超越低通动力学：具有共振神经元的频率选择性脉冲储备池

# Beyond Low-Pass Dynamics: Frequency-Selective Spiking Reservoirs with Resonant Neurons
Source: [https://arxiv.org/html/2610.37906](https://arxiv.org/html/2610.37906)
###### Abstract

Spiking reservoir computing combines the training efficiency of fixed recurrent reservoirs with the energy efficiency of event\-driven inference, where computation is triggered by sparse spikes rather than executed densely at every timestep\. However, existing spiking reservoirs almost exclusively rely on Leaky Integrate\-and\-Fire \(LIF\) neurons, whose low\-pass dynamics limit their ability to distinguish information encoded at different temporal frequencies\. We introduce HRF\-Res, a liquid state machine in which LIF neurons are replaced by a heterogeneous population of Harmonic Resonate\-and\-Fire \(HRF\) neurons\. These neurons exhibit intrinsic oscillatory dynamics and selectively respond to input frequencies, enabling the reservoir to capture frequency\-specific temporal patterns\. The result is more discriminative representations and higher classification accuracy than LIF\-based reservoirs, achieved without sacrificing the energy efficiency of spiking computation, with up to 19×\\timeslower energy consumption than an equivalent non\-spiking oscillatory reservoir\. Evaluated on ten benchmarks spanning time series, sequential images, audio spike trains, and neuromorphic vision, HRF\-Res achieves state\-of\-the\-art accuracy among spiking reservoir methods\. These results demonstrate that introducing frequency\-selective dynamics into spiking reservoirs improves representational quality and classification performance while preserving their inherent energy efficiency\.

## 1Introduction

Processing temporal sensor streams at the edge, from industrial vibration monitoring to event\-based vision and neuromorphic audio, requires models that are both computationally efficient and capable of operating under strict power constraints\. Modern sequence models such as LSTMs\[[1](https://arxiv.org/html/2610.37906#bib.bib8)\], GRUs\[[2](https://arxiv.org/html/2610.37906#bib.bib10)\], and Transformers\[[3](https://arxiv.org/html/2610.37906#bib.bib11)\]achieve strong performance but were not designed for this regime: they require backpropagation through time \(BPTT\) to train, and perform dense computation at every timestep during inference\. As a result, their energy consumption scales with model size rather than input activity, making them inefficient for low\-power, event\-driven settings where signals are sparse and computation should ideally be triggered only when new information arrives\.

Reservoir Computing \(RC\) addresses the training cost: a fixed, randomly initialised recurrent network projects inputs into a rich dynamical state and only a linear readout is trained\[[4](https://arxiv.org/html/2610.37906#bib.bib12),[5](https://arxiv.org/html/2610.37906#bib.bib13)\], eliminating BPTT entirely\. The inference cost, however, remains: RC reservoirs perform dense computation at every timestep, so energy still scales with network size rather than input activity\. Spiking Neural Networks \(SNNs\) offer a complementary solution\. By encoding information as discrete spike events, computation is triggered only by activity, making energy consumption proportional to firing rate rather than network size\[[6](https://arxiv.org/html/2610.37906#bib.bib15),[7](https://arxiv.org/html/2610.37906#bib.bib26)\]\. This event\-driven paradigm maps naturally onto neuromorphic hardware such as Intel Loihi\[[8](https://arxiv.org/html/2610.37906#bib.bib16)\], IBM TrueNorth\[[9](https://arxiv.org/html/2610.37906#bib.bib17)\], and SpiNNaker\[[10](https://arxiv.org/html/2610.37906#bib.bib22)\], and is particularly well suited to event\-based sensing modalities\[[11](https://arxiv.org/html/2610.37906#bib.bib23)\]\. Combining SNNs with RC yields Liquid State Machines \(LSMs\)\[[5](https://arxiv.org/html/2610.37906#bib.bib13)\], which inherit both training efficiency and activity\-driven inference\. Recent work has shown competitive LSM performance on temporal benchmarks\[[12](https://arxiv.org/html/2610.37906#bib.bib31),[13](https://arxiv.org/html/2610.37906#bib.bib38),[14](https://arxiv.org/html/2610.37906#bib.bib40)\]\.

Despite this promise, existing spiking reservoirs almost exclusively rely on leaky integrate\-and\-fire \(LIF\) neurons\. A LIF neuron acts as a temporal integrator\[[15](https://arxiv.org/html/2610.37906#bib.bib19)\]: inputs that vary rapidly over time are increasingly attenuated, resulting in an effectively low\-pass response\. A reservoir composed of such units therefore encodes how much input arrives, but not at which frequencies\. For signals with multiple temporal scales, this constitutes a fundamental representational limitation\[[16](https://arxiv.org/html/2610.37906#bib.bib46)\]\.

Oscillatory dynamics provide a natural mechanism to overcome this lack of sensitivity to temporal frequency structure\. Subthreshold membrane oscillations have been widely observed in biological neurons\[[17](https://arxiv.org/html/2610.37906#bib.bib20),[18](https://arxiv.org/html/2610.37906#bib.bib21)\]and are believed to play a key role in temporal coding\[[19](https://arxiv.org/html/2610.37906#bib.bib25),[20](https://arxiv.org/html/2610.37906#bib.bib6)\]\. In artificial systems, networks of oscillators form a powerful computational substrate for temporal processing\[[21](https://arxiv.org/html/2610.37906#bib.bib24)\], and non\-spiking oscillatory reservoirs such as RON\[[22](https://arxiv.org/html/2610.37906#bib.bib14)\]show that heterogeneous resonance frequencies can improve performance and stability\. Whether these benefits can be realised within a spiking reservoir remains an open question\.

In this work, we instantiate oscillatory dynamics in a spiking setting using Resonate\-and\-Fire neurons, which incorporate subthreshold oscillatory behaviour within a spike\-generating mechanism\[[23](https://arxiv.org/html/2610.37906#bib.bib18),[24](https://arxiv.org/html/2610.37906#bib.bib50)\]\. Specifically, we employ Harmonic Resonate\-and\-Fire \(HRF\) neurons, which exhibit damped oscillatory dynamics and intrinsic resonance frequencies\. Unlike LIF neurons, HRF neurons respond preferentially to inputs matching their resonance, effectively acting as band\-pass filters\. Intuitively, a heterogeneous population of HRF neurons behaves like a bank of tuning forks, each responding to its own frequency band, so that the reservoir state reflects not only how much input arrives but also at which temporal frequencies\. By replacing LIF units with a heterogeneous population of HRF neurons, we introduce HRF\-Res, a Liquid State Machine that combines the efficiency of spiking computation with frequency\-selective representations\.

Our aim is not the neuron substitution per se, but to establish frequency selectivity as a computational principle for spiking reservoirs\. Through a controlled comparison in which neuron dynamics are the only difference between models, we test whether band\-pass behaviour survives in a full recurrent spiking network, whether it makes reservoir representations more class\-discriminative, and whether this translates into higher accuracy at low energy cost\. Our contributions are as follows:

- •We show that HRF neurons act as distributed band\-pass filters, producing intrinsically more class\-discriminative representations than an equivalent LIF reservoir before any readout is trained\. A controlled ablation confirms that this representational advantage translates into improved classification accuracy across multiple datasets and modalities\.
- •HRF\-Res achieves up to 19×\\timeslower energy consumption compared to the non\-spiking RON baseline\[[22](https://arxiv.org/html/2610.37906#bib.bib14)\], with only modest accuracy loss\.
- •We evaluate the proposed model on ten benchmarks spanning univariate time series, sequential images, audio spike trains, and neuromorphic vision, achieving state\-of\-the\-art performance among spiking reservoir methods\.

Code to reproduce all experiments and results reported in this paper is available at[https://github\.com/AuroraMicheli/HRF\-Res](https://github.com/AuroraMicheli/HRF-Res)\.

## 2Related Work

Reservoir computing and oscillatory dynamics\.Reservoir Computing \(RC\)\[[4](https://arxiv.org/html/2610.37906#bib.bib12),[5](https://arxiv.org/html/2610.37906#bib.bib13)\]provides an efficient framework for temporal processing by decoupling representation learning from training\. Its dynamical properties and practical usage have been extensively studied\[[25](https://arxiv.org/html/2610.37906#bib.bib44),[26](https://arxiv.org/html/2610.37906#bib.bib28)\]\. A key factor in RC performance is the choice of reservoir dynamics\. Early work on Liquid State Machines already suggested that neuron models with resonant or oscillatory subthreshold dynamics can yield more separable representations than standard integrator neurons\[[16](https://arxiv.org/html/2610.37906#bib.bib46)\]\. More recently, oscillatory systems have been revisited as a powerful computational substrate, both from a theoretical perspective\[[21](https://arxiv.org/html/2610.37906#bib.bib24)\]and in practice: Ceni et al\.\[[22](https://arxiv.org/html/2610.37906#bib.bib14)\]showed that reservoirs composed of heterogeneous damped oscillators significantly outperform standard RC models\. However, this approach relies on continuous\-valued units, performing dense computation at every timestep, and therefore does not benefit from the energy savings of event\-driven inference\. Beyond reservoir computing, oscillatory dynamics have recently been explored as a general inductive bias in gradient\-trained deep networks, from coupled\-oscillator RNNs and GNNs\[[27](https://arxiv.org/html/2610.37906#bib.bib1),[28](https://arxiv.org/html/2610.37906#bib.bib2),[29](https://arxiv.org/html/2610.37906#bib.bib3)\]to Kuramoto\-oscillator units in modern architectures\[[30](https://arxiv.org/html/2610.37906#bib.bib4)\], building on earlier work on neural synchrony via complex\-valued units\[[31](https://arxiv.org/html/2610.37906#bib.bib5)\]\.

Spiking reservoir computing\.Liquid State Machines \(LSMs\)\[[5](https://arxiv.org/html/2610.37906#bib.bib13)\]extend RC to spiking neural networks, enabling event\-driven computation and compatibility with neuromorphic hardware\. A large body of work has explored architectural and parametrisation improvements, including heterogeneous time constants\[[12](https://arxiv.org/html/2610.37906#bib.bib31)\], efficient implementations and scaling strategies\[[13](https://arxiv.org/html/2610.37906#bib.bib38),[32](https://arxiv.org/html/2610.37906#bib.bib39)\], and ensemble or receptive\-field approaches\[[14](https://arxiv.org/html/2610.37906#bib.bib40)\]\. Some recent works also consider hybrid models that depart from the strict RC paradigm, such as Gaurav et al\.\[[33](https://arxiv.org/html/2610.37906#bib.bib41)\], which combine non\-spiking feature extractors with trainable spiking components\. Despite these advances, existing spiking reservoirs predominantly rely on leaky integrate\-and\-fire \(LIF\) neurons, which lack intrinsic frequency selectivity\.

Resonate\-and\-fire neurons\.Resonate\-and\-fire \(RF\) neurons\[[23](https://arxiv.org/html/2610.37906#bib.bib18)\]introduce oscillatory membrane dynamics into spiking models, enabling selective responses to input frequency\. Their computational properties have been explored in a variety of contexts, including short\-time Fourier transform computation on neuromorphic hardware\[[34](https://arxiv.org/html/2610.37906#bib.bib47)\], and for converting analog signals into spike representations for downstream SNNs\[[35](https://arxiv.org/html/2610.37906#bib.bib48),[36](https://arxiv.org/html/2610.37906#bib.bib49)\]\. RF neurons have also been used as harmonic oscillators in feedforward SNNs for image, audio, and time\-series classification, as well as related signal processing tasks\[[24](https://arxiv.org/html/2610.37906#bib.bib50),[37](https://arxiv.org/html/2610.37906#bib.bib45),[34](https://arxiv.org/html/2610.37906#bib.bib47),[38](https://arxiv.org/html/2610.37906#bib.bib9)\]\. Adaptation\-based mechanisms in spiking networks offer a related, though distinct, route to temporally structured dynamics beyond standard LIF\[[39](https://arxiv.org/html/2610.37906#bib.bib7)\]\. These works highlight the potential of RF dynamics for efficient temporal representation and frequency analysis\. However, their integration into fixed\-weight reservoir computing frameworks remains largely unexplored\. This work addresses this gap by investigating oscillatory spiking neurons within the reservoir computing paradigm\.

## 3Methods

### 3\.1Neuron Models

The LIF neuron is the standard building block of spiking reservoir computing\. Its membrane potentialvvdecays towards rest at a rate controlled by the time constantτm\\tau\_\{m\}, while integrating incoming inputI⁡\(t\)I\(t\):

v˙=−vτm\+I⁡\(t\),\\dot\{v\}=\-\\frac\{v\}\{\\tau\_\{m\}\}\+I\(t\),\(1\)wherev˙\\dot\{v\}denotes the time derivative ofvv\. Whenvvcrosses a thresholdθLIF\\theta\_\{\\text\{\\tiny LIF\}\}, the neuron emits a spikesLIF=1s^\{\\text\{\\tiny LIF\}\}=1andvvresets\. The decay term−v/τm\-v/\\tau\_\{m\}acts as a leaky integrator: it smooths out rapid fluctuations in the input, so the membrane potential reflects a low\-pass\-filtered version ofI⁡\(t\)I\(t\)\. As a consequence, the LIF neuron responds most strongly to low\-frequency inputs, regardless of the frequency content of the signal\.

