Neurons are the cellular units of our nervous system and can be found everywhere in our body, i.e. in the brain, muscles, and all organs. Neurons are constantly growing. These growth processes are governed by complex interactions of internal and external signaling pathways. During their growth, neurons continuously change their shape. Here, we present a spatially resolved model consisting of a set of ten reaction–diffusion–convection PDEs that describe the dynamics of signaling pathways, which will determine the changing morphology. To this end, we explicitly define a neuron cell and its extracellular surroundings as a domain, which we respectively divide into intracellular (tubulin, MAP-2 in free/phosphorylated/bound states, and calcium) and extracellular (Slit-like inhibitor) subdomains. The two subdomains are separated by a moving inner boundary, which gives the neuron its shape. Our goal is to simulate complex shape changes such as elongation, branching, and retraction, using the Level set Method to track the moving boundary separating the two subdomains, each with distinct PDE dynamics. Based on these the solutions of these PDE, we compute the kinetics of the interface deformation which we subsequently use with three signed distance functions to calculate the position of the inner boundary over time. The presented equations are discretized in space by a vertex-centered finite volume scheme on unstructured grids. The time discretization involves implicit Euler method. We use the geometric multigrid method for solving the linear systems arising from the linearizations of the discretized equations in the Newton's method. Parallelized implementation is based on the toolbox ug4. We present results of simulations of several benchmarks. This model unifies intracellular and extracellular signaling dynamics, allowing for the emergence of complex shape changes. Consequently, this model is a versatile tool for studying morphological changes and neuronal responses to diverse stimuli.