The variables he and ne represent the inactivation of Na+ and the activation of K+, respectively. \([Ca_^ + }} _\), \([Na_^ ]_\), \([Na_^ ]_\), \([K_^ ]_\), \([K_^ ]_\), \([Cl_^ ]_\), and \([Cl_^ ]_\) refer to extracellular or intracellular ion concentrations of Ca+, Na+, K+, and Cl−, respectively. The parameter Ce denotes the membrane capacitance. \(I_ = g_ m_^ h_ (V_ - E_)\), \(I_ = g_ m_^ (V_ - E_)\), \(I_ = g_ n_^ (V_ - E_ )\), \(I_ = g_ \frac^ ]_ }}^ ]_ + 1}}(V_ - E_)\), and \(I_ = \rho_ \frac }}\frac - [Na_^ ]_ }}}} }}\frac - [K_^ ]_ }} }}\) represent the currents of Na+, persistent Na+, K+, Ca2+-activated K+, and Na+-K+ pump, respectively.\(I_ = I_ + I_ + I_\) is leak current, with \(I_ = g_ (V_ - E_ )\), \(I_ = g_ (V_ - E_ )\), \(I_ = g_ (V_ - E_ )\),\(E_ = 26.64\ln \left( ^ ]_ }}^ ]_ }}} \right)\), \(E_ = 26.64\ln \left( ^ ]_ }}^ ]_ }}} \right)\), and \(E_ = 26.64\ln \left( ^ ]_ }}^ ]_ }}} \right)\). The electroneutrality is considered as follows:\([Na_^ ]_ = 144} - \beta_ ([Na_^ ]_ - 18})\),\([Cl_^ ]_ = 130} - \beta_ ([Cl_^ ]_ - 6})\), wherein 144 mM and 18 mM is the sodium concentration outside and inside the neuron, respectively, and 130 mM and 6 mM correspond to the normal resting \([Cl_^ ]_\), and \([Cl_^ ]_\), respectively. The parameters \(g_\), \(g_\), \(g_\), \(g_\), \(g_\), \(g_\), and \(g_\) represent the conductances, and ENae, EKe, and ECle the reversal potentials. The parameter \(\varphi_\) determines the inactivation of Na+ and the activation of K+. \(I_ = 0.3\ln \left( ^ } \right]_ \left[ ^ } \right]_ }}^ } \right]_ \left[ ^ } \right]_ }}} \right)\) is current for the K+/Cl− cotransporter, and \(I_ = 0.1\frac^ ]_ } \right)}} }}\left[ ^ ]_ [Cl_^ ]_ }}^ ]_ [Cl_^ ]_ }}} \right) + \ln \left( ^ ]_ [Cl_^ ]_ }}^ ]_ [Cl_^ ]_ }}} \right)} \right]\) is currents for the Na+/K+/2Cl− cotransporter. Especially, \(I_ = \varepsilon_ ([K_^ ]_ - K_ )\) and \(I_ = \varepsilon_ ([K_^ ]_ - K_ )\) are the loss of extracellular and intracellular potassium to the surrounding environment, respectively. In the study (Desroches et al. 2019), only sink of extracellular potassium ion (\(I_ = \varepsilon_ ([K_^ ]_ - K_ )\)) is considered. Such a model can be well used to reproduce the epileptiform firing observed in the experiment. However, there is no equilibrium point for the model. Inspired by the model with multiple sinks (Cui et al. 2018), a sink for the intracellular potassium (\(I_ = \varepsilon_ ([K_^ ]_ - K_ )\)) is considered in the present paper. Then, the model used in this paper has equilibrium and bifurcations.

The functions are as follows: \(m_ = \frac }} + \beta_ )}}\),\(\alpha_ = \frac + 54)}} + 54)/4)}} }}\),\(\beta_ = \frac + 27)}} + 27)/5)}} }}\),\(m_ = 1/\left[ + 25)/2.5)}} } \right]\),\(\, \alpha_ = \frac + 52)}} + 52)/5)}} }}\),\(\beta_ = 0.5e^ + 57)/40)}}\),\(\alpha_ = 0.128( - (V_ + 50)/18)\), and \(\, \beta_ = 4/\left[ + 27)/5)}} } \right]\).

The parameters are set as follows: \(g_\) = 100, \(g_\) = 80, \(g_\) = 100, \(g_\) = 1, \(g_\) = 0.015, \(g_\) = 0.0015, and \(g_\) = 0.05 with unit mS/cm2, ECae = 120 mV, \(C_\) = 1 (μF/cm2), τCae = 80 ms, εe = 0.002 mmol/(C cm), γe = 0.4442 mmol/(C cm), \(\rho_\) = 0.25 mM/s, \(Na_\) = 22 mM, \(K_\) = 3.5 mM, \(\varepsilon_\) = 0.4 \(}^\), \(K_\) = 3.5 mM, \(K_\) = 134 mM, \(\varepsilon_\) = 0.005 \(}^\), βe = 4, \(\varphi_\) = 1, and τe = 1000.