A four-dimensional neuronal model to describe the complex nonlinear dynamics observed in the firing patterns of a sciatic nerve chronic constriction injury model

Various bifurcation scenarios from period-1 bursting to period-1 spiking were observed with decreasing calcium concentrations in the extracellular solution in a sciatic nerve chronic constriction injury model. Some scenarios manifested complex processes beginning from the period-adding bifurcation with chaotic or stochastic burstings, which changed to a bursting pattern with many spikes per burst, and subsequently to chaotic spiking via a shrinkage phenomenon and period-1 spiking via an inverse period-doubling bifurcation. The longest interspike interval of the firing pattern increased gradually before the shrinkage and decreased drastically after the shrinkage. A four-dimensional model containing free luminal calcium concentration in the endoplasmic reticulum was used, which is slower than intracellular calcium concentration and has a negative feedback action with the intracellular calcium concentration. The bifurcation scenarios with complex processes simulated in the four-dimensional model closely match those obtained experimentally, and a negative feedback action of the slower variable during the genesis of complex nonlinear dynamics was identified. These results reveal a four-dimensional theoretical model with the luminal calcium concentration, which can accurately simulate the complexes processes of experimental bifurcation scenarios.