Competition and timescale separation in calcium-based plasticity

Among existing formulations, the calcium-based model of early long-term plasticity introduced by Graupner and Brunel in 2012 is still widely used and serves as a foundational reference for describing synaptic potentiation and depression. However, the model exhibits limitations such as abrupt changes, slow drift toward extreme potentiation or depression in the absence of calcium, and instantaneous molecular signaling from calcium concentration to synaptic changes, thus neglecting timescale separation and competing signaling mechanisms between potentiation and depression. Building on this seminal work, we propose a minimal and general framework for the signaling pathways underlying synaptic changes, which addresses the issues of the original model. Consistent with Graupner and Brunel's philosophy, our approach avoids committing to specific physiological mechanisms, which is advantageous given their substantial variability across brain regions and cell types. By introducing timescales for potentiation and depression pathways, our model strengthens the robustness and memory dependence of synaptic modification, and shares similarity with the sliding threshold model of Bienenstock, Cooper and Munro. Greater timescale separation shifts the Spike-Timing-Dependent Plasticity (STDP) kernel upward, allowing us to set a much higher potentiation threshold than in previous studies. Conversely, increased competition lowers the STPD kernel in regimes where the timescale separation is not too strong. Our formulation improves both the biological plausibility and stability of the synaptic model, while offering a flexible framework for future extensions, e.g. adding pathways with distinct timescales to capture short-term and late long-term plasticity within a unified model.