Longtime behaviour and bursting frequency, via a simple formula, of FitzHugh–Rinzel neurons

Abstract

AbstractThe longtime behaviour of the FitzHugh–Rinzel (FHR) neurons and the transition to instability of the FHR steady states, are investigated. Criteria guaranteeing solutions boundedness, absorbing sets, in the energy phase space, existence and steady states instability via oscillatory bifurcations, are obtained. Denoting by $$ \lambda ^{3} + \sum\nolimits_{{k = 1}}^{3} {A_{k} } (R)\lambda ^{{3 - k}} = 0 $$ λ 3 + ∑ k = 1 3 A k ( R ) λ 3 - k = 0 , with R bifurcation parameter, the spectrum equation of a steady state $$m_0$$ m 0 , linearly asymptotically stable at certain value of R, the frequency f of an oscillatory destabilizing bifurcation (neuron bursting frequency), is shown to be $$ f=\displaystyle \frac{\sqrt{A_2(R_\mathrm{H})}}{2\pi } $$ f = A 2 ( R H ) 2 π with $$R_\mathrm{H}$$ R H location of R at which the bifurcation occurs. The instability coefficient power (ICP) (Rionero in Rend Fis Acc Lincei 31:985–997, 2020; Fluids 6(2):57, 2021) for the onset of oscillatory bifurcations, is introduced, proved and applied, in a new version.