Vestibular reservoir computing

Smita Deb; Shirin Panahi; Mulugeta Haile; Ying-Cheng Lai

cs.LGApr 10, 2026

Vestibular reservoir computing

Authors: Smita Deb, Shirin Panahi, Mulugeta Haile, Ying-Cheng Lai

Organizations: School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA · Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523, USA · DEVCOM Army Research Office, 6340 Rodman Road, Aberdeen Proving Ground, MD 21005-5069, USA · Department of Physics, Arizona State University, Tempe, Arizona 85287, USA

Abstract

Reservoir computing (RC) is a computational framework known for its training efficiency, making it ideal for physical hardware implementations. However, realizing the complex interconnectivity of traditional reservoirs in physical systems remains a significant challenge. This paper proposes a physical RC scheme inspired by the biological vestibular system. To overcome hardware complexity, we introduce a designed uncoupled topology and demonstrate that it achieves performance comparable to fully coupled networks. We theoretically analyze the difference between these topologies by deriving a memory capacity formula for linear reservoirs, identifying specific conditions where both configurations yield equivalent memory. These analytical results are demonstrated to approximately hold for nonlinear reservoir systems. Furthermore, we systematically examine the impact of reservoir size on predictive statistics and memory capacity. Our findings suggest that uncoupled reservoir architectures offer a mathematically sound and practically feasible pathway for efficient physical reservoir computing.

Figures & tables

Figure 1: Schematic representation of the proposed vestibular reservoir computing (RC) scheme. The reservoir incorporates biomechanical structures (semicircular canals and otolith organs) that process external stimuli via endolymph movement, alongside neurophysiological components (hair cells) that transduce mechanical stimuli into neural signals. The computational framework models these interactions by coupling semicircular canal mechanics with FitzHugh–Nagumo (FHN) neural dynamics.

Figure 2: Performance comparison of coupled and uncoupled vestibular reservoir computers for Lorenz and chaotic food-chain systems. (a-d) Results for the coupled configuration trained on the Lorenz system, showing training, validation, closed-loop testing, and closed-loop attractor reconstruction, respectively. The validation phase evaluates generalization on unseen data in an open-loop setup, while attractor reconstruction demonstrates the autonomous system’s ability to reproduce long-term chaotic dynamics. (e-h) Corresponding results for the uncoupled vestibular reservoir computer trained on the Lorenz system. (i-l) Results for the coupled configuration trained on the chaotic food-chain system. (m-p) Corresponding results for the uncoupled configuration on the food-chain system. The time series plots display the variables $u_{1}$ and $u_{2}$ from Eq. ( 12 ) and Eq. ( 13 ). Hyperparameter values are listed in SI Sec. S2 , Tab. S1 .

Figure 3: Short-term predictive performance of coupled and uncoupled vestibular reservoir computers. The plots show the mean training and validation errors (quantified as Normalized Root-Mean-Square Error, NRMSE) as a function of reservoir network size for the (a-d) Lorenz and (e-h) chaotic food-chain systems. The shaded regions represent the standard deviation on a logarithmic scale, calculated over

100

independent trials.

Figure 4: Predictive statistics of coupled and uncoupled vestibular reservoir computers. The plots display the Kullback-Leibler (KL) divergence, largest Lyapunov exponent, and mean Deviation Value (DV) as a function of network size. Panels correspond to: (a-c) coupled reservoir on the Lorenz system; (d-f) uncoupled reservoir on the Lorenz system; (g-i) coupled reservoir on the food-chain system; and (j-l) uncoupled reservoir on the food-chain system. Shaded regions indicate the standard deviation calculated on a linear scale from 100 independent trials.

Figure 5: Methods for calculating the memory function. (a) A scalar stochastic signal $u(t)$ serves as both the input and the target signal $y(t)$ for training. (b) Schematic of the standard calculation method. The full reservoir state matrix $\mathbb{R}$ has dimensions $2N\times T$ . The reduced state matrix $\mathbb{R}_{\tau}$ (blue) is defined over the interval $[\tau+1,T]$ , while the time-delayed state matrix $\bar{\mathbb{R}}_{\tau}$ covers $[1,T-\tau]$ . The output weight vector $\mathbf{W}_{\rm out}^{\tau}$ is trained using $\mathbb{R}_{\tau}$ and the target series $\mathbf{y}_{\tau}$ (interval $[\tau+1,T]$ ). The predicted series $\hat{\mathbf{y}}_{\tau}$ is then computed as the product of $\mathbf{W}_{\rm out}^{\tau}$ and the time-delayed states $\bar{\mathbb{R}}_{\tau}$ . (c) Alternative calculation method. Here, $\mathbf{W}_{\rm out}^{\tau}$ is trained using the time-delayed target series $\mathbf{y}_{\tau}$ and the reduced state matrix $\mathbb{R}_{\tau}$ . The prediction $\hat{\mathbf{y}}_{\tau}$ is obtained by multiplying $\mathbf{W}_{\rm out}^{\tau}$ and $\mathbb{R}_{\tau}$ directly. (d) The refined method used to eliminate statistical imbalance (see text). The relevant time intervals are adjusted such that the reservoir state definition remains constant ( $\mathbb{R}_{\tau}\equiv\mathbb{R}$ ) for all $\tau$ .

