Figure 1: Modeling the dendritic spine geometry and its electro-diffusional properties.

a) Spine head contains ion channels. It is connected to the parent dendrite with a thin cylindrical neck. Membrane is modeled as an impermeable dielectric with small electrical permittivity compared to cytoplasm (). At resting state, the spine head is polarized with negative electrical potential ∈_m ≪ ∈_m compared to external potential $C_{head}(t)$ Ion concentration c_head(t) also varies compared to bulk concentration c₀. Bulk is electroneutral except for a thin boundary layer near the membrane- i.e. the Debye layer (inset), where positive (red line) and negative (blue line) ion concentrations differ due to local variation of the electrical potential (black line). Most of the electrical potential drops through the poorly conducting cell membrane. Equivalent circuit describing the dendritic spine electrostatics is represented: C_m denotes the capacitance of the head membrane, C_neck and R_neck denote the membrane capacitance and the longitudinal cytoplasmic resistance of the spine neck. C_dend and R_dend denote the membrane and resistance of the dendrite. An additional membrane resistance R_leak models the ion leaks through the dendrite membrane. All of which combine to determine the voltage dynamics inside the spine head. b) Geometrical determinants of passive spine neck resistance (ion diffusion coefficient and concentration are respectively fixed to $D=\frac{1}{4}{D}_0={0.510}^{-9}{\text{m}}^2{\text{s}}^{-1}\text{and}{c}_0=150\text{m}M).$ and c₀ = 150 mM). c) Physiological determinants of spine neck resistance (neck length and radius are respectively fixed to L = 1 μm and r₀ = 50 mn).