Analysis of Reservoir Computing
Using Oscillator Circuit
Kazuki Yasufuku, Yoko Uwate and Yoshifumi Nishio

2024 IEEE International Symposium on Circuits and Systems (ISCAS) | 979-8-3503-3099-1/24/$31.00 ©2024 IEEE | DOI: 10.1109/ISCAS58744.2024.10558653†Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {yasufuku, uwate, nishio}@ ee.tokushima-u.ac.jp
Abstract— This paper presents an investigation into
networked reservoir computing using interconnected oscillators.
Specifically, the Bonhoeffer-van der Pol (BVP) oscillator is usedas the basis for the reservoir, and its performance is evaluatedin speech recognition tasks. The results reveal that the BVP
reservoir exhibits enhanced fidelity to input waveforms andimproved accuracy in speech recognition compared to the vander Pol (VDP) reservoir. Furthermore, the impact of reservoir
configuration on accuracy is examined by varying the numberof oscillators, the connection probability, and the couplingstrength. The results indicate that accuracy improves as theparameters are increased. These results emphasize theimportance of carefully selecting and setting the dynamics of thephysical system and parameters of the reservoir to achieve
optimal performance.
Keywords— oscillator, reservoir computing, synchronization

I. INTRODUCTION
Recent advancements in AI (Artificial Intelligence),
particularly deep learning, recurrent networks, and largescale language modeling, have led to profound breakthroughs
across various applications such as speech recognition, image
classification, and natural language processing [1]-[5].
However, these conventional AI models face challenges due
to parameter tuning intricacies, learning costs, and electricity
consumption [6]. These demands pose limitations on
deploying AI systems in environments constrained byresources or on edge devices with restricted power andcomputational capacities.
Surprisingly, nature offers impressive examples of efficientcomputational systems. For instance, insects, despite theirsmall brains, demonstrate the ability to navigate intricate
environments, process sensory information, and display
intelligent behaviors using limited computational resources[7]. This raises the question of whether there are alternative
computational paradigms that can achieve high performance
while being energy-efficient and resource-friendly.
Reservoir computing (RC) is one such paradigm. It is amethod attracting significant attention in the AI domain,
promising efficient and precise computations while reducing
traditional AI models' energy and computational needs. RC
has its origins in echo-state networks, a special model ofrecurrent neural networks [8], [9] and liquid-state machines
based on neuroscientific findings [10]. Unlike conventional
neural networks, which require extensive parameter tuning, it
mainly requires only learning of the readout layer due to its
inherent dynamics of nonlinear projection of inputs into ahigh-dimensional space. This unique feature echo-state
simplifies learning while still delivering effective
performance. More interestingly, RC is not limited to neuralnetworks, but has been implemented using a variety ofphysical systems, including water systems, optoelectronic
devices, and photonic systems, and has attracted muchattention in recent years [11]-[13]. While RC has exhibited
promising results, our understanding of its potential and the
optimal architectures for maximizing computational accuracy
still have gaps. Future research should focus on exploring
innovative reservoir designs, enhancing training algorithms,
and investigating the fusion of reservoir computing with other
AI techniques.
This study delves into networked reservoir computing using
oscillators as computational elements. Coupled oscillator
models are not only engineering but also fundamental inmodeling and analyzing the synchronization behavior ofsystems with rhythmic behavior, including systems inecology, and neuroscience [14]-[17]. Among these, thesynchronization phenomena of van der Pol oscillators, simple
yet versatile coupled systems of nonlinear oscillators, havegarnered considerable attention. This system canapproximate various natural phenomena, enhancing its utility.
FitzHugh and Nagumo in 1962 derived the BVP model as asimplified version of the Hodgkin-Huxley equation. The
BVP equation, considered a reasonable extension of the VDP
equation, can be realized in a circuit with simple passive
elements and a single nonlinear conductor. Its simplicity
often positions it as a fundamental oscillation unit to explain
the behavior of the nervous system and circadian rhythms[18]-[20].
Interestingly, compared to VDP, BVP exhibit a wide arrayof nonlinear phenomena, including limit cycles, equilibrium
bifurcations, and hard oscillations. Therefore, we leverage
the unique dynamics of oscillators for computational tasks
and employ their synchronization properties for information
processing. This study contrasts two different circuit
configurations, VDP and BVP, as reservoirs to verify theirperformance in achieving superior calculation accuracy.
II. RESERVOIR COMPUTING WITH OSCILLATOR CIRCUITIn this study, we present a novel approach for reservoir
computing utilizing a networked configuration ofinterconnected oscillators. The proposed methodology
harnesses the unique dynamics of oscillators to enableefficient and accurate computations. Fig. 1 provides an
overview of the structure of the networked reservoir
computing system.

