SSM-Induced Kernel in SHaRe-SSM

• SSM-Induced Kernel is a temporal impulse response derived from second-order resonate-and-fire dynamics in SHaRe-SSM, offering a built-in mechanism for long-range dependency.
• The kernel is implicitly induced through discretized dynamics using IM or IMEX schemes, ensuring causal, stable, and efficient propagation of past inputs.
• It underpins a kernel-based regression approach that decodes spike histories into continuous outputs, while enabling parallel scan for scalable long-sequence evaluation.

Searching arXiv for the specified paper to ground the article in the current record. The term SSM-induced kernel in SHaRe-SSM denotes an implicit temporal impulse-response kernel generated by linear second-order state-space dynamics rather than a kernel in the classical RKHS or Gaussian-process sense. In this formulation, a harmonic resonate-and-fire oscillator defines a causal mapping from past inputs to current states and outputs, so that unrolling the recurrence yields a convolution-like dependence on prior events over time. Within SHaRe-SSM, this induced kernel underlies long-sequence modeling for both classification and regression, and in the regression setting it is complemented by a kernel-based regression with a learnable kernel convolved over time, initialized as a decaying LI filter (Agrawal et al., 16 Oct 2025).

1. Conceptual meaning of the induced kernel

In SHaRe-SSM, the induced kernel is the temporal response implied by the hidden dynamics of a second-order spiking state-space model. The central idea is that, because the hidden dynamics are linear between spikes, the state evolution can be written in a causal convolution-like form whose weights are determined by the transition operator of the discretized oscillator. The kernel is therefore not manually specified; it is induced by the dynamical system itself (Agrawal et al., 16 Oct 2025).

A key clarification is that the paper does not use the term kernel in the classical statistical sense. Instead, it uses the term in the sequence-modeling sense: the linear second-order SSM defines an impulse response that maps historical inputs to present hidden states and outputs. This distinguishes the construction from ANN feature-map kernels and from GP-style covariance kernels. The induced kernel is a property of the temporal dynamics, not an independently parameterized reproducing kernel (Agrawal et al., 16 Oct 2025).

This distinction matters because it locates SHaRe-SSM within the line of long-sequence SSMs rather than within kernel methods in the usual RKHS framework. A plausible implication is that interpretive emphasis should be placed on recurrence, discretization, and spectral behavior of the state transition rather than on positive-definite kernel design.

2. Second-order resonate-and-fire dynamics

The paper begins from a Harmonic Resonate-and-Fire neuron defined in continuous time as

$$u'(t) = - \omega^2 v(t) - 2b\,u(t) + x(t), \qquad v'(t) = u(t), \qquad z(t)=\Theta(u(t)-\theta).$$

Here, $v(t)$ is the hidden state, $u(t)$ is a derivative or velocity-like state, $x(t)$ is the input spike signal, $\omega$ is the oscillation frequency, $b$ is the damping coefficient, and $z(t)$ is the spike output (Agrawal et al., 16 Oct 2025).

The SHaRe-SSM formulation removes the damping term from the core SSM and writes

$$u'(t) = -\Omega v(t) + Bx(t), \qquad v'(t)=u(t), \qquad z(t)=\Theta(v(t)-\theta), \qquad y(t)=Cz(t)+Dx(t),$$

with

$$u(t),v(t)\in\mathbb{R}^p,\qquad \Omega\in\mathbb{R}^{p\times p}\ \text{diagonal},\qquad B\in\mathbb{R}^{p\times h},\qquad C\in\mathbb{R}^{h\times p},\qquad D\in\mathbb{R}^{h}.$$

This second-order structure introduces both a position-like state $v$ and a velocity-like state $v(t)$0, and the paper argues that such a form is better suited to oscillatory and long-range dynamics than first-order SSMs (Agrawal et al., 16 Oct 2025). The induced kernel emerges from precisely this two-state oscillator structure: every input spike perturbs the oscillator, and the subsequent ringing over time determines the temporal weighting applied to past inputs.

