Rotating neurons for all-analog implementation of cyclic reservoir computing

Abstract

Hardware implementation in resource-efficient reservoir computing is of great interest for neuromorphic engineering. Recently, various devices have been explored to implement hardware-based reservoirs. However, most studies were mainly focused on the reservoir layer, whereas an end-to-end reservoir architecture has yet to be developed. Here, we propose a versatile method for implementing cyclic reservoirs using rotating elements integrated with signal-driven dynamic neurons, whose equivalence to standard cyclic reservoir algorithm is mathematically proven. Simulations show that the rotating neuron reservoir achieves record-low errors in a nonlinear system approximation benchmark. Furthermore, a hardware prototype was developed for near-sensor computing, chaotic time-series prediction and handwriting classification. By integrating a memristor array as a fully-connected output layer, the all-analog reservoir computing system achieves 94.0% accuracy, while simulation shows >1000× lower system-level power than prior works. Therefore, our work demonstrates an elegant rotation-based architecture that explores hardware physics as computational resources for high-performance reservoir computing.

Introduction

Reservoir computing is a bioinspired machine learning paradigm introduced in the early 21st century^1,2,3. The randomly and recurrently connected nonlinear nodes in the reservoir layer provide efficient implementation platforms for recurrent neural networks with low training costs (Fig. 1a). In principle, the complex dynamics generated by the reservoir nonlinearly map the input data to spatiotemporal state patterns in a high-dimensional feature space, where the state vectors of different classes can be linearly separated^1,4. Furthermore, reservoir computing is a powerful approach for processing temporal signals due to the recurrent connections that create dependencies between current and past neuron states, which is also known as short-term memory or fading memory^2,5. In particular, reservoir computing has demonstrated excellent performance in complex time-series prediction and classification tasks^4,6.

Fig. 1: Reservoir computing architectures.

a A conventional reservoir computing architecture with random connections. b A simplified version of a reservoir, also known as a cyclic reservoir. The randomly connected neurons are replaced with a ring structure. c Illustration of the working principle of the proposed rotating neuron reservoir (RNR) that can be physically implemented. The input weights are uniformly distributed in the range of [−1, 1], and a pre-neuron rotor sends the signal to different neuron channels at different time steps. After flowing through the dynamic neurons, the signal is sent to different state channels via another post-neuron rotor, and the final states are read out through a fully connected layer and used in training. d Sketch of the working principle for the case of three neurons, where R denotes the rotation matrix. The legend for all subfigures is provided at the bottom.

Given the potential of reservoir computing, exploring physical dynamics as computational resources of reservoirs for highly efficient information processing has received considerable research attention in recent years. In 2011, a pioneer study⁷ introduced a delay-based reservoir and the concept of virtual nodes into a physical implementation of a cyclic reservoir (CR), as shown in Fig. 1b which is a simplified reservoir without performance degradation⁵. This compelling finding provided an effective method for performing hardware-based reservoir computing, making it an attractive candidate in the field of neuromorphic computing. In follow-up studies, various emerging devices and systems were investigated as physical reservoirs⁸, and they included spintronic devices⁹, photonic devices^{10,11,12,13,14}, quantum devices¹⁵, memristive devices^16,17,18, nanowire networks¹⁹, and even soft robotic arms²⁰. However, the main drawbacks associated with the use of delayed feedback and time-multiplexing are as follows: (i) delayed feedback is costly for hardware implementations using conventional complementary metal–oxide–semiconductor (CMOS) technology or optical approaches, which require additional digital components^7,21, such as analog-to-digital converters (ADCs) and random-access memory, or bulky optical fibers^10,11,22,23, respectively; (ii) in the absence of a delayed feedback line, a reservoir computing system cannot simultaneously maintain an appropriate memory capacity (MC) or satisfactory state richness. For example, previous research revealed that shortening the step size in time multiplexing could improve the MC but at the cost of reducing the state richness, or vice versa¹⁶. (iii) The serial operations in time multiplexing increase system complexity and latency for both input and readout, whereas parallel computing, which enhances the throughput, is more desirable in neuromorphic computing²⁴. These obstacles hinder further reductions in power and size when the cost for an entire reservoir computer, from the signal input to the computing output, is considered; thus, a knowledge gap associated with massive deployment in practical applications remains. There is an urgent need to develop a new architecture involving hardware-based reservoir computers of miniature size with low power consumption and high capability for large-scale integration^8,25.

In this work, we propose a rotating neuron-based architecture for physically performing reservoir computing in a more intuitive way, namely rotating neurons reservoir (RNR), whose rotation behavior matches with the neurons update in a CR, as rigorously proven through mathematical derivations. Compared with the existing implementations in reservoir computing^{17,19,20,21,23}, the RNR is hardware-friendly, resource-efficient, fully parallel, and explainable by standard CR. To verify the feasibility and potential of the RNR, an electrical RNR (eRNR) design based on CMOS circuits is introduced together with a simulator. Furthermore, a prototype eRNR composed of eight parallel reservoir circuits is built to perform analog near-sensor computing, and real-time Mackey–Glass time series prediction and real-time handwriting recognition are successfully performed in hardware experiments. To realize an all-analog reservoir computing system, the eRNR is further integrated with an analog memristor array that implements the fully connected output layer. Through the proposed noise-aware training method, the conductance variation of the memristor array is accommodated, and high classification accuracy of 94.0% is achieved for a handwritten vowel recognition task. Finally, a CMOS circuit simulation based on standard 65 nm technology indicated that the eRNR system is projected to consume as little as 32.7 μW of system power in the handwriting recognition task; this total would be more than three orders of magnitude lower than that achieved by literature-reported reservoir systems. These results highlight the tremendous potential of the proposed RNR, offering a promising paradigm for resource-efficient reservoir computers.

Results

Physical CR with rotating neurons

The rotation couples the physical RNR and software CR. The mathematical derivation of the RNR proves that a rotating neuron array is equivalent to a CR model (Fig. 1b) as detailed in the Methods section. Figure 1c illustrates the operation principle of the rotation-based reservoir: if the neuron array is fixed, the pre- and post-neuron rotors rotate in the same direction to periodically shift the connections, which is equivalent to rotating the neuron while fixing the pre- and post-neuron rotors. Figure 1d shows an example of a three-neuron RNR. The rotors shift the connections before and after the neurons. The channels on the right-side output the analog computing results equivalent to the neuron states in a CR model with the same input. We shall mention that the fundamental of RNR is widely applicable to various rotating components, not limited to CMOS implementations, that can be developed as a reservoir by embedding dynamic neurons.

