Dynamics from seconds to hours in Hodgkin-Huxley model with time-dependent ion concentrations and buffer reservoirs

The classical Hodgkin--Huxley (HH) model neglects the time-dependence of ion concentrations in spiking dynamics. The dynamics is therefore limited to a time scale of milliseconds, which is determined by the membrane capacitance multiplied by the resistance of the ion channels, and by the gating time constants. We study slow dynamics in an extended HH framework that includes time-dependent ion concentrations, pumps, and buffers. Fluxes across the neuronal membrane change intra- and extracellular ion concentrations, whereby the latter can also change through contact to reservoirs in the surroundings. Ion gain and loss of the system is identified as a bifurcation parameter whose essential importance was not realized in earlier studies. Our systematic study of the bifurcation structure and thus the phase space structure helps to understand activation and inhibition of a new excitability in ion homeostasis which emerges in such extended models. Also modulatory mechanisms that regulate the spiking rate can be explained by bifurcations. The dynamics on three distinct slow times scales is determined by the cell volume-to-surface-area ratio and the membrane permeability (seconds), the buffer time constants (tens of seconds), and the slower backward buffering (minutes to hours). The modulatory dynamics and the newly emerging excitable dynamics corresponds to pathological conditions observed in epileptiform burst activity, and spreading depression in migraine aura and stroke, respectively.Comment: 18 pages, 11 figure

Bistable dynamics underlying excitability of ion homeostasis in neuron models

When neurons fire action potentials, dissipation of free energy is usually not directly considered, because the change in free energy is often negligible compared to the immense reservoir stored in neural transmembrane ion gradients and the long-term energy requirements are met through chemical energy, i.e., metabolism. However, these gradients can temporarily nearly vanish in neurological diseases, such as migraine and stroke, and in traumatic brain injury from concussions to severe injuries. We study biophysical neuron models based on the Hodgkin-Huxley (HH) formalism extended to include time-dependent ion concentrations inside and outside the cell and metabolic energy-driven pumps. We reveal the basic mechanism of a state of free energy-starvation (FES) with bifurcation analyses showing that ion dynamics is for a large range of pump rates bistable without contact to an ion bath. This is interpreted as a threshold reduction of a new fundamental mechanism of 'ionic excitability' that causes a long-lasting but transient FES as observed in pathological states. We can in particular conclude that a coupling of extracellular ion concentrations to a large glial-vascular bath can take a role as an inhibitory mechanism crucial in ion homeostasis, while the Na$^+$/K$^+$ pumps alone are insufficient to recover from FES. Our results provide the missing link between the HH formalism and activator-inhibitor models that have been successfully used for modeling migraine phenotypes, and therefore will allow us to validate the hypothesis that migraine symptoms are explained by disturbed function in ion channel subunits, Na$^+$/K$^+$ pumps, and other proteins that regulate ion homeostasis.Comment: 14 pages, 8 figures, 4 table

Lokale Dynamik von Ionen basierten Neuronenmodellen bei kortikaler Streudepolarisation, Schlaganfall und Epilepsie

