Existence and stability of traveling wave solutions of neuronal network equations

In this thesis, based on some phenomena in neuronal networks, we derive certain system of integral-differential equations. Then we utilize various complicated analysis to demonstrate that the neuronal network equations possess traveling pulse solutions and these wave solutions are exponentially stable. We will use the shooting argument and the exchange lemma to prove the existence of wave solutions. We then use some fundamental analysis to characterize the asymptotic behavior of the solutions of some intermediate system as $z\to \pm \infty$. Then we use the method of variation of parameter to define solutions of the eigenvalue problem and we use the Evans function to find the eigenvalues of the operator. This is achieved by employing explicit solutions of the intermediate system to get accurate information of the Evans function. Since appropriate location of the spectrum of the operator implies linear stability and linear stability implies nonlinear stability in the sense of $L\infty$-norm, we assert the nonlinear stability of the traveling wave.