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𝑔1𝑣=√ 𝑥 ,
𝑔3

Fig.1. Configuration of the proposed oscillator reservoir.
To enable effective information processing, input signals
are introduced into the oscillator network and propagatedthrough the interconnections between the oscillators. The
resulting output signals are obtained from multiple output
terminals positioned within the network. Input signals are
input by adjusting the strength of the connection between the
input node and the reservoir layer. Data input to the oscillator
network is mapped to a high-dimensional feature space tofacilitate pattern recognition in readout. The oscillator
network used in this study exhibits key properties that makeit well-suited for reservoir computing. Firstly, it possesses
nonlinear characteristics, enabling the network to capture andprocess complex input patterns. Additionally, the network
operates in a high-dimensional space, enhancing its
computational capacity and enabling the representation of
intricate relationships within the data. Lastly, the network
demonstrates short-term memory properties, enabling it toretain and utilize past information to inform current
computations. It is important to note that the couplingstrength within the reservoir layer is fixed throughout thecomputations. We also conducted a comparative analysis toevaluate the impact of coupling strength on the network's
performance. Details of this analysis will be discussed laterin the paper. The output signals of the network are obtainedby measuring the voltage differences between each oscillator
and a reference oscillator.
Two different circuit configurations were utilized to assess
the effectiveness of the proposed networked reservoir
computing approach. The circuit models used in this study
are shown below in Fig. 2.

𝑔1 𝐶
𝑖=√
𝑦 , 𝑡 = √𝐿𝐶𝜏𝑔3 𝐿

𝜀 = 𝑔1 √

{

(2)

𝐿
1 𝐿
, 𝛾= √
𝐶
𝑅 𝐶
𝑁𝑑𝑥𝑛= 𝜀𝑥𝑛 (1 − 𝑥𝑛 2 ) − 𝑦𝑛 − ∑ 𝐾(𝑥𝑘 − 𝑥𝑛 )
𝑑𝜏
𝑘=1

(3)

𝑑𝑦𝑛
= 𝑥𝑛
𝑑𝜏

{

In the computer simulations, we assume that the voltage
and current characteristics of the nonlinear resistor in each
oscillator are given as follows.
(4)
𝑖𝑔 = −𝑔1 𝑣 + 𝑔3 𝑣 3(𝑔1 , 𝑔3 > 0).
The circuit equation for the BVP circuit is shown in Eq. (5).
𝑁

𝑑𝑣𝐴𝑛
1
𝐶
= −𝑖𝐿 − 𝑖𝐺 − ∑ (𝑣𝐴𝑛 − 𝑣𝐴𝑘 )
𝑑𝑡
𝑅
𝑘=1

𝑑𝑣𝐵𝑛
1
𝐶
= 𝑖𝐿 − 𝑣𝐵𝑑𝑡𝑟𝑑 𝑖𝐿 𝑛𝐶= 𝑣𝐴 − 𝑣𝐵
𝑑𝑡

(5)

{
R represents the resistance value connecting the circuits.
Normalize the circuit equations using the normalization
parameters as in VDP. The normalization parameters and thenormalized circuit equations are shown in Eqs. (6), (7).

𝑣𝐴 = √

𝑔1𝑥 ,𝑔3

𝑣𝐵 = √

𝑔1
𝑦 ,
𝑔3

𝐿
1 𝐿
𝜀 = 𝑔1 √ , 𝐾 = √ ,
𝐶
𝑅 𝐶
{

𝑔1 𝐶
𝑖𝐿 = √
𝑧 ,
𝑔3 𝐿

(6)

1 𝐿
𝜎 = √ , 𝑡 = √𝐿𝐶𝜏
𝑟 𝐶
𝑁

𝑑𝑥𝑛
= 𝜀𝑥𝑛 (1 − 𝑥𝑛 2 ) − 𝑧𝑛 + ∑ 𝐾(𝑥𝑘 − 𝑥𝑛 )
𝑑𝜏
𝑘=1

{

Fig.2. Circuit model (a)VDP. (b)BVP.
First, the circuit equation for the VDP circuit is shown inEquation (1).
𝑁
𝑑𝑣𝑛
1
𝐶
= 𝑖𝐿 − 𝑖𝐺 − ∑ (𝑣𝑘 − 𝑣𝑛 )
𝑑𝑡
𝑅
(1)
𝑘=1

𝑑𝑖𝑛
𝐿
= 𝑣𝑛
𝑑𝑡

{
R represents the resistance value connecting the circuits.
Normalize the circuit equations using the normalization
parameters. The normalization parameters and thenormalized circuit equations are shown in Eqs. (2), (3).

