The distinct stochastic and deterministic dynamics between period-adding and period-doubling bifurcations of neural bursting patterns

Abstract

Period-adding bifurcations and period-doubling bifurcations of neural firing patterns, which were both observed in the biological experiment on a neural pacemaker and simulated in a theoretical model (Chay model), manifested different stochastic dynamics near the bifurcation points. For period-adding bifurcations, a noise-induced stochastic bursting, whose behavior is stochastic transition between period-k and period-( $$k+1$$ ) bursts, lying between period-k and period-( $$k+1$$ ) burstings ( $$k=1,2,3$$ ). For period-doubling bifurcations, period-1 bursting is changed to period-2 bursting firstly and then to period-4 bursting. No stochastic firing patterns similar to those lying in the period-adding bifurcation were detected. Using the method of the fast–slow variables dissection, the deterministic burstings in both period-adding bifurcation and period-doubling bifurcation are classified into “fold/homoclinic” bursting with a saddle-node point and a saddle-homoclinic point, which behaves as a critical phase sensitive to noisy disturbance. For the bursting pattern near the period-doubling bifurcation point, the trajectory of burstings is far from the saddle-homoclinic point. Near the period-adding bifurcation points from period-k to period-( $$k+1$$ ) burstings, the trajectories of the bursting patterns pass through the neighborhood of the saddle-homoclinic point and exhibit a platform. The platform appears after the k-th spike for the period-k bursting and between the k-th spike and the ( $$k+1$$ )-th spike for the period-( $$k+1$$ ) bursting. For some bursts of period-k bursting, noise can induce a novel spike near the platform to form a burst with $$k+1$$ spikes, and for some bursts of period-( $$k+1$$ ) bursting, the last spike can be terminated by noise to form a burst with k spikes. It is the cause that the stochastic bursting whose behavior is stochastic transition between period-k burst and period-( $$k+1$$ ) burst is induced by noise, and the stochastic transition happens within or near the platform, i.e., the neighborhood of the saddle-homoclinic point. More detailed transition dynamics can be explained by the stable/unstable manifold of the saddle point. The underlying deterministic dynamics between period-adding and period-doubling bifurcation points that can lead to distinct stochastic dynamics are identified, which are helpful for understanding the roles of noise and provide critical phase to apply control strategy to modulate the firing patterns.