The Harmonic Resonate\-and\-Fire \(HRF\) neuron\[[24](https://arxiv.org/html/2610.37906#bib.bib50),[23](https://arxiv.org/html/2610.37906#bib.bib18)\]replaces this LIF integration with the dynamics of a damped harmonic oscillator\. Its state is described by two coupled variables, a velocityxxand a positionyy:

x˙\\displaystyle\\dot\{x\}=I⁡\(t\)−γ​y−ε​x,\\displaystyle=I\(t\)\-\\gamma\\,y\-\\varepsilon\\,x,\(2\)y˙\\displaystyle\\dot\{y\}=x\.\\displaystyle=x\.\(3\)Hereγ\>0\\gamma\>0is the stiffness, which sets the natural oscillation frequencyω=γ\\omega=\\sqrt\{\\gamma\}of the neuron, andε\>0\\varepsilon\>0is the damping, which controls how quickly oscillations decay\. The neuron emits a spikesHRF=1s^\{\\text\{\\tiny HRF\}\}=1wheny\>θRFy\>\\theta\_\{\\text\{\\tiny RF\}\}\. The key difference from the LIF neuron is that the restoring force−γ​y\-\\gamma yin Eq\. \([2](https://arxiv.org/html/2610.37906#S3.E2)\) causes the membrane to oscillate rather than simply decay: the neuron accumulates energy from inputs that arrive in phase with its natural frequencyω\\omegaand attenuates inputs at other frequencies\. This gives the HRF neuron band\-pass rather than low\-pass characteristics: it is selectively sensitive to a specific frequency band determined byγ\\gammaandε\\varepsilon, rather than responding indiscriminately to all slow input\.

### 3\.2The HRF\-Res Model

Our proposed model, HRF\-Res, organisesNNneurons into two interacting populations, a LIF encoder layer and an HRF oscillatory reservoir, all with fixed, randomly initialised weights, as illustrated in Figure[1](https://arxiv.org/html/2610.37906#S3.F1)\. Only a linear readout is optimised, following the RC paradigm\[[4](https://arxiv.org/html/2610.37906#bib.bib12),[5](https://arxiv.org/html/2610.37906#bib.bib13)\]\. Both neuron models are simulated in discrete time via the Euler method\[[40](https://arxiv.org/html/2610.37906#bib.bib27)\]with timestepΔ​t\\Delta t\. The complete discrete\-time update at each timestepttis

𝐯t\+1\\displaystyle\\mathbf\{v\}\_\{t\+1\}=𝐯t\+Δ​t​\(−𝐯tτm\+𝐔𝐮t\+𝐖𝐬tHRF\+𝐛\),\\displaystyle=\\mathbf\{v\}\_\{t\}\+\\Delta t\\left\(\-\\frac\{\\mathbf\{v\}\_\{t\}\}\{\\tau\_\{m\}\}\+\\mathbf\{U\}\\mathbf\{u\}\_\{t\}\+\\mathbf\{W\}\\mathbf\{s\}\_\{t\}^\{\\text\{\\tiny HRF\}\}\+\\mathbf\{b\}\\right\),\(4\)𝐱t\+1\\displaystyle\\mathbf\{x\}\_\{t\+1\}=𝐱t\+Δ​t​\(𝐕𝐬tLIF−𝜸⊙𝐲t−𝜺⊙𝐱t\),\\displaystyle=\\mathbf\{x\}\_\{t\}\+\\Delta t\\left\(\\mathbf\{V\}\\mathbf\{s\}\_\{t\}^\{\\text\{\\tiny LIF\}\}\-\\boldsymbol\{\\gamma\}\\odot\\mathbf\{y\}\_\{t\}\-\\boldsymbol\{\\varepsilon\}\\odot\\mathbf\{x\}\_\{t\}\\right\),\(5\)𝐲t\+1\\displaystyle\\mathbf\{y\}\_\{t\+1\}=𝐲t\+Δ​t​𝐱t,\\displaystyle=\\mathbf\{y\}\_\{t\}\+\\Delta t\\,\\mathbf\{x\}\_\{t\},\(6\)where𝐮t∈ℝNinp\\mathbf\{u\}\_\{t\}\\in\\mathbb\{R\}^\{N\_\{\\mathrm\{inp\}\}\}is the external input,𝐯t,𝐱t,𝐲t∈ℝN\\mathbf\{v\}\_\{t\},\\mathbf\{x\}\_\{t\},\\mathbf\{y\}\_\{t\}\\in\\mathbb\{R\}^\{N\}are the LIF membrane potentials and the HRF velocity and position states respectively\.𝐬tLIF,𝐬tHRF∈\{0,1\}N\\mathbf\{s\}\_\{t\}^\{\\text\{\\tiny LIF\}\},\\mathbf\{s\}\_\{t\}^\{\\text\{\\tiny HRF\}\}\\in\\\{0,1\\\}^\{N\}are the LIF and HRF spike vectors,𝜸,𝜺∈ℝN\\boldsymbol\{\\gamma\},\\boldsymbol\{\\varepsilon\}\\in\\mathbb\{R\}^\{N\}collect the per\-neuron stiffness and damping coefficients, and⊙\\odotdenotes element\-wise multiplication\.𝐔∈ℝN×Ninp\\mathbf\{U\}\\in\\mathbb\{R\}^\{N\\times N\_\{\\mathrm\{inp\}\}\}is the input matrix,𝐕∈ℝN×N\\mathbf\{V\}\\in\\mathbb\{R\}^\{N\\times N\}the LIF\-to\-HRF synaptic matrix,𝐖∈ℝN×N\\mathbf\{W\}\\in\\mathbb\{R\}^\{N\\times N\}the HRF\-to\-LIF feedback matrix, and𝐛∈ℝN\\mathbf\{b\}\\in\\mathbb\{R\}^\{N\}a bias vector\.

![Refer to caption](https://arxiv.org/html/2610.37906v1/figure1_FINAL_paper.png)Figure 1:Architecture of our proposed HRF\-Res\.The external input𝐮t\\mathbf\{u\}\_\{t\}is projected into a LIF encoder population \(left, pink\) via the fixed matrix𝐔\\mathbf\{U\}\. LIF spikes𝐬tLIF\\mathbf\{s\}\_\{t\}^\{\\text\{\\tiny LIF\}\}are passed to the HRF oscillatory reservoir \(right, blue\) via𝐕\\mathbf\{V\}; each HRF neuron oscillates at its own resonance frequency and feeds spikes𝐬tHRF\\mathbf\{s\}\_\{t\}^\{\\text\{\\tiny HRF\}\}back to the LIF layer via𝐖\\mathbf\{W\}\. All weights are fixed; only the linear readout𝐫=𝐖o​𝐲T\+𝐛o\\mathbf\{r\}=\\mathbf\{W\}\_\{o\}\\mathbf\{y\}\_\{T\}\+\\mathbf\{b\}\_\{o\}is trained\.All weight matrices are randomly initialised and kept fixed throughout training\. The input matrix𝐔\\mathbf\{U\}and bias𝐛\\mathbf\{b\}are scaled by a global input gainν\\nu, while the recurrent matrix𝐖\\mathbf\{W\}is rescaled to a target spectral radiusρ\\rhofollowing standard reservoir practice\[[4](https://arxiv.org/html/2610.37906#bib.bib12),[26](https://arxiv.org/html/2610.37906#bib.bib28)\]\(see Appendix[A\.1](https://arxiv.org/html/2610.37906#A1.SS1)for full initialisation details\)\. HRF neuron parameters\(γi,εi\)\(\\gamma\_\{i\},\\varepsilon\_\{i\}\)are drawn independently across neurons, inducing a diversity of intrinsic time constants\. This heterogeneity allows the reservoir to simultaneously represent temporal structure at multiple timescales, improving the expressivity of the fixed dynamics \(see Appendix[A\.1](https://arxiv.org/html/2610.37906#A1.SS1)\)\. The timestepΔ​t\\Delta tis treated as a hyperparameter rather than a fixed integration step\[[22](https://arxiv.org/html/2610.37906#bib.bib14)\]\. HRF\-Res is implemented in Python using standard machine learning libraries, and all hyperparameters are selected via random search on a held\-out validation set \(see Appendix[A\.4](https://arxiv.org/html/2610.37906#A1.SS4)for the evaluation protocol and Appendix[A](https://arxiv.org/html/2610.37906#A1)for full parameter ranges and selected configurations\)\.

HRF\-Res supports two families of readout strategies: membrane\-based readouts, which operate on the continuous HRF position state𝐲t\\mathbf\{y\}\_\{t\}, and spike\-based readouts, which operate directly on𝐬tHRF\\mathbf\{s\}\_\{t\}^\{\\text\{\\tiny HRF\}\}\. This flexibility reflects the practical deployment context: membrane\-based readouts extract richer temporal information from the oscillatory dynamics and are well\-suited to software or hybrid implementations, while spike\-based readouts yield a fully event\-driven pipeline compatible with neuromorphic hardware end\-to\-end\. By default we use the final HRF state𝐲T\\mathbf\{y\}\_\{T\}as the readout feature, mapped to the output via𝐫=𝐖o​𝐲T\+𝐛o\\mathbf\{r\}=\\mathbf\{W\}\_\{o\}\\mathbf\{y\}\_\{T\}\+\\mathbf\{b\}\_\{o\}, where𝐖o∈ℝNout×N\\mathbf\{W\}\_\{o\}\\in\\mathbb\{R\}^\{N\_\{\\mathrm\{out\}\}\\times N\}and𝐛o∈ℝNout\\mathbf\{b\}\_\{o\}\\in\\mathbb\{R\}^\{N\_\{\\mathrm\{out\}\}\}are the weights and biases of the output layer, optimised with regularisation strengthCCtreated as a hyperparameter\. These are the*only*trainable parameters of HRF\-Res\. We conduct a systematic ablation over membrane\-based and spike\-based readout strategies in Section[4\.3](https://arxiv.org/html/2610.37906#S4.SS3), showing that the choice can be adapted to the target deployment scenario without significant accuracy loss\.

### 3\.3Energy Cost Model

A central motivation for spiking reservoir computing is energy\-efficient inference\. To enable a consistent comparison between spiking and non\-spiking models, we quantify computational cost using multiply\-accumulate operations \(MACs\) for non\-spiking networks and accumulate operations \(ACs\) for spiking networks\[[41](https://arxiv.org/html/2610.37906#bib.bib29),[42](https://arxiv.org/html/2610.37906#bib.bib30)\]\. MACs correspond to dense multiplication\-and\-accumulation operations performed at every timestep over all connections, independent of neural activity\. In contrast, ACs occur only when presynaptic spikes arrive and consist of adding synaptic contributions to the postsynaptic state, without runtime multiplication\. We measure spiking computation using synaptic operations \(SOPs\), defined as the number of spike\-triggered synaptic transmissions, with each SOP corresponding directly to one AC event\. This enables a unified comparison of spiking and non\-spiking models under a common hardware energy model\. We convert MACs and ACs into energy using standard CMOS estimates \(EMACE\_\{\\mathrm\{MAC\}\}andEACE\_\{\\mathrm\{AC\}\}\)\. Following standard practice in the SNN literature, energy figures count synaptic operations, which scale as𝒪⁡\(N2\)\\mathcal\{O\}\(N^\{2\}\)and dominate the cost of recurrent computation; the activity\-independent𝒪⁡\(N\)\\mathcal\{O\}\(N\)neuron state updates, together with full details, are reported in Appendix[A\.2](https://arxiv.org/html/2610.37906#A1.SS2)\.

### 3\.4Reservoir Characterisation

A key hypothesis of this work is that oscillatory neuron dynamics in a spiking reservoir produce frequency\-selective temporal representations that are more aligned with class structure than those of integrative dynamics, leading to improved classification\. We test this through two complementary analyses on frozen reservoirs, using a controlled baseline that isolates the effect of neuron dynamics\.

As a controlled baseline we introduce LIF\-Res: a model identical to HRF\-Res in every respect except that the HRF population \(blue in Figure[1](https://arxiv.org/html/2610.37906#S3.F1)\) is replaced by LIF neurons with heterogeneous membrane time constantsτm,i\\tau\_\{m,i\}and thresholdsθi\\theta\_\{i\}, drawn per neuron from log\-uniform distributions\. This mirrors theγi/εi\\gamma\_\{i\}/\\varepsilon\_\{i\}heterogeneity of the HRF reservoir, following prior work showing that heterogeneous time constants improve performance\[[12](https://arxiv.org/html/2610.37906#bib.bib31),[43](https://arxiv.org/html/2610.37906#bib.bib32)\]\. An independent hyperparameter search is conducted for LIF\-Res on each dataset using the same protocol as for HRF\-Res, ensuring the comparison reflects the peak performance of each neuron model rather than an arbitrary parameter transfer \(see Appendix[A\.5](https://arxiv.org/html/2610.37906#A1.SS5)\)\. Any observed difference therefore has a single cause: the oscillatory versus integrative nature of the reservoir dynamics\. The first analysis tests whether HRF neurons behave as band\-pass filters in a full spiking reservoir, beyond the continuous\-time, single\-neuron regime of Eqs\. \([2](https://arxiv.org/html/2610.37906#S3.E2)\)–\([3](https://arxiv.org/html/2610.37906#S3.E3)\)\. This property is not guaranteed to persist under discrete\-time integration, recurrent coupling, and heterogeneous random connectivity\. The second analysis evaluates whether such frequency selectivity yields representations that are intrinsically more discriminative than those of LIF\-Res, even without readout training\. This is a stricter criterion: frequency\-selective responses do not necessarily translate into features that improve classification\. Together, these analyses provide both an empirical characterisation of the HRF reservoir and evidence that this property is relevant for downstream tasks\.

#### Frequency tuning and Q\-factor\.

We measure the preferred input frequencyf∗f^\{\*\}of each reservoir neuron by sweeping 30 log\-spaced sinusoidal inputs through the full network and identifying the frequency maximising steady\-state response power via the Discrete Fourier Transform \(DFT\)\. Filter sharpness is quantified by the Q\-factorQ=f∗/Δ​f−3​dBQ=f^\{\*\}/\\Delta f\_\{\-3\\,\\mathrm\{dB\}\}, whereΔ​f−3​dB\\Delta f\_\{\-3\\,\\mathrm\{dB\}\}is the half\-power bandwidth around the peak\. A neuron is band\-pass resonant \(Q\>0Q\>0\) if both a lower and upper−3\-3dB crossing exist, and integrative \(Q=0Q=0\) otherwise\. Being dimensionless and independent ofΔ​t\\Delta t,QQis directly comparable across models and datasets\. Full sweep parameters are provided in Appendix[B](https://arxiv.org/html/2610.37906#A2), where we also verify that applying a Hann window before the DFT, which reduces spectral leakage that could broaden the measured peaks, leaves all results unchanged\.

