A mathematical model of potential spreading along neuron dendrites is proposed to describe the synaptic transmissions in the so-called cerebellar granule cells, which consist of a nearly spherical soma emitting a finite number of dendrites. The model accounts for the nonlinear dependence of the NMDA receptors, located at the virtual ends of any dendrite, upon the voltage. The corresponding initial-boundary value problem is formulated in the framework of Sobolev spaces. Existence and uniqueness of a weak solution are proved along with regularity results ensuring that the solution is classical.