A Time-Variant Spiking Neuron from a Unified State-Dynamics Perspective

Jing Yang; Qianhui Liu; Gang Pan; Zhanglu Yan

cs.NEOct 14, 2026

A Time-Variant Spiking Neuron from a Unified State-Dynamics Perspective

Authors: Jing Yang, Qianhui Liu, Gang Pan, Zhanglu Yan

Organizations: School of Artificial Intelligence, Shandong University, Jinan, China · College of Computer Science and Technology, Zhejiang University, Hangzhou, China · Department of Computer Science, National University of Singapore, Singapore

Abstract

Spiking neural networks (SNNs) have increasingly incorporated rich neuronal dynamics to enhance temporal processing. However, richer dynamics do not necessarily enable neurons to adapt how information is processed over time. To examine this issue systematically, we introduce a unified state-dynamics framework that characterizes diverse spiking neurons through a common formulation and enables direct comparison of their dynamical designs. The framework reveals a common limitation of existing neurons: although they enhance temporal processing through richer internal states and state evolution, the dynamical rules determining how inputs are integrated, how states evolve, and how they are read out generally remain unchanged throughout temporal processing. Such time-invariant rules limit the ability of neurons to adjust information processing as information relevance changes over time. Inspired by time-varying regulation in biological neurons, we propose the Time-Variant Multi-Compartment Neuron (TV-MCN). TV-MCN builds on multiscale compartment dynamics and makes the rules for input integration, state evolution, and state readout time-varying according to the current input. Specifically, an input-dependent internal time regulates the rate of state evolution, while the current input dynamically modulates input integration and state readout. Extensive experiments show that TV-MCN achieves strong overall performance, with particularly large gains on tasks requiring selective retention and retrieval of information, while ablation studies further support the effectiveness of its time-varying mechanisms.

Figures & tables

Figure 1: From biological neuronal computation to the unified state-dynamics framework. Left: Synaptic inputs drive internal states, corresponding to input integration through $B$ and state evolution through $A$ . Middle: Dendritic states are read out at the soma to form the membrane potential. Right: Spike generation is followed by ionic currents that drive post-spike state recovery.

Neuron	$h$	$A$	$B$	$C$
LIF ( Burkitt, 2006 )	$u$	$-\tau_{m}^{-1}$	$W\tau_{m}^{-1}$	$1$
ad-LIF ( Baronig et al., 2025 )	$[u,w]^{\mathsf{T}}$	$\left[\begin{smallmatrix}-\tau_{u}^{-1}&-\tau_{u}^{-1}\\ a\tau_{w}^{-1}&-\tau_{w}^{-1}\end{smallmatrix}\right]$	$\operatorname{diag}(\tau_{u}^{-1},b\tau_{w}^{-1})$	$e_{1}^{\mathsf{T}}$
TC-LIF ( Zhang et al., 2024 )	$[U^{D},U^{S}]^{\mathsf{T}}$	$\left[\begin{smallmatrix}1&\beta_{1}\\ \beta_{2}&1+\beta_{1}\beta_{2}\end{smallmatrix}\right]$	$\left[\begin{smallmatrix}1&-\gamma\\ \beta_{2}&-(\beta_{2}\gamma+V_{\rm th})\end{smallmatrix}\right]$	$e_{2}^{\mathsf{T}}$
DGN ( Bai et al., 2026 )	$[\mathbf{D}^{\mathsf{T}},V]^{\mathsf{T}}$	$\left[\begin{smallmatrix}-\tau_{s}^{-1}I_{N}&0\\ \mathbf{w}^{\mathsf{T}}&-(g_{l}+\mathbf{c}^{\mathsf{T}}\mathbf{D})\end{smallmatrix}\right]$	$\left[\begin{smallmatrix}\tau_{s}^{-1}I_{N}\\ 0\end{smallmatrix}\right]$	$e_{N+1}^{\mathsf{T}}$
DH-LIF ( Zheng et al., 2024 )	$[\mathbf{i}^{\mathsf{T}},u]^{\mathsf{T}}$	$\left[\begin{smallmatrix}\Lambda_{\alpha}&0\\ (1-\beta)\bm{\alpha}^{\mathsf{T}}&\beta\end{smallmatrix}\right]$	$\left[\begin{smallmatrix}I_{D}-\Lambda_{\alpha}\\ (1-\beta)(\mathbf{1}-\bm{\alpha})^{\mathsf{T}}\end{smallmatrix}\right]$	$e_{D+1}^{\mathsf{T}}$
HetSyn ( Deng et al., 2025 )	$[\mathbf{I}^{\mathsf{T}},J]^{\mathsf{T}}$	$-\operatorname{diag}(\tau_{s,1}^{-1},\ldots,\tau_{s,M}^{-1},\tau_{J}^{-1})$	$\operatorname{diag}(w_{1},\ldots,w_{M},\vartheta)$	$[\mathbf{1}_{M}^{\mathsf{T}},-1]$

