---
title: SSM-Induced Kernel in SHaRe-SSM
url: https://www.emergentmind.com/topics/ssm-induced-kernel
type: topic
---

# SSM-Induced Kernel in SHaRe-SSM

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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** [2510.14386].

## 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 [2510.14386].

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 [2510.14386].

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 [2510.14386].

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 $u$, and the paper argues that such a form is better suited to oscillatory and long-range dynamics than first-order SSMs [2510.14386]. 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
$$
u_n = u_{n-1} + \Delta t\left(-\Omega v_\star + Bx_n\right), \qquad
v_n = v_{n-1} + \Delta t\,u_n,
$$
where $v_\star=v_n$ for IM and $v_\star=v_{n-1}$ for IMEX [2510.14386].

By concatenating the state into $s_n$, the dynamics take the form
$$
s_n = M s_{n-1} + F_n.
$$
Unrolling the recurrence yields
$$
s_n = M^n s_0 + \sum_{k=1}^{n} M^{n-k}F_k.
$$
This expression is the direct origin of the induced kernel: the response to input $x_k$ is governed by powers of $M$, so the temporal weighting over the past is the impulse response defined by repeated application of the transition matrix [2510.14386].

For the **implicit (IM)** discretization, the paper derives
$$
M^{\text{IM}}=
\begin{pmatrix}
S & -S\Delta t\,\Omega \\
S\Delta t & S
\end{pmatrix},
\qquad
F_n^{\text{IM}}=
\begin{pmatrix}
S\Delta t\,Bx_n \\
S\Delta t^2\,Bx_n
\end{pmatrix},
$$
where
$$
S=(I+\Delta t^2\Omega)^{-1}.
$$
Hence the recursion becomes
$$
s_n = M^{\text{IM}} s_{n-1} + F_n^{\text{IM}}.
$$

For the **implicit-explicit (IMEX)** discretization, the paper gives
$$
M^{\text{IMEX}}=
\begin{pmatrix}
I & -\Delta t\,\Omega \\
\Delta t\,I & I-\Delta t^2\Omega
\end{pmatrix},
\qquad
F_n^{\text{IMEX}}=
\begin{pmatrix}
\Delta t\,Bx_n \\
\Delta t^2\,Bx_n
\end{pmatrix},
$$
and also writes the factorized form
$$
M^{\text{IMEX}}=
\begin{pmatrix}
S & -\Delta t\,\Omega S \\
\Delta t\,S & S
\end{pmatrix},
\qquad
S=(I+\Delta t^2\Omega)^{-1}.
$$

In both cases, the kernel is the causal temporal response generated by repeated powers of the relevant transition matrix. Because $\Omega$ is diagonal and the structured matrix form is preserved, the paper emphasizes that repeated propagation is efficient and stable [2510.14386]. 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** [2510.14386].

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 [2510.14386].

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 [2510.14386]. 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
$$
(a_1,a_2)\bullet(b_1,b_2) = (b_1\cdot a_1,\; b_1\cdot a_2 + b_2),
$$
which is used to compute the recurrence
$$
s_n = Ms_{n-1}+F_n
$$
in parallel over time [2510.14386].

This is significant because the kernel is defined by repeated application of $M$, 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: $\Omega$ 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 [2510.14386].

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 [2510.14386].

For the **IM** discretization, the paper proves that the eigenvalues of $M^{\text{IM}}$ lie inside the unit circle:
$$
|\lambda_j|\le 1.
$$
It derives
$$
\lambda_{j_{1,2}} = s_j \pm i\,\Delta t\,s_j\sqrt{\Omega_j},
\qquad
s_j=\frac{1}{1+\Delta t^2\Omega_j},
$$
and therefore
$$
|\lambda_{j_{1,2}}|^2 = \frac{1}{1+\Delta t^2\Omega_j}\le 1.
$$
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 [2510.14386].

For the **IMEX** discretization, the paper states that the eigenvalues satisfy
$$
|\lambda_j|=1,
$$
under the stated conditions. It gives
$$
\lambda_{j_{1,2}} =
\frac{1}{2}(2-\Delta t^2\Omega_j)
\pm
\frac{1}{2}\sqrt{\Delta t^2\Omega_j(4-\Delta t^2\Omega_j)}.
$$
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 [2510.14386].

The appendix, as summarized in the paper synthesis, also provides an expression for the expected eigenvalue magnitude moment under $\Omega_j\sim\mathcal U([0,\Omega_{\max}])$ and concludes that the magnitude remains large enough to support long-range dependencies even when $\Omega_{\max}=1$ and $\Delta t=1$ [2510.14386]. 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 [2510.14386].

## 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 [2510.14386].

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 [2510.14386]. 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 [2510.14386]. 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 [2510.14386]. 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
$$
\Omega,\ B,\ C,\ D,\ \Delta t,\ \theta_C,\ \theta_D,\ \theta_{\text{encoder}}.
$$
It reports that homogenizing thresholds and dynamics generally hurts performance, whereas heterogeneous initialization improves robustness and performance [2510.14386]. 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
$$
s_n = Ms_{n-1}+F_n.
$$
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 [2510.14386].

Source: https://www.emergentmind.com/topics/ssm-induced-kernel