The influence of sodium and potassium dynamics on excitability,seizures, and the stability of persistent states: I. Single neuron dynamics

In these companion papers, we study how the interrelated dynamics of sodium and potassium affect the excitability of neurons, the occurrence of seizures, and the stability of persistent states of activity. In this first paper, we construct a mathematical model consisting of a single conductance-based neuron together with intra- and extracellular ion concentration dynamics. We formulate a reduction of this model that permits a detailed bifurcation analysis, and show that the reduced model is a reasonable approximation of the full model. We find that competition between intrinsic neuronal currents, sodium-potassium pumps, glia, and diffusion can produce very slow and large-amplitude oscillations in ion concentrations similar to what is seen physiologically in seizures. Using the reduced model, we identify the dynamical mechanisms that give rise to these phenomena. These models reveal several experimentally testable predictions. Our work emphasizes the critical role of ion concentration homeostasis in the proper functioning of neurons, and points to important fundamental processes that may underlie pathological states such as epilepsy.

Dynamics of hydrocephalus: a physical approach

As brain ventricles lose their ability to regulate the cerebrospinal fluid (CSF) pressure, serious brain conditions collectively named hydrocephalus can appear. By modelling ventricular dynamics with the laws of physics, dynamical instabilities are evidenced, caused by either CSF transport dysregulations or abnormal properties of the elasticity of the ependyma. We show that these instabilities would lead, in most cases, to dilation of the ventricles, establishing a close connection to hydrocephalus, or in some other cases to a ventricular contraction as observed in the slit ventricle syndrome. Signs seem to indicate the possibility of phase transitions occurring as a result of these instabilities, which might have important clinical consequences, such as the inability to recover a healthy state. Even so, our dynamical approach could allow the development of a unified view of these complex intracranial conditions along with a classification that might be clinically relevant.