Wir untersuchen die Phasenraumstruktur biophysikalischer Nervenzellmodelle mit dynamischen Ionenkonzentrationen. Wir beginnen mit dem klassischen Hodgkin-Huxley-Modell zur Beschreibung elektrisch anregbarer Zellmembranen mit kontstanten intra- und extrazellulären Ionenkonzentrationen. Die Erweiterung des Modells zur Beschreibung von Ionendynamik wird erläutert. Diese Erweiterung führt zu einem geschlossenen System ohne Teilchenaustausch mit der Umgebung, jedoch mit Energiezufluss zur Versorgung der Ionenpumpen, die das System fern vom thermodynamischen Gleichgewicht halten. Unter Ausnutzung aller Symmetrien und durch eine häufig verwendete sowie eine neuartigen Näherung der Gatingdynamik erhalten wir ein ionenbasiertes Neuronenmodell vier dynamischen Variablen. Anschließend untersuchen wir die Dynamik dieses geschlossenen Systems. Wir finden einen neuartigen stabilen Fixpunkt, der neben dem physiologischen Ruhezustand koexistiert. Simulationen zeigen, dass dieser Fixpunkt starke Ähnlichkeit mit dem Donnan-Gleichgewicht des Systems hat. Wir bezeichnen den neuen stabilen Systemzustand mit "free energy-starvation" (FES). Es handelt sich dabei um die erste jemals gefundene Bistabilität von Zuständen in Neuronenmodellen mit unterschiedlichen Ionenkonzentrationen. Zu den Störungen, die zum Übergang von dem physiologischen Ruhezustand zu FES führen, zählen Stimulationen mit angelegten Strömen und eine Schwächung der Ionenpumpen. Wir führen wir die erste Bifurkationsanalyse eines vereinheitlichenden Modells für Aktionspotentiale und Ionendynamik durch und variieren die Stärke der Ionenpumpen. Wir zeigen die Koexistenz eines physiologischen Ruhezustands und eines FES-artigen Zustands für eine Vielzahl von reduzierten ionenbasierten Modellvarianten. Wir reproduzieren dieses Resultat mit einem biophysikalisch detaillierteren Modell, das Kager et al. entwickelt haben. Ein wichtiges Ergebnis der Analyse ist, dass ein abgeschlossenes Neuron sich nur von FES erholen kann, wenn die Pumpstärke drastisch erhöht wird. Wir schlussfolgern: Ionenhomöostase in Neuronen kann nicht nur auf den Pumpen beruhen. "Spreading depression" (SD) stellt ein wichtiges Extrembeispiel für neuronale Ionendynamik dar, bei dem vorrübergehend FES eintritt. Die bisherige Vermutung, dass die Erholung von FES durch die Ionenpumpen erklärt werden kann, wird durch unsere Ergebnisse widerlegt. Danach betrachten wir das Neuron als ein offenes System, das Kaliumionen mit externen Reservoirs austauschen kann. Die vorherige Bistabilität wird dadurch zu einer ionischen Anregbarkeit, d.h. einer Anregungsdynamik mit großen Veränderungen in den Ionenkonzentrationen. Erstmalig werden die Zeitskalen aller Prozesse in einem solchen System herausgearbeitet. Die Zu- bzw. Abnahme des Kaliumionengehalts ist die langsamste dynamische Variable. Sie bewirkt die Übergänge zwischen physiologischen Bedingungen und FES, und hat damit eine zentrale, bisher nicht erkannte Bedeutung. Durch eine weitere Bifurkationanalyse, lassen sich die einzelnen dynamischen Prozesse, die während SD zu beobachten sind, explizit anhand der Phasenraumstruktur erklären. Die Auslösung von SD, die lang andauernde Depolarisation, die anschließende abrupte Repolarisation und die langsame abschließende Erholung können nun mit Hilfe der Bifurkationsstruktur verstanden werden. Außerdem untersuchen wir stationäre Oszillationen. Das dynamische Repertoire umfasst dabei epilepsieartige neuronale Aktivität, tonisches Feuern und periodische SD, die jeweils mit bestimmten Bifurkationen zusammenhängen. Diese Analyse ist die erste mathematische Unterscheidung von SD und epilepsieartiger Dynamik. Im Anschluss untersuchen wir das Anschwellen von Zellen durch Osmose während extremer Veränderungen in den Ionenkonzentrationen. Wir kombinieren existierende Modelle für dynamische Volumenänderungen und erhalten so ein neues Modell, das auf rein physikalischen Prinzipien beruht. Wir identifizieren die für Zellschwellen entscheidenden Parameter. Insbesondere erkennen wir erstmalig die besondere Wichtigkeit der Leitfähigkeit der Anionenkanäle. Außerdem zeigen unsere Simulationen, dass neuronale Volumendynamik adiabatisch approximiert werden kann. Wir zeigen schließlich, dass die Phasenraumstruktur durch Volumendynamik nicht verändert wird. Abschließend demonstrieren wir in einer physiologischen Anwendung, dass die Modellierung aller Zeitskalen notwendig ist um neuronale Dynamik theoretisch zu untersuchen. Um den Einfluss eines genetischen Defekts des Natriumkanals auf die SD-Suszeptibilität des Neurons zu untersuchen vergleichen wir Simulationen von einem reinen Aktionspotential-Modell mit dem ionenbasierten Ansatz. Ersteres führt zu widersprüchlichen Schlussfolgerungen. Die Ergebnisse für das ionenbasierte Modell sind hingegen konsistent mit experimentellen Daten und belegen eine erhöhte Suszeptibilität.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 ion dynamics, in which transient FES is observed. Recovery from FES during SD was long believed to be due to the pumps - our results disprove this hypothesis. The next part deals with open neuron systems, in which potassium ions can be exchanged with an external glial or a vascular reservoir. This changes the dynamics from bistability to what we call ionic excitability, i.e., excitability with large changes in the ion concentrations. For the first time all of the many different time scales of dynamics in open neuron systems are identified. This leads to a new slow-fast description of ion dynamics during SD. The gain or loss of potassium ions is the slowest-changing variable, and it causes the transition of the neuron from physiological conditions to FES or vice versa. The importance of this quantity was not realized before and a bifurcation analysis, in which it is varied as a parameter provides a phase space explanation of each process involved in local SD dynamics: the ignition of SD, the subequent prolonged depolarized phase, the abrupt repolarization, and the final recovery. Furthermore we investigate oscillatory dynamics obtained for coupling the neuron to a bath with an elevated potassium ion concentration. The dynamical repertoire contains seizure-like activity (SLA), tonic firing and periodic SD - each associated with specific bifurcations. This is the first time seizure-like dynamics and SD are mathematically described and categorized as genuinely different dynamics. Thereafter we direct our attention to osmosis-driven cell swelling caused by extreme ion dynamics. We combine earlier models of neural volume dynamics to derive a description based on first physical principles - as opposed to the phenomenological approaches normally used in SD modeling. This enables us to identify critical system parameters, and, in particular, to understand the central role of anion channels in volume dynamics. Moreover, we conclude from numerical simulations that an adiabatic approximation is valid for neural volume dynamics. The general phase space structure of the system is shown not to change when cell swelling in included. Finally, the necessity to model all time scales of neural dynamics is demonstrated in a direct physiological application. To assess the effect of a sodium channel mutation on the SD susceptibility of tissue, we perform simulations with a pure spiking model and with the reduced ion-based model. In the context of neural mass models the former would lead to partly contradicting conclusions, while only the results for the ion-based model are consistent with experiments and show an increased SD susceptibility

