Optimal Memory Encoding Through Fluctuation-Response Structure

Reservoir computing exploits nonlinear dynamical systems to encode temporal inputs into high-dimensional state space representations. Although reservoir performance is often characterized through memory, nonlinearity, and their tradeoff, such aggregate measures do not reveal how task-relevant information is organized within the state space. Here, we introduce an eigen-spectral decomposition framework linking the degree-wise information processing capacity to the corresponding state space modes. As a result, we are able to quantify the degree-wise representation energy, and show that in some cases, substantial amounts of information processing capacity may reside in low-energy modes that are vulnerable to experimental noise. These results suggest that useful reservoir computation depends not only on dimensionality expansion, but also on the geometric organization of task-relevant information, with direct implications on physical reservoir computers.

Reservoir Computing (RC) designs Recurrent Neural Networks around a fixed, i.e., untrained, recurrent layer, and is a natural candidate for neuromorphic hardware. Memristive-friendly reservoirs derive the neuron dynamics from memristive-device kinetics, but still rely on dense recurrent matrices, which are expensive to realize physically. In this paper, we replace the dense matrix with a structured orthogonal operator, built from sign diagonals, a permutation, and a fast Walsh-Hadamard transform. The operator is multiplier-free, requires $O(N)$ parameters and $O(N\log N)$ operations per step, and is never materialized as a matrix. We instantiate it in a standard and in a memristive-friendly Echo State Network, with one binary input connection per unit. Our mathematical analysis shows that exact orthogonality yields an echo state condition that is tight in the recurrent scaling, and a noise response that is predictable at design time. Moreover, the operator mixes the whole state in a single application. Experiments on twenty classification and seven regression benchmarks, at reservoir sizes up to $N = 8192$, show that the structured models match dense orthogonal reservoirs, and achieve better mean performance than the cycle reservoir by a margin that widens with size. Furthermore, we time the recurrent step on three hardware platforms, where it is up to $50\times$ faster than a dense product and $10^4\times$ smaller in memory. Finally, we ablate the operator and measure the response to noise, quantization, device mismatch and discrete faults.

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.