Spike-Kal Algorithm

• Spike-Kal is a spiking neural network–assisted Kalman filtering algorithm that uses two-layer LIF networks to approximate the Kalman gain and enhance noise adaptability.
• It leverages innovation features and reward-modulated STDP for online adaptation, eliminating the need for explicit matrix inversion and pre-specified noise models.
• Spike-Kal achieves reduced computational complexity and significant error reductions in high-dimensional tracking tasks, as demonstrated on linear motion, Lorenz systems, and UAV trajectories.

Spike-Kal is a spiking neural network–assisted Kalman filtering algorithm that integrates the predictive estimation capabilities of the Kalman filter with the adaptive, nonlinear representational power of two-layer leaky integrate-and-fire (LIF) spiking neural networks (SNNs). Developed to address difficulties in modeling and adapting to time-varying noise, Spike-Kal directly leverages data-driven SNNs to approximate the Kalman gain matrix, achieving both improved robustness in noise uncertainty and reduced computational complexity in high-dimensional inference contexts (Xiao et al., 17 Apr 2025).

1. Mathematical Formalism and Algorithmic Workflow

The traditional linear Kalman filter operates via recursive state estimation, using the following equations:

• State prediction: $x_{k|k-1} = F_k x_{k-1|k-1} + B_k u_k$
• Covariance prediction: $P_{k|k-1} = F_k P_{k-1|k-1} F_k^\top + Q_k$
• Kalman gain: $$K_k = P_{k|k-1} H_k^\top (H_k P_{k|k-1} H_k^\top + R_k )^{-1}$$
• State update: $x_{k|k} = x_{k|k-1} + K_k (z_k - H_k x_{k|k-1})$
• Covariance update: $P_{k|k} = (I - K_k H_k) P_{k|k-1}$

Spike-Kal eliminates the explicit matrix inversion for $K_k$ in favor of data-driven estimation:

• Feature Extraction: $\Delta x_{k-1} = x_{k-1|k-1} - x_{k-1|k-2}$ (system prediction error), $\Delta y_k = z_k - H_k x_{k|k-1}$ (observation residual). These are concatenated into $v_k = [\Delta x_{k-1}; \Delta y_k]$.
• SNN Gain Approximation: $K_k \approx f_{\mathrm{SNN}}(v_k; W)$, where $P_{k|k-1} = F_k P_{k-1|k-1} F_k^\top + Q_k$0 are the synaptic weights. Each output neuron produces a spike pattern decoded to an entry of $P_{k|k-1} = F_k P_{k-1|k-1} F_k^\top + Q_k$1.

2. SNN Architecture and Integration Mechanism

The SNN in Spike-Kal utilizes LIF neurons whose subthreshold membrane potential $P_{k|k-1} = F_k P_{k-1|k-1} F_k^\top + Q_k$2 and synaptic current $P_{k|k-1} = F_k P_{k-1|k-1} F_k^\top + Q_k$3 evolve as:

• $P_{k|k-1} = F_k P_{k-1|k-1} F_k^\top + Q_k$4
• $P_{k|k-1} = F_k P_{k-1|k-1} F_k^\top + Q_k$5

The network configuration consists of:

• Input layer: $P_{k|k-1} = F_k P_{k-1|k-1} F_k^\top + Q_k$6 neurons
• Output layer: $P_{k|k-1} = F_k P_{k-1|k-1} F_k^\top + Q_k$7 neurons encoding the entries of $P_{k|k-1} = F_k P_{k-1|k-1} F_k^\top + Q_k$8
• Encoding: Continuous vector $P_{k|k-1} = F_k P_{k-1|k-1} F_k^\top + Q_k$9 is injected as input current into $$K_k = P_{k|k-1} H_k^\top (H_k P_{k|k-1} H_k^\top + R_k )^{-1}$$0 spiking neurons, producing spike trains over a fixed simulation window.
• Topology: Fully connected from input to output layer via trainable synapses $$K_k = P_{k|k-1} H_k^\top (H_k P_{k|k-1} H_k^\top + R_k )^{-1}$$1.
• Decoding: The sequence of output layer spikes is translated to $$K_k = P_{k|k-1} H_k^\top (H_k P_{k|k-1} H_k^\top + R_k )^{-1}$$2 for use in the filter update.