#### Class discriminability without readout training \(CKA\)\.

We measure how well reservoir states are intrinsically aligned with the structure of the data using Centred Kernel Alignment \(CKA\)\[[44](https://arxiv.org/html/2610.37906#bib.bib33),[45](https://arxiv.org/html/2610.37906#bib.bib34)\], without training any readout\. Since CKA compares fixed\-length representations, we summarise each trajectory by its temporal mean reservoir state: for each test sampleiiwe extract the temporal mean reservoir state𝐲¯i=1T​∑t=1T𝐲t\(i\)∈ℝN\\bar\{\\mathbf\{y\}\}\_\{i\}=\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}\\mathbf\{y\}\_\{t\}^\{\(i\)\}\\in\\mathbb\{R\}^\{N\}, normalise each row to unit norm and stack these into𝐇∈ℝn×N\\mathbf\{H\}\\in\\mathbb\{R\}^\{n\\times N\}\. Because the HRF position oscillates around an equilibrium, averaging partially cancels the very oscillatory component that distinguishes HRF from LIF dynamics, making this a conservative choice for HRF\-Res\. The feature kernel is𝐊H=𝐇𝐇⊤\\mathbf\{K\}\_\{H\}=\\mathbf\{H\}\\mathbf\{H\}^\{\\top\}and the label kernel is\(𝐊Y\)i​j=𝟏\[yi=yj\]\(\\mathbf\{K\}\_\{Y\}\)\_\{ij\}=\\mathbf\{1\}\[y\_\{i\}=y\_\{j\}\]\. Both are centred with𝐊~=𝐇c​𝐊𝐇c\\tilde\{\\mathbf\{K\}\}=\\mathbf\{H\}\_\{c\}\\mathbf\{K\}\\mathbf\{H\}\_\{c\}where𝐇c=𝐈n−1n​𝟏𝟏⊤\\mathbf\{H\}\_\{c\}=\\mathbf\{I\}\_\{n\}\-\\frac\{1\}\{n\}\\mathbf\{1\}\\mathbf\{1\}^\{\\top\}, and CKA is computed as

CKA⁡\(𝐊H,𝐊Y\)=⟨𝐊~H,𝐊~Y⟩F⟨𝐊~H,𝐊~H⟩F​⟨𝐊~Y,𝐊~Y⟩F,\\mathrm\{CKA\}\(\\mathbf\{K\}\_\{H\},\\mathbf\{K\}\_\{Y\}\)=\\frac\{\\langle\\tilde\{\\mathbf\{K\}\}\_\{H\},\\tilde\{\\mathbf\{K\}\}\_\{Y\}\\rangle\_\{F\}\}\{\\sqrt\{\\langle\\tilde\{\\mathbf\{K\}\}\_\{H\},\\tilde\{\\mathbf\{K\}\}\_\{H\}\\rangle\_\{F\}\\,\\langle\\tilde\{\\mathbf\{K\}\}\_\{Y\},\\tilde\{\\mathbf\{K\}\}\_\{Y\}\\rangle\_\{F\}\}\},\(7\)where⟨⋅,⋅⟩F\\langle\\cdot,\\cdot\\rangle\_\{F\}is the Frobenius inner product\. A score of 1 indicates perfect class clustering; 0 indicates none\. Further implementation details, and controls showing that the results are robust to the number of samplesnnand to discarding an initial transient of the trajectory, are provided in Appendix[B\.2](https://arxiv.org/html/2610.37906#A2.SS2)\.

## 4Results

### 4\.1HRF Reservoirs Function as Distributed Band\-Pass Filters

We apply the analyses of Section[3\.4](https://arxiv.org/html/2610.37906#S3.SS4)to compare frequency tuning and representational discriminability in HRF\-Res and LIF\-Res across four datasets chosen to span diverse input modalities and temporal regimes: sMNIST\[[46](https://arxiv.org/html/2610.37906#bib.bib42)\]\(sequential images,T=784T=784,N=800N=800\), FordA\[[47](https://arxiv.org/html/2610.37906#bib.bib35)\]\(univariate time series,T=500T=500,N=800N=800\), SHD\[[48](https://arxiv.org/html/2610.37906#bib.bib36)\]\(audio spike trains,T=250T=250, 700 channels,N=3,000N=3\{,\}000\), and DVS Gesture\[[49](https://arxiv.org/html/2610.37906#bib.bib37)\]\(event\-based vision,T=200T=200, 2,048 channels,N=3,000N=3\{,\}000\)\.

#### Frequency tuning and Q\-factor \(Figures[2](https://arxiv.org/html/2610.37906#S4.F2)and[3](https://arxiv.org/html/2610.37906#S4.F3)B\)\.

The difference between the two reservoirs is immediate\. The fraction of resonant neurons \(Q\>0Q\>0\) is 98%, 100%, 62%, and 83% for HRF\-Res \(sMNIST, FordA, SHD, DVS Gesture\), compared to 23%, 9%, 15%, and 4% for LIF\-Res \(mean over three seeds; standard deviations at most 2 percentage points\)\. Beyond this expected gap, a key result is that many HRF neurons retain sharp, well\-defined resonance within the full recurrent network\. Among resonant units, Q\-factors are consistently higher for HRF \(4\.2 vs\. 3\.6, 4\.6 vs\. 2\.0, 2\.5 vs\. 0\.6, 3\.4 vs\. 1\.4\)\. Preferred\-frequency distributions further highlight this contrast: HRF neurons span a broad range of tuned frequencies \(standard deviation off∗f^\{\*\}: 0\.049, 0\.041, 0\.021, 0\.016 Hz\), whereas LIF neurons always collapse to the lowest frequency \(standard deviation: 0\.002–0\.008 Hz\), consistent with pure integrative behaviour\. On FordA, HRF responses concentrate around 0\.42 Hz, suggesting tuning to a dominant temporal scale in the signal\.

Figure 2:Band\-pass properties of HRF\-Res vs\. LIF\-Res\.\(A\)Fraction of resonant neurons \(Q\>0Q\>0\) per dataset\. HRF\-Res has 4–20×\\timesmore resonant neurons than LIF\-Res, confirming that band\-pass dynamics survive in a full spiking reservoir setting\.\(B\)Mean Q\-factor among resonant neurons\. HRF neurons form sharper band\-pass filters on all datasets, with the largest gaps on FordA \(4\.64\.6vs\.2\.02\.0\), DVS Gesture \(3\.43\.4vs\.1\.41\.4\) and SHD \(2\.52\.5vs\.0\.60\.6\)\. Bars show mean±\\pmstd over three seeds\.
#### Class discriminability \(Figure[3](https://arxiv.org/html/2610.37906#S4.F3)A\)\.

HRF\-Res achieves higher mean CKA than LIF\-Res on all four datasets\. The gap is largest on FordA \(\+322%\+322\\%\), followed by DVS Gesture \(\+13%\+13\\%\) and SHD \(\+9%\+9\\%\); on sMNIST \(\+9%\+9\\%\) the difference lies within the seed\-to\-seed variability\. Absolute scores are low by design, as untrained reservoirs are not optimised for class discrimination\. Notably, the ranking by CKA gap matches the ranking by absolute Q\-factor gap \(FordA\>\>DVS Gesture≈\\approxSHD\>\>sMNIST\), consistent with frequency selectivity driving representational discriminability\.

Taken together, these analyses confirm that HRF\-Res functions as a distributed bank of band\-pass filters that project temporal inputs onto frequency\-structured, class\-discriminative trajectories, a property that LIF\-Res cannot produce despite being given equally heterogeneous parameters and an independent hyperparameter search\. Section[4\.2](https://arxiv.org/html/2610.37906#S4.SS2)tests whether this representational advantage translates to a classification improvement\.

Figure 3:HRF\-Res produces more class\-discriminative representations\.\(A\)CKA between temporal mean reservoir states and category labels \(mean±\\pmstd over three seeds\)\. HRF\-Res achieves higher CKA than LIF\-Res on all four datasets\.\(B\)Distribution of per\-neuron preferred frequencies \(log scale\)\. HRF neurons show distributed tuning across a range of frequencies, consistent with a heterogeneous bank of band\-pass filters; LIF neurons collapse to the lowest sweep frequency, consistent with pure integration\.

### 4\.2HRF Neurons Improve Classification Across Modalities

We test whether the representational advantage of Section[4\.1](https://arxiv.org/html/2610.37906#S4.SS1)translates to classification accuracy, using the same four datasets and reservoir sizes, and the final reservoir state at timeTTas the readout feature\. HRF\-Res outperforms LIF\-Res on three of four datasets, with accuracy gains consistent with the predictions of Section[4\.1](https://arxiv.org/html/2610.37906#S4.SS1)\(Table[1](https://arxiv.org/html/2610.37906#S4.T1)\)\. The largest gap is on sMNIST \(\+26\.6%\+26\.6\\%,93\.64%93\.64\\%vs\.67\.00%67\.00\\%\), where HRF resonance is highest \(98% vs\. 23%\)\. SHD follows with\+10\.5\+10\.5\(83\.48%83\.48\\%vs\.72\.95%72\.95\\%\) and FordA with\+2\.6\+2\.6\(69\.72%69\.72\\%vs\.67\.15%67\.15\\%\)\. DVS Gesture is the exception, where the two models perform comparably \(71\.81±1\.59%71\.81\\pm 1\.59\\%vs\.73\.48±1\.12%73\.48\\pm 1\.12\\%\) and the overlapping standard deviations indicate no significant difference; we discuss this case in Section[5](https://arxiv.org/html/2610.37906#S5)\. HRF\-Res is also more energy\-efficient than LIF\-Res on most datasets despite achieving higher accuracy \(energy estimated via the framework of Section[3\.3](https://arxiv.org/html/2610.37906#S3.SS3)\), as LIF neurons appear to require higher firing rates to encode temporal information\. DVS Gesture is again the exception\. These results confirm the hypothesis put forward in Section[4\.1](https://arxiv.org/html/2610.37906#S4.SS1): equipping a spiking reservoir with oscillatory HRF neurons produces measurably richer representations than an equivalent LIF reservoir, and this representational advantage translates directly to classification accuracy across diverse temporal data modalities\.

Table 1:HRF\-Res vs\. LIF\-Res\.Both architectures are otherwise identical; hyperparameters for LIF\-Res are independently re\-tuned per dataset\. Results are mean±\\pmstd over 5 seeds with final reservoir state readout\. HRF\-Res outperforms or matches LIF\-Res on all four datasets, with accuracy gains up to\+26\.6%\+26\.6\\%and lower energy consumption on most datasets\. See Appendix[A\.5](https://arxiv.org/html/2610.37906#A1.SS5)for LIF\-Res architecture details and Appendix[A\.4](https://arxiv.org/html/2610.37906#A1.SS4)for the evaluation protocol\.DatasetModelAcc \(%\)Energy \(J\)sMNISTHRF\-Res93\.64±0\.25\\mathbf\{93\.64\\pm 0\.25\}\(1\.676±0\.034\)×10−4\(1\.676\\pm 0\.034\)\\times 10^\{\-4\}LIF\-Res67\.00±1\.0467\.00\\pm 1\.04\(2\.352±0\.048\)×10−4\(2\.352\\pm 0\.048\)\\times 10^\{\-4\}FordAHRF\-Res69\.72±0\.68\\mathbf\{69\.72\\pm 0\.68\}\(0\.790±0\.015\)×10−4\(0\.790\\pm 0\.015\)\\times 10^\{\-4\}LIF\-Res67\.15±0\.6667\.15\\pm 0\.66\(1\.478±0\.038\)×10−4\(1\.478\\pm 0\.038\)\\times 10^\{\-4\}SHDHRF\-Res83\.48±0\.44\\mathbf\{83\.48\\pm 0\.44\}\(1\.044±0\.021\)×10−3\(1\.044\\pm 0\.021\)\\times 10^\{\-3\}LIF\-Res72\.95±0\.5372\.95\\pm 0\.53\(1\.280±0\.210\)×10−3\(1\.280\\pm 0\.210\)\\times 10^\{\-3\}DVS GestureHRF\-Res71\.81±1\.5971\.81\\pm 1\.59\(1\.265±0\.025\)×10−3\(1\.265\\pm 0\.025\)\\times 10^\{\-3\}LIF\-Res73\.48±1\.12\\mathbf\{73\.48\\pm 1\.12\}\(0\.953±0\.074\)×10−3\(0\.953\\pm 0\.074\)\\times 10^\{\-3\}

### 4\.3Readout Strategy Ablation

The choice of readout strategy is known to significantly affect performance in reservoir computing\[[26](https://arxiv.org/html/2610.37906#bib.bib28)\], and carries practical implications for deployment: membrane\-based readouts operating on𝐲t\\mathbf\{y\}\_\{t\}suit software or hybrid implementations, while spike\-based readouts on𝐬tHRF\\mathbf\{s\}\_\{t\}^\{\\text\{\\tiny HRF\}\}yield a fully event\-driven neuromorphic pipeline\. Even within membrane\-based readouts, it is not clear which summary statistic of𝐲t\\mathbf\{y\}\_\{t\}best captures class\-relevant information\. We therefore compare six strategies \(Figure[4](https://arxiv.org/html/2610.37906#S4.F4)\): five membrane\-based operating on𝐲t\\mathbf\{y\}\_\{t\}\(the final state𝐲T\\mathbf\{y\}\_\{T\}, the temporal mean, the temporal root mean square, the temporal standard deviation, and a combined3​N3N\-dimensional vector concatenating the RMS, the standard deviation and the final state\), and one spike\-based readout using the mean firing rate of𝐬tHRF\\mathbf\{s\}\_\{t\}^\{\\text\{\\tiny HRF\}\}\. HereNNis the number of reservoir neurons \(Section[3\.2](https://arxiv.org/html/2610.37906#S3.SS2)\)\. The first four membrane\-based modes and the spike\-based mode all produceNN\-dimensional features, enabling a fair comparison\. On every dataset, the best membrane\-based readout outperforms the spike\-based alternative, indicating that the continuous oscillatory dynamics of𝐲t\\mathbf\{y\}\_\{t\}carry richer temporal information than mean spike rates\. Among membrane\-based modes, performance is stable across statistics on sMNIST and SHD, while temporal averaging outperforms the final state by∼\\sim4% on FordA, suggesting that aggregating over time reduces noise in univariate vibration signals\. The combined readout achieves the best or near\-best accuracy on all four datasets, at the cost of tripling the parameter count\.

