The spatial component of input signals often carries information crucial to a neuron's function, but models which map synaptic inputs to the transmembrane potential can be computationally expensive. Existing reduced models of the neuron either merge compartments, thereby sacrificing the spatial specificity of inputs, or apply model reduction techniques which sacrifice the biological interpretation of the model. We use Krylov subspace projection methods to construct reduced models of the passive and quasi-active neurons which preserve both the spatial specificity of inputs and the biological interpretation as an RC and RLC circuit, respectively. Each reduced model accurately computes the potential at the spike initiation zone (siz) given a much smaller dimension and simulation time, as we show numerically and theoretically. The structure is preserved through the similarity in the circuit representations, for which we provide circuit diagrams and mathematical expressions for the circuit elements. Furthermore, the transformation from the full to the reduced system is straightforward and depends on the intrinsic properties of the dendrite. As each reduced model is accurate and has a clear biological interpretation, the reduced models can be used not only to simulate morphologically accurate neurons but also to examine the underlying functions performed in dendrites.