Heriot-Watt Mathematics Report Series
HWM03-35, 8 December 2003

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

S Coombes, G J Lord and M R Owen

Abstract

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 kernel. 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 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 that require both advanced and retarded terms.

For cases in which local PDEs are derived, we investigate traveling wave solutions in a comoving frame by numerically computing global heteroclinic connections. We then calculate exact solutions, parameterized by axonal conduction velocity, for sigmoidal firing rate functions in the limit of infinite gain, for a variety of spatial connectivities and synaptic responses. 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.

Our numerical results are shown to be consistent with exact calculations. Thus we combine analytical and numerical approaches to provide a thorough exploration of the effect of synaptic processing and adaptation, dendritic and axonal delays and patterns of axo-dendritic connectivity on one dimensional network dynamics

Google Scholar Search: links, citations and journal (if available)

Publication

S Coombes, G J Lord, M R Owen , Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Physica D, 178, No 3-4, 219-241, (2003).

Contact Details | 2003 Reports Index | Full Index