STDP as Probabilistic Attribution: An Exact-Balance Continuous Kernel for Normalized Temporal Credit Assignment

TMLR Paper9261 Authors

Abstract: We introduce a unified continuous kernel for spike-timing-dependent plasticity (STDP) that connects local spike-timing updates to normalized probabilistic attribution in convergent circuits. Classical phenomenological STDP models describe long-term potentiation (LTP; the strengthening of synapses when a presynaptic spike precedes a postsynaptic spike) and long-term depression (LTD; the weakening of synapses in the reverse order) using piecewise timing windows, while standard simulator implementations commonly realize such rules with local traces. Our contribution is therefore not a claim of asymptotic speedup over trace-based STDP, but a single differentiable trace-interaction kernel whose induced learning window can be analyzed in closed form. The model represents presynaptic and postsynaptic events by dimensionless exponentially decaying traces and defines synaptic change by their cooperative and competitive interaction. For an isolated pre–post spike pair, we derive the closed-form STDP window and prove that the total integrated potentiation and depression areas are exactly balanced for all positive decay rates. We further summarize parameter sweeps and component ablations showing how the two decay rates tune window morphology and why both the multiplicative gating term and competitive difference term are required for a biphasic timing-sensitive window. A fit to the classical data of Bi and Poo (1998) gives $R^2=0.63$ and reveals a narrow near-coincident positive-update regime for small post-before-pre lags. This regime is a structural consequence of fitting a continuous kernel with mismatched decay rates; however, the raw observations in the corresponding interval are positive, so we treat it as a data-consistent near-synchronous attribution hypothesis rather than a confirmed biological mechanism or a mere fitting artifact. Forcing the zero crossing to $\Delta t = 0$ substantially worsens the fit (Appendix D). At the network level, we show that when the additive kernel is combined with multiplicative afferent normalization, the mean-field dynamics reduce to a delta-rule-like update whose fixed point is a normalized event-rate target, $w_i^* = \nu_i q_i/\sum_j \nu_j q_j$. Under a strict causal-window approximation, $q_i=P(\mathrm{Post}\mid \mathrm{Pre}_i)$, and under the corresponding partition assumptions this target can be interpreted as posterior attribution, $P(\mathrm{Pre}_i\mid \mathrm{Post})$. Without those assumptions, the fixed point should be read as normalized conditional event-rate attribution. Simulations confirm convergence in sparse regimes, document progressive degradation under dense firing, and show that the proposed kernel outperforms classical STDP baselines, including variants matched for area balance, trace mode, step size, and temporal footprint, under identical normalization. An iso-rate control confirms the network tracks attribution probability independent of firing rate, a decorrelated control confirms it tracks the product $\nu_i q_i$, and a heterogeneous-delay condition confirms robustness to non-uniform causal timing.

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Joao_Sacramento1

Submission Number: 9261