Figure 4:Readout strategy ablation\.Test accuracy \(%, mean±\\pmstd over 5 trials\) for six readout strategies across four datasets\. Blue bars: membrane\-based readouts operating on the HRF position state𝐲t\\mathbf\{y\}\_\{t\}, using the final state \(𝐲T\\mathbf\{y\}\_\{T\}\), temporal mean, root mean square \(RMS\), or standard deviation \(Std\)\. Green bar: combined membrane\-based readout concatenating RMS, Std, and final state \(3​N3N\-dimensional\)\. Orange bar: spike\-based readout using the mean HRF spike rate\. Bold outline marks the best accuracy per dataset\. The best membrane\-based readout outperforms the spike\-based alternative on every dataset, and the combined readout achieves the best or near\-best accuracy on all datasets\.
### 4\.4Energy Efficiency of the Spiking Formulation

Having established that oscillatory spiking dynamics outperform integrative ones, we ask whether the spiking formulation itself is justified relative to a non\-spiking oscillatory reservoir, which is the relevant tradeoff when targeting deployment on neuromorphic hardware\. We compare HRF\-Res against RON\[[22](https://arxiv.org/html/2610.37906#bib.bib14)\], which shares the same core design but replaces LIF neurons with tanh\-activated units and HRF neurons with continuous harmonic oscillators\. For RON we use the hyperparameters reported in the original paper; for HRF\-Res we use the configurations selected via the random search described in Appendix[A](https://arxiv.org/html/2610.37906#A1)\. Some accuracy cost is expected, as continuous activations carry more information per timestep than binary spikes; the question is whether the energy saving, quantified using the framework of Section[3\.3](https://arxiv.org/html/2610.37906#S3.SS3), is large enough to justify the spiking formulation\. Both models are evaluated at matched reservoir size \(N=800N=800\) and readout \(final reservoir state𝐲T\\mathbf\{y\}\_\{T\}\) on four of the benchmarks from the original RON evaluation\[[22](https://arxiv.org/html/2610.37906#bib.bib14)\]: sequential images \(sMNIST, psMNIST, npCIFAR\-10\) and univariate time series \(FordA\)\.

As shown in Table[2](https://arxiv.org/html/2610.37906#S4.T2), HRF\-Res achieves 9–19×\\timesenergy reduction over RON across all four benchmarks, with accuracy losses of only 1\.8–4\.9 percentage points\. The saving is structural: the dominant𝒪⁡\(N2\)\\mathcal\{O\}\(N^\{2\}\)recurrent cost is gated by spiking activity rather than executed densely at every timestep, so it is expected to hold broadly across tasks and deployment scenarios\. Including the activity\-independent𝒪⁡\(N\)\\mathcal\{O\}\(N\)neuron state updates of both models, which add less than 10% to the energy of HRF\-Res, the reduction remains 9–17×\\times\(Appendix[A\.2](https://arxiv.org/html/2610.37906#A1.SS2)\)\. These results demonstrate that the spiking formulation of HRF\-Res achieves the energy efficiency required for edge deployment without sacrificing the representational power of oscillatory reservoir dynamics, offering a practical path to neuromorphic time series classification at a fraction of the inference cost of non\-spiking alternatives\.

Table 2:Energy efficiency: HRF\-Res vs\. RON\.All experiments withN=800N=800and final readout\. RON energy is computed from MAC counts \(grey\); HRF\-Res energy from SOPs \(green\), which cost∼\\sim5×\\timesless per operation than MACs in 45 nm CMOS\[[41](https://arxiv.org/html/2610.37906#bib.bib29)\]\. HRF\-Res achieves 9–19×\\timesenergy reduction over RON with accuracy losses of only 1\.8–4\.9 percentage points\.RON\[[22](https://arxiv.org/html/2610.37906#bib.bib14)\]HRF\-Res \(ours\)DatasetAcc \(%\)MACs \(×108\\times 10^\{8\}\)Energy \(×10−4\\times 10^\{\-4\}J\)Acc \(%\)SOPs \(×108\\times 10^\{8\}\)Energy \(×10−4\\times 10^\{\-4\}J\)ReductionsMNIST95\.44±0\.1695\.44\\pm 0\.165\.0245\.02423\.11±0\.6923\.11\\pm 0\.6993\.64±0\.2593\.64\\pm 0\.251\.862±0\.0371\.862\\pm 0\.0371\.676±0\.0421\.676\\pm 0\.04213\.8×13\.8\\timespsMNIST91\.33±0\.4791\.33\\pm 0\.474\.9214\.92122\.64±0\.6822\.64\\pm 0\.6887\.96±0\.3387\.96\\pm 0\.332\.093±0\.0422\.093\\pm 0\.0421\.884±0\.0461\.884\\pm 0\.04612\.0×12\.0\\timesFordA73\.15±7\.6773\.15\\pm 7\.673\.2043\.20414\.74±0\.4414\.74\\pm 0\.4469\.72±0\.6869\.72\\pm 0\.680\.878±0\.0250\.878\\pm 0\.0250\.790±0\.0150\.790\\pm 0\.01518\.7×18\.7\\timesnpCIFAR\-1038\.50±0\.3238\.50\\pm 0\.327\.1687\.16832\.97±0\.9932\.97\\pm 0\.9933\.57±0\.2733\.57\\pm 0\.273\.830±1\.2003\.830\\pm 1\.2003\.450±1\.0803\.450\\pm 1\.0809\.6×9\.6\\times

### 4\.5Comparison to Spiking Reservoir Baselines

We evaluate HRF\-Res against prior spiking reservoir methods on seven benchmarks chosen to match those reported by each baseline\[[12](https://arxiv.org/html/2610.37906#bib.bib31),[13](https://arxiv.org/html/2610.37906#bib.bib38),[32](https://arxiv.org/html/2610.37906#bib.bib39),[14](https://arxiv.org/html/2610.37906#bib.bib40)\], enabling direct comparison: univariate time series \(FordA, FordB, Wafer, Earthquakes\[[47](https://arxiv.org/html/2610.37906#bib.bib35)\]\), audio spike trains \(SHD\[[48](https://arxiv.org/html/2610.37906#bib.bib36)\]\), and neuromorphic vision \(DVS Gesture\[[49](https://arxiv.org/html/2610.37906#bib.bib37)\], N\-MNIST\[[50](https://arxiv.org/html/2610.37906#bib.bib43)\]\)\. Reservoir sizes are matched to each baseline where a direct comparison exists; smaller reservoirs are used elsewhere to demonstrate efficiency\. We use the combined readout for SHD, DVS Gesture, N\-MNIST, and larger\-reservoir FordA; for FordB, Wafer, and Earthquakes we use the final\-state readout, which achieves competitive results with fewer parameters\.

HRF\-Res achieves competitive or state\-of\-the\-art accuracy across all modalities \(Table[3](https://arxiv.org/html/2610.37906#S4.T3)\)\. On SHD, HRF\-Res with 6,000 neurons \(89\.78%89\.78\\%\) surpasses the 16,000\-neuron ELSM\[[12](https://arxiv.org/html/2610.37906#bib.bib31)\]at nearly3×3\\timesfewer neurons\. On FordB, Wafer, and Earthquakes, HRF\-Res with only 800 neurons outperforms all Dey et al\.\[[13](https://arxiv.org/html/2610.37906#bib.bib38)\]baselines using up to6×6\\timesmore neurons\. On FordA, HRF\-Res \(large,2,5002\{,\}500neurons,82\.05%82\.05\\%\) outperforms the Dey et al\. baseline at matched size, while even the small 800\-neuron variant \(73\.91%73\.91\\%\) remains competitive\. On N\-MNIST, HRF\-Res \(97\.13%97\.13\\%\) outperforms Patiño\-Saucedo et al\.\[[32](https://arxiv.org/html/2610.37906#bib.bib39)\]\(94\.43%94\.43\\%\) and approaches Biswas et al\.\[[14](https://arxiv.org/html/2610.37906#bib.bib40)\]\(98\.10%98\.10\\%\), who use a more complex ensemble architecture\. On DVS Gesture, HRF\-Res achieves81\.06%81\.06\\%versus85\.2%85\.2\\%for Biswas et al\. at matched neuron count; the gap is attributable to their spatial and temporal ensemble architecture combining multiple reservoirs, rather than neuron count alone\. All HRF\-Res results are averaged over 5 seeds with standard deviation, confirming consistent performance across a simple fixed random reservoir with no task\-specific adaptations\. Together, these results establish HRF\-Res as a competitive spiking reservoir model that generalises across diverse temporal modalities and input dimensionalities, whereas individual baselines typically cover only a subset of these settings\.

Table 3:Comparison with spiking reservoir baselines\.Neurons: number of reservoir units\. HRF\-Res results: mean±\\pmstd over 5 seeds\. HRF\-Res achieves competitive or state\-of\-the\-art accuracy across all modalities, in most cases with fewer neurons than prior methods\.DatasetMethodNeuronsAcc \(%\)SHDHRF\-Res6,00089\.78±0\.35\\mathbf\{89\.78\\pm 0\.35\}Deckers et al\.\[[12](https://arxiv.org/html/2610.37906#bib.bib31)\]16,00089\.3089\.30Biswas et al\.\[[14](https://arxiv.org/html/2610.37906#bib.bib40)\]3,00077\.8077\.80FordAHRF\-Res80073\.91±0\.7473\.91\\pm 0\.74HRF\-Res\(large\)2,50082\.05±0\.55\\mathbf\{82\.05\\pm 0\.55\}Dey et al\.\[[13](https://arxiv.org/html/2610.37906#bib.bib38)\]2,50080\.3780\.37FordBHRF\-Res80070\.55±0\.14\\mathbf\{70\.55\\pm 0\.14\}Dey et al\.\[[13](https://arxiv.org/html/2610.37906#bib.bib38)\]2,50064\.3264\.32WaferHRF\-Res80099\.33±0\.04\\mathbf\{99\.33\\pm 0\.04\}Dey et al\.\[[13](https://arxiv.org/html/2610.37906#bib.bib38)\]5,00098\.8598\.85EarthquakesHRF\-Res80077\.22±1\.10\\mathbf\{77\.22\\pm 1\.10\}Dey et al\.\[[13](https://arxiv.org/html/2610.37906#bib.bib38)\]4,50071\.9471\.94N\-MNISTHRF\-Res3,60097\.13±0\.1297\.13\\pm 0\.12Patiño\-Saucedo et al\.\[[32](https://arxiv.org/html/2610.37906#bib.bib39)\]4,09694\.4394\.43Biswas et al\.\[[14](https://arxiv.org/html/2610.37906#bib.bib40)\]3,60098\.10\\mathbf\{98\.10\}DVS GestureHRF\-Res4,00081\.06±0\.5581\.06\\pm 0\.55Biswas et al\.\[[14](https://arxiv.org/html/2610.37906#bib.bib40)\]4,00085\.20\\mathbf\{85\.20\}

## 5Conclusion

We introduced HRF\-Res, a spiking reservoir computing model based on Harmonic Resonate\-and\-Fire neurons, replacing conventional LIF dynamics with a heterogeneous population of oscillatory units\. We showed that HRF neurons act as distributed band\-pass filters within the reservoir, producing intrinsically more class\-discriminative representations prior to readout training\. This translates into consistent accuracy gains across diverse temporal benchmarks \(up to\+26\.6\+26\.6% over LIF\-Res\), while achieving up to19×19\\timeslower inference energy than a non\-spiking oscillatory baseline\. Together, these results highlight frequency\-selective dynamics as a powerful and underexplored computational principle for spiking reservoirs, enabling richer temporal representations without sacrificing efficiency\.

Several limitations remain: energy estimates are model\-based and not yet validated on neuromorphic hardware, the readout is trained offline, and a rigorous theoretical characterisation of HRF reservoir dynamics in terms of memory capacity, separation properties, and stability remains an open question\. Additionally, the representational advantage of HRF neurons is expected to be most pronounced for signals with identifiable frequency structure; performance gains may be smaller for tasks where class\-relevant information is not encoded in temporal frequency content\. This is consistent with DVS Gesture, where HRF\-Res does not improve over LIF\-Res, and where class identity plausibly depends more on spatial trajectories than on stable temporal frequency content\. Frequency selectivity also cuts both ways under distribution shift: HRF\-Res degrades more gracefully than LIF\-Res under additive noise, but is more sensitive to changes in the input sampling rate, which move the input spectrum away from the neurons’ preferred bands \(Appendix[C](https://arxiv.org/html/2610.37906#A3)\)\. All results are based on classification benchmarks and generalisation to other settings remains to be explored\. Two natural extensions are to learn the resonance parameters\(γi,εi\)\(\\gamma\_\{i\},\\varepsilon\_\{i\}\), for instance with surrogate gradients, trading part of the training efficiency of reservoir computing for task\-adapted frequency tuning, and to reintroduce the dynamic synapses of the original LSM formulation\[[5](https://arxiv.org/html/2610.37906#bib.bib13)\], which we omitted, as is common in recent spiking reservoirs, to isolate the effect of neuron dynamics\. Since short\-term synaptic plasticity filters inputs at the synapse while HRF neurons filter them at the membrane, the two mechanisms may well be complementary\.

## Acknowledgments and Disclosure of Funding

This research received no specific external funding\. The authors declare no competing interests\.

