Macro-Micro Inference: Robust Synaptic Classification via Spike-Triggered Extrapolation

Abstract

This work introduces a framework for reconstructing the interaction graph of neuronal networks modeled as multivariate point processes. The methodology performs bivariate inference, identifying synaptic links exclusively from the spike trains of a pair of neurons, without requiring observations of the remaining network activity. We propose a Macro-Micro Extrapolation algorithm to address data sparsity at the micro-scale, inferring synaptic interactions in the limit $Δ\to 0^+$. A key contribution is the Spike-Triggered Estimator, which leverages the local reset property of Galves-Löcherbach dynamics to decouple local synaptic jumps from higher-order network contributions, significantly reducing estimation variance and eliminating spurious dependencies on baseline firing intensities. By employing an adaptive hybrid logic that switches between sample averaging and our novel Pyramid Extrapolation, we ensure robust classification of excitatory, inhibitory, and null connections even in low signal-to-noise regimes. The framework's scalability and precision are validated by numerical results on dense cliques and structured layered networks, achieving perfect classification accuracy across diverse topological motifs.

Figures (5)

• Figure 1: Classification map for topology identification. The decision space is partitioned by thresholds $\zeta_{\pm} = \pm 1/2$. The blue gaps represent the safety margins that accommodate the statistical error of the estimator $\hat{\mathcal{G}}^{j \to i}(\Delta_A)$, ensuring it remains within the correct decision region relative to the theoretical bounds.
• Figure 2: Minimal network architecture for validation.
• Figure 3: Synaptic extrapolation results. (Top Left) Excitatory, (Top Right) Null, (Bottom) Inhibitory. Blue marks: raw gains; Open red circles: pyramid meta-points; Dashed line: extrapolation to $\Delta \to 0$. Red dotted lines indicate the classification thresholds.
• Figure 4: Network Architecture and Metrological Performance ($N=60$). Top: Proximity matrix showing layered connectivity with dictatorial pacemaker ($L_0$), feed-forward blocks, and inhibitory feedback ($L_3 \to L_1$). Bottom: Scatterplot comparing recovered functional index $I$ against ground truth $w$. The Pyramid method (red circles) follows the theoretical gain $I=2w_{i,j}$ with high linearity, neutralizing the bias of the sample mean (blue crosses). Data points collected via $N_D=2000$ fixed-event protocol.
• Figure 5: Asymptotic resilience stress-test under extreme sampling regimes ($8\times \Delta_1$). The plot illustrates the de-biasing performance of the Pyramid estimator when the observation window is deliberately expanded up to eight times the theoretical limit. Colored circles: measured indices $G(\Delta)$, exhibiting significant decay (up to 43% for excitatory links) due to network-induced macroscopic interference. Dashed lines: geometric extrapolation derived from the Pyramid iterative logic. Solid points at $\Delta=0$: recovered microscopic estimates. Despite the severe damping of the raw signal, the method reconstructs the instantaneous gain with high fidelity (e.g., recovering an index of $1.53$ from a degraded $0.70$ baseline for $w=0.95$), confirming that the geometric projection effectively compensates for the non-linear curvature induced by macroscopic observation scales.

Theorems & Definitions (11)

• Remark 1
• Lemma 1: Baseline Probability Estimates
• proof
• Corollary 1
• Lemma 2: Probability Ratios for Connectivity
• proof
• Corollary 2
• proof
• Theorem 1: Stability of Interaction Estimators
• proof