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In this paper, we study the bounded traveling wave solutions to the Nagumo nerve conduction equation by using the theory of dynamical system and undetermined assumption method. We make a detailed qualitative analysis on the dynamical system corresponding to the traveling wave solution to this equation and obtain a threshold value c ∗ that when the wave velocity is less than c ∗ , the bounded traveling wave solution to Nagumo equation exhibits oscillatory and damped properties, and when the wave velocity is higher than c ∗ , it usually takes the form of a wavefront solution. In addition, we obtain several new pulse solutions and wavefront solutions to Nagumo equation by using undetermined assumption method. In particular, we obtain the approximate oscillatory damped solutions when the wave velocity less than the threshold c ∗. Further, by establishing the integral equation between the approximate solution and the exact solution, and examining the asymptotic state of the solution at infinity, the error estimation between the approximate solution and the exact solution of the oscillatory solution is obtained as an infinitesimal quantity decreasing exponentially. Finally, we compare the approximate solutions of the oscillatory solutions with the numerical solutions, and they are in agreement. • We focus on the bounded traveling wave solutions of Nagumo equation. • We find a critical value c ∗ , which characterizes magnitude of propagation velocity. • We obtain the analytical approximation solutions of the oscillatory damped solutions. • We give the global error estimations for analytical approximation solutions. [ABSTRACT FROM AUTHOR]