A Generalized Framework for the Calcium Control Hypothesis Describes Weight-Dependent Synaptic Changes in Behavioral Time Scale Plasticity

The brain modifies synaptic strengths to store new information via long-term potentiation (LTP) and long-term depression (LTD). Evidence has mounted that long-term plasticity is controlled via concentrations of calcium ([Ca2+]) in postsynaptic spines. Several mathematical models describe this phenomenon, including those of Shouval, Bear, and Cooper (SBC) (Shouval et al., 2002, 2010) and Graupner and Brunel (GB)(Graupner & Brunel, 2012). Here we suggest a generalized version of the SBC and GB models, based on a fixed point - learning rate (FPLR) framework, where the synaptic [Ca2+] specifies a fixed point toward which the synaptic weight approaches asymptotically at a [Ca2+]-dependent rate. The FPLR framework offers a straightforward phenomenological interpretation of calcium-based plasticity: the calcium concentration tells the synaptic weight where it is going and how fast it goes there. The FPLR framework can flexibly incorporate various experimental findings, including the existence of multiple regions of [Ca2+] where no plasticity occurs, or plasticity in cerebellar Purkinje cells, where the directionality of calcium-based synaptic changes is thought to be reversed relative to cortical and hippocampal neurons. We also suggest a modeling approach that captures the dependency of late-phase plasticity stabilization on protein synthesis. We demonstrate that due to the asymptotic, saturating nature of synaptic changes in the FPLR rule, the result of frequency- and spike-timing-dependent plasticity protocols are weight-dependent. Finally, we show how the FPLR framework can explain plateau potential-induced place field formation in hippocampal CA1 neurons, also known as behavioral time scale plasticity (BTSP).