Thus, the main challenge of implementing a hardware RNR is the construction of the physical rotors and dynamic neurons based on the above approaches. Figure 2a illustrates a schematic of an N-neuron eRNR designed using CMOS circuits. The implementation of the input layer using binary weights is important because it allows the system to directly interface with analog sensory signals. W_in is taken to be a matrix consisting of a randomly generated uniform distribution of -1 and 1 values, which have been proven to be effective as multilevel weights²⁶. Assuming that the signal source is u(t), for each neuron, the driving signal should be γu(t) or –γu(t) during one-time step, where γ is the input scaling factor. W_in can be configured by changing the switches (S₁ to S_N). Note that the W_in should remain unchanged while the RNR is operating so that the switches can be replaced with fixed connections.

Fig. 2: Implementation of the eRNR.

a Schematic of an N-neuron eRNR. Given an input u(k), first, an operational amplifier generates another signal source −u(k) or negative input. The switch array S₁ to S_N determines the input weights W_in by selecting a positive or negative source for each multiplexer. The multiplexers m₁ to m_N and m₁’ to m_N’ are involved in the electrical implementation of pre- and post-neuron rotors, respectively. The log₂N-bits counter outputs an address signal to sequentially activate the channels of each multiplexer at switch intervals τ_r. Based on the distinct sequence of neuron connections (in₁ to in_N for the input and out₁ to out_N for the output), the behavior of the multiplexer array is equivalent to that of a rotor cyclically shifting connections between neurons and input/output channels. The sequence for output channels is a mirror version of that of input channels, which complies with the common-directional rotation principle in RNR theory. b General schematic of the dynamic properties required for a neuron in an RNR. When a neuron input ${\gamma({{{{\bf{R}}}}^{{{{\rm{k}}}}-1}})^{{{\rm{T}}}}{{{{\bf{W}}}}_{{{\rm{in}}}}{{{\bf{u(k)}}}}}}$ that has been processed by a pre-neuron rotor and input weights are provided, the neuron performs nonlinear transform f, integration (feedback line), and leakage (decay factor d) operations on the signal. a(k) is the neuron output at the kth step. c A dynamic neuron in the eRNR. C_int and R_int serve as integrators. The rectifying diode D_ReLU provides an activation function similar to a nonlinear ReLU function. Finally, high resistance R_leakage is added to control the current leakage rate, that is, the decay factor d in Eq. (6). d, e The nonlinear properties (d) and dynamic integration (e) of the neuron for R_int = 10 kΩ, C_int = 1 µF, and R_leakage = 100 kΩ. D_ReLU is a germanium diode with a forward voltage of approximately 0.3 V. f Schematic of a complete eRNR system that includes M parallel N-neuron RNRs. The total length of the state matrix is M × N. The voltage signal of each state channel is multiplied by the trained output weights stored in a memristor array to yield the final computing result.

Next, the pre-neuron rotor is implemented using N N-channel multiplexers composed of transmission gates. All multiplexers share a common address line from a log₂N (for N = 2, 4, 8, 16 …) bit counter but different channel sequences for neuron connections, as illustrated in Fig. 2a. A driving clock with a period of τ_r is used to sequentially increase the counter address from 0 to N − 1 and then reset it to 0. This address is used to control the activated channels of all the multiplexers. Because the sequence of neuron connections is inconsistent, every multiplexer is connected to a different neuron during one τ_r. Such a configuration ensures that every input channel transmitting γu(t) or −γu(t) continues to poll every neuron during every rotation cycle τ_r × N, which corresponds to the transformation ${\gamma({{{{\bf{R}}}}^{{{{\rm{k}}}}-1}})^{{{\rm{T}}}}{{{{\bf{W}}}}_{{{\rm{in}}}}{{{\bf{u(k)}}}}}}$ as described in the Methods section, where R^k-1 denotes (k-1)-time-shifting. Upon receiving the neuron input ${\gamma({{{{\bf{R}}}}^{{{{\rm{k}}}}-1}})^{{{\rm{T}}}}{{{{\bf{W}}}}_{{{\rm{in}}}}{{{\bf{u(k)}}}}}}$ and adding to its current value, the resulting neuron output a(k) is represented by the voltage level measured at the right side of the neuron circuit. The final step is to employ another post-neuron rotor at the output to convert a(k) to a state vector s(k). The post-neuron rotor performs an operation that is a mirror of that implemented by the input multiplexer array to obtain the forward rotation R.

In addition to the rotors, dynamic neurons are also crucial elements in nonlinear computing. Based on the fundamental RNR characteristics described in the Methods section, a neuron in the RNR should possess three important characteristics: nonlinearity, integration ability, and leakage ability (Fig. 2b). Figure 2c illustrates a dynamic neuron specifically for the eRNR. Figure 2d and e plot the nonlinearity (a rectified linear unit (ReLU) that can be implemented with a diode) and integration characteristics (with a time constant τ_n = R_int × C_int for the neuron), respectively. In the absence of the diode, the activation function becomes linear. The design and modeling of the dynamic neuron used in the eRNR are detailed in the Methods section. As discussed in Fig. 2b and the Methods section, most of the recently reported devices and materials for physical reservoir computing could also be used as the neuron in the RNR architecture^9,16,17. Finally, an eRNR can be built by combining rotors and neurons. Multiple parallel RNRs can simultaneously connect to a common input signal but use different W_in configurations to increase the state richness. Figure 2f illustrates a complete eRNR computing architecture that includes M parallel N-neuron eRNRs. The output weights are obtained through training and mapped in a memristor array to calculate the final results.