Table 1: Parameterizations of spiking neurons under the unified state-dynamics framework.

Figure 2: Input-dependent dynamics of TV-MCN. At

t_{1}

(left), the smaller

\Delta_{t_{1}}

slows internal time (bottom axis) and state evolution (red dots on blue curves at equal external-time intervals). At

t_{2}

(right), the larger

\Delta_{t_{2}}

accelerates both. Different inputs at these moments generate different

B_{t}

and

C_{t}

; blue and purple arrow widths indicate input-integration and state-readout strengths, respectively.

Neuron	GSC	SSC	SHD	DVS128 Gesture
LIF ( Burkitt, 2006 )	78.97	53.98	74.73	87.85
ad-LIF ( Baronig et al., 2025 )	46.51	47.42	71.51	89.24
TC-LIF ( Zhang et al., 2024 )	82.31	55.59	69.43	85.76
DH-LIF ( Zheng et al., 2024 )	82.47	61.04	82.60	88.89
HetSyn ( Deng et al., 2025 )	90.00	66.76	88.60	89.58
DGN ( Bai et al., 2026 )	71.73	63.88	73.85	84.72

Table 2: Comparison of spiking neurons on temporal classification tasks. Accuracies are reported in %. Bold and underlining indicate the best and second-best results, respectively.

Figure 3: Generalization to sequence length and interference. Left: Length generalization on Induction. The same trained model is evaluated at different sequence lengths, with the dashed line indicating the training length. Right: Generalization across noise fractions

\rho

on MAD Noisy ICR. Models trained at

\rho=0.2

are evaluated at all four noise fractions without retraining.

Figure 4: Adding predictions for nine neuron models under the shared two-layer fully connected network.

Task	Model	Params (M)	Est. FLOPs (G/seq)	Performance
MAD ICR Acc. (%)	PMSN ( Chen et al., 2026 )	0.291	0.098	73.97
	D-RF ( Zhang et al., 2025 )	3.298	1.059	99.87
	SpikingSSMs ( Shen et al., 2025 )	35.804	10.491	98.98
	S5-RF ( Huber et al., 2024 )	1.612	0.887	39.18
	Spiking LMUFormer ( Liu et al., 2024 )	0.802	0.207	57.65
	TV-MCN	0.062	0.026	100.00

Table 4: Comparison with state-of-the-art methods on synthetic sequence tasks. Bold and underlining indicate the best and second-best results, respectively.

Model Variant	Token Acc. (%)	Exact Acc. (%)
TV-MCN	93.46	37.72
w/o internal time modulation	7.20	0.00
w/o dynamic $B/C$ coupling	61.24	0.02

Table 5: Component ablation on Selective Copying. Exact accuracy requires all target tokens to be correct. Bold indicates the best results.

Figure 5: Recall behavior after component removal. Left: Target and outputs of the three variants for the same input; red indicates errors. Right: Token accuracy after inserting additional irrelevant inputs before recall for the same samples.

Figure 6: Effect of internal-time modulation on instruction-dependent retention. (a) The input content is identical; only the STORE instruction changes, requiring the model to remember either A or B and recall it later. (b) The input-dependent rate of internal-state evolution under the two instructions. (c–d) The full TV-MCN and the model without internal-time modulation are compared to show how strongly A, B, and irrelevant content are retained in the internal states. The left and right columns correspond to remembering A and B, respectively.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Task	Preset training budget	Batch size	Base learning rate	Optimizer	Normalization
GSC	200 epochs	128	0.001	AdamW	LayerNorm
SSC	128 epochs	128	0.001	Adam	LayerNorm
SHD	100 epochs	40	0.01	AdamW	LayerNorm
DVS128 Gesture	200 epochs	32	0.005	AdamW	Masked BatchNorm
MAD ICR	200 epochs	128	0.0005	AdamW	LayerNorm
MAD Noisy ICR	200 epochs	128	0.0005	AdamW	LayerNorm

Appendix

Table 6: Task-level training configurations for the shared-network comparisons.