Anions Govern Cell Volume: A Case Study of Relative Astrocytic and Neuronal Swelling in Spreading Depolarization

Cell volume changes are ubiquitous in normal and pathological activity of the brain. Nevertheless, we know little about the dynamics of cell and tissue swelling, and the differential changes in the volumes of neurons and glia during pathological states such as spreading depolarizations (SD) under ischemic and non–ischemic conditions, and epileptic seizures. By combining the Hodgkin–Huxley type spiking dynamics, dynamic ion concentrations, and simultaneous neuronal and astroglial volume changes into a comprehensive model, we elucidate why glial cells swell more than neurons in SD and the special case of anoxic depolarization (AD), and explore the relative contributions of the two cell types to tissue swelling. Our results demonstrate that anion channels, particularly Cl−, are intrinsically connected to cell swelling and blocking these currents prevents changes in cell volume. The model is based on a simple and physiologically realistic description. We introduce model extensions that are either derived purely from first physical principles of electroneutrality, osmosis, and conservation of particles, or by a phenomenological combination of these principles and known physiological facts. This work provides insights into numerous studies related to neuronal and glial volume changes in SD that otherwise seem contradictory, and is broadly applicable to swelling in other cell types and conditions

SD dynamics in model system consisting of a neuron, glial compartment, and the ECS.

<p>SD is initiated by interrupting pump activity and glial ion regulation for 20 sec (shaded or hatched region). In <b>(a)</b> the evolution of potentials is shown. During pump interruption the cell depolarizes and differences between <i>V</i>, <i>E</i>_<i>K</i> and <i>E</i>_<i>Na</i> become small. The system repolarizes abruptly and potential differences are rebuilt after about 150 sec (repolarization point indicated by vertical pink line and a star). The volume of the whole system and the respective portion of the neuron, ECS, and the glia cell are shown in <b>(b)</b>. Relative changes of the three compartments and the full system are shown in <b>(c)</b> with an inset for finer resolution between 40 sec and 110 sec.</p

Physiological resting conditions.

<p>Physiological resting conditions.</p

The same transition as in Fig 3 with blocked Cl⁻ channels.

<p>The pumps are switched off after 50 sec (marked by the star). The main plots show <b>(a)</b> potentials and <b>(b)</b> ion concentrations. Volumes are shown in the inset in <b>(b)</b>.</p

Simulation of the neuronal response to KCl perfusion.

<p>For the slow perfusion case in <b>(a)</b> 20 fmol of KCl are added within 200 sec (pink curve). The beginning and end of KCl addition are marked by the vertical lines (with stars). The neuron retains its polarization and volume (inset). For fast perfusion (turquoise curve) the same amount of KCl is added between 20 and 70 sec (vertical lines marked with stars). This induces depolarization and swelling. In <b>(b)</b> this is related to the fixed point structure. The black and red section of the fixed point curve indicate stable physiological conditions and FES, respectively. With slow perfusion the neuron remains on the physiological branch, with fast perfusion it goes into FES. <i>K</i>_<i>e</i> and <i>ω</i>_<i>i</i> (inset) are significantly increased.</p

Cl⁻ fluxes and uptake factor <i>χ</i>.

<p>In <b>(a)</b>, changes in the Cl⁻ content in the three compartments during SD with <i>χ</i> = 0.8 (as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0147060#pone.0147060.g007" target="_blank">Fig 7</a>) are shown. Pump interruption and the recovery point are indicated as before. In <b>(b)</b>, we used <i>χ</i> = 0.2 which means less Cl⁻ uptake and more Na⁺ release. The cell does not recover from depolarization, but remains in FES.</p

Resting values and parameters for the glia model.

<p>Resting values and parameters for the glia model.</p