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## Appendix AExperimental Details and Hyperparameter Search

This appendix is self\-contained: it provides all implementation details, data preprocessing steps, training and evaluation procedures, and hyperparameter configurations needed to reproduce the results in the paper\.

### A\.1Architecture Details

#### Weight initialisation\.

The input matrix𝐔∈ℝN×Ninp\\mathbf\{U\}\\in\\mathbb\{R\}^\{N\\times N\_\{\\mathrm\{inp\}\}\}is sampled entry\-wise from𝒰⁡\(0,1\)\\mathcal\{U\}\(0,1\)and scaled by the input scaling hyperparameterν\>0\\nu\>0; the bias𝐛∈ℝN\\mathbf\{b\}\\in\\mathbb\{R\}^\{N\}is sampled from𝒰⁡\(−1,1\)\\mathcal\{U\}\(\-1,1\)and scaled by the sameν\\nu\. The HRF\-to\-LIF feedback matrix𝐖∈ℝN×N\\mathbf\{W\}\\in\\mathbb\{R\}^\{N\\times N\}is sampled from𝒰⁡\(−2,2\)\\mathcal\{U\}\(\-2,2\)and rescaled to a target spectral radiusρ\>0\\rho\>0\[[4](https://arxiv.org/html/2610.37906#bib.bib12),[26](https://arxiv.org/html/2610.37906#bib.bib28)\]\. The LIF\-to\-HRF synaptic matrix𝐕∈ℝN×N\\mathbf\{V\}\\in\\mathbb\{R\}^\{N\\times N\}is sampled from𝒰⁡\(−2,2\)\\mathcal\{U\}\(\-2,2\)without spectral normalisation\. All matrices are fixed after initialisation and never updated\. HRF\-Res is implemented in PyTorch with a logistic regression readout from scikit\-learn\. Each input sequence is processed starting from a zero\-initialised reservoir state\.

#### Heterogeneous HRF parameters\.

Per\-neuron stiffness and damping are drawn asγi∼𝒰⁡\(γ−Δ​γ/2,γ\+Δ​γ/2\)\\gamma\_\{i\}\\sim\\mathcal\{U\}\(\\gamma\-\\Delta\\gamma/2,\\,\\gamma\+\\Delta\\gamma/2\)andεi∼𝒰⁡\(ε−Δ​ε/2,ε\+Δ​ε/2\)\\varepsilon\_\{i\}\\sim\\mathcal\{U\}\(\\varepsilon\-\\Delta\\varepsilon/2,\\,\\varepsilon\+\\Delta\\varepsilon/2\), with values clamped to be strictly positive\. The natural resonance frequency of neuroniiisωi=γi\\omega\_\{i\}=\\sqrt\{\\gamma\_\{i\}\}\. The centres\(γ,ε\)\(\\gamma,\\varepsilon\)and widths\(Δ​γ,Δ​ε\)\(\\Delta\\gamma,\\Delta\\varepsilon\)reported in Tables[6](https://arxiv.org/html/2610.37906#A1.T6)–[7](https://arxiv.org/html/2610.37906#A1.T7)are treated as hyperparameters, inducing a diversity of intrinsic oscillation frequencies across the reservoir\. The timestepΔ​t\\Delta tis treated as a hyperparameter rather than a fixed integration step, as we are interested in the computational properties of the discrete\-time system in its own right\[[22](https://arxiv.org/html/2610.37906#bib.bib14)\]\.

#### Spike generation\.

The positionyyis not reset after a spike, so that the subthreshold oscillation continues undisturbed\. To prevent a neuron from firing at every consecutive timestep whileyyremains above threshold, the HRF threshold is made refractory: neuroniispikes whenyi\>θRF\+riy\_\{i\}\>\\theta\_\{\\text\{\\tiny RF\}\}\+r\_\{i\}, whererir\_\{i\}is incremented by one at each spike and decays asri←rie−Δt/τrefr\_\{i\}\\leftarrow r\_\{i\}\\,e^\{\-\\Delta t/\\tau\_\{\\mathrm\{ref\}\}\}withτref=0\.25\\tau\_\{\\mathrm\{ref\}\}=0\.25\.

#### Sparse connectivity\.

When the LIF→\\toHRF connectivitypV<1p\_\{V\}<1, a binary mask is sampled i\.i\.d\. withPr⁡\(entry=1\)=pV\\Pr\(\\text\{entry\}=1\)=p\_\{V\}and applied entry\-wise to𝐕\\mathbf\{V\}\. The HRF→\\toLIF connectivity is kept fully dense \(pW=1\.0p\_\{W\}=1\.0\) in all experiments\.

#### Sparse input projection\.

For high\-dimensional inputs \(SHD: 700 channels; DVS Gesture: 2 048 channels; N\-MNIST: 578 channels\), a sparse random projection is applied to𝐔\\mathbf\{U\}: a binary mask is sampled with each entry set to 1 with probabilityρinp\\rho\_\{\\mathrm\{inp\}\}\(the input density hyperparameter\), and the masked weights are rescaled by1/ρinp1/\\sqrt\{\\rho\_\{\\mathrm\{inp\}\}\}to preserve the expected norm of the input drive\. At least one active connection per neuron and per input channel is ensured by rejection sampling\.

### A\.2Energy Estimation Details

To quantify inference\-time efficiency, we estimate computational cost using synaptic operations \(SOPs\)\[[41](https://arxiv.org/html/2610.37906#bib.bib29),[42](https://arxiv.org/html/2610.37906#bib.bib30)\]\. SOPs measure the number of spike\-driven synaptic events and provide a hardware\-relevant proxy for energy consumption in spiking systems\.

On neuromorphic hardware, each presynaptic spike crossing a synapse corresponds to a single accumulate \(AC\) operation with energy costEAC=0\.9​pJE\_\{\\mathrm\{AC\}\}=0\.9\\,\\mathrm\{pJ\}\(45 nm CMOS\)\. In contrast, non\-spiking neural networks require multiply\-accumulate \(MAC\) operations at every timestep, with costEMAC=4\.6​pJE\_\{\\mathrm\{MAC\}\}=4\.6\\,\\mathrm\{pJ\}\[[41](https://arxiv.org/html/2610.37906#bib.bib29)\]\.

For a two\-population reservoir, the total number of synaptic operations per input sequence is given by:

SOPs=rHRF⋅N⋅T⋅Ch→l\+rLIF⋅N⋅T⋅Cl→h,\\mathrm\{SOPs\}=r\_\{\\text\{\\tiny HRF\}\}\\cdot N\\cdot T\\cdot C\_\{h\\to l\}\\;\+\\;r\_\{\\text\{\\tiny LIF\}\}\\cdot N\\cdot T\\cdot C\_\{l\\to h\},\(8\)whererHRFr\_\{\\text\{\\tiny HRF\}\}andrLIFr\_\{\\text\{\\tiny LIF\}\}denote the mean firing rates of HRF and LIF populations,NNis the number of reservoir neurons,TTis the sequence length, andCh→lC\_\{h\\to l\}andCl→hC\_\{l\\to h\}denote the number of active synaptic connections in each direction\.

This formulation reflects the event\-driven nature of spiking computation: synaptic cost scales with spike activity rather than with full connectivity\. In contrast, non\-spiking models perform dense matrix multiplications at every timestep, leading to computation that scales with network size and sequence length regardless of input sparsity\.

The internal HRF state dynamics require𝒪⁡\(N\)\\mathcal\{O\}\(N\)operations per timestep independent of spike activity, while recurrent synaptic communication is gated by sparse spiking events\. As a result, the dominant recurrent cost scales with effective firing activity rather than fullN2N^\{2\}connectivity, which is the primary source of energy savings in spiking reservoir models\.

All energy comparisons in the main paper are derived from SOP counts using the above formulation and the hardware constants defined above\.

#### Neuron state updates\.

The SOP count above covers the event\-driven synaptic cost\. Neuron state updates are instead paid at every timestep, regardless of spiking activity\. With the constant factors\(1−Δ​t​εi\)\(1\-\\Delta t\\,\\varepsilon\_\{i\}\),Δ​t​γi\\Delta t\\,\\gamma\_\{i\}and\(1−Δ​t/τm,i\)\(1\-\\Delta t/\\tau\_\{m,i\}\)precomputed, each HRF neuron requires 3 MACs per timestep \(Eqs\. \([5](https://arxiv.org/html/2610.37906#S3.E5)\)–\([6](https://arxiv.org/html/2610.37906#S3.E6)\)\), as it updates two coupled state variables, while each LIF neuron requires 1 MAC \(Eq\. \([4](https://arxiv.org/html/2610.37906#S3.E4)\)\); bias additions and threshold comparisons are neglected\. Per timestep, this amounts to4​N4NMACs for HRF\-Res \(HRF reservoir and LIF encoder\),2​N2NMACs for LIF\-Res, and3​N3NMACs for RON, whose oscillators follow the same update \(thetanh\\tanhevaluation is not counted\)\. These costs scale as𝒪⁡\(N\)\\mathcal\{O\}\(N\), whereas the synaptic cost scales as𝒪⁡\(N2\)\\mathcal\{O\}\(N^\{2\}\)\.

Table[4](https://arxiv.org/html/2610.37906#A1.T4)reports the resulting estimates for the comparison of Table[2](https://arxiv.org/html/2610.37906#S4.T2)\. State updates add less than 10% to the energy of HRF\-Res on all four datasets, and the reduction over RON remains 9–17×\\times\. For the comparison of Table[1](https://arxiv.org/html/2610.37906#S4.T1), including state updates increases the energy of HRF\-Res by at most0\.12×10−40\.12\\times 10^\{\-4\}J \(sMNIST\) and does not change the ranking between HRF\-Res and LIF\-Res on any dataset\.

Table 4:Energy including neuron state updates\(×10−4\\times 10^\{\-4\}J per sequence,N=800N=800\)\. “Synaptic” is the energy reported in Table[2](https://arxiv.org/html/2610.37906#S4.T2); “State” is the cost of the𝒪⁡\(N\)\\mathcal\{O\}\(N\)neuron state updates, computed as MACs\.RONHRF\-ResDatasetSynapticStateTotalSynapticStateTotalReductionsMNIST23\.110\.0923\.201\.6760\.1151\.79113\.0×13\.0\\timespsMNIST22\.640\.0922\.731\.8840\.1151\.99911\.4×11\.4\\timesFordA14\.740\.0614\.800\.7900\.0740\.86417\.1×17\.1\\timesnpCIFAR\-1032\.970\.1133\.083\.4500\.1473\.5979\.2×9\.2\\times

### A\.3Data Preprocessing

sMNIST / psMNIST\.MNIST images \(28×2828\\times 28pixels\) are read pixel\-by\-pixel to form sequences of lengthT=784T=784\. For psMNIST, a fixed random permutation \(seed 0, independent of model seed\) is applied to the pixel order before each forward pass\.

npCIFAR\-10\.Each32×3232\\times 32RGB image is read row\-by\-row, producingT=32T=32steps of 96\-dimensional input \(32×332\\times 3channels\)\. Following\[[22](https://arxiv.org/html/2610.37906#bib.bib14)\], 968 steps of fixed random Gaussian noise \(drawn once, shared across all trials\) are appended, givingT=1000T=1000total steps\.

FordA / FordB / Wafer / Earthquakes\.Raw univariate sequences from the UCR archive\[[47](https://arxiv.org/html/2610.37906#bib.bib35)\]are used without modification\. Sequence lengths are: FordA and FordB:T=500T=500; Wafer:T=152T=152; Earthquakes:T=512T=512\.

SHD\.Event streams are binned intoT=250T=250frames over a 1\.4 s window usingtonic, yielding sparse binary tensors of shape\(250×700\)\(250\\times 700\)\.

DVS Gesture\.Events are binned intoT=200T=200frames; spatial downsampling by factor 4 reduces128×128128\\times 128to32×3232\\times 32per polarity, giving 2 048 input channels per step\.

N\-MNIST\.Events are binned intoT=30T=30frames; spatial downsampling by factor 2 reduces34×3434\\times 34to17×1717\\times 17per polarity, giving 578 input channels\.

### A\.4Training and Evaluation Protocol

#### Readout\.

The only trained component is the linear readout: a logistic regression classifier \(scikit\-learn\) fitted with the L\-BFGS solver \(max\. 2 000 iterations\) andℓ2\\ell\_\{2\}regularisation strengthCC\. For SHD \(20 classes\) the multinomial objective is used; all other datasets use one\-vs\-rest\.

#### Data splits\.

For UCR datasets \(FordA, FordB, Wafer, Earthquakes\), the full provided training partition is used for readout training\. For all other datasets the standard train/test splits are used\. Hyperparameter search uses a held\-out validation split of 10% of the training set \(stratified\); the test set is never accessed during search\.

#### Evaluation\.

All results are mean±\\pmstd over five independent trials\. Each trial uses a different random seed controlling all fixed weight matrices \(𝐔\\mathbf\{U\},𝐕\\mathbf\{V\},𝐖\\mathbf\{W\},𝐛\\mathbf\{b\}\) and, for neuromorphic datasets, the sparse input projection mask\. The readout is re\-fitted from scratch in each trial\. The same five seeds are used for HRF\-Res and LIF\-Res to ensure a controlled comparison\.

#### Hyperparameter search\.

Each configuration is evaluated on the validation split with a single seed; the best configuration is then re\-evaluated with five seeds on the full train/test split\. Parameters spanning multiple orders of magnitude \(Δ​t\\Delta t,γ\\gamma,ε\\varepsilon,sinps\_\{\\mathrm\{inp\}\},θenc\\theta\_\{\\mathrm\{enc\}\},θres\\theta\_\{\\mathrm\{res\}\},ρinp\\rho\_\{\\mathrm\{inp\}\}\) are sampled log\-uniformly; all others uniformly\. The regularisationCCis sampled from a small discrete set\. The number of search trials per dataset is reported in the corresponding table caption\.