Neuron	Internal state size and initialization
LIF	The decay coefficient is 0.9, and the firing threshold is 1.
ad-LIF	Membrane time constants are initialized from $\mathcal{N}(20,5^{2})$ , and adaptation time constants from $\mathcal{N}(200,5^{2})$ . The base threshold is 1.6.
TC-LIF	The two raw coupling parameters are initialized to 0, corresponding to an effective value of 0.5 after the sigmoid transformation. The firing threshold is 1, and the dendritic reset coefficient is 0.5.
DH-LIF	Four dendritic branches are used. Membrane parameters are initialized from $\mathcal{U}(0,4)$ and branch parameters from $\mathcal{U}(2,6)$ . The firing threshold is 1.
HetSyn	Synaptic weights are initialized from $\mathcal{N}(0,0.01^{2})$ . Synaptic time constants are initialized to 20 ms, the time step is 1 ms, and the firing threshold is 1.
DGN	The time step is 1, and the synaptic time constant is 20. Conductance and current weights are both initialized from $\mathcal{N}(0.01,0.005^{2})$ .

Appendix

Table 7: Internal state sizes and baseline initialization schemes of the neuron models.

Neuron and task	Adopted configuration
LIF – STORE–RECALL	The decay coefficient is changed from 0.9 to 0.99.
ad-LIF – SSC	Membrane time constants are initialized from $\mathcal{N}(5,1.25^{2})$ .
ad-LIF – Adding	Membrane time constants are initialized from $\mathcal{N}(5,1^{2})$ , and adaptation time constants from $\mathcal{N}(200,50^{2})$ . The base threshold is 0.4.
ad-LIF – Induction	Membrane time constants are initialized from $\mathcal{N}(5,1.25^{2})$ , and adaptation time constants from $\mathcal{N}(200,50^{2})$ . The base threshold is 0.4.
ad-LIF – STORE–RECALL	Membrane time constants are initialized from $\mathcal{N}(5,1.25^{2})$ , and adaptation time constants from $\mathcal{N}(200,50^{2})$ . The base threshold remains 1.6.
TC-LIF – STORE–RECALL	The initial effective coupling value is changed from 0.5 to 0.125.

Appendix

Table 8: Method- and task-specific configurations for the shared-network comparisons.

Method	Architecture
PMSN	Two stages, each containing three spiking layers, with widths of 128 and 256, respectively. Each neuron has four compartments. Average pooling is applied along the feature dimension between stages and at the network output.
D-RF	Six independent residual blocks of width 512, with a complex-state dimension of 8 and a causal convolution at the input. Following the adopted forward implementation, the first block is executed twice, giving six independent sets of block parameters and seven block calls.
SpikingSSMs	Sixteen layers of width 1024 and state dimension 64, with a pre-LayerNorm residual structure. The authors’ pretrained SDN modules are loaded and frozen.
S5-RF	Four layers of width 512, with HiPPO spectral initialization divided into 16 blocks. The RF core and LI readout are retained.
Spiking LMUFormer	One native block with hidden width 256, memory size 512, a feedforward expansion factor of 4, and memory time-scale parameter $\theta=128$ .
TV-MCN	Two layers of 128 neurons, with four compartments per neuron. Each layer includes a depthwise causal convolution with kernel size 4. The convolutions are initialized as identity mappings and add 1,280 parameters in total.

Appendix

Table 9: Architectures used in the complete-network comparison.

Method	MAD ICR	MAD Noisy ICR	Selective Copying	STORE– RECALL	Induction	Adding
PMSN	0.01	0.01	0.01	0.01	0.001	0.01
D-RF	0.005	0.005	0.005	0.005	0.005	0.005
SpikingSSMs	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005
S5-RF	0.004	0.004	0.004	0.004	0.004	0.004
Spiking LMUFormer	0.0025	0.0025	0.0025	0.0025	0.0025	0.0025
TV-MCN	0.0005	0.001	0.0001	0.001	0.001	0.001

Appendix

Table 10: Global base learning rates for the complete-network comparison.

