Spiking Neural Networks for Robust Fitting

• Spiking neural networks are bio-inspired models that use sparse, event-driven binary spikes to achieve robust performance under noise and hardware variability.
• Robust fitting is enabled through methodologies like temporal coding, noise-inclusive training, and certified bounds that enhance adversarial resistance.
• Neuromorphic implementations leverage event-driven algorithms and discrete spike computations to ensure energy efficiency and reliable operation in real-world applications.

Spiking neural networks (SNNs) are bio-inspired computational models that utilize sparse, event-driven communication via binary spike events and exhibit strong potential for robust fitting under noise, perturbations, and hardware variability. Robust fitting in this context refers to the system’s ability to perform accurate parameter estimation, classification, or prediction despite input variability, adversarial disturbances, or hardware-induced errors, thus ensuring reliable deployment in real-world and low-power neuromorphic environments. Robustness in SNNs emerges from their temporal coding, nonlinear spike generation, noise adaptation, network dynamics, and the architectural choices that exploit or regulate such features for inference stability.

1. Robust Fitting Methodologies in Spiking Neural Networks

Distinct methodologies underpin robust fitting in SNNs:

• Temporal Coding and Spike-Time Learning: Transforming real-valued signals into spike times, with early and late spikes representing binary values, enables the encoding of information in the temporal domain. For example, input encoding may use

$$f(x) = \frac{x - \text{Min}}{\text{Max} - \text{Min}} \cdot L,$$

where $L$ sets the coding interval (Yang et al., 2018). SNNs trained with spike-time error backpropagation (e.g., SpikeProp) adjust synaptic weights such that the correct output spike times are associated with input patterns, even when those inputs are contaminated by sinusoidal or Gaussian perturbations.

• Noise-Inclusive Training: Deliberate inclusion of channel noise or device-induced noise in the neuron or synaptic model (e.g., noisy integrate-and-fire, stochastic channel models) leads to smoother transitions at parameter boundaries. Compared to deterministic networks, these stochastic SNNs exhibit gradual, probabilistic activation changes, thereby improving tolerance to synaptic inaccuracies, quantization, or analog device noise (Olin-Ammentorp et al., 2019).
• Adaptive Neuronal Dynamics and Bifurcation: Bifurcation SNNs introduce a set of learnable parameters ($\lambda$) that decouple the control rate ($\gamma$) from neuron activation. This bifurcation enables a range of dynamic responses per neuron and renders performance robust to hyperparameter choices and input regimes, as the effective eigenvalues ($\rho_i = \gamma + \beta_i$) can adapt during training (Zhang et al., 2019).
• Regularization via Activity Summary Statistics: In optimizing network fit to neural recordings or dynamical tasks, summary statistics (e.g., peristimulus time histograms, noise correlations) are included in the loss function, alongside log-likelihood, using differentiable simulators with straight-through gradient estimators. This joint objective encourages the SNN’s simulated outputs to match the desired statistical structure, essential for robust system identification even in the presence of hidden neurons (Bellec et al., 2021).

2. Input Perturbation and Robustness Mechanisms

SNNs demonstrate explicit robustness properties under structured input perturbations:

• Sinusoidal and Gaussian Perturbations: Input-level robustness is characterized by the network's ability to sustain classification accuracy following structured (periodic) perturbations, such as $\tilde{x} = \tilde{x}_0 + A \sin(2\pi y)$ (sinusoidal perturbation), or proximity-skewed shifts via

$$\tilde{x} = \tilde{x}_0 + \tilde{x}_0 \left(I - \exp\left(-\frac{r^2}{2}\right) \cdot \mathrm{sgn}(l)\right)$$

for Gaussian perturbations. Empirical experiments show that, even as the amplitude of the sinusoidal component increases, SNNs exhibit only modest drops in classification, and Gaussian noise—by clustering most perturbed points near the target—induces minimal performance loss (Yang et al., 2018).

• Adversarial Noise and Discretization: Poisson input encoding and short-integration windows impose discrete, quantized input representations in SNNs. The effect of this discrete encoding is magnified by reducing the number of timesteps, which increases adversarial resistance: Small perturbations are less likely to shift the binary spike train representation, and adversarial accuracy can increase by reducing temporal integration (Sharmin et al., 2020). The spike-generation nonlinearity and LIF neuron’s leak parameter ($\lambda$) further dampen the propagation of adversarial noise.
• Certified Boundary Methods: Formal robustness guarantees for SNNs draw upon interval bound propagation (S-IBP) and linear relaxations (S-CROWN), which tightly bound the nonlinear “fire” and temporal update mechanisms in SNN neurons under both digital and binary spike input uncertainty. These methods lead to rigorously certified output bounds that guarantee the output remains correct under bounded input perturbations ($37.7\%$ attack error reduction with $3.7\%$ accuracy loss in certain configurations) (Liang et al., 2022).

3. Robust Fitting on Neuromorphic Hardware

Neuromorphic implementations of SNNs introduce challenges due to limited numerical precision, inherent device mismatch, and restricted instruction sets:

• Event-Driven Algorithms and Sampling: The NeuroRF architecture (“Event-driven Robust Fitting on Neuromorphic Hardware” (Nguyen et al., 13 Aug 2025)) directly maps robust fitting (e.g., RANSAC-style linear regression) into a layered SNN design. RandomSampling neurons select data points probabilistically; an auxiliary layer performs the “lifting” operation that couples data selection and model estimation. Candidate models are refined iteratively, using:

$$\theta^{(t)} = \theta^{(t-1)} - \alpha (Q' \theta'^{(t-1)} + P' z)$$

with $z$ encoding random sampling and $\theta'$ representing the joint effect of data selection and model update.

