Stochasticity in Ca$^{2+}$ increase in spines enables robust and sensitive information coding

A dendritic spine is a very small structure (~0.1 {\mu}m$^3$) of a neuron that processes input timing information. Why are spines so small? Here, we provide functional reasons; the size of spines is optimal for information coding. Spines code input timing information by the probability of Ca$^{2+}$ increases, which makes robust and sensitive information coding possible. We created a stochastic simulation model of input timing-dependent Ca$^{2+}$ increases in a cerebellar Purkinje cell's spine. Spines used probability coding of Ca$^{2+}$ increases rather than amplitude coding for input timing detection via stochastic facilitation by utilizing the small number of molecules in a spine volume, where information per volume appeared optimal. Probability coding of Ca$^{2+}$ increases in a spine volume was more robust against input fluctuation and more sensitive to input numbers than amplitude coding of Ca$^{2+}$ increases in a cell volume. Thus, stochasticity is a strategy by which neurons robustly and sensitively code information.