Moreover, a noise-free simulator was developed to evaluate the performance of the eRNR under different configurations and demonstrate its equivalence to a CR (as proven analytically in the Methods section). The first simulation was designed to confirm the consistency between the RNR and the CR and emphasize the role of rotation in the RNR. The key network characteristics based on different parameters, nonlinearities, and rotation directions were investigated. Before comparing the network characteristics of the software CR and the hardware RNR, a numerical method was developed to calculate the software CR parameters, such as the input scaling factor α and recurrent strength β, from the RNR behaviors to find the CR counterpart for a hardware RNR (see Methods). The prime task-independent network characteristic for a reservoir is the MC, which indicates its capability to retain the fading memory of the previous input^8,27 and plays a critical role in the reservoir’s performance in temporal signal processing. The standard MC measurement is introduced in Supplementary Note 1. Figure 3a plots the MC as a function of reservoir size N in different scenarios. We observed excellent agreement in the MC between the eRNR and its CR counterpart for both ReLU and linear activation functions. The ReLU neurons yielded a lower MC because the nonlinearity suppressed the fading information for previous inputs, as also observed in earlier studies^27,28. For the RNR, we investigated the effect of the rotating direction to validate the design of the two rotors. The four lines at the bottom of Fig. 3a show the MC when the two rotors stopped or co-directionally rotated. The near-zero MC suggests that in cases with no rotation and counter-directional rotation, the RNR failed to implement reservoir computing functionalities since there was no MC for processing the temporal signal. In addition to MC, the other three important network characteristics are computing ability (CA), kernel quality (KQ), and generalization rank (GR)²⁹ (see Supplementary Note 1). These factors were analyzed by varying the time constant of neurons τ_n, which also changed the parameter matching result for the CR counterpart. As shown in Fig. 3b, the network characteristics of the physical eRNR again matched that of its CR counterpart. Here, the minor difference may be attributed to the imperfect diode characteristics as a ReLU function. The results presented in Fig. 3a, b corroborate the finding that a properly configured RNR (rotation in a common direction) is equivalent to a software-based CR and hence can be used for implementing physical reservoir computing.

Fig. 3: eRNR simulation results for network characteristics and nonlinear system approximation.

a Memory capacity (MC) versus the reservoir size N for different scenarios. The two blue lines plot the MC of the eRNR using dynamic linear and ReLU neurons, respectively. The purple and green dots are obtained from the parameter-matched CR counterparts. The remaining four lines show the MCs of dysfunctional RNRs (counter-directional rotation and no rotation). The symbols ‘↓↓’, ‘↓↑’ and ‘→’ indicate that the pre- and post-neuron rotors perform common-direction rotation, counter-directional rotation, and no rotation, respectively. The parameters are τ_n = 1 s, τ_r = 0.125 s, γ = 0.5, and M = 1. b. The computing ability (CA), generalization rank (GR), and kernel quality (KQ) as a function of τ_n for the dynamic neurons. For every τ_n value, the properties of the RNR are first calculated. Then, the CR counterpart is calculated through the parameter matching method, and the results are analyzed. The obtained parameters are τ_r = 0.125 s, γ = 0.5, N = 200, and M = 1, and nonlinearity is provided by the diode. c NRMSE result for the NARMA10 system approximation task based on the two key parameters: the time constant τ_n and input scaling factor γ. The other parameters are N = 400 and M = 1. d NRMSE result for the NARMA10 modeling task when varying the reservoir size N and the number of parallel reservoirs M. The parameters are τ_n = 1 s, τ_r = 0.125 s, and γ = 0.05. e An example prediction result y’(k) and the ground truth y(k) when NRMSE = 0.055, that is, for the best result obtained in (d). The parameters are τ_n = 1 s, τ_r = 0.125 s, γ = 0.05, N = 388, and M = 50 in this case, and a diode with a ReLU function is used.

The performance benchmark for the eRNR

As an implementation of reservoir computing, the eRNR should be able to approximate a nonlinear system, for which a nonlinear autoregressive moving average system (NARMA) is a widely recognized benchmark for testing reservoir computing performance. A standard tenth-order NARMA system can be expressed by the following formula:

$$y\left(k+1\right)=0.3y\left(k\right)+0.05y\left(k\right)\mathop{\sum }\limits_{i=0}^{9}y\left(k-i\right)+1.5x\left(k\right)x\left(k-9\right)+0.1$$

(1)

where x(k) is a randomly generated white noise input in the range of [0, 0.5] and y(k + 1) is the target number. As can be observed in Eq. (1), the recursive configuration demands both nonlinear fitting and MC for the prediction model. In this task, an eRNR model was used to receive the x(k) input and then predict the y(k + 1) output after training. In total, 4000 data samples (x(k) and y(k)) for NARMA10 were generated to train (3000 samples) and test (1000 samples) the eRNR model. Given the same x(k), the normalized root mean square error (NRMSE) of the predicted result y’(k) versus y(k) calculated with the NARMA10 model based on Eq. (1) was used to quantify modeling performance. In the first trial, two key parameters of the eRNR, the input scaling factor γ and time constant of dynamic neurons τ_n, were assessed while other parameters were fixed to obtain the optimal NRMSE for a single 400-neuron eRNR. The input scaling factor changes the effective range of nonlinearity, and the time constant affects the decay factor d. The noise-free simulation result is plotted in Fig. 3c, where the optimal value (NRMSE = 0.078) was found at γ = 0.061 and τ_n = 1.1 s. It is worth mentioning that in a neuromorphic computing system, the electronic devices directly interacting with the environment and natural signals could exhibit a much longer time constant (e.g., greater than millisecond scale) compared with that of typical digital systems³⁰. A fast time constant could result in an insufficient MC for retaining historical information. Such biologically realistic time constant values (τ_n and τ_r, from milliseconds to seconds scale) were used throughout the explored hardware implementation and simulation processes. The performance can be further improved by increasing the number of parallel reservoirs M with different input weights W_in as illustrated in Fig. 2f. As shown in Fig. 3d, the resulting NRMSE can be clearly reduced by increasing M or N. The minimum NRMSE achieved in this experiment is 0.055 at N = 388 and M = 50. Figure 3e shows an instance of the predicted value y’(t) in comparison with the ground truth y(t) when NRMSE = 0.055. To the best of our knowledge, the NRMSE values for both the single eRNR (0.078) and parallel eRNRs (0.055) are lower than those reported in the previous studies^7,31 in the field of reservoir computing. Notably, the exponential form of nonlinearity in the transition region of the diode (different from the ideal ON/OFF form in the ReLU function used by the software) enhances the state representation of the NARMA10 system. This result demonstrates the tremendous potential of the eRNR in high-order nonlinear system approximation due to the rich physical dynamics of electronics devices.

