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

The 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.

The structure of the bounded trajectories set of a scalar convex differential equation

The present paper describes the topological and ergodic structure of the set of bounded trajectories of the flow defined by a scalar convex differential equation. We characterize the minimal subsets, the ergodic measures concentrated on them, and study the longtime behaviour of the bounded trajectories in terms of the Lyapunov exponents of the linearized equations. In particular, we obtain conditions that guarantee the existence of almost-periodic, almost-automorphic and recurrent solutions.