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We present a phase space analysis to explore the potential of single neuron local arithmetic operations on its input conductances. This analysis was conducted first by deriving a rational function model of local spatial summation by using the equivalent circuits for steady-state membrane potentials. It is shown that developed functional phases exist in the space of input conductances, where a single neuron's local operation on input conductances can be described in terms of a set of well-defined arithmetic functions. It is further suggested that this single neuron local rational arithmetic is programmable, in the sense that the selection of these functional phases can be effectively instructed by presynaptic activities. This programmability adds the degree of freedom in a single neuron's ability to process the input information.

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