𝑑𝑦𝑛
= 𝑧𝑛 − 𝜎𝑦𝑛𝑑𝜏𝑑𝑧𝑛= 𝑥𝑛 − 𝑦𝑛𝑑𝜏(7)

N is the number of oscillators to be connected to. Also, K is
coupling strength.
The tasks conducted in this study encompassed speech
recognition. Detailed procedures for these tasks will bepresented in the Results section.
Simulation Method:
For simulation purposes, the Runge-Kutta method was used,enabling accurate and efficient computation of the system
dynamics.
Learning Method:
Ridge regression was used for the learning process. The aim

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was to determine the optimal output weights to achieve desired
performance. The output weights were determined using theequation:(8)
Ŵout = D𝑋 𝑇 (X 𝑋 𝑇 + βI)−1
Here, Ŵout refers to the output weights, 𝐷 represents theteacher signal matrix, 𝑋 is the output matrix of the reservoir,
and 𝑋 † = 𝑋 𝑇 (X𝑋 𝑇 )−1 denotes the pseudo-inverse of 𝑋. The
parameter 𝛽 > 0 is the regularization parameter, and 𝐼 denotes
the unit matrix. By including 𝛽 in the linear regression,
overlearning is prevented.
For the evaluation of the speech recognition task, the Word
Error Rate (WER) metric was used. WER is a measure thatquantifies the percentage of misclassifications in therecognized speech output compared to the ground truth. A
lower WER indicates a higher level of accuracy in the speech
recognition task.
III. SIMULATION RESULTS
A. Speech Recognition Task
In the subsequent step, a speech recognition task was
conducted. An overview diagram is shown in Fig. 3.

Fig.3. Speech Recognition Overview.
For the input dataset, a total of 500 data samples were
utilized, consisting of recordings from five different speakers
enunciating each of the numbers from zero to nine, repeated
10 times. The input data used in this study is adapted from
Lyon's auditory model. After removing the silent parts, thespeech signal is transformed into a cochleargram
representation. This conversion results in time series datawith dimensions corresponding to different frequency
channels. In our case, the cochleargram data is characterized
by 77 dimensions, reflecting the spectral characteristics of the
speech signal. The dataset was divided such that half of thedata, encompassing five pronunciations of each number
spoken by each of the five speakers, was allocated for training
purposes, while the remaining data was reserved for testing[21]. Similar to the input waveform generation task, the initialtransient response of 10,000τ was disregarded in this task as
well. Since the available data was limited, no data was fed
into the reservoir during the initial 10,000τ period, and thetraining and test data were introduced from 10,001τ onwards.
The training process involved observing the output of the
VDP and BVP reservoirs, as shown in Fig. 4, respectively.

Fig.4. Output during training of the reservoir. (a) VDP. (b)
BVP. The reservoir consists of N = 100, with a connection
probability of 1.0 (indicating perfect coupling) in thereservoir layer. As parameter values, ε=0.1 and K=0.03 wereused for the reservoir layer. Only 15 responses are listed forvisibility.
The ridge regularization parameter β was set to 1.0 for
verification purposes. The outcomes of the speech
recognition task are presented in Figs. 5 and 6. Also, therespective training and test errors (WER) are also shown in
TABLE Ⅰ.

Fig. 5. Confusion Matrix for Speech Recognition using VDP
Reservoir (a) Training Prediction (b) Test Prediction.

Fig. 6. Confusion Matrix for Speech Recognition using BVPReservoir (a) Training Prediction (b) Test Prediction
TABLE I.

train
test

ERROR EVALUATION OF SPEECH RECOGNITION TASKVDP AND BVP (WER)
without reservoir
0.72800.7320

VDP
0.000
0.044

BVD
0.012
0.06

Figure 4 demonstrates the input of the speech waveform at
the 10000τ stage, and a comparison is made between thereservoir outputs of the VDP and BVP circuits. It is observedthat the reservoir output of the VDP circuit does not fullyreflect the input waveform, while the BVP circuit exhibits abetter correspondence between the reservoir output and the
input waveform. This difference in waveform representation
is believed to have a direct impact on the accuracy of speech

