Abstract The longtime behaviour of the FitzHugh–Rinzel (FHR) neurons and transition to instability FHR steady states, are investigated. Criteria guaranteeing solutions boundedness, absorbing sets, in energy phase space, existence states via oscillatory bifurcations, obtained. Denoting by $$ \lambda ^{3} + \sum\nolimits_{{k = 1}}^{3} {A_{k} } (R)\lambda ^{{3 - k}} 0 <mml:math xmlns:mml="http://w...