#### Compute\.

All experiments were run on a single NVIDIA A100 GPU \(40 GB\)\. Feature extraction per trial ranges from∼\\sim10 s \(sMNIST,N=800N=800\) to∼\\sim15 min \(N\-MNIST,N=3600N=3600\)\. Total compute across all experiments \(search and final evaluation\) is approximately 500 GPU\-hours\.

### A\.5LIF\-Res Baseline

LIF\-Res is architecturally identical to HRF\-Res except that the HRF reservoir is replaced by a second LIF population\. It consists of a LIF encoder and a LIF reservoir, both of sizeNN, connected by the same weight matrices𝐔\\mathbf\{U\},𝐕\\mathbf\{V\},𝐖\\mathbf\{W\}with the same initialisation and sparsity conventions as HRF\-Res\. The reservoir LIF neurons have heterogeneous membrane time constantsτm,i\\tau\_\{m,i\}and thresholdsθres,i\\theta\_\{\\mathrm\{res\},i\}, drawn per neuron from log\-uniform distributions centred on hyperparametersτm±Δ​τm/2\\tau\_\{m\}\\pm\\Delta\\tau\_\{m\}/2andθres±Δ​θres/2\\theta\_\{\\mathrm\{res\}\}\\pm\\Delta\\theta\_\{\\mathrm\{res\}\}/2respectively, mirroring theγi/εi\\gamma\_\{i\}/\\varepsilon\_\{i\}heterogeneity of HRF\-Res\. Soft reset is used: upon spiking, the membrane potential is decremented by the threshold rather than reset to zero\. The encoder time constant is fixed toτenc=20\\tau\_\{\\mathrm\{enc\}\}=20for all datasets, identical to HRF\-Res\. An independent hyperparameter search is conducted for LIF\-Res on each dataset using the same protocol as for HRF\-Res, with 80 trials for SHD and 100 trials for all other datasets, all using the final reservoir state for the readout\. Full search ranges and selected configurations are in Table[5](https://arxiv.org/html/2610.37906#A1.T5)\.

Table 5:Hyperparameter search ranges and selected configurations for theLIF\-Resbaseline\. All datasets usereadout\_mode = finaland the same logistic\-regression readout as HRF\-Res \(Appendix[A\.4](https://arxiv.org/html/2610.37906#A1.SS4)\)\. For each*searched*parameter, the search range is shown initalicsabove the selected value \(bold\)\.*Fixed*parameters are reported without a range\. Ranges are log\-uniform for parameters spanning orders of magnitude, and uniform otherwise\. The encoder membrane time constant is fixed toτenc=20\\tau\_\{\\mathrm\{enc\}\}=20for all datasets\. The search comprised 80 random trials for SHD and 100 trials for all other datasets\.DatasetFixed parametersSearched parameters\(range /selected\)NNStepsρinp\\rho\_\{\\mathrm\{inp\}\}LIF→\\toHRFconn\.HRF→\\toLIFconn\.Δ​t\\Delta tρ\\rhosinps\_\{\\mathrm\{inp\}\}θenc\\theta\_\{\\mathrm\{enc\}\}τm\\tau\_\{m\}Δ​τm\\Delta\\tau\_\{m\}θres\\theta\_\{\\mathrm\{res\}\}Δ​θres\\Delta\\theta\_\{\\mathrm\{res\}\}CCsMNIST8007841\.01\.00\.20\.21\.01\.0\[0\.03, 0\.15\]0\.034\[0\.85, 1\.05\]0\.976\[0\.5, 5\.0\]0\.847\[0\.01, 0\.20\]0\.151\[5, 50\]36\.19\[3, 40\]3\.60\[10−310^\{\-3\}, 0\.05\]0\.00254\[10−310^\{\-3\}, 0\.04\]0\.01180\{10−310^\{\-3\},10−210^\{\-2\},10−110^\{\-1\}, 1, 10\}0\.01FordA8005001\.01\.00\.20\.21\.01\.0\[0\.02, 0\.30\]0\.068\[0\.60, 1\.00\]0\.875\[0\.3, 5\.0\]4\.036\[0\.01, 0\.30\]0\.146\[10, 80\]36\.68\[5, 60\]25\.85\[0\.005, 0\.20\]0\.157\[0\.002, 0\.15\]0\.01254\{10−310^\{\-3\},10−210^\{\-2\},10−110^\{\-1\}, 1, 10\}0\.1SHD30002500\.0360\.0360\.20\.21\.01\.0\[0\.05, 1\.0\]0\.084\[0\.70, 1\.30\]1\.172\[0\.02, 1\.0\]0\.196\[0\.03, 0\.50\]0\.419\[5, 60\]13\.01\[3, 50\]2\.38\[10−310^\{\-3\}, 0\.10\]0\.09598\[10−310^\{\-3\}, 0\.04\]0\.00375\{10−310^\{\-3\},10−210^\{\-2\},10−110^\{\-1\}, 1\}0\.01DVS Gesture30002000\.0310\.0311\.01\.01\.01\.0\[0\.05, 0\.40\]0\.056\[0\.85, 1\.30\]0\.866\[0\.01, 0\.15\]0\.031\[0\.10, 3\.0\]0\.533\[2, 100\]58\.59\[1, 80\]23\.17\[10−310^\{\-3\}, 0\.10\]0\.02224\[10−310^\{\-3\}, 0\.08\]0\.00194\{10−410^\{\-4\}, 3×10−4\\times 10^\{\-4\},10−310^\{\-3\}, 3×10−3\\times 10^\{\-3\},10−210^\{\-2\}\}0\.01
Notation\.NN: reservoir size\. Steps: sequence length \(timesteps\)\.ρinp\\rho\_\{\\mathrm\{inp\}\}: input spike density \(fraction of active input channels per step\)\. LIF→\\toHRF / HRF→\\toLIF conn\.: connection density of𝐕\\mathbf\{V\}and𝐖\\mathbf\{W\}respectively \(see text above\)\.Δ​t\\Delta t: simulation timestep\.ρ\\rho: spectral radius of𝐖\\mathbf\{W\}\.sinps\_\{\\mathrm\{inp\}\}: input scaling factorν\\nu\.θenc\\theta\_\{\\mathrm\{enc\}\}: encoder LIF firing threshold\.τm\\tau\_\{m\}: centre of the log\-uniform distribution of reservoir membrane time constants\.Δ​τm\\Delta\\tau\_\{m\}: width of that distribution \(heterogeneity\)\.θres\\theta\_\{\\mathrm\{res\}\}: centre of the log\-uniform distribution of reservoir firing thresholds\.Δ​θres\\Delta\\theta\_\{\\mathrm\{res\}\}: width of that distribution \(heterogeneity\)\.CC: inverseℓ2\\ell\_\{2\}regularisation strength of the readout \(selected from the discrete candidate set shown\)\.

### A\.6HRF\-Res: Hyperparameter Configurations

Tables[6](https://arxiv.org/html/2610.37906#A1.T6)and[7](https://arxiv.org/html/2610.37906#A1.T7)report full search ranges and selected configurations for HRF\-Res\. Table[6](https://arxiv.org/html/2610.37906#A1.T6)covers the six datasets used in the ablation studies \(Sections[4\.2](https://arxiv.org/html/2610.37906#S4.SS2)and[4\.4](https://arxiv.org/html/2610.37906#S4.SS4)\): sMNIST, FordA, SHD, DVS Gesture, psMNIST, and npCIFAR\-10\. Table[7](https://arxiv.org/html/2610.37906#A1.T7)covers the datasets used in the comparison against spiking reservoir baselines \(Section[4\.5](https://arxiv.org/html/2610.37906#S4.SS5)\): FordB, Wafer, Earthquakes, and N\-MNIST, together with the larger\-reservoir configurations of FordA, SHD, and DVS Gesture\.

For all datasets, hyperparameters were selected by random search on a held\-out validation set \(log\-uniform sampling for parameters spanning orders of magnitude, uniform otherwise\)\. The LIF→\\toHRF connectivity \(fraction of non\-zero entries in𝐕\\mathbf\{V\}\) and, for high\-dimensional\-input datasets, the input density \(ρinp\\rho\_\{\\mathrm\{inp\}\}\) were included as searched parameters\. The HRF→\\toLIF connectivity was kept fully dense \(1\.01\.0\) throughout, and the encoder membrane time constant fixed toτenc=20\\tau\_\{\\mathrm\{enc\}\}=20\.

For datasets with two reservoir sizes, oscillator parameters \(Δ​t\\Delta t,γ\\gamma,ε\\varepsilon,Δ​γ\\Delta\\gamma,Δ​ε\\Delta\\varepsilon,θres\\theta\_\{\\mathrm\{res\}\}\) are transferred from the smaller\-NNsearch without modification \(marked†\\dagger\); only size\-sensitive parameters \(sinps\_\{\\mathrm\{inp\}\},ρ\\rho,θenc\\theta\_\{\\mathrm\{enc\}\},pVp\_\{V\},CC\) are re\-searched\. FordB, Wafer, and Earthquakes share the search space of FordA \(N=800N=800\), differing only inθenc\\theta\_\{\\mathrm\{enc\}\}\(FordB, Wafer: 1\.0; Earthquakes: 0\.2\)\. The Combined readout \(3​N3N\-dimensional concatenation of RMS, standard deviation, and final state of𝐲t\\mathbf\{y\}\_\{t\}; Section[4\.3](https://arxiv.org/html/2610.37906#S4.SS3)\) is used for larger\-reservoir configurations and N\-MNIST; the Final readout for all ablation datasets\. For each searched parameter we report the search range in square brackets \(continuous range, sampled log\-uniformly or uniformly as described in Appendix[A\.4](https://arxiv.org/html/2610.37906#A1.SS4)\) or curly braces \(discrete set\), followed by the selected value in bold\.

Table 6:HRF\-Reshyperparameter search ranges and selected configurations for the ablation datasets\. Each searched parameter is shown asrange/selected\. Fixed parameters have no range\.DatasetFixedSearched\(range /selected\)NNStepsρinp\\rho\_\{\\mathrm\{inp\}\}HRF→\\toLIFconn\.ReadoutLIF→\\toHRFconn\.Δ​t\\Delta tγ\\gammaε\\varepsilonΔ​γ\\Delta\\gammaΔ​ε\\Delta\\varepsilonsinps\_\{\\mathrm\{inp\}\}ρ\\rhoθenc\\theta\_\{\\mathrm\{enc\}\}θres\\theta\_\{\\mathrm\{res\}\}CCsMNIST8007841\.01\.0Final\{\.1,\.2,\.5,1\} /\.2\[\.03,\.15\]/\.042\[1,5\]/2\.7\[\.02,\.3\]/\.08\[\.5,3\.5\]/2\.0\[\.1,1\.5\]/1\.0\[\.5,5\]/2\.0\[\.85,1\.05\]/\.99\[\.01,\.2\]/\.05\[10−310^\{\-3\},\.05\]/\.005\{\.001,\.01,\.1,1,10\} /1\.0FordA \(NN​=​800\)8005001\.01\.0Final\{\.1,\.2,\.5,1\} /1\.0\[\.005,\.5\]/\.051\[\.01,10\]/7\.01\[10−310^\{\-3\},1\]/\.153\[\.1,5\]/3\.01\[0,\.5\]/\.419\[\.01,10\]/\.625\[\.3,1\.1\]/\.749\[\.005,1\]/\.082\[5​e−45\\mathrm\{e\}\{\-4\},\.5\]/\.001\{\.001,\.01,\.1,1,10\} /\.1SHD \(NN​=​3000\)3000250\[\.02,\.3\]/\.0361\.0Final\{\.1,\.2,\.5,1\} /\.2\[\.2,\.8\]/\.223\[\.01,\.5\]/\.036\[\.02,\.12\]/\.060\[\.02,\.5\]/\.268\[0,\.08\]/\.063\[\.02,1\.8\]/\.230\[\.7,1\.5\]/1\.16\[\.03,1\]/1\.00\[10−310^\{\-3\},\.02\]/\.013\{\.001,\.01,\.1,1\} /\.01DVS \(NN​=​3000\)3000200\[\.01,\.15\]/\.0311\.0Final\{\.1,\.2,\.5,1\} /1\.0\[\.15,\.4\]/\.259\[\.01,\.15\]/\.046\[\.003,\.08\]/\.035\[\.01,\.15\]/\.130\[\.01,\.10\]/\.099\[\.01,\.12\]/\.113\[1,1\.6\]/1\.58\[\.3,3\]/2\.97\[\.005,\.05\]/\.036\{10−410^\{\-4\},…,\.01\} /\.01psMNIST8007841\.01\.0Final\{\.1,\.2,\.5,1\} /\.2\[\.03,\.15\]/\.047\[1,6\]/2\.62\[\.05,\.8\]/\.240\[1,5\]/3\.84\[\.5,3\]/1\.86\[1,8\]/3\.67\[1,2\]/1\.55\[\.01,\.2\]/\.050\[10−310^\{\-3\},\.05\]/\.005\{\.001,\.01,\.1,1,10\} /1\.0npCIFAR\-1080010001\.01\.0Final\{\.1,\.2,\.5,1\} /\.2\[\.01,\.2\]/\.055\[\.01,2\]/\.115\[\.01,2\]/\.178\[\.1,2\]/1\.42\[0,1\]/\.963\[10−310^\{\-3\},\.5\]/\.058\[\.5,1\.5\]/1\.47\[\.01,1\]/\.124\[10−310^\{\-3\},\.5\]/\.154\{\.001,\.01,\.1,1,10\} /1\.0
Notation\.NN: reservoir size\. Steps: sequence length\.ρinp\\rho\_\{\\mathrm\{inp\}\}: input density \(fraction of input channels per neuron;1\.01\.0= dense; shown as range/selected when searched\)\. LIF→\\toHRF / HRF→\\toLIF: connection density of𝐕\\mathbf\{V\}and𝐖\\mathbf\{W\}\.Δ​t\\Delta t: timestep\.γ\\gamma/ε\\varepsilon: centre of HRF stiffness/damping distributions \(ω=γ\\omega\\\!=\\\!\\sqrt\{\\gamma\}\)\.Δ​γ\\Delta\\gamma/Δ​ε\\Delta\\varepsilon: widths \(heterogeneity\)\.sinps\_\{\\mathrm\{inp\}\}: input scaling\.ρ\\rho: spectral radius of𝐖\\mathbf\{W\}\.θenc\\theta\_\{\\mathrm\{enc\}\}/θres\\theta\_\{\\mathrm\{res\}\}: LIF encoder / HRF reservoir thresholds\.CC: inverseℓ2\\ell\_\{2\}regularisation strength of the readout\.

