Local Dynamics of Ion-Based Neuron Models for Cortical Spreading Depression, Stroke and Seizures

Abstract

In this thesis we investigate the phase space structure of biophysical neuron models with dynamic ion concentrations. We start with the classical Hodgkin–Huxley model of an electrically excitable neural membrane with constant intra– and extracellular ion concentrations. The extension of this model to include changes in ion concentrations that result from transmembrane currents is carefully reviewed. This extension describes a closed system, in which no particle exchange with the surroundings is considered, however the neuron contains ion pumps that dissipate energy to keep it far from the thermodynamic equilibrium. Exploiting all symmetries of the model and applying one commonly used and one novel approximation of the gating dynamics, we obtain a reduced ion–based neuron model with only four dynamical variables. The dynamics of the closed neuron system is investigated in the next part. A new stable fixed point that coexists with the physiological resting state is found. Numerical simulations show that the new fixed point is extremely close to the Donnan equilibrium, i.e., the thermodynamic equilibrium of the system. The neuron cannot fire action potentials in this state, because the electric energy that is usually stored in the ion gradients is almost fully dissipated. We refer to this condition as free energy–starvations (FES). This is the first bistability of neuron states with completely different intra– and extracellular ion concentrations ever reported. Perturbations that cause the transition from the physiological resting state to FES are, for example, long stimulations with applied currents or a temporary interruption of the pump activity. We perform the first bifurcation analysis of a unified model for action potentials and ion dynamics. We vary the pump rate as a bifurcation parameter, and thereby prove the coexistence of a physiological resting state and FES in a large number of reduced ion–based model variants. The result is also replicated for a very biophysically detailed model developed by Kager et al.. Most importantly, the bifurcation analysis shows that a closed neuron system can only recover from FES if the rate of the ion pumps is extremely enhanced. This leads to our first major conclusion: ion homeostasis cannot rely on the pumps alone. Spreading depression (SD) is an important example of neural