• Instruction Set-Aware Arithmetic: Precision constraints and lack of floating-point support on neuromorphic hardware (e.g., Intel Loihi 2) are addressed by emulating gradient descent steps with integer arithmetic and right shifts, approximating the learning rate scaling by $\alphā = \lceil \alpha \cdot 2^{\beta}\rceil$ and replacing division operations.
• Energy Efficiency: The event-driven asynchronous computation (neurons only fire when needed) reduces power consumption—NeuroRF achieves robust fitting with only $15\%$ of the energy used by CPU-based methods at equivalent accuracy (Nguyen et al., 13 Aug 2025). This suggests that, while runtime may be increased due to hardware clock constraints, resource-constrained applications (e.g., edge devices) can leverage neuromorphic models for robust inference with significant energy savings.

4. Biological and Theoretical Drivers of Robustness

Key biological processes and theoretical analyses underpin SNN robustness:

• Excitation-Inhibition Balance via Short-Term Depression: Biologically, robust information flow is attributed to a balance between excitatory and inhibitory currents, refined by mechanisms such as short-term synaptic depression (STD). The effect is to dynamically scale the effective excitatory influence, yielding a nonlinear response and maintaining irregular, fluctuation-driven firing even without strong external drive (Politi et al., 23 Jan 2024).
• Biological E:I Ratios and Inhibitory Diversity: Explicitly enforcing a biologically consistent 80:20 excitatory-inhibitory ratio, coupled with low initial firing rates and high diversity among inhibitory spiking patterns, leads to robust and efficient classification even in noisy settings. Inhibitory diversity helps maintain decorrelated activation patterns and improved feature separation. The Van Rossum distance, computed as

$$d(a, b) = \sqrt{\frac{1}{\tau} \int_0^\infty (g_a(t) - g_b(t))^2 dt}$$

where $g_a(t)$ is an exponentially filtered spike train, provides an effective robustness metric (Kilgore et al., 24 Apr 2024).

• Input-Output Stability in Membrane Dynamics: Robustness at the level of single neurons and entire networks is further supported by theoretical guarantees; LIF or dynamically modified (DLIF) neurons with perturbation recurrence

$$\epsilon^l[t] = \lambda \epsilon^l[t-1] + \sum_j w^l_{ij} \Delta s^{l-1}_j[t]$$

satisfy $L_2$ input-output stability, such that a small input disturbance yields proportionally bounded membrane perturbations, with the bound $\gamma^l = \sqrt{1/(1-\lambda)} \|W^l\|$ (Ding et al., 31 May 2024).

5. Experimental Evidence and Task Performance

Empirical studies across multiple tasks demonstrate the breadth of SNN robustness:

• XOR and Benchmark Data: SNNs maintain high classification rates under increasing perturbation amplitudes on the XOR problem and real-world datasets such as Iris, breast cancer, and Landsat, especially under Gaussian perturbations that keep inputs clustered near targets (Yang et al., 2018).
• Temporal Dynamical Systems: Supervised SNNs trained via knowledge distillation from non-spiking RNNs (with adaptive, local learning rules) accurately track the desired state for temporal XOR and audio-based wake-phrase detection, even under process mismatch, quantization, and thermal noise, obviating the need for per-device calibration in hardware deployment (Büchel et al., 2021).
• Adversarially Robust Vision Models: SNNs trained with adversarially robust ANN-to-SNN conversion, followed by robust finetuning (TRADES-inspired objectives), outperform direct SNN adversarial training methods when evaluated under an ensemble of white-box and black-box attacks, while maintaining efficient coding (short spike trains) and low inference latency (Özdenizci et al., 2023). Certified training methods (S-IBP/S-CROWN) further ensure provable resilience for both digital and spiking inputs (Liang et al., 2022).
• Hardware-Implemented Robust Fitting: In practical neuromorphic settings, spiking architectures for robust fitting (e.g., on Loihi 2) achieve equivalent estimation accuracy with a substantial reduction in energy costs compared to CPU-based routines (Nguyen et al., 13 Aug 2025).

6. Architectural Extensions and Future Prospects

Current progress—embodied in SNNs with adaptive neuron dynamics, biologically-informed architecture (e.g., bifurcation parameters, E/I balance), event-driven hardware mapping, and new robust training objectives—lays the groundwork for several directions:

• Robust model fitting is now tractable under hardware and analog constraints, opening the door to embedded and mobile applications where power limitations are severe.
• Certified robustness and adversarial resistance are attainable without significant drops in clean accuracy.
• Inclusion of higher-order summary statistics and biologically-motivated regularization provides a path to realistic simulation of cortical networks from data, particularly when only partial observation is available.
• Open questions remain around scaling to deeper or recurrent SNNs, optimizing backward-pass efficiency on hardware, and fully harnessing noise as a resource for stability and generalization.

Robust fitting in spiking neural networks thus emerges as an intersectional advance—combining algorithmic, theoretical, and hardware innovation—to produce models and systems capable of reliable operation across a spectrum of adversarial, noisy, and hardware-constrained scenarios.