Source-linked AI summary
The Influence of Sodium and Potassium Dynamics on Excitability, Seizures, and the Stability of Persistent States: I. Single Neuron Dynamics
John R. Cressman, Ghanim Ullah, Jokubas Ziburkus, Steven J. Schiff, Ernest Barreto
TL;DR
The paper asks how coupled sodium and potassium dynamics affect neuronal excitability and seizure-like behavior. It models a conductance-based single neuron with intra- and extracellular ion dynamics, then reduces the model for bifurcation analysis. The reduction reasonably approximates the full model and identifies Hopf bifurcations that produce slow ion-concentration oscillations and seizure-like behavior.
Problem
The paper investigates how local sodium and potassium concentration fluctuations may modulate single-neuron excitability and contribute to seizure behavior.
Method
The authors construct a conductance-based single-neuron model with ion concentration dynamics and reduce it by replacing fast spiking with empirical fits to time-averaged currents.
Results
The reduced model reasonably approximates the full model and identifies Hopf bifurcations that lead to slow ion-concentration oscillations and seizure-like behavior.
Takeaways & Limitations
Ion concentration homeostasis is a critical component of neuronal function, and the identified mechanisms may underlie pathological states such as epilepsy.
Takeaways & Limitations
The approximation can produce substantial errors when internal potassium is decoupled from sodium dynamics, and glial saturation occurs over thousands of seconds.
Abstract
from arXiv · showhide
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.
INTRODUCTION
The paper examines how coupled sodium and potassium concentration dynamics modulate single-neuron excitability and contribute to seizure-like behavior. It introduces a conductance-based model incorporating cellular mechanisms that regulate the extracellular micro-environment.
- Research motivation: Previous work examined extracellular micro-environmental effects, but paid little attention to cellular control of those factors as a way to modulate neuronal responses.The paper addresses this gap by focusing on local ion concentration dynamics.
- Physiological motivation: Ion reversal potentials depend on intra- and extracellular concentrations, linking sodium and potassium dynamics to neuronal excitability.Activity increases extracellular potassium and intracellular sodium, producing competing effects on excitability.
- Study aim: The study investigates whether interrelated sodium and potassium dynamics can generate seizure-like behavior in a single neuron.Network-level effects are reserved for the companion article.
- Physiological motivation: Modest increases in extracellular potassium are known to increase neuronal excitability and may contribute to spontaneous activity.The relatively small extracellular space can make potassium changes particularly influential.
- Study approach: The authors model a conductance-based single neuron embedded within extracellular and glial compartments, including pumps, diffusion, glial buffering, and ion channels.A reduced formulation supports detailed bifurcation analysis and is reported to reasonably approximate the full model’s dynamics.
METHODS
The full model couples a conductance-based neuron to dynamic intracellular sodium and extracellular potassium concentrations. Ion concentrations are updated through membrane currents, pumps, glial uptake, diffusion, and Nernst reversal potentials.
- Neuron model: The full model contains a single-compartment conductance-based neuron with sodium, potassium, calcium-gated potassium, and leak currents.Its dynamic variables include voltage, gating variables, intracellular calcium, extracellular potassium, and intracellular sodium.
- Ion dynamics: The model couples ion concentrations to membrane voltage through Nernst reversal potentials.The potassium concentration equation includes a volume-fraction correction and a current-to-concentration conversion factor.
- Model assumptions: The model uses an instantaneous equilibrium approximation for the fast sodium gating variable m.This assumes m reaches m∞ immediately because it is fast relative to voltage changes.
- Ion dynamics: Extracellular potassium dynamics combine neuronal K+ currents, Na+-K+ pumps, glial uptake, and lateral diffusion.These processes determine the rate of change of local extracellular potassium concentration.
- Homeostatic mechanisms: The sodium-potassium pump is represented by sigmoidal dependence on intracellular sodium and extracellular potassium and saturates at high concentrations.Normal resting conditions use a pump rate of 1.25mM/sec.
Iglia
The concentration model represents glial buffering, diffusion, sodium conservation, and concentration-dependent reversal potentials. It also uses simplifying assumptions and omits several structural and cellular features of mammalian neurons.
- Glial buffering: Glial uptake is modeled as a simplified combination of passive and active potassium removal from the extracellular space.The model assumes the large glial network provides a nearly insatiable extracellular buffer.
- Diffusion: Potassium diffusion is directed toward the concentration in a nearby large reservoir representing bath solution or brain vasculature.Normal conditions use ko,∞=4.0 mM and a diffusion constant of ε=1.2Hz.