Neuron	Trainable Params (K)
LIF ( Burkitt, 2006 )	33.024
ad-LIF ( Baronig et al., 2025 )	33.536
TC-LIF ( Zhang et al., 2024 )	33.028
DH-LIF ( Zheng et al., 2024 )	133.376
HetSyn ( Deng et al., 2025 )	65.536
DGN ( Bai et al., 2026 )	65.536

Appendix

Table 11: Trainable parameter counts of the shared two-layer network. K denotes

10^{3}

parameters. Bold and underlining mark the smallest and second-smallest counts, respectively.

Neuron	Sel. Copy. Exact	MAD ICR Exact	MAD Noisy Exact	STORE–RECALL Word
LIF ( Burkitt, 2006 )	0.00	0.00	0.00	0.00
ad-LIF ( Baronig et al., 2025 )	0.00	0.00	0.00	0.00
TC-LIF ( Zhang et al., 2024 )	0.00	0.00	0.00	0.00
DH-LIF ( Zheng et al., 2024 )	0.00	0.00	0.00	86.27
HetSyn ( Deng et al., 2025 )	0.00	24.38	0.31	0.00
DGN ( Bai et al., 2026 )	0.00	0.00	0.00	0.00

Appendix

Table 12: Exact-sequence and word accuracies for the shared-network neuron comparison. All values are in %. Bold indicates the best results, and underlining marks nonzero second-best results.

Model	Sel. Copy. Exact	MAD ICR Exact	MAD Noisy Exact	STORE–RECALL Word
PMSN ( Chen et al., 2026 )	0.00	0.00	0.00	63.46
D-RF ( Zhang et al., 2025 )	0.00	93.13	94.22	100.00
SpikingSSMs ( Shen et al., 2025 )	75.66	56.80	25.55	99.82
S5-RF ( Huber et al., 2024 )	0.00	0.00	0.00	67.38
Spiking LMUFormer ( Liu et al., 2024 )	0.00	0.00	0.00	0.00
TV-MCN	49.93	99.92	99.69	99.82

Appendix

Table 13: Exact-sequence and word accuracies for the complete-network comparison. All values are in %. Bold indicates the best results, and underlining marks nonzero second-best results.

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Multi-Timescale Conductance Spiking Networks: A Sparse, Gradient-Trainable Framework with Rich Firing Dynamics for Enhanced Temporal Processing

Alex Fulleda-Garcia, Saray Soldado-Magraner, Josep Maria Margarit-Taulé

Instituto de Microelectrónica de Barcelona (IMB-CNM), Consejo Superior de Investigaciones Cientíﬁcas (CSIC) Cerdanyola del Vallès, Spain · Department of Neurobiology, University of California Los Angeles (UCLA) Los Angeles, USA · Joint Senior authors

Spiking Neural Networks Neuromorphic Computing

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Spiking neural networks (SNNs) promise low-power event-driven computation for temporally rich tasks, but commonly used neuron models often trade off gradient-based trainability, dynamical richness, and high activity sparsity. These limitations are acute in regression, where approximation error, noise and spike discretization can severely degrade continuous-valued outputs. Indeed, many state-of-the-art (SOTA) SNNs rely on simple phenomenological dynamics trained with surrogate gradients and offer limited control over spiking diversity and sparsity. To overcome such limitations, we introduce multi-timescale conductance spiking networks, a gradient-trainable framework in which neural dynamics emerge from shaping the current-voltage (I-V) curve by tuning fast, slow and ultra-slow conductances. This parametrization allows systematic control over excitability, can be implemented efficiently in analog circuits, and yields rich firing regimes including tonic, phasic and bursting responses within a single model. We derive a discrete-time formulation of these differentiable dynamics, enabling direct backpropagation through time without surrogate-gradient approximations. To probe both trainability and accuracy, we evaluate feedforward networks of these neurons at the predictability limit of Mackey-Glass time-series regression and compare them to baseline LIF and SOTA AdLIF networks. Our model outperforms LIF and AdLIF networks, while exhibiting substantially sparser activity from both communication and computational perspectives. These results highlight multi-timescale conductance spiking neurons as a promising building block for energy-aware temporal processing and neuromorphic implementation.

Abstract

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