Physical eRNR implementation: real-time chaotic signal prediction

The eRNR design can be implemented using commercial off-the-shelf components. Here, we developed a proof-of-concept prototype with τ_n = 1 s, N = 8, and M = 8, as shown in Fig. 4a. The eight parallel eRNRs shared common power, counter, positive input, and negative input characteristics. The input weight W_in varied for every eRNR to create diverse neuron dynamics and increase the state richness. More details about the prototype can be found in Supplementary Note 2. To evaluate the state generation performance, the first experiment with the 8 × 8 eRNR system was a multistep ahead prediction for Mackey–Glass chaotic system, which has been used in various reservoir computing studies as a benchmark task^1,17,32. The Mackey–Glass system is defined by

$$\frac{{dy}}{dt}=\beta \frac{y\left(t-\tau \right)}{1+{y\left(t-\tau \right)}^{n}}-\gamma y(t)$$

(2)

where the system parameters γ, β, and n were set to the widely used values 0.1, 0.2, and 10, respectively. Additionally, the system is chaotic when τ > 16.8, and predictions become correspondingly more difficult. In this experiment, we set τ = 17 and the initial value y(0) = 1.2 following previous works. The samples generated based on the Mackey–Glass system were input into the 8 × 8 eRNR system with a sampling rate of 8 Hz. This sampling rate should be the same as the driving frequency of the counter to ensure that every sample point is captured; that is, τ_r = 0.125 s. Based on this configuration, the 64 parallel output channels produce state values of the measured voltage for postprocessing. With our customized demonstration platform (the description of this platform is available in Supplementary Note 2), the Mackey–Glass chaotic signal y(k) was continuously fed into the eRNR system. The training state matrix s(k) with a length of 64 based on y(k) was used for output weight W_out training through linear regression, and the target value was input into the Mackey–Glass dataset shifted by i steps (y(k + i)). Here, the number of shifted steps i depended on how many steps ahead of y(k) the system could predict. The system continuously received y(k) without any preprocessing and produced 64 state outputs, which were multiplied by W_out to predict the value y’(k + i). This process was performed in real-time with the demonstration platform, and all the data, including y(k), y’(k + i), and s(k), were visualized (see Supplementary Movie 1).

Fig. 4: 8 × 8 eRNR prototypes for Mackey–Glass time series prediction.

a An eRNR prototype consisting of eight 8-neuron eRNRs (i.e., M = 8 and N = 8). b NRMSE result for multistep-ahead Mackey-Glass time series prediction. The state matrix used in this experiment was obtained from the parallel output channels of the eRNR hardware. c, d. Two cases of one-step-ahead prediction with the Mackey–Glass time series result compared with the ground truth using c one eRNR (NRMSE = 0.17) and d eight parallel eRNRs (NRMSE = 0.03). e, f Phase space of the prediction compared with the ground truth using (c) one eRNR and (d) eight parallel eRNRs. The phase diagram was created by plotting the predicted and ground truth series y(t) for the x-axis and y(t − τ) for the y-axis.

To better understand how the number of parallel RNRs (i.e., M) affected the prediction performance of the system, the states within 360 s (2880 × 64 samples, half for training and half for testing) were collected with the platform. Again, the NRMSE was used to quantify the difference between the actual values y(k + i) and the predicted values y’(k + i). The result is shown in Fig. 4b. As i increased, the time series became increasingly difficult to predict, resulting in a higher NRMSE; however, this NRMSE increase can be alleviated by using additional parallel reservoirs to enhance computational performance. Two examples of one-step-ahead prediction using one reservoir (NRMSE = 0.17) and eight parallel reservoirs (NRMSE = 0.03) are plotted in Fig. 4c, d, respectively. The traces of y(k + i) and y’(k + i) in the phase space were also examined (Fig. 4e, f). The traces of eight eRNRs exhibited excellent consistency with the true values compared with the traces for the one-reservoir system. These experimental results suggest that the 8 × 8 eRNR prototypes can be used to make accurate predictions of variables in the Mackey-Glass chaotic system after training. Even with the inevitable noise introduced by the analog circuits, the eRNR can successfully emulate the chaotic system, with a low NRMSE of 0.03. Moreover, our experiment revealed that the eRNR prototype can properly predict one-step-ahead for more chaotic signals (τ > 17) (Supplementary Fig. 1a–f). In comparison, the system performance could degrade as τ increases in multistep-ahead prediction (Supplementary Fig. 1g).

Demonstration of near-sensor computing: handwriting recognition

In the literature, some previously reported reservoir computing demonstrations achieved relatively low power consumption for certain parts inside systems using emerging devices and materials^9,16,17. However, the operations for entire systems are usually overlooked. An interface between a sensory signal and the reservoir input is usually necessary, and assistive techniques, such as converting between digital and analog data, memory buffering, preprocessing and feature extraction, are also often required^7,9,17. These sophisticated operations increase system complexity and power consumption but are necessary for conventional physical reservoir computing and remain a key challenge for practical deployment⁸. In this work, a prime advantage of our eRNR prototype is that it can directly receive analog sensory signals and produce the parallel state output without any digital memory use or preprocessing, which could considerably reduce the power consumption of the overall system. In fact, this strength is highly attractive for emerging applications in analog near-sensor computing; notably, the processor can act as a direct interface for sensory signals for cognitive computing purposes³³.

To demonstrate analog near-sensor computing, a resistive touch screen was employed to provide an analog sensory signal for a handwritten vowel recognition task. In the experimental setup, a front-end circuit converted the resistive variations into two continuous signals representing the X and Y coordinates of the activated pixel on the screen. The 8 × 8 eRNR system used in the Mackey–Glass task was divided into two 4 × 8 eRNR subsystems (i.e., N = 8 and M = 4) to process X and Y temporal signals, and the total length of the state channel was still 64. In this case, the two subsystems still shared common power and counter but had different positive and negative inputs from the X- and Y-axes. A photograph of the hardware is shown in Fig. 5a. This experiment demonstrates that five different handwritten vowels (A, E, I, O, and U) can be distinguished after high-dimensional nonlinear mapping in the eRNR. Additionally, one important advantage of using reservoir computing systems is that their short-term memory property allows the network to retain the fading information of previous inputs in the state matrix at each time step. Thus, the state matrix obtained at the end of a handwritten event contains the information for the entire handwritten trace. After training, the eRNR system can perform point-by-point analog reservoir state generation without accessing digital memory. Consequently, the memory unit for storing a certain length of data, such as the data in a sliding window or segmented signal, in conventional machine learning approaches can be eliminated by making full use of the MC. Further advancement of this system involves the analog output weights stored in a memristor crossbar array to realize all-analog signal processing^34,35, for which the power consumption can be further reduced by taking advantage of the computing-in-memory capability of memristors. Thus, from the sensory signal to the classification result, the entire system can perform near-sensor computing in the analog domain, as shown in Fig. 5b.

