This paper will focus on a phase analysis to explore the potential of single neuron local arithmetic and logic operations on their input conductances. The analysis is based on a rational function model of local spatial summation with the equivalent circuits for steadystate membrane potentials. The prototypes of arithmetic and logic operations are then constructed with their input and output range by analyzing the conditions for performing these operations. A mapping from a partition of input conductance space into functionally distinct phases is depicted, and the multiple mode models for arithmetic and logic are then established. This indicates that the single neuron local rational arithmetic and logic is programmable, and the selection of these functional phases can be effectively instructed by presynaptic activities. This programmability makes the single neuron more free to process the input information.