Table 7:HRF\-Reshyperparameter search ranges and selected configurations for the baseline comparison datasets \(Section[4\.5](https://arxiv.org/html/2610.37906#S4.SS5)\)\. Larger\-reservoir variants of FordA, SHD, and DVS Gesture reuse oscillator parameters from Table[6](https://arxiv.org/html/2610.37906#A1.T6)\(marked†\\dagger\) and re\-search only size\-sensitive parameters\. FordB, Wafer, and Earthquakes share the same search space as FordA \(NN​=​800\)\. Each searched parameter is shown asrange/selected\. Fixed parameters have no range\.†Transferred from smaller\-NNsearch\.*Combined*:3​N3N\-dimensional readout \(RMS \+ Std \+ Final of𝐲t\\mathbf\{y\}\_\{t\}; Section[4\.3](https://arxiv.org/html/2610.37906#S4.SS3)\)\.DatasetFixedSearched\(range /selected\)NNStepsρinp\\rho\_\{\\mathrm\{inp\}\}HRF→\\toLIFconn\.ReadoutLIF→\\toHRFconn\.Δ​t\\Delta tγ\\gammaε\\varepsilonΔ​γ\\Delta\\gammaΔ​ε\\Delta\\varepsilonsinps\_\{\\mathrm\{inp\}\}ρ\\rhoθenc\\theta\_\{\\mathrm\{enc\}\}θres\\theta\_\{\\mathrm\{res\}\}CCFordA \(NN​=​2500\)25005001\.01\.0Combined\[\.001,\.5\]/\.001\.051†\.051^\{\\dagger\}7\.01†7\.01^\{\\dagger\}\.153†\.153^\{\\dagger\}3\.01†3\.01^\{\\dagger\}\.419†\.419^\{\\dagger\}\[\.2,4\]/2\.13\[\.6,1\.05\]/\.985\[\.5,2\.5\]/1\.21\.001†\.001^\{\\dagger\}\[5​e−45\\mathrm\{e\}\{\-4\},\.05\]/\.0005SHD \(NN​=​6000\)6000250\.036†\.036^\{\\dagger\}1\.0Combined1\.0†1\.0^\{\\dagger\}\.223†\.223^\{\\dagger\}\.036†\.036^\{\\dagger\}\.060†\.060^\{\\dagger\}\.268†\.268^\{\\dagger\}\.063†\.063^\{\\dagger\}\.230†\.230^\{\\dagger\}1\.16†1\.16^\{\\dagger\}1\.00†1\.00^\{\\dagger\}\.013†\.013^\{\\dagger\}\.01†\.01^\{\\dagger\}DVS \(NN​=​4000\)4000200\[\.01,\.10\]/\.0111\.0Combined\{\.1,\.2,\.5,1\} /\.2\.259†\.259^\{\\dagger\}\.046†\.046^\{\\dagger\}\.035†\.035^\{\\dagger\}\.130†\.130^\{\\dagger\}\.099†\.099^\{\\dagger\}\[\.02,\.3\]/\.112\[\.85,1\.6\]/1\.38\[\.5,5\]/4\.44\[\.02,\.2\]/\.093\{10−410^\{\-4\},3e\-4,10−310^\{\-3\},3e\-3,\.01,\.03\} /\.01FordB8005001\.01\.0Final\{\.1,\.2,\.5,1\} /\.2\[\.005,\.5\]/\.2\[\.01,10\]/1\.88\[10−310^\{\-3\},1\]/\.022\[\.1,5\]/2\.64\[0,\.5\]/\.068\[\.01,10\]/1\.76\[\.3,1\.1\]/\.95\[\.005,1\]/1\.0\[5​e−45\\mathrm\{e\}\{\-4\},\.5\]/\.005\{\.001,\.01,\.1,1,10\} /1\.0Wafer8001521\.01\.0Final\{\.1,\.2,\.5,1\} /\.2\[\.005,\.5\]/\.2\[\.01,10\]/1\.88\[10−310^\{\-3\},1\]/\.022\[\.1,5\]/2\.64\[0,\.5\]/\.068\[\.01,10\]/1\.76\[\.3,1\.1\]/\.95\[\.005,1\]/1\.0\[5​e−45\\mathrm\{e\}\{\-4\},\.5\]/\.005\{\.001,\.01,\.1,1,10\} /1\.0Earthquakes8005121\.01\.0Final\{\.1,\.2,\.5,1\} /\.2\[\.005,\.5\]/\.2\[\.01,10\]/1\.88\[10−310^\{\-3\},1\]/\.022\[\.1,5\]/2\.64\[0,\.5\]/\.068\[\.01,10\]/1\.76\[\.3,1\.1\]/\.95\[\.005,1\]/\.2\[5​e−45\\mathrm\{e\}\{\-4\},\.5\]/\.005\{\.001,\.01,\.1,1,10\} /1\.0N\-MNIST360030\[\.01,\.20\]/\.0181\.0Combined\{\.1,\.2,\.5,1\} /\.2\.109\.109\.021\.362\.083\[\.02,\.5\]/\.210\[\.8,1\.6\]/1\.05\[\.05,\.5\]/\.494\[\.01,\.15\]/\.058\{\.001,\.005,\.01,\.05,\.1,\.5\} /\.05
Notationas in Table[6](https://arxiv.org/html/2610.37906#A1.T6)\. For SHD \(NN​=​6000\), all parameters are transferred from theNN​=​3000 search; only reservoir size and readout mode changed\. FordB, Wafer, and Earthquakes share the same search space as FordA \(NN​=​800\);θenc\\theta\_\{\\mathrm\{enc\}\}differs:1\.01\.0for FordB and Wafer,0\.20\.2for Earthquakes\. For DVS \(NN​=​4000\) and N\-MNIST,ρinp\\rho\_\{\\mathrm\{inp\}\}was also searched \(shown in the fixed column as range/selected\)\. N\-MNIST uses spatial downsampling factor 2 \(17×\\times17×\\times2 = 578 input channels\)\.

## Appendix BReservoir Characterisation: Supplementary Details

### B\.1Frequency Sweep Parameters

The frequency sweep analysis sweeps 30 log\-spaced frequencies per dataset\. For each frequencyff, a sinusoidal inputx⁡\(t\)=sin⁡\(2​π​f⋅t⋅Δ​t\)x\(t\)=\\sin\(2\\pi f\\cdot t\\cdot\\Delta t\)is broadcast identically across allNinpN\_\{\\mathrm\{inp\}\}input channels and the network is run forTsweepT\_\{\\mathrm\{sweep\}\}timesteps\. The first half of the trajectory is discarded as transient; each neuron’s preferred frequencyf∗f^\{\*\}is identified as the frequency maximising its steady\-state response power, estimated via the Discrete Fourier Transform \(DFT\) of the second half\. The same sweep range\[fmin,fmax\]\[f\_\{\\min\},f\_\{\\max\}\]is applied to both HRF\-Res and LIF\-Res for each dataset; each model uses its own independently tunedΔ​t\\Delta t\.

Table 8:Frequency sweep parameters and modelΔ​t\\Delta tvalues per dataset\.Datasetfminf\_\{\\min\}\(Hz\)fmaxf\_\{\\max\}\(Hz\)TsweepT\_\{\\mathrm\{sweep\}\}Δ​t\\Delta t\[HRF\-Res\]Δ​t\\Delta t\[LIF\-Res\]sMNIST0\.012\.06,0000\.0420\.034FordA0\.012\.06,0000\.0510\.068SHD0\.0012\.050,0000\.2230\.084DVS Gesture0\.0012\.050,0000\.2590\.056The largerTsweepT\_\{\\mathrm\{sweep\}\}and smallerfminf\_\{\\min\}for SHD and DVS Gesture reflect the slower intrinsic timescales of these datasets: the hyperparameter search selects largerΔ​t\\Delta tvalues to match the longer temporal dependencies present in audio spike trains and event\-camera streams\. Using a shorter sweep or higherfminf\_\{\\min\}for these datasets would fail to resolve the low\-frequency resonances of the HRF neurons, underestimating their Q\-factors\.

The Q\-factor is computed per neuron from the response power matrixR∈ℝNfreqs×NR\\in\\mathbb\{R\}^\{N\_\{\\mathrm\{freqs\}\}\\times N\}as follows\. For each neuronjj, the peak responsePpeakP\_\{\\mathrm\{peak\}\}is identified at its preferred frequencyf∗f^\{\*\}\. The half\-power bandwidthΔ​f−3​dB\\Delta f\_\{\-3\\,\\mathrm\{dB\}\}is estimated by finding the lower and upper frequency crossings whereRf​j=Ppeak/2R\_\{fj\}=P\_\{\\mathrm\{peak\}\}/2, using linear interpolation on the log\-frequency axis\.Q=f∗/Δ​f−3​dBQ=f^\{\*\}/\\Delta f\_\{\-3\\,\\mathrm\{dB\}\}is set to zero if either crossing does not exist, indicating purely integrative \(low\-pass\) behaviour with no well\-defined resonance peak\. We emphasise that the sweep provides a mechanistic characterisation of each reservoir’s dynamical repertoire and does not constitute a direct model\-to\-model comparison of absolute frequency scales, since the two models have independently tunedΔ​t\\Delta tvalues\.

#### Effect of spectral windowing\.

The results in Section[4\.1](https://arxiv.org/html/2610.37906#S4.SS1)compute the DFT on the steady\-state trajectory without an explicit window\. To verify that spectral leakage does not affect our conclusions, we repeated the analysis applying a Hann window before the DFT \(Table[9](https://arxiv.org/html/2610.37906#A2.T9)\)\. Windowing shifts the estimates slightly but leaves every qualitative conclusion unchanged: on all four datasets, HRF\-Res has a far larger fraction of resonant neurons, higher Q\-factors, and a broader distribution of preferred frequencies than LIF\-Res\.

Table 9:Frequency\-selectivity analysis with a Hann window applied before the DFT \(mean±\\pmstd over three seeds\)\.DatasetModelResonant \(%\)MeanQQStd off∗f^\{\*\}\(Hz\)sMNISTHRF\-Res89\.0±1\.189\.0\\pm 1\.14\.38±0\.044\.38\\pm 0\.040\.0829±0\.00290\.0829\\pm 0\.0029LIF\-Res16\.7±2\.816\.7\\pm 2\.83\.67±0\.143\.67\\pm 0\.140\.0018±0\.00010\.0018\\pm 0\.0001FordAHRF\-Res100\.0±0\.1100\.0\\pm 0\.14\.62±0\.034\.62\\pm 0\.030\.0395±0\.00120\.0395\\pm 0\.0012LIF\-Res3\.8±0\.63\.8\\pm 0\.61\.02±0\.031\.02\\pm 0\.030\.0030±0\.00050\.0030\\pm 0\.0005SHDHRF\-Res60\.5±0\.760\.5\\pm 0\.72\.49±0\.022\.49\\pm 0\.020\.0238±0\.00010\.0238\\pm 0\.0001LIF\-Res5\.7±0\.15\.7\\pm 0\.10\.63±0\.010\.63\\pm 0\.010\.0075±0\.00050\.0075\\pm 0\.0005DVS GestureHRF\-Res80\.3±1\.080\.3\\pm 1\.03\.41±0\.013\.41\\pm 0\.010\.0157±0\.00010\.0157\\pm 0\.0001LIF\-Res1\.6±0\.21\.6\\pm 0\.21\.49±0\.071\.49\\pm 0\.070\.0036±0\.00040\.0036\\pm 0\.0004

### B\.2CKA: Implementation Details

We use the linear kernel throughout, following\[[45](https://arxiv.org/html/2610.37906#bib.bib34)\]\. For each test sampleii, the temporal mean reservoir state𝐲¯i=1T​∑t=1T𝐲t\(i\)∈ℝN\\bar\{\\mathbf\{y\}\}\_\{i\}=\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}\\mathbf\{y\}\_\{t\}^\{\(i\)\}\\in\\mathbb\{R\}^\{N\}is computed over the full sequence, starting from a zero\-initialised reservoir state\. Each row is then normalised to unit norm before kernel computation, removing differences in overall activation magnitude between the two models and ensuring the comparison reflects representational geometry rather than scale\. The feature kernel is𝐊H=𝐇𝐇⊤\\mathbf\{K\}\_\{H\}=\\mathbf\{H\}\\mathbf\{H\}^\{\\top\}where𝐇∈ℝn×N\\mathbf\{H\}\\in\\mathbb\{R\}^\{n\\times N\}stacks the normalised mean states, and the label kernel is\(𝐊Y\)i​j=𝟏\[yi=yj\]\(\\mathbf\{K\}\_\{Y\}\)\_\{ij\}=\\mathbf\{1\}\[y\_\{i\}=y\_\{j\}\]\. Both kernels are centred with the centering matrix𝐇c=𝐈n−1n​𝟏𝟏⊤\\mathbf\{H\}\_\{c\}=\\mathbf\{I\}\_\{n\}\-\\frac\{1\}\{n\}\\mathbf\{1\}\\mathbf\{1\}^\{\\top\}and CKA is computed as in Eq\. \([7](https://arxiv.org/html/2610.37906#S3.E7)\)\. If either⟨𝐊~H,𝐊~H⟩F\\langle\\tilde\{\\mathbf\{K\}\}\_\{H\},\\tilde\{\\mathbf\{K\}\}\_\{H\}\\rangle\_\{F\}or⟨𝐊~Y,𝐊~Y⟩F\\langle\\tilde\{\\mathbf\{K\}\}\_\{Y\},\\tilde\{\\mathbf\{K\}\}\_\{Y\}\\rangle\_\{F\}is smaller than10−1010^\{\-10\}, CKA is set to zero to avoid numerical instability\.