Fig. 5: Analog near-sensor computing for handwriting recognition.

a The hardware used in this experiment: a handwriting sensor (resistive touch screen), a front-end circuit, and two 4 × 8 eRNR circuits for the x- and y-axes of the sensor. b A conceptual schematic of analog near-sensor computing without any digital memory. The front-end circuit drives the resistive touch screen and allows it to collect the handwriting information, which is then converted into two-dimensional x- and y-analog signals. These signals are then input into two 4 × 8 parallel eRNR circuits. The trained analog weights W_out in the memristor array are used to obtain the classification output for the five handwritten vowels. c–f The signal flows measured from the eRNR hardware for different handwritten patterns, including c the five handwritten vowels, d the sensory signals for the x- and y-axes x(k), e the 64 channel reservoir states s(k) of the eRNRs, and (f) the output y(k) computed based on s(k) and the trained weights. g Confusion matrix using W_out without noise-aware training. The overall accuracy is 97.1%. h Classification accuracy as a function of simulated memristor conductance variation with and without the noise-aware training method. The measured average variation of the memristor array was 0.368 μS. i Confusion matrix using analog W_out stored in the memristor array. The overall accuracy was 94.0%, with a standard deviation of 0.8%.

In our experiment, handwritten vowel data from eight participants were collected (see Methods), and typical handwritings are displayed in Fig. 5c. For different handwritten vowels, Fig. 5d shows the X and Y signals input into the eRNRs, and Fig. 5e shows the resulting state output of the 64 channels. Using the labeling, training, and testing procedure introduced in the Methods section, 683 handwritten vowels (of a total of 703 in the test set) were correctly recognized, yielding a high accuracy of 97.1%. Examples of the point-by-point outputs for the five classes are illustrated in Fig. 5f, and the confusion matrix is shown in Fig. 5g. The errors mainly occurred when predicting ‘O’, which was misclassified as ‘U’ in some cases since these two classes are associated with similar writing traces. Here the software-trained W_out was deployed with the demonstration platform to perform real-time near-sensor handwriting recognition (see Supplementary Movie 2).

The next experiment further integrated the eRNR system with a memristor crossbar array that served as the output layer. In this experiment, a differential pair of two memristors was used to represent one synaptic weight, so 640 memristors were used to represent all the weights in the above W_out (see Methods and Supplementary Fig. 2). It is noted that the analog weights in a memristor array usually suffer from conductance variation issues (e.g., read noise) due to the nonideal device characteristics, leading to certain performance degradation compared with the floating-point digital weights in software³⁵. The next simulation evaluated the effect of memristor conductance noise on the classification performance of the system to establish a proper training scheme. Figure 5h shows the result of directly mapping W_out without noise-aware training; notably, the accuracy decreased significantly as the noise level increased. In our experiment, the intrinsic noise of the memristor was the dominant noise source in the all-analog system. To achieve high accuracy, we adopted a noise-aware training method to obtain a robust W_out in the presence of memristor conductance variation^36,37. In the noise-aware training scheme, Gaussian white noise with a standard deviation of ±0.03 was added to the normalized training state data before regression, and the resulting accuracy is plotted in Fig. 5h. The comparisons between digital W_out, target analog W_out, and the average values of the measured W_out after mapping are visualized in Supplementary Fig. 3. Most of the weight values can be successfully mapped to the memristor array with acceptable device variation, and the standard deviation (target conductance minus measured conductance) is approximately 0.368 μS. Finally, the confusion matrix using analog W_out measured from the memristor array is shown in Fig. 5i. Using the noise-aware training method and the measured analog W_out, the classification accuracy was improved from 29.2 ± 0.9% (without noise-aware training) to 94.0 ± 0.8% (with noise-aware training). The recognition result for each participant is summarized in Supplementary Fig. 4.

System-level power estimation and benchmark testing

The power consumption for the whole eRNR-based reservoir computing system can be divided into two parts: eRNR circuit consumption and memristor array consumption. For the eRNR circuit, an 8-neuron eRNR was designed and simulated using a standard 65 nm CMOS process based on the parameters used in the handwriting recognition task. The power estimation process and simulation are described in the Methods, where the power of eRNR was estimated by the simulation of the CMOS circuit using the foundry-provided library. The result indicates that the eRNR method can reduce the system power consumption for the handwriting task and chaotic signal prediction to 32.7 μW. The simulation also suggests that the static power, mainly associated with the dynamic neurons and the leakage current of transistors, plays a dominant role when the processing rate (1/τ_r) is lower than 100 kHz (for which the power consumption was estimated to be 79.1 μW). This striking advantage is associated with the unique all-analog computing capability of our eRNR-implemented reservoir computing system, which saves the energy for frequent data conversion between digital and analog domains. It should also be highlighted that our all-analog eRNR provides more than three orders of magnitude lower system-level power consumption compared with previous cutting-edge reservoir computing systems, whose power are in the ranges of 83 mW to 150 W using different implementation methods (see Supplementary Table 1)^10,38,39,40.

As we can see, in contrast to conventional digital systems, the electronics’ intrinsic dynamics were fully explored as computational resources in the all-analog eRNR architecture. A complete rotation-based reservoir computing system can be implemented by designing pre- and post-neuron rotors and dynamic neurons; this approach uses highly simplified hardware and is endorsed by the CR theory. Additional discussion and comparison of the power efficiency of the eRNR can be found in Supplementary Note 3.

Discussion

In summary, we developed a hardware-friendly RNR architecture for all-analog neuromorphic computing; the resulting structure represents a fundamentally different reservoir architecture than those used in conventional hardware implementations. The proposed RNR has been validated in theory, simulation, and experimental analyses. The theoretical analysis of RNR rigorously mapped the CR algorithm onto the physical rotation of dynamic neuron array, providing a solid foundation for hardware implementation. Such an RNR can be embedded into natural rotating components in various electronics, mechanical systems, or even nanorobotics and empower them with computing capability. In the simulation using the eRNR model, the NARMA10 prediction task was performed to benchmark the system with varying hyperparameters, and record-low NRMSE values of 0.078 for a single eRNR and 0.055 for parallel eRNRs were achieved. It was found that the additional nonlinearity provided by the hardware-based dynamic neurons enhanced system performance in the approximation of the NARMA10 system, thus highlighting the computing potential of the proposed RNR. Furthermore, an 8 × 8 eRNR prototype was developed based on RNR theory for near-sensor analog computing. The prototype successfully demonstrated multistep-ahead prediction of chaotic time series, and eight parallel reservoirs were found to reduce the prediction NRMSE from 0.17 to 0.03 for the studied Mackey–Glass chaotic system. This experimental result further validates the computing capability of our eRNR prototype under different experimental configurations. By further integrating the eRNR with an analog memristor array as the fully connected output layer, an all-analog reservoir computing system was realized to perform handwriting recognition tasks. A noise-aware training method was used to accommodate the conductance variation of the memristor array and improved the classification accuracy to 94.0%. In the simulation of the eRNR circuit, the overall system power consumption was estimated to be as low as 32.7 μW for the handwriting tasks operating at 10 Hz (τ_r = 0.1 s), reflecting an advantage of more than three orders of magnitude compared to the consumption reported for reservoir computing systems in the literature. Additionally, further power analysis suggested that the static power, mainly dissipated by the dynamic neurons, dominates the system at processing rates below 100 kHz, while the overall system power remains at a low level for high processing rates (>100 kHz) (see Supplementary Table 1). This result can be explained by the fact that most computations occur in the analog domain that only contribute to static power, which is a general advantage of analog neuromorphic computing. Dynamic power, mainly attributed to logic switches and memristor arrays, starts to dominate the system at processing rates higher than 100 kHz (see Supplementary Table 2). Further discussion on the low-power advantage of eRNRs can be found in Supplementary Note 3.

