In this paper, we study the Bogdanov–Takens singularity in the Hindmarsh–Rose neuron model with time delay. We use the center manifold reduction and the normal form method, by means of which the dynamics near this nonhyperbolic equilibrium can be reduced to the study of the dynamics of the corresponding normal form restricted to the associated two-dimensional center manifold.

We show that changes in the time delay length can lead to the saddle-node bifurcation, to the Hopf bifurcation, and to the homoclinic bifurcation.

Y. Li, Z. Wei, W. Zhang, M. Perc, R. Repnik, Bogdanov–Takens singularity in the Hindmarsh–Rose neuron with time delay, Applied Mathematics and Computation 354 (2019) 180-188