© 2026, Society for Industrial and Applied Mathematics PublicationsIn the present work, we investigate the canard explosion occurring in a neurodynamics model-the van der Pol system with a time-delayed feedback. We assume that the feedback gain is small (on the same order as the time scale difference) such that the delay differential equation (DDE) can be reduced to a planar ordinary differential equation (ODE) on a non-local center manifold. We then expand the ODE flow on this center manifold in the small parameter and apply a nonlinear time transformation to obtain high-order approximations of the critical value and the critical manifold simultaneously. A significant advantage of the present method is that the tedious computations of the center manifold reduction with normal form usually involved in solving delay differential equa-tions are avoided. We prove that each perturbation order of the critical manifold can be expressed as a polynomial of a spatial variable. This greatly simplifies the computations and makes high-order computations possible. We also compare the proposed approach with the existing small-delay ex-pansion of which DDEs are reduced to ODEs by assuming a small delay value. While the latter cannot predict the critical manifold due to the discontinuity, our approach provides accurate and continuous approximations. More significantly, the analytical results obtained by the nonlinear time transformation method go far beyond the existing first-order results, showing an excellent agreement with the numerical simulations even for large delay values. The procedure developed in this work is efficient but simple. Because our method predicts both the critical value and the critical manifold accurately, it has great potential for practical applications in dynamical systems with time-delayed coupling and small coupling coefficients, such as controlling the occurrence of the neuronal spiking and its amplitude, in the future.