- Concentration approximation: The reduced potassium balance assumes sodium influx is compensated by potassium efflux, allowing intracellular potassium to be approximated from intracellular sodium.The approximation uses normal resting concentrations of 140.0 mM intracellular potassium and 18.0 mM intracellular sodium.
- Sodium dynamics: Only one sodium differential equation is used because total sodium is assumed to be conserved.The normal resting extracellular sodium concentration is 144.0mM.
- Scope and limitations: The model omits dendritic and axonal geometry, spatial channel distributions, cotransporters, and immobile anions, while retaining the essential dynamics targeted by the study.The authors state that the validity of constant ion concentrations is debated outside the isolated squid giant axon.
2. Reduced model
The reduced model removes fast spiking dynamics in favor of slower ion concentration dynamics and supports bifurcation analysis. It reproduces the full model’s qualitative concentration behavior, including stable equilibria and oscillations.
- Reduction method: The reduction replaces the fast Hodgkin-Huxley spiking mechanism with empirical fits to time-averaged sodium and potassium currents.Currents are averaged over one second after the model reaches a resting or spiking asymptotic state.
- Reduction method: The reduced model consists of concentration equations with sodium and potassium currents replaced by fitted sigmoidal functions.The fitted currents are functions of sodium and potassium concentration ratios.
- Bifurcation analysis: The paper identifies bifurcations in the reduced model and analyzes their implications for the full model’s behavior.Bifurcation diagrams were obtained using XPPAUT.
- Dynamical regimes: The reduced model predicts either stable equilibria with constant ion concentrations or stable periodic orbits with oscillatory concentrations.Changes in parameters alter solution stability through bifurcations.
1. Overview
The model examines how elevated extracellular potassium produces seizure-like activity and uses time-scale separation to analyze slow ion dynamics. The reduced model qualitatively reproduces the full model’s concentration behavior while enabling numerical bifurcation analysis.
- At normal potassium concentrations near 4 mM, the resting potential is maintained, whereas 8 mM can produce spontaneous bursts and seizure-like events.
- The full model generates events lasting tens of seconds, containing spikes lasting about 1 ms and therefore spanning four orders of magnitude in time scales.
- Slow periodic sodium and potassium concentration dynamics underlie the overall modulation of fast Hodgkin-Huxley spiking.
- The reduced model removes fast Hodgkin-Huxley spiking to focus on slow ion-concentration dynamics and support numerical bifurcation analysis.
- Although reduced-model traces are not identical to full-model traces, the reduction captures the qualitative behavior of ion concentrations.
2. Analysis of the reduced model
The reduced model identifies a region of oscillation governed by bifurcations and shaped by potassium reservoir concentration, diffusion, and glial buffering. These mechanisms determine whether ion concentrations oscillate, settle to equilibrium, or exhibit large-amplitude behavior.
- As ko,∞ rises to approximately 1.9, an unstable periodic orbit collapses onto the equilibrium and a coexisting large-amplitude stable periodic orbit attracts [K]o.
- For sufficiently high ko,∞, the periodic behavior ends and [K]o approaches equilibrium values.
- At ko,∞≈2.13, a subcritical Hopf bifurcation creates an unstable periodic orbit that collides with a stable orbit near ko,∞≈2.15.
- Increasing diffusion or glial buffering terminates oscillations through a subcritical Hopf followed by a saddle-node bifurcation of periodic orbits.
- Reducing diffusion or glial strength increases extracellular potassium oscillation amplitude because potassium removal from the extracellular space is impeded.
- The region of oscillation is bounded by Hopf bifurcations and contains parameter values where the only stable attractor is a periodic orbit.
3. Analysis of the full model
The full model reproduces tonic firing, slow bursting, depolarization block, and continuous firing patterns corresponding to reduced-model dynamics. Varying diffusion-related parameters changes burst frequency, envelope shape, and potassium excursions.
- High extracellular potassium depolarizes the neuron beyond threshold, producing tonic firing while [K]o remains nearly constant apart from perturbations of about 0.1 mM per spike.
- Across the region of oscillation, increasing ε while holding glial strength fixed makes bursts more frequent and reduces the amplitude of [K]o oscillations.
- A [K]o peak near 40 mM briefly drives the neuron into depolarization block.
- Stable ion-concentration equilibria in the reduced model can correspond to rapid tonic firing in the full model, while higher reservoir potassium eventually produces depolarization block.
- At elevated extracellular potassium, quiescent periods can represent depolarization block rather than resting behavior, and bursts have rounded rather than square envelopes.
- The full model also supports continuous firing with a wavy envelope caused by oscillating ion concentrations.