We usen=500n=500randomly sampled test examples for all datasets except DVS Gesture, for which we use the full test set \(n=264n=264\)\. The choice ofn=500n=500is a practical constraint: computing the fulln×nn\\times nkernel matrix becomes memory\-intensive for largernn\. Both models are evaluated at their respective best hyperparameter configurations as determined by the random search described in Appendix[A](https://arxiv.org/html/2610.37906#A1), ensuring the comparison reflects the best achievable representational quality for each neuron model\.

#### Choice of summary statistic\.

CKA\[[45](https://arxiv.org/html/2610.37906#bib.bib34)\]compares fixed\-length representations, so applying it to reservoir trajectories requires a summary statistic\. We use the temporal mean as a conservative choice: since the HRF position𝐲t\\mathbf\{y\}\_\{t\}oscillates around an equilibrium, averaging partially cancels the oscillatory component that distinguishes HRF from LIF dynamics\. Statistics that retain more of this component, such as the RMS or the combined statistic of Section[4\.3](https://arxiv.org/html/2610.37906#S4.SS3), would be expected to accentuate rather than diminish the differences reported in Section[4\.1](https://arxiv.org/html/2610.37906#S4.SS1)\.

#### Sensitivity to sample size\.

Linear CKA estimates can be noisy for smallnn\. For a fixed reservoir per model, we recomputed CKA over repeated random subsamples of the test set of increasing size, up to the full test set \(Table[10](https://arxiv.org/html/2610.37906#A2.T10)\)\. The variability decreases markedly withnn, the estimate at our reported sample sizes is close to the full\-test\-set value, and the HRF\-Res–LIF\-Res gap is positive and of comparable magnitude at every sample size, including the smallest\.

Table 10:CKA as a function of the number of test samplesnn\(mean±\\pmstd over repeated random subsamples; one fixed reservoir per model\)\. “Full”: entire test set \(1,320 / 10,000 / 2,264 / 264 samples for FordA / sMNIST / SHD / DVS Gesture\), for which only one sample is possible\. DVS Gesture has fewer than 500 test samples\.FordAsMNISTSHDDVS GesturennHRF\-ResLIF\-ResHRF\-ResLIF\-ResHRF\-ResLIF\-ResHRF\-ResLIF\-Res100\.0555±\.0217\.0555\\pm\.0217\.0211±\.0104\.0211\\pm\.0104\.2628±\.0244\.2628\\pm\.0244\.2238±\.0267\.2238\\pm\.0267\.2632±\.0236\.2632\\pm\.0236\.2380±\.0359\.2380\\pm\.0359\.3552±\.0333\.3552\\pm\.0333\.3245±\.0286\.3245\\pm\.0286200\.0357±\.0115\.0357\\pm\.0115\.0101±\.0044\.0101\\pm\.0044\.2298±\.0244\.2298\\pm\.0244\.1922±\.0264\.1922\\pm\.0264\.2294±\.0149\.2294\\pm\.0149\.2049±\.0174\.2049\\pm\.0174\.3253±\.0125\.3253\\pm\.0125\.2989±\.0120\.2989\\pm\.0120500\.0274±\.0039\.0274\\pm\.0039\.0053±\.0025\.0053\\pm\.0025\.2217±\.0101\.2217\\pm\.0101\.1781±\.0135\.1781\\pm\.0135\.2063±\.0084\.2063\\pm\.0084\.1838±\.0137\.1838\\pm\.0137––1000\.0268±\.0035\.0268\\pm\.0035\.0034±\.0009\.0034\\pm\.0009\.2166±\.0081\.2166\\pm\.0081\.1761±\.0102\.1761\\pm\.0102\.2018±\.0072\.2018\\pm\.0072\.1804±\.0069\.1804\\pm\.0069––Full\.0263\.0263\.0029\.0029\.2111\.2111\.1729\.1729\.1959\.1959\.1732\.1732\.3203\.3203\.2914\.2914

#### Sensitivity to the initial transient\.

Since the temporal mean includes the initial transient from the zero\-initialised state, we recomputed CKA after discarding the first 10%, 25%, and 50% of each trajectory \(Table[11](https://arxiv.org/html/2610.37906#A2.T11)\)\. CKA changes only marginally with the washout length, and HRF\-Res retains a higher score than LIF\-Res in every setting\. The classification experiments with the final\-state readout are unaffected by washout by construction\.

Table 11:CKA after discarding an initial fraction of each trajectory \(washout; one fixed reservoir per model\)\. Washout steps at 10/25/50%: FordA 50/125/250; sMNIST 78/196/392; SHD 25/62/125; DVS Gesture 20/50/100\.FordAsMNISTSHDDVS GestureWashoutHRF\-ResLIF\-ResHRF\-ResLIF\-ResHRF\-ResLIF\-ResHRF\-ResLIF\-Res0%\.0399\.0078\.2708\.2144\.2143\.1836\.3509\.283510%\.0414\.0084\.2707\.2144\.2145\.1829\.3509\.283225%\.0365\.0086\.2723\.2155\.2143\.1832\.3509\.282550%\.0301\.0069\.2720\.2200\.2142\.1881\.3507\.2819

## Appendix CRobustness to Temporal Perturbations

The band\-pass mechanism of Section[4\.1](https://arxiv.org/html/2610.37906#S4.SS1)makes two opposite predictions under distribution shift: frequency selectivity should suppress broadband perturbations, but should make the reservoir more sensitive to shifts of the input spectrum\. We test both on FordA and sMNIST, keeping trained reservoirs and readouts fixed and perturbing inputs only at test time \(three independently initialised reservoirs per model\)\. We consider additive zero\-mean Gaussian noise with standard deviationσ\\sigma, and sampling\-rate shifts implemented by linear interpolation, where a rate below \(above\)1\.01\.0slows down \(speeds up\) the input sequence\. Time warping is not considered, as it lacks a single standard operationalisation\.

Under additive noise \(Table[12](https://arxiv.org/html/2610.37906#A3.T12)\), HRF\-Res degrades more gracefully than LIF\-Res: atσ=0\.5\\sigma=0\.5, HRF\-Res loses 5\.6 percentage points on FordA versus 14\.4 for LIF\-Res, and on sMNIST HRF\-Res retains above\-chance accuracy at noise levels at which LIF\-Res drops to chance\. Under sampling\-rate shifts \(Table[13](https://arxiv.org/html/2610.37906#A3.T13)\) the trend reverses: on sMNIST, HRF\-Res drops to near\-chance accuracy for every shift tested, whereas LIF\-Res degrades less, and on FordA the two models degrade similarly\. Both observations follow from the same mechanism: a bank of band\-pass filters rejects energy outside its preferred bands, but a rate change moves the informative content of the signal away from those bands\. Neither model, however, is robust to sampling\-rate mismatch, and both would require retraining or adaptation in that setting\.

Table 12:Test accuracy \(%\) under additive Gaussian input noise of standard deviationσ\\sigma\(mean±\\pmstd over three reservoirs\)\. Theσ=0\\sigma=0row reproduces the clean results of Table[1](https://arxiv.org/html/2610.37906#S4.T1)\(five seeds\)\.FordAsMNISTσ\\sigmaHRF\-ResLIF\-ResHRF\-ResLIF\-Res0\.0069\.72±0\.6869\.72\\pm 0\.6867\.15±0\.6667\.15\\pm 0\.6693\.64±0\.2593\.64\\pm 0\.2567\.00±1\.0467\.00\\pm 1\.040\.0569\.42±0\.9069\.42\\pm 0\.9067\.07±0\.9567\.07\\pm 0\.9577\.22±4\.0777\.22\\pm 4\.0712\.46±1\.1912\.46\\pm 1\.190\.1070\.10±0\.8170\.10\\pm 0\.8166\.39±0\.9566\.39\\pm 0\.9573\.82±3\.0173\.82\\pm 3\.0112\.16±0\.6312\.16\\pm 0\.630\.2068\.03±0\.7568\.03\\pm 0\.7562\.67±0\.8062\.67\\pm 0\.8067\.14±1\.5267\.14\\pm 1\.5212\.15±0\.9112\.15\\pm 0\.910\.3067\.25±1\.1267\.25\\pm 1\.1257\.73±0\.2757\.73\\pm 0\.2759\.90±1\.3459\.90\\pm 1\.3412\.04±0\.7012\.04\\pm 0\.700\.5064\.09±0\.9764\.09\\pm 0\.9752\.80±0\.2252\.80\\pm 0\.2246\.18±1\.4146\.18\\pm 1\.4112\.08±0\.6912\.08\\pm 0\.69Table 13:Test accuracy \(%\) under sampling\-rate shifts \(rate=1\.0=1\.0: original sampling rate; mean±\\pmstd over three reservoirs\)\. The rate=1\.0=1\.0row reproduces the clean results of Table[1](https://arxiv.org/html/2610.37906#S4.T1)\(five seeds\)\.FordAsMNISTRateHRF\-ResLIF\-ResHRF\-ResLIF\-Res0\.7052\.02±0\.5552\.02\\pm 0\.5549\.80±1\.3349\.80\\pm 1\.337\.37±2\.837\.37\\pm 2\.8311\.22±1\.4111\.22\\pm 1\.410\.8053\.11±3\.0453\.11\\pm 3\.0450\.66±1\.6050\.66\\pm 1\.609\.04±1\.429\.04\\pm 1\.4213\.48±2\.8613\.48\\pm 2\.860\.9055\.23±1\.0855\.23\\pm 1\.0853\.64±3\.7353\.64\\pm 3\.7313\.06±4\.2013\.06\\pm 4\.2018\.68±1\.6518\.68\\pm 1\.651\.0069\.72±0\.6869\.72\\pm 0\.6867\.15±0\.6667\.15\\pm 0\.6693\.64±0\.2593\.64\\pm 0\.2567\.00±1\.0467\.00\\pm 1\.041\.1057\.80±1\.5057\.80\\pm 1\.5055\.68±5\.8455\.68\\pm 5\.8410\.22±0\.4610\.22\\pm 0\.4625\.35±8\.2525\.35\\pm 8\.251\.2052\.25±0\.7852\.25\\pm 0\.7853\.38±4\.6053\.38\\pm 4\.609\.80±0\.009\.80\\pm 0\.0018\.53±3\.6718\.53\\pm 3\.671\.3050\.66±0\.3750\.66\\pm 0\.3753\.06±4\.5153\.06\\pm 4\.519\.73±0\.109\.73\\pm 0\.1015\.69±2\.4415\.69\\pm 2\.44
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58. Answer:\[Yes\]
59. Justification: All datasets are properly cited\. All are publicly available benchmarks for research use\. The software libraries used \(PyTorch, scikit\-learn\) are open\-source under BSD\-style licenses\. Explicit license names are not stated in the paper but all assets are standard research benchmarks with well\-established terms of use\.
60. Guidelines: - •The answer\[N/A\]means that the paper does not use existing assets\. - •The authors should cite the original paper that produced the code package or dataset\. - •The authors should state which version of the asset is used and, if possible, include a URL\. - •The name of the license \(e\.g\., CC\-BY 4\.0\) should be included for each asset\. - •For scraped data from a particular source \(e\.g\., website\), the copyright and terms of service of that source should be provided\. - •If assets are released, the license, copyright information, and terms of use in the package should be provided\. For popular datasets,[paperswithcode\.com/datasets](https://paperswithcode.com/datasets)has curated licenses for some datasets\. Their licensing guide can help determine the license of a dataset\. - •For existing datasets that are re\-packaged, both the original license and the license of the derived asset \(if it has changed\) should be provided\. - •If this information is not available online, the authors are encouraged to reach out to the asset’s creators\.
61. 13\.New assets
62. Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
63. Answer:\[N/A\]
64. Justification: No new datasets, pretrained models, or other assets are released with this paper\. Code is publicly available\.
65. Guidelines: - •The answer\[N/A\]means that the paper does not release new assets\. - •Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates\. This includes details about training, license, limitations, etc\. - •The paper should discuss whether and how consent was obtained from people whose asset is used\. - •At submission time, remember to anonymize your assets \(if applicable\)\. You can either create an anonymized URL or include an anonymized zip file\.
66. 14\.Crowdsourcing and research with human subjects
67. Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation \(if any\)?
68. Answer:\[N/A\]
69. Justification: The paper does not involve crowdsourcing or research with human subjects\.
70. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper\. - •According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector\.
71. 15\.Institutional review board \(IRB\) approvals or equivalent for research with human subjects
72. Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board \(IRB\) approvals \(or an equivalent approval/review based on the requirements of your country or institution\) were obtained?
73. Answer:\[N/A\]
74. Justification: The paper does not involve crowdsourcing or research with human subjects, so IRB approval was not required\.
75. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Depending on the country in which research is conducted, IRB approval \(or equivalent\) may be required for any human subjects research\. If you obtained IRB approval, you should clearly state this in the paper\. - •We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution\. - •For initial submissions, do not include any information that would break anonymity \(if applicable\), such as the institution conducting the review\.
76. 16\.Declaration of LLM usage
77. Question: Does the paper describe the usage of LLMs if it is an important, original, or non\-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does*not*impact the core methodology, scientific rigor, or originality of the research, declaration is not required\.
78. Answer:\[N/A\]
79. Justification: LLMs were used only for writing and editing purposes and did not impact the core methodology, scientific rigor, or originality of the research\.
80. Guidelines: - •The answer\[N/A\]means that the core method development in this research does not involve LLMs as any important, original, or non\-standard components\. - •Please refer to our LLM policy in the NeurIPS handbook for what should or should not be described\.