Slow rhythmic activity from an interplay of voltage and extracellular concentration dynamics: a minimal biophysical mechanism for neuronal bursting

Abstract

Slow brain rhythms, for example during slow-wave sleep or pathological conditions like seizures and spreading depolarization, can be accompanied by synchronized oscillations in extracellular potassium concentration. Slow brain rhythms typically have longer periods than tonic action-potential firing. They are assumed to arise from network-level mechanisms, involving synaptic interactions and delays, or from intrinsically bursting neurons equipped with ion channels of slow dynamics. Here, we demonstrate that both mechanisms are not necessarily required and that slow rhythms can also be generated from an interplay of fast neuronal voltage dynamics and changes in extracellular ionic concentrations alone in any neuron with type excitability. The coupling of fast-spiking neuron dynamics and a slow extracellular potassium transient is regulated by the Na+/K+-ATPase. We use bifurcation analysis and the slow-fast method to reveal that this coupling suffices to generate a hysteresis loop organized around a bistable region that emerges from a saddle-node loop bifurcation - a common feature of type excitable neurons. Moreover, the Na+/K+-ATPase not only plays a key role in burst generation by shearing the bifurcation diagram but also modulates tonic spiking and depolarization block by its density and pump rate. These dynamics of bursting, tonic spiking and depolarization block, accompanied by the fluctuation of extracellular potassium, are likely to be relevant for pathological conditions. We suggest that these dynamics can result from any disturbance in extracellular potassium regulation, such as glial malfunction or hypoxia. The identification of a minimal mechanistic requirement for producing these dynamics adds to a better understanding of pathologies in brain rhythms may direct attention to alternative pharmacological targets for therapy. Author SummaryThe brain can produce slow rhythms, such as those observed during sleep or epilepsy. These rhythms are much slower than the neuronal electrical signals, and their origins are still under debate. Mechanisms discussed so far are based on the connection delays in neural networks or on neuronal ion channels with particularly slow kinetics. We show that neurons with specific spiking dynamics - allowing them to fire at arbitrarily low frequencies (type neurons) - can produce slow rhythmic patterns without requiring synaptic connectivity or special ion channels. In these cells, slow rhythmic activity arises from the interplay of slow changes in extracellular potassium concentration and the cells voltage dynamics, mediated by the Na+/K+-ATPase pump. The latter, found in all neurons, regulates the concentrations of sodium and potassium ions across the cell membrane. The core mechanism is not idiosyncratic, rather mathematical analysis shows under which conditions slow rhythmic activity can arise generically from the pump-based coupling in a broad class of neurons. We demonstrate that the pump is relevant for the creation of different firing patterns, which can be associated with various diseases. A better understanding these complex dynamics is important for the development of more effective treatments for concentration-dependent pathologies.