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The Influence of Sodium and Potassium Dynamics on Excitability, Seizures, and the Stability of Persistent States: II. Network and Glial Dynamics
Ghanim Ullah, John R. Cressman, Ernest Barreto, Steven J. Schiff
TL;DR
The paper examines how network and glial dynamics relate to seizure-like activity and the stability of persistent neuronal activity. It models ionic conditions and finds that balanced extracellular K+ and glial control support stability, whereas glial dysfunction can promote transitions toward seizures.
Problem
The study addresses the lack of a unifying dynamical definition of seizures and asks how network dynamics can produce seizure patterns.
Method
The paper uses an ionic-current network model incorporating neuronal populations, extracellular K+ conditions, glial control, and network excitability.
Results
Balanced extracellular K+ controlled by the glial syncytium supports persistent activity that remains stable to perturbations, while glial dysfunction can trigger transitions from stable neuronal activity toward seizures.
Takeaways & Limitations
Glial activity is implicated in the stability of persistent network states and in whether perturbation responses decay or grow across transient phenomena.
Takeaways & Limitations
Diffusion between network elements is beyond the scope of the study.
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. We seek to study these dynamics with respect to the following compartments: neurons, glia, and extracellular space. We are particularly interested in the slower time-scale dynamics that determine overall excitability, and set the stage for transient episodes of persistent oscillations, working memory, or seizures. In this second of two companion papers, we present an ionic current network model composed of populations of Hodgkin-Huxley type excitatory and inhibitory neurons embedded within extracellular space and glia, in order to investigate the role of micro-environmental ionic dynamics on the stability of persistent activity. We show that these networks reproduce seizure-like activity if glial cells fail to maintain the proper micro-environmental conditions surrounding neurons, and produce several experimentally testable predictions. Our work suggests that the stability of persistent states to perturbation is set by glial activity, and that how the response to such perturbations decays or grows may be a critical factor in a variety of disparate transient phenomena such as working memory, burst firing in neonatal brain or spinal cord, up states, seizures, and cortical oscillations.
INTRODUCTION
The paper asks how neuronal networks can keep physiological persistent states stable against perturbations and when those perturbations instead produce seizure-like activity. It focuses on extracellular ions and glia as regulators of this stability and addresses these questions with a mathematical network model.
- Persistent neural activity supports functions such as short-term working memory, but its stability requires a controlled balance of network dynamics.
- The study asks which network properties stabilize persistent states against perturbations and under what conditions perturbations cause seizure-like activity.
- Existing accounts commonly associate seizure transitions with a shift from balanced or dominant inhibition toward dominant excitation, but the perturbation conditions governing these transitions remain unclear.
- The authors hypothesize that a network’s response to perturbations is a dynamical signature of functional stability and an operational test of excitatory-inhibitory balance.
- The model examines how glial uptake of extracellular K+ and glial neurotransmitter release may alter neuronal network activity, an interaction described as largely unstudied.
- The paper constructs a Hodgkin-Huxley network with dynamic extracellular and intracellular K+ and Na+ concentrations governed by glia and active pumps.
METHODS
The methods implement a spatial network of excitatory and inhibitory single-compartment neurons coupled to dynamic ionic concentrations, pumps, diffusion, and glial uptake. Synaptic interactions include depolarization-dependent transmission changes, while neuronal and ionic equations are integrated numerically.
- Network architecture: The network contains 100 excitatory pyramidal cells and 100 inhibitory interneurons represented as single-compartment neurons.
- Network architecture: Neuronal membrane dynamics combine Hodgkin-Huxley ionic currents with intrinsic and synaptic inputs for excitatory and inhibitory populations.
- Synaptic interactions: Depolarization block reduces a cell’s synaptic outputs toward zero through a voltage-dependent factor in the synaptic current model.
- Synaptic interactions: Synaptic footprints differ spatially: PC-to-PC connections are narrow, whereas PC-to-IN and IN-to-IN connections are wider.
- Ion concentration dynamics: Extracellular K+ is updated from neuronal K+ currents, Na+-K+ pumps, glial uptake, and lateral diffusion, while intracellular and extracellular Na+ concentrations also evolve dynamically.
- Ion concentration dynamics: The model includes active pumps, bath diffusion, and glial buffering, with glial buffering strength represented by Gglia.
- Numerical implementation: The equations are integrated with fourth-order Runge-Kutta using a 0.01ms time step, while extracellular K+ diffusion uses a forward-difference method on a 10.0μm grid.
