Spreading depression and several other pathological conditions are associated with changes in ion concentrations in the extracellular space (ECS) of the brain. To understand such pathologies, we need to understand the interplay between the mechanisms that govern ECS concentration dynamics. The dynamics of an ion species depends on (i) cellular output as well as ECS transport due to (ii) diffusion and (iii) electrical migration. To estimate the electrodiffusive transports, one must keep track the concentrations of all present ion species and the electric potential simultaneously. This is experimentally unfeasible, which makes computational modeling an attractive tool. Existing electrodiffusive models have typically been based on the computationally demanding Poisson-Nernst-Planck equations, and studies have been limited either to phenomena on very small spatiotemporal scales (micrometers and milliseconds), or to simplified and idealized 1-dimensional (1-D) transport processes. Here, we present the 3-D Kirchhoff-Nernst-Planck (KNP) framework, tailored to simulate electrodiffusive effects on large spatiotemporal scales in the ECS surrounding active neurons, and use it to simulate the dynamics of ion concentrations and the electrical potential surrounding a morphologically detailed pyramidal cell simulated with the NEURON simulator. The KNP framework is based on the assumptions that the ECS bulk solution is strictly electroneutral, and that the only net charge density in the tissue is the membrane charge associated with the local neuronal membrane potential. We show that these assumptions allow simulation time to shortened by several orders of magnitude compared to the more physically detailed PNP schemes, without loss of accuracy at spatiotemporal scales of micrometers and microseconds and beyond. About 100 seconds of simulated activity in about 1 mm3 of tissue was performed in less than a day on a standard desktop computer. So far, we have used this framework to explore how ionic diffusion can have an impact on extracellular potentials in small model systems. However, we envision applications to more complex and biologically realistic systems will be useful in future studies addressing pathological conditions associated with large concentration variations in the ECS.