To further enhance the eRNR system capabilities when performing complex tasks, a useful approach is to increase the number of neurons (N) or the number of parallel eRNRs (M) to expand the network size. Furthermore, a deep eRNR, consisting of multiple eRNR cells in series, could enhance the classification performance for inputs of different classes. From a hardware perspective, dynamic neurons could be replaced by recently reported emerging devices (e.g., dynamic memristors^16,17 and spintronic devices⁹) to further reduce the system size and power consumption. Different configurations of neurons could be beneficial for enhancing state richness and improving system performance. In addition, the eRNR design can be miniaturized and monolithically integrated onto chips to reduce power requirements and promote ultrafast computing. It is also worth mentioning that various rotational hardware could be explored for constructing efficient pre- and post-neuron rotors, which are the key to implementing the RNR. Our work demonstrates that the RNR is well-suited for large-scale and high-speed neuromorphic computing systems and has tremendous potential for use in applications involving the Internet of Things and edge computing, among others.

Methods

Fundamentals of the RNR

For a typical reservoir computing with an m-dimensional input, an n-dimensional output, and N neurons (Fig. 1a), the input coefficients W_in (m × N) and reservoir weights W_res (N × N) are randomly generated¹. The complex dynamics stemming from the massive and random connections in the reservoir layer aid in nonlinearly mapping the m-dimensional input to the N-dimensional feature space where different input classes can be linearly separated. For n output classes, only the output weights W_out (N × n) need to be trained by using linear regression, which is relatively efficient compared to other recurrent neural network methods^1,2,41. Note that linear ridge regression is used for training throughout this work. The neuron dynamics in the reservoir layer play an important role in signal mapping based on the following equation:

$${{{{{\boldsymbol{s}}}}}}\left({{{{{\boldsymbol{k}}}}}}+{{{{{\bf{1}}}}}}\right)=f\left[{{\alpha {{{{{\bf{W}}}}}}}_{{{{{{\bf{in}}}}}}}{{{{{\boldsymbol{u}}}}}}\left({{{{{\boldsymbol{k}}}}}}+{{{{{\bf{1}}}}}}\right)+\beta {{{{{\bf{W}}}}}}}_{{{{{{\bf{res}}}}}}}{{{{{\boldsymbol{s}}}}}}\left({{{{{\boldsymbol{k}}}}}}\right)\right]$$

(3)

where s(k) denotes the neuron state matrix with length N at the kth time step, u(k) is the m-dimensional input, α and β are the scaling factors for the input and recurrent weights, respectively, and f(x) is a nonlinear transform function. In reservoir computing, the reservoir layer W_res can be designed in a deterministic manner rather than being based on random connections⁵. In this case, the W_res becomes a shifted identity matrix R

$${{{{\bf{W}}}}_{res}}={{{\bf{R}}}}=\left[\begin{array}{cccccc}{0} & {0} & {\cdots} & {0} & {0} & {1} \\ {1} & {0} & \ddots& {0} & {0} & {0} \\ {0} & {1} & {\ddots} & {0} & {0} & {0} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} & {\vdots} & {\vdots} \\ {0} & {0} & {\ddots} & {1} & {0} & {0} \\ {0} & {0} & {\cdots} & {0} & {1} & {0}\end{array}\right]$$

(4)

As a result, W_res is significantly simplified, and the network topology becomes CR, as shown in Fig. 1b. Previous research concluded that CR could achieve comparable results to those of conventional reservoir computing⁵. Then, the matrix R corresponds to one-time shifting in a ring structure, and R^k indicates a k-time cyclic shift analogous to physically rotating an object. As illustrated in Fig. 1d, it is assumed that (i) the post- and pre-neuron rotors are described by R and its transpose matrix R^T, respectively; (ii) a(k) is the dynamic neuron output at the kth step; and (iii) s_r(k) is the state matrix of the RNR at the kth step measured at the end of each rotor’s channel (before the output weights). Considering the rotation of the neuron output, the state s_r(k) updating formula can be written as

$${{{{{{\bf{R}}}}}}}^{{{{{{\rm{k}}}}}}-1}{{{{{\boldsymbol{a}}}}}}\left({{{{{\boldsymbol{k}}}}}}\right)={{{{{{\boldsymbol{s}}}}}}}_{{{{{{\boldsymbol{r}}}}}}}({{{{{\boldsymbol{k}}}}}})$$

(5)

which indicates that, at the kth step, the state matrix s_r(k) is obtained by rotating the neuron output a(k) for (k-1) times. Furthermore, the output of dynamic neurons is determined based on both an input shift and the previous states

$${{{{{\boldsymbol{a}}}}}}\left(k+1\right)={f}_{r}[{\gamma ({{{{{{\bf{R}}}}}}}^{{{{{{\rm{k}}}}}}})}^{T}{{{{{{\bf{W}}}}}}}_{{{{{{\bf{in}}}}}}}{{{{{\boldsymbol{u}}}}}}\left(k+1\right)+d{{{{{\boldsymbol{a}}}}}}(k)]$$

