Modulation of multi-timescale compound Ca–NMDA–Na oscillations in pyramidal neuron by extracellular electric fields

Evidence shows that there exist dendritic Ca²⁺-spike-dependent and NMDA-spike-dependent multi-timescale compound oscillations in epileptiform activity, and the electric field (EF) plays a significant role in the propagation of compound oscillations. However, it is still unclear how the EF-induced spatial polarization modulates the interaction between dendritic Ca²⁺ oscillations and NMDA oscillations, and subsequently influences somatic Na⁺ spikes. To address this issue, we built a biophysical pyramidal neuron model with complex dendritic morphology, which is capable of reproducing multi-timescale neuronal oscillations observed in epileptiform discharges. By investigating the EF stimulation thresholds for triggering dendritic Ca²⁺ and NMDA spikes as well as somatic Na⁺ discharges, we found that the dendritic depolarization first activates dendritic Ca²⁺ oscillations, subsequently leading to the generation of dendritic NMDA oscillations, which together facilitate Na⁺ spike generation by counteracting somatic hyperpolarization. Finally, we proposed a minimal three-compartment neuronal model that successfully reproduces the Ca–NMDA–Na compound oscillations. Through singular perturbation and bifurcation analysis, we demonstrated the modulatory influence of EF on multi-timescale neuronal compound oscillations. Additionally, our results indicate that the EF-induced depolarization at the apical dendrite causes the system equilibrium point to experience an invariant circle saddle-node bifurcation to trigger dendritic Ca²⁺ oscillations. These oscillations then drive the basal dendrite to generate dendritic NMDA oscillations by experiencing a subcritical Hopf bifurcation. In this case, the soma experiences a subcritical Hopf bifurcation to produce Na⁺ spikes. These results provide valuable insights into the mechanisms underlying the generation of epileptiform discharges in the brain, which is helpful for developing therapeutic strategies for epilepsy.