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Bursting is one of the primary activity regimes of neurons. Our study is focused on determining a generic biophysical mechanism underlying the coexistence of the bursting and silent regimes observed in a neuron model. We show that the main ingredient for this mechanism is a saddle periodic orbit. The stable manifold of the orbit sets a threshold between the regimes of activity. Thus, the range of the controlling parameters, where the coexistence is observed, is limited by the bifurcations' values at which the saddle orbit appears and disappears. We show that it appears through the subcritical Andronov-Hopf bifurcation, where the equilibrium representing the silent regime loses stability, and disappears at the homoclinic bifurcation. Correspondingly, the bursting regime disappears in close proximity to the homoclinic bifurcation.

BackgroundMultistability of oscillatory and silent regimes is a ubiquitous phenomenon exhibited by excitable systems such as neurons and cardiac cells. Multistability can play functional roles in short-term memory and maintaining posture. It seems to pose an evolutionary advantage for neurons which are part of multifunctional Central Pattern Generators to possess multistability. The mechanisms supporting multistability of bursting regimes are not well understood or classified.Methodology/principal findingsOur study is focused on determining the bio-physical mechanisms underlying different types of co-existence of the oscillatory and silent regimes observed in a neuronal model. We develop a low-dimensional model typifying the dynamics of a single leech heart interneuron. We carry out a bifurcation analysis of the model and show that it possesses six different types of multistability of dynamical regimes. These types are the co-existence of 1) bursting and silence, 2) tonic spiking and silence, 3) tonic spiking and subthreshold oscillations, 4) bursting and subthreshold oscillations, 5) bursting, subthreshold oscillations and silence, and 6) bursting and tonic spiking. These first five types of multistability occur due to the presence of a separating regime that is either a saddle periodic orbit or a saddle equilibrium. We found that the parameter range wherein multistability is observed is limited by the parameter values at which the separating regimes emerge and terminate.ConclusionsWe developed a neuronal model which exhibits a rich variety of different types of multistability. We described a novel mechanism supporting the bistability of bursting and silence. This neuronal model provides a unique opportunity to study the dynamics of networks with neurons possessing different types of multistability.

Our study is focused on modulation of dynamics of single leech heart interneurons (HNs). We consider two models of HNs representing these neurons under two different pharmacological treatments: (1) blocking of Ca2+ currents and inhibitory coupling with the Ca2+-containing saline and partial blocking of K+ currents; (2) decoupling HNs with bicuculline. In (1), an HN demonstrates slow plateau-like oscillations [1,2]. In (2), an HN demonstrates endogenously bursting activity [3]. We analyze how the interburst interval and burst duration could be controlled by manipulating hyperpolarization-activated current, Ih, and persistent Na+ current, IP, namely by variation of their conductances and the half-activation voltages, V1/2. For example, burst duration increases greatly from 1.7 s to 8.9 s as Vh,1/2 increased from -30 mV to 4 mV. The interburst interval grows from 0.6 s to 125 s as the Vh,1/2 decreases from 4 [mV] to -56 [mV] in accordance with a saddlenode bifurcation. In (2), we similarly show that the variation of Vh,1/2 could be a target for modulation of the bursting. In both cases, we show co-existence of bursting and silence. Interestingly, the co-existence is sensitive to gh (and to maximal conductance of fast Ca2+ current, gCaF too in (2)) and is not sensitive to the maximal conductances of other currents. In (1), if gh is increased from 4 nS to 8 nS, the bistability is then observed in an almost five-fold larger range of the leak conductance values, gleak. In (2), if either gh is changed from 4 nS to 8 nS or gCaF is changed from 5 nS to 0 nS, the bistability is observed in an almost two-fold larger range of gleak. If the bistability is an indication of a dysfunctional dynamics, this observation describes a new, potentially pathological role of overexpression of Ih and ICaF.