(6)

where d denotes the decay factor resulting from the dynamic property of the neuron (see the next subsection in the Methods), γ is the scaling factor for the input, and f_r(x) is the nonlinear transform implemented by the dynamic neurons. Equation (6) describes the signal flow through the neurons. Given an input u(k), it is first multiplied by the input weights W_in. After k reverse rotations of the input connections, the signal is fed into the dynamic nonlinear neurons, which output a(k + 1). If both sides of Eq. (5) are multiplied by R^k, we can obtain

$${{{\bf{R}}}}^{{{{{{\rm{k}}}}}}}{{{{{\boldsymbol{a}}}}}}\left(k+1\right)={{{{{{\bf{R}}}}}}}^{{{{{{\rm{k}}}}}}}{f}_{r}\left[{\gamma ({{{{{{\bf{R}}}}}}}^{{{{{{\rm{k}}}}}}})}^{T}{{{{{{\bf{W}}}}}}}_{{{{{{\bf{in}}}}}}}{{{{{\boldsymbol{u}}}}}}\left(k+1\right)+d{{{{{\boldsymbol{a}}}}}}\left(k\right)\right]$$

(7)

Using Eq. (5), Eq. (7) can be simplified as

$${{{{{{\boldsymbol{s}}}}}}}_{{{{{{\boldsymbol{r}}}}}}}\left({{{{{\boldsymbol{k}}}}}}+{{{{{\bf{1}}}}}}\right)={f}_{r}\left[\gamma {{{{{{\bf{W}}}}}}}_{{{{{{\bf{in}}}}}}}{{{{{\boldsymbol{u}}}}}}\left(k+1\right)+d{{{{{\bf{R}}}}}}{{{{{{\boldsymbol{s}}}}}}}_{{{{{{\boldsymbol{r}}}}}}}\left({{{{{\boldsymbol{k}}}}}}\right)\right]$$

(8)

Here, the excellent consistency between Eq. (3) and Eq. (8) reveals that the proposed physical RNR architecture (Fig. 1c) is equivalent to a software CR. Thus, a rotating object with dynamic neurons can act as a reservoir computer without using extra control units, ADC or memory, which remarkably reduces the system complexity and power consumption compared with those in conventional hardware implementation (see Supplementary Note 3).

Design and modeling of dynamic neurons

By observing Eq. (3), it appears that a dynamic neuron for the proposed RNR should satisfy three important characteristics as shown in Fig. 2b: provide a nonlinear activation function f(x); support integration ability for the summation between the current input and previous state a(k − 1); and support leakage, as related to the decay factor d, to avoid saturation caused by the integration process. Any passive element that exhibits these three characteristics could essentially be used as a dynamic neuron in the RNR architecture by fine-tuning the time constants of neurons and rotors. A dynamic node working in a physical reservoir may suffer from device variation issues, which impact system performance. Previous studies have revealed that a certain degree of device variation may be beneficial to system performance by enhancing state richness^16,17, but determining how to precisely control device variability warrants future explorations.

In implementations using standard electronics (Fig. 2c), a ReLU-type nonlinear transform can be provided by a diode, and the resistor R_int and capacitor C_int can act as integrators. Leakage can be considered by connecting the system to the ground via a large resistance R_leakage. In the simulation, this neuron can be modeled as follows:

$$\dot{{V}_{o}(t)}=\frac{1}{{R}_{{{{{\rm{int}}}}}}{C}_{{{{{\rm{int}}}}}}}{V}_{i}\left(t\right)-\frac{{R}_{{int}}+{R}_{{{{{\rm{leakage}}}}}}}{{R}_{{{{{\rm{int}}}}}}{R}_{{{{{\rm{leakage}}}}}}{C}_{{{{{\rm{int}}}}}}}{V}_{o}\left(t\right)+\frac{1}{{C}_{{{{{\rm{int}}}}}}}{I}_{s}({e}^{-\frac{{V}_{o}\left(t\right)}{{V}_{T}}}-1)$$

(9)

where V_o(t) and V_i(t) denote the input and output voltages, respectively. The saturation current I_s and thermal voltage V_T stem from the Shockley diode equation $I={I}_{s}({e}^{\frac{{V}_{D}}{{V}_{T}}}-1)$. The typical values for germanium diodes I_s = 25 × 10⁻⁹ A and V_T = 0.026 V were used in the simulation. In the case of linear neurons, the last term $\frac{1}{{C}_{{int}}}{I}_{s}({e}^{-\frac{{V}_{o}\left(t\right)}{{V}_{T}}}-1)$ should be removed from Eq. (9).

In our simulation, Eq. (9) was solved in MATLAB/Simulink. The discrete neuron output in Eq. (3) becomes a(k) = V_o(kτ_r). The pre- and post-neuron rotors can be modeled by continuously shifting W_inu(k) and the neuron output a(k). Since R_leakage is a large resistance, the time constant associated with this neuron is mainly determined by the integrator τ_n = R_intC_int. For the rate of rotation τ_r, we normally use an empirical value of τ_r = τ_n/8.

Parameter matching

It has been analytically proven that a physical RNR can perform the same functionality as a CR (Eq. (8)). Therefore, given a properly configured RNR, its CR counterpart should exist and exhibit similar network characteristics. Parameter matching provides a numerical method to determine the CR counterpart. The main difference between a hardware RNR and a software CR is associated with nonideal dynamic neurons, which result in different amplitude ranges for integration and nonlinearity. Therefore, the objective is to find the appropriate scaling coefficients for the software activation function to approximate the hardware neuron output under the same input W_inu(k). An arbitrary u(k) was generated as an input to the RNR, and the neuron output a(k) was obtained. Assuming that this a(k) is generated by a software CR, a comparative neuron update vector can be defined

$${{{{{{\boldsymbol{a}}}}}}}_{{{{{{\boldsymbol{p}}}}}}}\left({{{{{\boldsymbol{k}}}}}}+{{{{{\bf{1}}}}}},{{{{{\boldsymbol{\alpha }}}}}},{{{{{\boldsymbol{\beta }}}}}},{{{{{{\boldsymbol{V}}}}}}}_{{{{{{\boldsymbol{c}}}}}}}\right)={ReLU}(\beta {{{{{\boldsymbol{a}}}}}}\left({{{{{\boldsymbol{k}}}}}}\right)+\alpha {{{{{{\bf{W}}}}}}}_{{{{{{\bf{in}}}}}}}u\left(k\right),{V}_{c})$$

