards more specialized
configurations? Overall, understanding the evolutionary constraints is
essential in evaluating its potential and limitations both in biological
systems and engineering applications.

###  7.6 Hybrids and New Foundations

Last but not least, some recent works from 2019 have merged RC with other
systems or paradigms such as deep learning [49, 50, 218, 239, 92, 117, 324,
325, 326, 262, 263]. Moreover, some authors have started investigating
modifying the foundations (e.g., NG-RC [208]). In fact, research on random
networks [336, 96, 252, 337] shows that there is a wide range of methods
taking inspiration from some core mechanics of RC to build novel approaches
that might challenge reservoir computing’s current foundations in the near
future.

## Acknowledgment

This work was supported by JSPS Grant-in-Aid for Challenging Exploratory
Research—Grant Number JP22534665, JST Strategic Basic Research Promotion
Program (AIP Accelerated Research)—Grant Number 22584686, JSPS Research on
Academic Transformation Areas (A)—Grant Number 22572551, JST SPRING—Grant
Number JPMJSP2136.

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