Model- and Layer-Level Closed-Loop Forecasting with SSM-Based Spiking Neural Networks

Erik Stolt; Sanja Karilanova; Ayca Ozcelikkale

cs.LGOct 20, 2026

Model- and Layer-Level Closed-Loop Forecasting with SSM-Based Spiking Neural Networks

Authors: Erik Stolt, Sanja Karilanova, Ayca Ozcelikkale

Organizations: Department of Electrical Engineering, Uppsala University, Uppsala, Sweden

Abstract

Deep State-Space Models (Deep SSMs) have shown promising performance and computational benefits in processing of time-series data with complex temporal dependencies. Spiking neural networks (SNNs) constitute a biologically inspired class of deep SSMs, where computing blocks communicate through spikes, discrete single-bit events rather than continuous activation functions. A particularly important task in time-series processing is forecasting, i.e. predicting future values of a sequence from past observations alone. However, how to best perform forecasting with SNNs is an open question, where the standard autoregressive nature of the data processing with SNNs requires values that are not available during the forecasting phase. In this article, we address this challenge. We explore two closed-loop forecasting strategies, in which the network's own generated values are fed back into the network as if they were observed past values: (i) model-level and (ii) layer-level closed-loop forecasting. We evaluate these approaches on the ETT-small datasets, a standard benchmark for time-series forecasting, under both spike-based and continuous activation functions. We compare the proposed closed-loop approaches against state-of-the-art methods such as SpaceTime, as well as the zero-masking baseline in S4. Both of the proposed approaches perform comparably to or better than zero-masking with S4, and in most cases match SpaceTime. These findings highlight the trade-off between continuous and low-bit quantized activations, and demonstrate that the proposed Deep SSM architectures provide a competitive framework for forecasting.

Figures & tables

Fig. 1 : Deep SSM architecture with an input layer consisting of

m

input features, followed by

L

SSM layers, and an output layer consisting of

q

output features.

TABLE I : Univariate time-series forecasting results on three ETT-small datasets. Results are averaged over five random initializations; subscripts denote the standard deviation. Results for S4 and SpaceTime are taken from [ 6 ] . Bold denotes the best and underline the second-best model per row.

Fig. 2 : Average elapsed time in seconds per epoch when training on the ETTh1 dataset. All models scale approximately linear with context length.

Panel of Figure 4 — Fig. 3 : Example forecasting samples for all strategies on the ETT-small datasets with context length $P=720$ , forecast horizon $F=96$ , and with Signed-Spk. activation .

Description	Parameter Value
Input dimension per SSM neuron	$n_{\text{in}}=1$
Output dimension per SSM neuron	$n_{\text{out}}=1$
Number of stacked SSM neurons	$h=64$
Number of SSM layers	$L=3$
State dimension per neuron	$n=128$
Dropout rate	$0.0$

TABLE II : Model parameters used in the univariate forecasting of ETT-small.

Parameter	Description
$m$	Number of input features in the input layer
$q$	Number of output features in the output layer
$n_{\text{in}}$	Input dimension per SSM neuron
$n_{\text{out}}$	Output dimension per SSM neuron
$h$	Number of parallel SSM neurons
$L$	Number of SSM layers

TABLE III : Notation used in the model architecture and in network layer

\ell

Parameter	Number of Trainable Parameters
$\bm{W}^{(\ell)}\in\mathbb{R}^{d_{\text{in}}\times d_{\text{out}}}$	$d_{\text{in}}d_{\text{out}}$
$\vec{b}^{(\ell)}\in\mathbb{R}^{d_{\text{in}}}$	$d_{\text{in}}$
$\bm{A}\in\mathbb{R}^{n}$ (diag)	$hn\,/\,2$
$\bm{A}\in\mathbb{C}^{n}$ (diag)	$hn$
$\bm{B}\in\mathbb{R}^{n\times n_{\text{in}}}$	$n_{\text{in}}n\,/\,2$
$\bm{C}\in\mathbb{R}^{n_{\text{out}}\times n}$	$n_{\text{out}}n\,/\,2$

TABLE IV : Number of trainable parameters for the components of the Deep SSM.

\bm{W}^{(\ell)}

denotes the mixing matrix between layer

\ell

and

\ell+1

with the corresponding bias vector

\vec{b}^{(\ell)}

Panel of Figure 8 — Fig. 4 : Example forecasting samples for all strategies on the ETT-small datasets with context length $P=720$ , forecast horizon $F=96$ , and with GELU activation .

Time Series Forecasting

Time series forecasting predicts future values from ordered observations using statistical, neural, and probabilistic models to support decisions under uncertainty.

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SpikeLite: Lightweight Spiking Neural Networks for Time-Series Forecasting

Bang Hu, Changze Lv, Mingjie Li, Xiaoqing Zheng+2

School of Computer Science, Fudan University, Shanghai, China

Multivariate Time Series Forecasting Efficient Neural Network Inference Time Series Forecasting Spiking Neural Networks Sparse Attention

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Spiking neural networks (SNNs) offer an energy-efficient paradigm for time-series forecasting through spike-driven computation. However, recent SNN forecasters often pursue higher accuracy through increasingly complex attention mechanisms, or specialized neuronal dynamics, weakening the lightweight motivation of SNNs. We introduce SpikeLite, a spiking forecasting framework built around two modules: a Frequency-Selective Spiking Encoder (FSSE) for frequency-sensitive temporal encoding and a Sparse Spiking Channel Attention (SSCA) module for selective cross-channel interaction. FSSE exploits the low-pass filtering behavior of LIF dynamics to reorganize each input sequence into frequency-sensitive components while collectively preserving the input at the decomposition stage. SSCA then learns a binary mask from encoded channel representations and uses it to selectively exchange information within spike-driven self-attention, retaining informative cross-channel interactions while suppressing redundant ones. When explicit channel interaction is unnecessary, SpikeLite uses the lighter FSSE-only channel-independent path. Experiments under the SeqSNN and SpikF protocols cover four standard multivariate and eight long-term forecasting benchmarks. SpikeLite achieves the best aggregate performance under both protocols, with an average

R^2

of 0.790 and RSE of 0.440, and lowest average MSE/MAE of 0.343/0.345 in long-term forecasting. Moreover, evaluation on the ECL dataset shows that SpikeLite achieves the lowest reported energy consumption, further demonstrating its potential for energy-efficient time-series forecasting.

Abstract

Time Series Forecasting

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