3. Discretization and explicit kernel induction

The paper discretizes the second-order system using either implicit (IM) or implicit-explicit (IMEX) schemes. The generic update is

$v(t)$1

where $v(t)$2 for IM and $v(t)$3 for IMEX (Agrawal et al., 16 Oct 2025).

By concatenating the state into $v(t)$4, the dynamics take the form

$v(t)$5

Unrolling the recurrence yields

$v(t)$6

This expression is the direct origin of the induced kernel: the response to input $v(t)$7 is governed by powers of $v(t)$8, so the temporal weighting over the past is the impulse response defined by repeated application of the transition matrix (Agrawal et al., 16 Oct 2025).

For the implicit (IM) discretization, the paper derives

$v(t)$9

where

$u(t)$0

Hence the recursion becomes

$u(t)$1

For the implicit-explicit (IMEX) discretization, the paper gives

$u(t)$2

and also writes the factorized form

$u(t)$3

In both cases, the kernel is the causal temporal response generated by repeated powers of the relevant transition matrix. Because $u(t)$4 is diagonal and the structured matrix form is preserved, the paper emphasizes that repeated propagation is efficient and stable (Agrawal et al., 16 Oct 2025). This suggests that the induced kernel is not only a conceptual description of memory but also the object through which practical long-range computation is organized.

4. Kernel-based spiking regression

In the regression setting, the paper identifies a specific difficulty: spike outputs are binary and therefore are not ideal for direct continuous target prediction. To address this, it proposes a kernel-based regression head in which the spike sequence is decoded by a learnable convolution kernel over time, initialized as a decaying LI filter (Agrawal et al., 16 Oct 2025).

The described workflow is explicit. First, SHaRe-SSM produces spike-based hidden representations. Second, instead of using a simple linear projection, the model applies a learnable temporal convolution kernel. Third, this kernel aggregates spike history over long horizons. Fourth, kernel size is tuned to avoid oversmoothing while preserving long-range dependence (Agrawal et al., 16 Oct 2025).

The resulting regression signal is a temporally smoothed weighted sum of past spikes, yielding a continuous output. In this use, the learned decoding kernel is distinct from the SSM-induced kernel of the hidden oscillator, although the two are coupled in practice: the hidden dynamics determine the structure of long-term spike activity, and the learned convolutional kernel transforms that spike history into a continuous prediction. A plausible implication is that SHaRe-SSM separates long-horizon memory formation from continuous-value readout.

For the 50k-length PPG-DaLiA task, the paper reports that a kernel size of 64 performed best, outperforming sizes 32 and 128 (Agrawal et al., 16 Oct 2025). The stated interpretation is that this choice offers the best tradeoff between expressivity and oversmoothing.

5. Parallel scan and long-sequence evaluation

The same recurrence that induces the kernel also enables parallel scan. The paper introduces the associative binary operator

$u(t)$5

which is used to compute the recurrence

$u(t)$6

in parallel over time (Agrawal et al., 16 Oct 2025).

This is significant because the kernel is defined by repeated application of $u(t)$7, so evaluating the state evolution over very long sequences requires efficient and numerically controlled propagation of matrix powers and affine updates. The paper emphasizes three computational properties: $u(t)$8 is diagonal, the structured matrix form permits linear-time hidden-dimension operations, and parallel scan reduces sequential recurrence costs while helping with long-range stability (Agrawal et al., 16 Oct 2025).

In this framework, the induced kernel is not merely an analytical byproduct. It is the object being computed when the recurrence is scanned across time. This establishes a close link between the mathematical form of the kernel and the implementation strategy used to make very long-range sequence processing feasible.

6. Stability, dissipation, and conservation

A major theoretical theme is the balance between stability and memory retention. The paper characterizes the induced kernel through the eigenstructure of the transition matrix and distinguishes a dissipative regime from a more conservative one (Agrawal et al., 16 Oct 2025).

