Mathematical study of a nonlinear neuron model with active dendrites

n this work, we have studied an extended version of the cable equation that includes both active and passive membrane properties, under the so-called sealed-end boundary condition. We have thus proved the existence and uniqueness of the weak solution, and defined a novel mathematical form of the somatic cable equation. In particular, we have manipulated the equation set to demonstrate that the diffusion term in the somatic equation is equivalent to the first-order space derivative of the membrane potential in the proximal dendrites. Our conclusion therefore clues how the somatic potential depends on the dynamic of the proximal dendritic segments, and provides the basis for the morphological reduction of neurons without any significant loss of computational properties.