Neuronal bursting: interactions of the persistent sodium and CAN currents

The pre-Botzinger complex (pBC) is a heterogeneous neuronal network within the mammalian brainstem and has been experimentally found to generate robust, synchronous bursts [1]. Significant modeling research has been conducted on characterizing the dynamics of individual neurons within the pBC. [2, 3] It is well known that the persistent sodium current (I_NaP) contributes to square-wave bursting seen in the pBC [4]. Recent experimental work within the pBC identified a signaling cascade that starts with presynaptic glutamate and ends with the release of intracellular calcium that activates a nonspecific cationic current (I_CAN) [5]. A subsequent model demonstrated that I_CAN may contribute to bursts within the pBC that exhibit depolarization block [6]. With these two mechanisms for generating bursts present within the pBC, an open question is how do they combine to generate the robust bursts seen in the network? The present work seeks to analyze the result of including both I_NaP and I_CAN within the same model. We consider the effects of heterogeneity in the conductance g_NaP of I_NaP and the conductance g_CAN of I_CAN; with this heterogeneity in mind, the model cell may be quiescent, tonically active, have only square-wave bursts, have only depolarization-block exhibiting bursts, or may show both types of bursting. Using the mathematical tools of bifurcation analysis and slow-fast decomposition, we illuminate the mechanisms underlying the transitions of a model cell between the types of dynamics listed above. Our results show that, in cases where g_CAN is relatively high, increasing g_NaP increases the range of g_CAN where the resultant cell has depolarization-block exhibiting bursts. On the other hand, when g_CAN is relatively low, increasing g_NaP may cause the cell to transition from quiescence, to square wave bursting, to tonic activity, to square wave bursts with high duty cycles, and finally further increase of g_NaP causes the cell to again be tonically active. The latter two transitions do not occur if I_CAN is absent. The interactions of I_CAN and I_NaP are relevant to many systems beyond the pBC. Individually, I_CAN and I_NaP have been focused on as important to rhythmic burst generation in other systems such as the entorhinal cortex [7]; however, it is likely that both currents are present in these systems. Thus, a detailed account for the interaction of I_CAN and I_NaP may help explain the rhythm generation encountered in other systems beyond the pBC.