Bistability at the onset of neuronal oscillations.

ACTION potentials; OSCILLATIONS; BIFURCATION theory; HOPF bifurcations; HYSTERESIS; FLOW visualization

The Hodgkin–Huxley (HH) model and squid axon (bathed in reduced Ca2+) fire repetitively for steady current injection. Moreover, for a current-range just suprathreshold, repetitive firing coexists with a stable steady state. Neuronal excitability, as such, shows bistability and hysteresis providing the opportunity for the system to perform as switchable between firing and non-firing states with transient input and providing the backbone as a dynamical mechanism for bursting oscillations. Some conditions for bistability can be derived by intricate analysis (bifurcation theory) and characterized by simulation, but conditions for emergence and robustness of such bistability do not typically follow from intuition. Here, we demonstrate with a semi-quantitative two-variable, V–w, reduction of the HH model features that promote/reduce bistability. Visualization of flow and trajectories in the V–w phase plane provides an intuitive grasp for bistability. The geometry of action potential recovery involves a late phase during which the dynamic negative feedback of I Na inactivation and I K activation over/undershoot, respectively, their resting values, thereby leading to hyperexcitabilty and an intrinsically generated opportunity to by-pass the spiral-like stable rest state and initiate the next spike upstroke. We illustrate control of bistability and dependence of the degree of hysteresis on recovery timescales and gating properties. Our dynamical dissection reveals the strongly attracting depolarized phase of the spike, enabling approximations like the resetting feature of adapting integrate-and-fire models. We extend our insights and show that the Morris–Lecar model can also exhibit robust bistability. [ABSTRACT FROM AUTHOR]

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