Integration steps:

1. Prediction: $$K_k = P_{k|k-1} H_k^\top (H_k P_{k|k-1} H_k^\top + R_k )^{-1}$$3
2. Feature vector: $$K_k = P_{k|k-1} H_k^\top (H_k P_{k|k-1} H_k^\top + R_k )^{-1}$$4
3. SNN inference: $$K_k = P_{k|k-1} H_k^\top (H_k P_{k|k-1} H_k^\top + R_k )^{-1}$$5 spike trains $$K_k = P_{k|k-1} H_k^\top (H_k P_{k|k-1} H_k^\top + R_k )^{-1}$$6 decoded $$K_k = P_{k|k-1} H_k^\top (H_k P_{k|k-1} H_k^\top + R_k )^{-1}$$7
4. State update: $$K_k = P_{k|k-1} H_k^\top (H_k P_{k|k-1} H_k^\top + R_k )^{-1}$$8

3. Learning and Optimization Methods

Unlike L2-loss minimization, Spike-Kal applies reward-modulated spike-timing-dependent plasticity (R-STDP) for synaptic learning:

• R-STDP Rule: $$K_k = P_{k|k-1} H_k^\top (H_k P_{k|k-1} H_k^\top + R_k )^{-1}$$9
  • $x_{k|k} = x_{k|k-1} + K_k (z_k - H_k x_{k|k-1})$0 for $x_{k|k} = x_{k|k-1} + K_k (z_k - H_k x_{k|k-1})$1, $x_{k|k} = x_{k|k-1} + K_k (z_k - H_k x_{k|k-1})$2 otherwise.
  • $x_{k|k} = x_{k|k-1} + K_k (z_k - H_k x_{k|k-1})$3 set scales for long-term potentiation (LTP) and depression (LTD); typical value: $x_{k|k} = x_{k|k-1} + K_k (z_k - H_k x_{k|k-1})$4.
  • $x_{k|k} = x_{k|k-1} + K_k (z_k - H_k x_{k|k-1})$5 are STDP time constants, e.g., $x_{k|k} = x_{k|k-1} + K_k (z_k - H_k x_{k|k-1})$6ms.
  • $x_{k|k} = x_{k|k-1} + K_k (z_k - H_k x_{k|k-1})$7 is the teaching reward, proportional to the Frobenius norm $x_{k|k} = x_{k|k-1} + K_k (z_k - H_k x_{k|k-1})$8 between the classical Kalman gain and the SNN output.

Training procedure:

• The standard Kalman filter runs in parallel with the SNN during training to provide $x_{k|k} = x_{k|k-1} + K_k (z_k - H_k x_{k|k-1})$9 and reward $P_{k|k} = (I - K_k H_k) P_{k|k-1}$0.
• Weight updates via R-STDP occur online for $P_{k|k} = (I - K_k H_k) P_{k|k-1}$1 steps (typically a few thousand steps).
• After convergence, weights $P_{k|k} = (I - K_k H_k) P_{k|k-1}$2 are frozen and the SNN provides the gain in inference mode.

Data sources include simulated linear motion, Lorenz attractor systems with additive Gaussian process/observation noise, and real-world UAV trajectory estimation with pixel-level, a priori unknown noise statistics.

4. Computational Complexity and Performance

Time complexity for gain computation:

• Classical approach: $P_{k|k} = (I - K_k H_k) P_{k|k-1}$3 per step due to matrix inversion.
• Spike-Kal SNN approach: $P_{k|k} = (I - K_k H_k) P_{k|k-1}$4, typically $P_{k|k} = (I - K_k H_k) P_{k|k-1}$5 assuming $P_{k|k} = (I - K_k H_k) P_{k|k-1}$6, where $P_{k|k} = (I - K_k H_k) P_{k|k-1}$7 is the SNN time window.

