Multi-Timescale Compound Oscillations in Pyramidal Neurons: Insights from a Three-Compartment Model

Oscillation activities are widely observed in neurons and are critical to the brain's physiological functions. A significant phenomenon among these activities is the simultaneous presence of multiple oscillations at different frequencies, referred to as compound oscillations, which are particularly evident in epileptiform activity. However, the dynamical mechanisms underlying compound oscillations remain unknown. In this study, we develop and analyze a three-compartment pyramidal neuron model to address this problem. The model incorporates somatic Na+, dendritic Ca2+ and N-methyl-D-aspartate (NMDA)-mediated firing behaviors, capturing the multi-timescale dynamics essential for understanding compound oscillations. Phase-plane analysis reveals that Ca2+ firing behaviors in apical dendrite are relaxation oscillations, characterized by two timescales. Using geometric singular perturbation theory, we show that NMDA and Na+ firing behaviors involve three timescales. Specifically, NMDA firing behaviors in basal dendrite manifest as square-wave bursting, accompanied by Na+ action potentials in soma. Moreover, bifurcation analysis indicates that Ca2+ firing behaviors lead to the alternation of homoclinic bifurcation and saddle-node bifurcation on an invariant cycle (SNIC), which cause the occurrence and disappearance of bursting in basal dendrite. We also explore the impact of NMDA receptor conductance on these oscillatory patterns. Simulation results validate that impairment or excessive activation of NMDA receptors can lead to pathological compound oscillations, which have significant physiological implications. These insights gained from this model not only advance our understanding of neuronal function but also have potential implications for developing therapeutic strategies for neurological disorders characterized by compound oscillations.