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recognition. This has also been confirmed in reservoirs using
spin torque oscillators, and it has been confirmed thatmicrowave application provides a quick response (fastrelaxation) and improves reservoir performance [22].
Furthermore, Figs. 5, 6 and TABLE Ⅰprovide clear evidenceof the disparity in speech recognition results. Despite thesimilarity in experimental conditions, except for thedifferences in circuit configurations, a significant
discrepancy in accuracy is observed. This indicates that the
choice of the circuit has a profound influence on the overall
performance and accuracy of the speech recognition task.
These findings emphasize the importance of selecting theappropriate circuit architecture for specific applications, asdifferent circuit configurations can lead to substantial
variations in the accuracy and performance of cognitive tasks.
Further investigation and analysis are necessary tounderstand the underlying factors contributing to thesedifferences and optimize circuit designs for specificcomputational tasks. While the flexibility in material
selection for reservoir computing holds great appeal, ourresults highlight the importance of meticulous material
selection tailored to the specific task. This strategic selectionof suitable materials in reservoir computing has the potential
to significantly improve accuracy and overall performance inapplications.
B. Effect of Reservoir Configuration
This section presents a comprehensive investigation into theaccuracy of oscillator reservoirs, with a particular focus on
their structural configurations. The reservoir utilized in thisstudy adopts a network-type structure, offering distinct
advantages compared to other types such as medium-based
reservoirs (e.g., water [11]) or soft material reservoirs (e.g.,octopus’ legs [23]). Specifically, the network structure allows
for more precise design control over the reservoir layer.
Building upon the BVP used in the previous chapter, we
subjected the reservoir to a speech recognition task.
The primary objective of this investigation was to assess
how variations in the reservoir configuration impact accuracy.
To achieve this, we examined the accuracy of speech
recognition while manipulating key parameters such as thenumber of oscillators N, coupling strength K, and coupling
probability P within the reservoir layer. The following
section presents the results obtained from these experiments,
shedding light on the relationship between reservoir
configuration and accuracy.
The results obtained from the experiments reveal important
insights into the relationship between reservoir configuration
and accuracy. Fig. 7 (a) shows that as N increases, theaccuracy of the reservoir improves. This trend is particularly
evident, showing significant enhancements in accuracy up toa certain point, approximately N=125. Additionally, Fig. 8 (b)highlights the influence of connection probability on
accuracy during both the learning and verification stages. It
demonstrates that a higher connection probability leads toimproved accuracy in the reservoir's performance.
Furthermore, Fig. 7 (c) demonstrates that increasing K also
contributes to enhanced accuracy, albeit to a lesser extent
compared to the impact of N and P. These findings
underscore the significance of reservoir configuration inachieving optimal accuracy levels for the speech recognition
task.

Fig. 7. Accuracy of the speech recognition task in BVP
reservoirs with different configurations. We take the averageof the three times WERs. (a) Variation of the number ofoscillators N, (b) Variation of the coupling probability P, (c)variation of the coupling strength K.
C. Overall Assessment of Results
Using BVP instead of VDP improved accuracy in the task.
We consider there are two reasons for this. One reason, as
mentioned earlier, is that BVP exhibits a fast response andadequately reproduces the input wave during output.
Another reason is attributed to obtaining responses at theedge of chaos. BVP exhibits both periodic and chaotic
solutions depending on the parameters. This condition aligns
well with the desirable properties of a reservoir that performs
better at the edge of chaos. The results, where accuracy
improved when varying the parameters shown in Fig. 8,
suggest that the circuit's response has approached closer tothe edge of chaos.
IV. CONCLUSIONS
We have investigated the accuracy and performance of
networked reservoirs using interconnected oscillators. The
proposed approach offers advantages in terms of flexible
reservoir layer design compared to other types of reservoirs.
Specifically, we focused on the BVP oscillator as the physicalsystem and conducted the task of speech recognition.
Increasing the number of oscillators (N) and the coupling
probability (P) led to improved accuracy in the speech
recognition task. Although the effect of varying the couplingstrength (K) on accuracy was less pronounced compared to N
and P, it still contributed to improved performance. Thesefindings highlight the importance of careful selection andconfiguration of parameters for achieving higher accuracy inreservoir computing tasks. The flexibility in designing thereservoir structure offers potential for further advancementsin the field. Furthermore, the results from our study
emphasize the significance of considering the structuralcharacteristics of the reservoir when designing efficient andaccurate reservoir computing systems.
Overall, our research contributes to the understanding of
networked reservoirs and their applications in artificial
intelligence. The insights gained from this study can informthe development of more efficient and high-performing
computing systems, paving the way for advancements incognitive computing and machine learning.

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