Waves and bumps in neuronal networks with axo-dendritic synaptic interactions

We consider a firing rate model of a neuronal network continuum that incorporates axo-dendritic synaptic processing and the finite conduction velocities of action po-tentials. The model equation is an integral one defined on a spatially extended do-main. Apart from a spatial integral mixing the network connectivity function with space-dependent delays, arising from non-instantaneous axonal communication, the integral model also includes a temporal integration over some appropriately identi-fied distributed delay kernel. These distributed delay kernels are biologically moti-vated and represent the response of biological synapses to spiking inputs. They are interpreted as Green’s functions of some linear differential operator. Exploiting this Green’s function description we discuss formal reductions of this non-local system to equivalent partial differential equation (PDE) models. We distinguish between those spatial connectivity functions that give rise to local PDE models and those that give rise to PDE models with delayed non-local terms. For cases in which lo-cal PDEs are derived, we investigate traveling wave solutions in a comoving fram