(10)

where a_p is the neuron output sequence of recurrent factor α, input scaling factor β, and the ReLU cutoff value V_c. For certain values of α, β and V_c, the CR should match the RNR if the resulting a_p(k) is close to a(k) for any k. Hence, the CR counterpart of an RNR can be found by matching the three parameters. First, V_c is the threshold voltage of the diode, unlike that in the ideal ON/OFF case in the software ReLU function (V_c = 0). This value can be obtained based on the minimum value of the a(k) sequence. Second, α and β are determined by searching the potential values and finding those that minimize the NRMSE between a(k) and a_p(k), which can be described as $\mathop{{{\min }}}\nolimits_{\alpha ,\beta }{NRMSE}\left({{{{{\boldsymbol{a}}}}}}\left({{{{{\boldsymbol{k}}}}}}\right),{{{{{{\boldsymbol{a}}}}}}}_{{{{{{\boldsymbol{p}}}}}}}\left({{{{{\boldsymbol{k}}}}}}\,{{{{{\boldsymbol{+}}}}}}\,{{{{{\boldsymbol{1}}}}}}{{{{{\boldsymbol{,}}}}}}\,\alpha ,\beta {{{{{\boldsymbol{,}}}}}}{V}_{c}\right)\right)$. For example, for an RNR with τ_n = 1 s, τ_r = 0.125 s, and γ = 0.5, the matched CR parameters are α = 0.87, β = 0.12, and V_c = −0.18, and the corresponding MC values are compared in Fig. 3a.

Handwritten vowel recognition using an eRNR

The parameters of the eRNR used in the handwritten vowel recognition task are τ_n = 1 s, τ_r = 0.1 s, N = 8, and M = 4 (for each X and Y channel). All data were collected with our customized platform. In total, 66 channel data streams, including the two-axis signals and signals from 64 reservoir state channels, were collected at each time step. During the data collection process, eight participants were asked to write the five vowels on a resistive touch screen, and repeat at least 20 times for each vowel. Data for 1103 handwritten vowels (2802 s) were successfully collected. The location and class of each handwritten vowel were labeled at the final rising/falling edge of the X and Y raw data. We labeled the end of each handwritten vowel (the blue square in Fig. 5d) where the state matrix at this time step contains the information of the handwritten trace because of MC. Specifically, the 64 × 1 state matrix collected at the time denoted by the green dot can be considered a feature vector for the corresponding handwritten trace.

After data collection and labeling, the database was divided into a training set (400 handwritten vowels; 1025.8 s) and a testing set (703 handwritten vowels; 1776.2 s). According to the point-by-point computation introduced above, the size of the training label matrix Y_train for the five classes should be a five-dimensional data stream in which only the locations of green squares are set to 1, and values of 0 are assigned at other points. For training W_out (64 × 5), ridge regression with the target = Y_train (five-dimensional label for 1025.8 s) and variables = S_train (64-dimensional state vector for 1025.8 s) was used. Next, W_out was multiplied by the test state matrix (Y_test’ = S_test × W_out) to obtain a five-dimensional output representing the possibility of five potential classes at each time step, which corresponded to the graphs in Fig. 5f. To quantify the classification accuracy, the predicted output for the testing set Y_test’ was compared with the manually labeled locations Y_test. For every location in a handwritten event, for example, y_test(k)|k = k_x, the actual output was investigated to find the maximum value in the range of y_test(k_x − 7) to y_test(k_x + 3). The corresponding channel that output the maximum value was considered the predicted class.

Memristor-based output layer

Memristor-based analog computing has displayed excellent potential in neuromorphic computing. While the input and reservoir layer are generally established based on eRNR design, the output layer, which employs standard vector-matrix multiplication operations, can be effectively implemented by a memristor array for end-to-end all-analog computing^42,43. The memristor array has a unit cell of one-transistor-one-resistor (1T1R). Each 1T1R consists of a resistive switching memristor with a material stack of TiN/HfO_x/TaO_y/TiN connected to a Si transistor that is fabricated using a standard 130 nm Si CMOS process^44,45. The description of the memristor array can be found in Supplementary Fig. 2. As described in the main text, we used 640 memristors in total to represent 320 weights in the output layer. The computation principles of memristor-based analog computing can be expressed as I = V × G = V × (G_p − G_n), where G represents the weight matrix W, and G_p and G_n are the positive and negative conductance matrices, respectively. Furthermore, we use a standard write-with-verify scheme to map the weight matrix W_out to the conductance of the memristor array³⁴.

Power estimation

As shown in Fig. 2a, the neurons, as passive components, are driven by the negative and positive sensory signals, providing a power source P_s. Also, the energy consumed by the counter and transmission gates depends on not only the static power but also the rate of rotation τ_r. The total power consumption P of the system consisting of M 8-neuron eRNRs (where the number of neurons N is fixed at 8) can be expressed as

$${P=P}_{c}+\left({P}_{s}+{P}_{t}+\frac{{E}_{c}^{{dyn}}+{E}_{t}^{{dyn}}}{{\tau }_{r}}\right)\times M+\frac{{E}_{m}^{{dyn}}}{{\tau }_{r}}$$

(11)

where P_c and P_t represent the static power of the counter and transmission gates, respectively, and ${E}_{c}^{{dyn}}$ and ${E}_{t}^{{dyn}}$ represent the dynamic energy dissipated in the transition region driven by the rate of rotation 1/τ_r. ${E}_{m}^{{dyn}}$ is the energy consumed in the output layer (memristor array) for one inference. The M parallel eRNRs can share one counter, but the power for the other components increases with the number of parallel eRNRs M. For our application involving real-time handwritten signals, the operation period τ_r is relatively slow (0.1 s) to match the time scale of human operations.

The simulation result shows that P_s = 3.27 μW, P_c = 0.93 μW, and P_t = 0.70 μW, regardless of how fast the rotors are operating. Moreover, the energy-related to the rotation rate is ${E}_{c}^{{dyn}}$ = 0.31 pJ and ${E}_{t}^{{dyn}}$ = 0.07 pJ. For the memristor-based output layer, the power dissipated by the voltage buffer driving the memristor array and the memristor array itself is 144 and 0.8 μW, respectively. During every τ_r, the only one-time inference is needed since all state channels are monotonously increased or decreased. The memristor array takes ~50 ns to respond to the state voltage. Therefore, the dynamic energy of the memristor array for every inference step is ${E}_{m}^{{dyn}}$ = (144 μW + 0.8 μW) × 50 ns × 64 = 463.36 pJ/class. The total power consumption of an 8 × 8 eRNR can then be calculated using Eq. (11). The simulated power breakdown at different frequencies is shown in Supplementary Table 2. Notably, this result also reveals that the power would not considerably increase at rates of rotation (1/τ_r) below 100 kHz since static power dissipation dominates the system.