Abstract

We extend our previous paper on deriving an approximate analytical solution of a nonlinear cable equation by including other ion channels in neurons and calcium dynamics based on reaction-diffusion dynamics that lead to a system of nonlinear cable equations. Here, excitable dendrite possesses clusters of voltage-activated ion channels that are discretely distributed as point sources or hotspots of transmembrane current along a continuous cable structure of fixed length. Single and/or trains of action potentials and spatially distributed synaptic inputs drive the depolarisation and activate sparsely distributed voltage-dependent calcium channels. This leads to calcium influx and diffusion in the cable. Here, time-dependent analytical solutions were obtained by applying a perturbation expansion of the non-dimensional voltage (Φ) and non-dimensional calcium (ΦCa) and then solving the resulting set of integral equations. We use this framework to gain insights into calcium-driven synaptic plasticity in dendrites. Many previous studies have traditionally focused on the local impact of calcium on whether the synapse's strength is increased (potentiated) or decreased (depressed). Only recently have studies focusing on heterosynaptic plasticity been gaining popularity, and here we ask the question of how a local plasticity rule is influenced by the spatially and temporally distributed synaptic inputs. Specifically, we focus on how synaptic inputs and calcium influx impact a calcium-derived temporal learning window for spike-timing-dependent plasticity (STDP) at nearby sites to assess the nature of the resulting distance-dependent interaction on the associated learning window at the synapse of interest.