RESULTS
Persistent activity is supported only within a balanced excitation-inhibition regime and becomes less stable as extracellular K+ rises or perturbations increase excitability. When glial regulation fails to maintain extracellular K+, the model produces seizure-like transitions and recurrent activity patterns.
- Persistent-state regime: Persistent, spatially restricted activity occurs between parameter boundaries where excitation and inhibition are sufficiently balanced.
- Persistent-state regime: Increasing extracellular K+ shifts the stability region toward smaller excitatory synaptic strength and narrows the range supporting stable activity.
- Perturbation stability: Excitatory perturbations restore the persistent state for αee = 0.215-0.216, but destroy it at αee ≤ 0.214 or leave the network in widespread firing at αee = 0.217.
- Seizure-like dynamics: As extracellular K+ rises, an initially silent cell develops tonic seizure-like firing, progresses to depolarization block, and resumes spiking as K+ falls.
- Seizure-like dynamics: Higher extracellular K+ can drive the whole network into depolarization block, followed by high-frequency, lower-amplitude seizure-like spiking during K+ oscillations.
DISCUSSION
The one-dimensional two-layer network model links persistent activity and its perturbation stability to excitatory–inhibitory balance, extracellular K+, and glial regulation. It reproduces persistent activity and seizure-like transitions, while identifying glial function and perturbation responses as determinants of network state.
- Persistent activity: The one-dimensional two-layer network model reproduces persistent activity resembling up states and working-memory delay activity.Persistent activity can last several seconds, with model pyramidal-cell firing rates similar to delay responses observed in vivo.
- Conditions for stability: Persistent activity requires balanced excitatory and inhibitory synaptic inputs together with sufficiently low extracellular K+.The study adds glial control of extracellular K+ to synaptic balance as a condition for stable activity.
- Glial regulation: Glial dysfunction can shift normal stable neuronal activity into seizure-like uncontrolled activity.The model associates this transition with impaired glial regulation and experimentally supported glial involvement in epilepsy.
- Perturbation stability: Depending on network excitability, perturbations can decay back to the persistent state, expand through the network, or destroy persistent activity.The response varies with synaptic efficacy and extracellular K+ level.
- Perturbation stability: A persistent activity packet remains stable when excitatory synaptic strength, extracellular K+, and perturbation strength stay within appropriate ranges.Small perturbations can destabilize networks already characterized by increased excitability or relatively high extracellular K+.
- Broader implications: The model proposes that perturbation responses help determine transient phenomena including seizures, working memory, up states, cortical oscillations, and neonatal burst firing.The authors connect decaying or growing responses to the emergence of transient activity patterns.
ACKNOWLEGEMENTS
The authors acknowledge valuable discussions with named colleagues and report support from NIH grants.
- The authors thank Jokubas Ziburkus, Andrew J Trevelyan, Maxim Bazhenov, and Partha Mitra for valuable discussions.
- The work was funded by NIH Grants K02MH01493, R01MH50006, F32NS051072, and CRCNS-R01MH079502.
FIGURE LEGENDS
The figure legends describe a ring network of excitatory and inhibitory neurons with diffusive extracellular potassium, and show how excitation, potassium, perturbations, and glial parameters shape persistent, seizure-like, and silent states.
- Network topology: The network contains pyramidal cells and interneurons arranged in layers on a ring, with Gaussian synaptic connections within and across layers.Potassium around each neuron diffuses to nearest neighbors in the same and adjacent layers.
- Persistent activity: A localized excitatory stimulus can generate persistent, spatially restricted activity that is extinguished by a strong synchronizing stimulus.The stimulus is applied to a subset of pyramidal cells, and the synchronizing pulse is 100 μA/cm2 for 1 ms.
- Persistent activity: Persistent-focus existence and stability depend on excitation, inhibition, steady-state extracellular potassium, and stimulus strength.Increasing ko,∞ lowers the αee range supporting a stable focus and decreases the fraction stable to moderate perturbations; weak stimuli fail to create the focus.
- Perturbation responses: After a second perturbation, the network may return to the stable focus, remain globally active, or lose persistent activity depending on excitation and extracellular potassium.Higher extracellular potassium can preserve enhanced activity after the stimulus vanishes, while baseline αee that is too small or too large prevents recovery to the stable focus.
- Potassium and glial dynamics: Spontaneous potassium-driven activity can progress from silence or regular spiking to seizure-like bursting, depolarization block, and silence.With stronger glial parameters, higher extracellular potassium produces high-frequency seizure-like firing rather than depolarization block, followed by a silent state.