Discussion On Existence And Uniqueness Of Traveling Wave Solution Of Some Integral Differential Equations Arising From Neuronal Networks

The existence and uniqueness of traveling wave solution of an integraldifferential equation arising from neuronal networks are discussed. This issue wasstudied by Linghai zhang in [1] where Leray-Schauder's fixed point theorem wasemployed. But the requirements of the theorem were not verified thoroughly so thatthe main conclusion of [1] does not soundly hold. In this paper, the proof in [1]is checked and the defects are pointed out. Then the original equation is transformedinto a autonomous system with a parameter on the phase-plane and the portrait of thesystem is studied. Once appearing six equilibrium points, it is possible to obtaina traveling wave solution which is different from the one proposed in [1]. Actually,by choosing certain group of parameters which meet demands of [1] and using MATLABsoftware, the portrait in phase-plane shows that the traveling wave solution assumedby [1] does not exist. At the end of the paper, by improving the condition of synapseconstantα, correct result is proved.