Empirical benchmarks:

• When deployed on custom RISC-V neuromorphic hardware, Spike-Kal achieved a $P_{k|k} = (I - K_k H_k) P_{k|k-1}$8 reduction in end-to-end latency for high-dimensional filtering compared to matrix-inverse–based methods, with the exact speedup contingent on filter dimension and SNN simulation duration.

5. Quantitative Evaluation and Baselines

Evaluation across tasks yields the following:

• Linear Motion (4D state): Spike-Kal achieves MSE$P_{k|k} = (I - K_k H_k) P_{k|k-1}$9 $K_k$0 vs. KF $K_k$1, SNN-KF $K_k$2; MSE$K_k$3 $K_k$4 vs. $K_k$5, $K_k$6.
• Lorenz System (3D state): Spike-Kal MSE$K_k$7 vs. EKF $K_k$8 (−20%), SNN-KF $K_k$9; MSE$\Delta x_{k-1} = x_{k-1|k-1} - x_{k-1|k-2}$0 vs. $\Delta x_{k-1} = x_{k-1|k-1} - x_{k-1|k-2}$1 (−57%); MSE$\Delta x_{k-1} = x_{k-1|k-1} - x_{k-1|k-2}$2 vs. $\Delta x_{k-1} = x_{k-1|k-1} - x_{k-1|k-2}$3 (−18%).
• UAV Tracking (2D position): Mean absolute error is reduced by $\Delta x_{k-1} = x_{k-1|k-1} - x_{k-1|k-2}$4–$\Delta x_{k-1} = x_{k-1|k-1} - x_{k-1|k-2}$5.

Comparative baselines:

• Standard KF (with slightly mis-specified noise covariances $\Delta x_{k-1} = x_{k-1|k-1} - x_{k-1|k-2}$6)
• Extended Kalman Filter (EKF) where applicable
• SNN-Kalman approach using classical STDP (Juárez-Lora et al.)

Spike-Kal maintains low error even when process and observation noise statistics drift, attributed to adaptation on the features $\Delta x_{k-1} = x_{k-1|k-1} - x_{k-1|k-2}$7 and $\Delta x_{k-1} = x_{k-1|k-1} - x_{k-1|k-2}$8 rather than relying on fixed $\Delta x_{k-1} = x_{k-1|k-1} - x_{k-1|k-2}$9.

6. Algorithmic Summary and Implementation Outline

The fundamental algorithmic pipeline is encapsulated in the following pseudocode:

$\Delta y_k = z_k - H_k x_{k|k-1}$4

This operational outline demonstrates the substitution of the traditional analytic gain matrix calculation with SNN-based inference and online R-STDP–guided learning during training.

7. Key Properties, Innovations, and Significance

Spike-Kal introduces several core advances:

• Removal of dependence on a priori specification or tuning of process and observation noise covariances ($\Delta y_k = z_k - H_k x_{k|k-1}$0), with real-time adaptation to shifting statistical profiles.
• Reduction of computational complexity by leveraging neuromorphic SNN hardware for gain inference, replacing matrix inversion.
• Empirical performance improvements of $\Delta y_k = z_k - H_k x_{k|k-1}$1–$\Delta y_k = z_k - H_k x_{k|k-1}$2 lower mean squared error versus EKF/KF in the presence of noise uncertainty, while preserving rapid convergence.
• Essential innovations include the direct use of innovation features ($\Delta y_k = z_k - H_k x_{k|k-1}$3) as SNN input, a fully connected two-layer LIF network for gain approximation, reward-modulated STDP for robust online learning, and hybrid teacher–student training where the traditional Kalman gain supervises the early SNN adaptation (Xiao et al., 17 Apr 2025).