  We consider a generalised diffusion equation in two dimensions for modeling
diffusion on a comb-like structures. We analyse the probability distribution
functions and we derive the mean squared displacement in $x$ and $y$
directions. Different forms of the memory kernels (Dirac delta, power-law, and
distributed order) are considered. It is shown that anomalous diffusion may
occur along both $x$ and $y$ directions. Ultraslow diffusion and some more
general diffusive processes are observed as well. We give the corresponding
continuous time random walk model for the considered two dimensional
diffusion-like equation on a comb, and we derive the probability distribution
functions which subordinate the process governed by this equation to the Wiener
process.
