Control of epileptiform bursting in the leech heart interneuron

The network controlling heartbeat in the medicinal leech contains leech heart interneurons (HNs). We modeled them under specific pharmacological conditions. The Ca$^{2+}$ currents were blocked by Co$^{2+}$. The K$^{+}$ currents, apart from the non-inactivating current, I$_{K2}$, were blocked by 4AP. The hyperpolarization-activated current, I$_{h}$, was blocked by Cs$^{+}$. Under these conditions, epileptiform bursting characterized by long interburst intervals (IBI) has been shown. We considered three distinct cases. Model 1 included I$_{K2}$, I$_{h}$, and the fast Na$^{+}$ current, I$_{Na}$. Model 2 was characterized by I$_{Na}$, I$_{K2}$, and the persistent Na$^{+}$ current, I$_{NaP}$. Model 3 consisted of I$_{Na,}$ I$_{K2}$, I$_{h}$, and I$_{NaP}$. We also investigated the bi-stability of bursting and silence as the leak conductance, g$_{leak}$, was varied. We showed that in 1 and 3, model HNs demonstrated bi-stability of silence and bursting. We analyzed how IBI and burst duration are controlled by the manipulation of I$_{h}$ and I$_{NaP}$. In 1, as V$_{1/2}$ of I$_{h}$ decreased, IBI grew towards infinity one over the square root of the parameter difference. In 2, we showed that as g$_{NaP}$ decreased from 6.156 nS to 6.155 nS, IBI grew in accordance with the one over square root law. The system underwent a saddle-node bifurcation just below 6.155 nS. Supported by NSF PHY-0750456.