DISCUSSION
The models show that ion concentration dynamics can generate seizure-like activity in a single neuron and identify Hopf bifurcations as a mechanism for slow ionic oscillations. The reduced model supports bifurcation analysis and reproduces the full model qualitatively, but its accuracy is limited on extremely long timescales and in some parameter regions.
- Experimental relevance: A broad range of bath potassium concentrations yields seizure-like behavior under otherwise normal conditions.The resulting activity is reported as both qualitatively and quantitatively similar to experimental models.
- Ion homeostasis: Competition among neuronal currents, pumps, glia, and diffusion produces stable periodic extracellular potassium oscillations and slow ionic dynamics.These effects may be an important mechanism underlying epileptic seizures.
- Model limitations: The model approximations are reliable over timescales longer than individual spikes, bursts, and seizures but can incur substantial errors over thousands of seconds.Glial saturation and decoupling internal potassium from sodium dynamics limit the reduced description at very long times.
- Long-term behavior: Over extremely long timescales, seizure-like events and tonic firing appear to be transients, while internal potassium may drift upward or downward depending on parameters and initial conditions.The model cell can ultimately reach a fixed point with extracellular potassium nearly equal to the bath concentration.
- Reduced-model validity: The reduced model is a good qualitative approximation to the full spiking model, with especially good agreement in the reported two-parameter region.The models disagree in some regions because the reduced model uses simple fits for time-averaged Hodgkin-Huxley currents.
- Implications: Ion concentration dynamics may play an important role in understanding neuronal dynamics, including pathological activity such as seizures.The authors frame ion homeostasis as relevant to both normal and pathological neuronal behavior.
- Single-cell seizure dynamics: The model demonstrates that recurring seizure-like events can occur in a single cell with intra- and extracellular ion concentration dynamics.The reported behavior is qualitatively and quantitatively similar to experimental seizure models.
- Dynamical mechanism: Hopf bifurcations lead to slow oscillations in ion concentrations that can produce seizure-like events.The mechanism links qualitative changes in system behavior to ionic concentration oscillations.
FIGURE LEGENDS
The figures use bifurcation diagrams and full-model traces to characterize oscillatory ion-concentration dynamics and compare reduced- and full-model behavior. They show how bath potassium, diffusion, and glial strength shape the oscillatory region.
- Bifurcation diagrams: Bifurcation diagrams track extracellular potassium as a function of bath potassium, pump strength, diffusion, and glial strength.Equilibria and periodic orbits are distinguished, with stability indicated by filled or open symbols.
- Bifurcation diagrams: Hopf bifurcation curves bound a region in which extracellular potassium is obliged to oscillate.Within this region, ion concentrations exhibit oscillatory behavior.
- Parameter effects: The one-dimensional diffusion diagram uses k_o,∞=2.0 and glial strength G=1.75 to compare bifurcation behavior across diffusion values.Dashed lines connect these one-dimensional diagrams to the corresponding points in the multidimensional bifurcation figures.
- Model dynamics: Full-model examples show membrane voltage together with extracellular potassium and intracellular sodium traces at selected parameter values.The quiescent states in one example correspond to depolarization block.
- Parameter effects: Increasing bath potassium from k_o,∞=1.77 to k_o,∞=1.9 moves the oscillatory region rightward until it intersects normal diffusion and glial-strength values.At k_o,∞=2.0, normal values lie inside the region; at k_o,∞=2.1, the region has moved farther right and normal values approach its left boundary.
APPENDIX A
Appendix A develops concentration dynamics and reduced-model equations for a conductance-based neuron, linking membrane currents to intra- and extracellular ion changes. The appendix also presents figure references for the model analyses.
- Ion concentration dynamics: The model assumes an intracellular-to-extracellular volume ratio of β=7.0, corresponding to 87.5% intracellular and 12.5% extracellular volume.
- Ion concentration dynamics: Ion conservation relates concentration changes across intracellular and extracellular volumes for membrane currents.The concentrations and volumes are indexed by intra- and extracellular subscripts.
- Current-to-concentration conversion: Membrane current is converted into concentration change by relating transported charge, ion number, Avogadro’s number, and the relevant flow volume.The appendix gives ΔQ = itotalΔt, ΔN = itotalΔt/q, and Δci = ΔN/(NAVoli).
- Current-to-concentration conversion: The concentration-current rate depends on the cell’s surface-area-to-volume ratio, while increased cell volume produces slower dynamics.For a sphere of radius 7 μm, the appendix gives α = 21 mcoul/M·cm2.
- Reduced model equations: The reduced model uses empirical fits of average Hodgkin–Huxley membrane currents as functions of sodium and potassium concentrations.The appendix lists fitted expressions g1 through g4 and associated parameter values.