Figure 6: Memory function

{\rm M}_{F}

for a linear reservoir computer driven by a scalar stochastic input. (a, b)

{\rm M}_{F}

versus time delay for an uncoupled and a coupled linear reservoir network, respectively. Orange and blue traces represent theoretical predictions and numerical simulation results, respectively. (c) Numerical computation of

{\rm M}_{F}

for an uncoupled linear reservoir (orange) configured to share the same eigenvalue spectrum as the coupled linear reservoir (blue). All simulations are averaged over 100 independent trials with a spectral radius

\rho=0.9

Figure 7: Memory function

{\rm M}_{F}

and memory capacity

{\rm M}_{C}

of the vestibular reservoir computer. (a-d) Memory function

{\rm M}_{F}

versus time delay for stochastic input. Panels (a, c) use hyperparameters optimized for the Lorenz system; panels (b, d) use hyperparameters optimized for the chaotic food-chain system. In panels (a) and (b), the green trace represents the baseline coupled reservoir; red points denote an uncoupled network constrained to share the same eigenvalue spectrum, while blue points denote an uncoupled network sharing only the spectral radius

\rho

(with random eigenvalues). In panels (c) and (d), the green trace represents the baseline uncoupled reservoir, compared against coupled networks with identical eigenvalues (red) or identical spectral radius (blue). (e–f) Total memory capacity

{\rm M}_{C}

as a function of reservoir size for coupled (blue) and uncoupled (red) configurations, using hyperparameters tuned on the Lorenz (e) and chaotic food-chain (f) systems. All results represent averages over 100 independent trials.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1: Schematic representation of the vestibular system and its functionality. (a) Anatomical schematic of the vestibular system. (b) Overview of functional applications, including postural stability, motion control, eye movement stabilization, robotics, and autonomous systems.

Physical reservoir computer	Model	$\gamma$	$\rho$	$\lambda$	$d$
Coupled	Lorenz	1	0.8	$10^{-4}$	0.4
$\cdots$	food-chain	3.5	0.2	$10^{-5}$	0.4
Uncoupled	Lorenz	1	0.5	$10^{-4}$	0.4
$\cdots$	food-chain	3.5	0.7	$10^{-5}$	0.4

Appendix

Table S1: Hpyerparameters of the coupled and uncoupled vestibular reservoir computer for Lorenz and chaotic food-chain systems.

Figure S2: Performance of the vestibular reservoir computer: Closed-loop attractor reconstruction. Comparison of ground truth (blue) and predicted (orange) attractors for all state variables of the Lorenz and chaotic food-chain systems. Results are shown for the (a-c) coupled and (d-f) uncoupled vestibular reservoir configurations. The autonomous reservoir successfully reproduces the underlying chaotic dynamics, validating its ability to capture the long-term behavior of the original systems. Simulation parameters are identical to those used in Fig. 2 of the main text.

Figure S3: Performance of the vestibular reservoir computer for the Lorenz system. (a) Training phase time series showing the ground truth (blue) alongside predicted outputs for the coupled (cyan) and uncoupled configurations. Note that the uncoupled network is configured to share the same eigenvalue spectrum as the coupled one. (b) Validation phase performance, evaluating the trained reservoir’s generalization accuracy on unseen data in an open-loop setup. (c-d) Testing phase time series and closed-loop attractor reconstruction. The autonomous reservoir successfully reproduces the underlying chaotic dynamics, confirming its ability to capture long-term system behavior. Both coupled and uncoupled configurations exhibit comparable performance.

Statistics	Coupled	Uncoupled with same eigenvalue
Training error	0.0111	0.0115
Validation error	0.0114	0.0117
Deviation value	0.331	0.352
KL Divergence	0.0005	0.0009
Largest Lyapunov exponent (predicted)	0.0308	0.0307

Appendix

Table S2: Comparison of mean predictive statistics for the Lorenz system. Results represent an ensemble average over 50 realizations for coupled and uncoupled (sharing the same eigenvalue spectrum) vestibular reservoir computers with

N=30

nodes. The theoretical largest Lyapunov exponent for the system is approximately

0.03

Figure S4: Performance of the vestibular reservoir computer for the chaotic food-chain system. (a) Training phase time series displaying the ground truth (blue) alongside predicted outputs for the coupled (cyan) and uncoupled (red) configurations, where the uncoupled network shares the same eigenvalue spectrum as the coupled one. (b) Validation phase performance, assessing generalization accuracy on unseen data in an open-loop setup. (c-d) Testing phase time series and closed-loop attractor reconstruction. The autonomous reservoir successfully reproduces the underlying chaotic dynamics, confirming its ability to capture long-term system behavior. Both configurations exhibit comparable performance.

Statistics	Coupled	Uncoupled with same eigenvalue
Training error	0.009	0.0098
Validation error	0.0095	0.0099
Deviation value	0.34	0.375
KL Divergence	0.0005	0.0008
Largest Lyapunov exponent (predicted)	0.0237	0.0225

Appendix

Table S3: Comparison of mean predictive statistics for the chaotic food-chain system. Results represent an ensemble average over 50 realizations for coupled and uncoupled (sharing the same eigenvalue spectrum) vestibular reservoir computers with

N=30

nodes. The theoretical largest Lyapunov exponent for the system is approximately

0.021

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