Waves and bumps in neuronal networks with axo-dendritic synaptic interactions. (English) Zbl 1013.92006

Summary: We consider a firing rate model of a neuronal network continuum that incorporates axo-dendritic synaptic processing and the finite conduction velocities of action potentials. The model equation is an integral one defined on a spatially extended domain. 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 identified distributed delay kernels. These distributed delay kernels are biologically motivated and represent the response of biological synapses to spiking inputs. They are interpreted as Green’s functions of some linear differential operators.
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 local PDEs are derived, we investigate traveling wave solutions in a co-moving frame by numerically computing global heteroclinic connections for sigmoidal firing rate functions.
We also calculate exact solutions, parameterized by axonal conduction velocity, for the Heaviside firing rate function (the sigmoidal firing rate function in the limit of infinite gain). The inclusion of synaptic adaptation is shown to alter traveling wave fronts to traveling pulses, which we study analytically and numerically in terms of a global homoclinic orbit. Finally, we consider the impact of dendritic interactions on waves and on static spatially localized solutions. Exact analysis for infinite gain shows that axonal delays do not affect the stability of single bumps. Furthermore, numerical continuation for finite gain leads to multiple bump solutions, and it is demonstrated that such localized multi-bumps are lost (in favor of global patterns) when a stable \(N\)-bump and an unstable \((N+2)\)-bump coalesce.
In essence, the work in this paper illustrates how physiologically significant features of synaptic processing, synaptic adaptation and patterns of axo-dendritic connectivity may be analyzed within a neural field firing rate framework.


92C20 Neural biology
35Q92 PDEs in connection with biology, chemistry and other natural sciences


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