For the IM discretization, the paper proves that the eigenvalues of $u(t)$9 lie inside the unit circle:

$x(t)$0

It derives

$x(t)$1

and therefore

$x(t)$2

The paper interprets this as stability with dissipation: the system remains bounded, but eigenvalues strictly below unit magnitude cause gradual loss of energy and memory over time (Agrawal et al., 16 Oct 2025).

For the IMEX discretization, the paper states that the eigenvalues satisfy

$x(t)$3

under the stated conditions. It gives

$x(t)$4

This is presented as an energy-preserving or unit-circle regime, in which oscillatory energy is conserved more effectively, making the induced kernel more suitable for long-range modeling (Agrawal et al., 16 Oct 2025).

The appendix, as summarized in the paper synthesis, also provides an expression for the expected eigenvalue magnitude moment under $x(t)$5 and concludes that the magnitude remains large enough to support long-range dependencies even when $x(t)$6 and $x(t)$7 (Agrawal et al., 16 Oct 2025). This supports the claim that the induced kernel is neither exploding nor vanishing too quickly.

The conceptual consequence is direct: IM generates a stable but more dissipative kernel, whereas IMEX generates a more conservative kernel whose oscillatory traces persist longer. The paper explicitly frames IMEX as the more “conservative” kernel generator and IM as the more dissipative one (Agrawal et al., 16 Oct 2025).

7. Empirical role and comparative position

Empirically, the paper connects kernel behavior to performance on long-sequence classification and regression. SHaRe-SSM is compared against first-order SSMs, Mamba, S5/S6, and ANN-based second-order models such as LinOSS. The reported finding is that it performs better than first-order SSMs on average, remains close to or competitive with LinOSS and D-LinOSS, and that IMEX tends to exhibit lower variance and more stability-consistent behavior (Agrawal et al., 16 Oct 2025).

For regression on 50k sequences, the paper states that SHaRe-SSM-IMEX outperforms all first-order spiking SSMs on PPG-DaLiA, and it identifies the kernel-based regression head as important because direct spike decoding is too crude for continuous outputs (Agrawal et al., 16 Oct 2025). This ties the induced-kernel perspective to a practical decoding mechanism for continuous targets.

The paper also contrasts SHaRe-SSM with standard ANN-based second-order SSMs. In ANN-style models such as LinOSS, hidden dynamics are followed by continuous nonlinear blocks including GeLU or GLU. SHaRe-SSM is described instead as fully spike-based, multiplication-free at the communication level, and not using GeLU, GLU, or GSU (Agrawal et al., 16 Oct 2025). Accordingly, its kernel is not an ANN feature-map kernel but a dynamical impulse-response kernel embedded in a spiking second-order oscillator.

Against transformers, the paper emphasizes that attention scales quadratically in sequence length, whereas the kernelized SSM runs in linear time with scan-based recurrence, avoids quadratic attention cost, and is more suitable for extremely long sequences such as 18k to 50k steps (Agrawal et al., 16 Oct 2025). The broader significance is that the induced kernel serves as the mechanism through which SHaRe-SSM retains long-horizon information without attention.

The paper further studies heterogeneity in

$x(t)$8

It reports that homogenizing thresholds and dynamics generally hurts performance, whereas heterogeneous initialization improves robustness and performance (Agrawal et al., 16 Oct 2025). This suggests that the induced kernel benefits from a richer spread of oscillatory time constants and thresholds, yielding a more diverse temporal basis for long-range dependence.

In summary, the SSM-induced kernel in SHaRe-SSM is the temporal impulse response generated by the discrete second-order resonate-and-fire dynamics

$x(t)$9

Its form depends on the discretization, its computational realization is enabled by parallel scan, and its long-range behavior is governed by the spectral properties of the transition matrix. For regression, the model augments this dynamical kernel with a learnable temporal convolution over spikes, producing continuous outputs while preserving the long-horizon structure of the underlying spiking SSM (Agrawal et al., 16 Oct 2025).