Draft:Epileptor-2

• Comment: Needing more inline citation. Currently with a poor style and reads like original research. -Lemonaka 02:41, 7 July 2025 (UTC)

• Comment: While this may be a notable topic, there is no way to determine that from this draft alone. The only sourced part of the entire draft is the title. Everything that follows does not have any listed reference and looks more like original research. This draft looks more like a scholarly assignment or paper, not like a Wikipedia article. Articles in an encyclopedia are written for a wide audience, and currently, your draft lacks any context for those unfamiliar with the subject. Additional sourced text should be added to provide this context. Also, Wikipedia is not a place to post program scripts. cyberdog958^Talk 01:05, 2 March 2025 (UTC)

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In computational neuroscience, Epileptor-2 is a mathematical model that describes in a simple manner the epileptic discharges observed in the brain. It is named after its predecessor, the Epileptor model, and is designed to replicate both brief interictal discharges (spike-and-wave complexes) and prolonged ictal discharges that constitute seizures. In the model, interictal events appear as short clusters of action potentials in individual neurons, lasting fractions of seconds or seconds, while ictal events are represented as longer clusters of short discharges, lasting about a minute.

The model is particularly useful for studying the role of ionic dynamics in seizure generation. According to Epileptor‑2, brief interictal discharges arise from stochastic oscillations of the membrane potential and synaptic resources, whereas ictal discharges emerge from slower oscillations in the extracellular concentration of potassium ions and the intracellular concentration of sodium ions. Both Epileptor and Epileptor‑2 demonstrate that ionic homeostasis is crucial for the transition from normal to pathological activity. The model has been successfully used to simulate the effects of electrical stimulation on epileptiform activity in brain slice preparations in vitro, often in the hippocampus

Model behaviour and comparison with experiment

Comparison of experimental and simulated ictal discharges. Top (black box): simultaneous experimental recordings of the membrane potential of a single neuron (upper trace) and the extracellular potassium concentration (lower trace) from a rat hippocampal slice during induced epileptiform activity; scale bars indicate 40 s, 40 mV and 3 mM. Bottom: the five state variables of the Epileptor‑2 model over the same kind of activity (time axis in seconds) – membrane potential of the quadratic‑integrate‑and‑fire neuron $U$, extracellular potassium $[K]_{o}$, intracellular sodium $[Na]_{i}$, population‑averaged polarisation $V$ and synaptic resource $x$. Two ictal discharges occur in the trace, each consisting of a dense burst of action potentials accompanied by a transient rise in extracellular potassium and intracellular sodium and a depletion of the synaptic resource.

In a typical simulation (Figure), the model generates recurring ictal discharges, a rhythm set by the slow accumulation and clearance of ions rather than by fast membrane dynamics. Each ictal discharge consists of interictal-like discharges. Each interictal discharge consists of a few action potentials. The Figure illustrates this behaviour and compares it with experimental data; over roughly 400 seconds of activity two ictal discharges occur, at approximately 0–50s and 270–350s.

Experimental recording

The upper panels of the Figure show two signals recorded simultaneously from a rat hippocampal slice: the membrane potential of a single neuron and the extracellular potassium concentration $[K]_{o}$ in its vicinity. Each ictal discharge begins abruptly with a dense, high‑frequency burst of action potentials; as the event progresses the spikes become sparser and more widely spaced before firing ceases. Concurrently, $[K]_{o}$ rises in a stepwise, saw‑tooth fashion from its resting level of about 3–4 mM to a peak of roughly 8–10 mM, then decays slowly back towards baseline as ionic homeostasis is restored. Between the two discharges both signals return to rest, interrupted only by a small transient rise in potassium.

Simulated variables

The lower panels display the model output, with each of the five state variables shown on its own axis (time in seconds):

• Membrane potential $U$ – the quadratic integrate and fire neuron rests near −60mV and fires clusters of action potentials (reaching about +20mV) during each ictal event, reproducing the burst‑then‑sparse spiking pattern seen in the experiment.
• Extracellular potassium $[K]_{o}$ – rises stepwise from about 3 mM to a peak near 7mM at the onset of each event and then decays; immediately after the discharge it dips slightly below its resting level before recovering slowly, reflecting clearance of potassium by the sodium–potassium pump.
• Intracellular sodium $[Na]_{i}$ – accumulates more slowly than potassium, climbing from about 10mM to around 23mM over the course of an event and decaying over tens of seconds afterwards.
• Mean polarisation $V$ – the population‑averaged polarisation fluctuates around 0 mV and shows large positive excursions (up to roughly 100mV) during the discharges.
• Synaptic resource $x$ – depletes from its maximal value of 1.0 down to about 0.65 during the high‑frequency firing and recovers to 1.0 in the interval between events, representing short‑term synaptic depression and recovery.

The simulated membrane‑potential and potassium traces closely match the experimental ones in both timing and waveform. This agreement supports the central hypothesis of the model: that slow potassium accumulation and sodium–potassium pump activity are largely sufficient to account for the onset and termination of seizure‑like events. In addition, the model predicts the intracellular sodium concentration and the synaptic resource – quantities that are difficult to measure experimentally – which can in principle be tested in future experiments and may help refine the understanding of the mechanisms underlying seizure generation.

Mathematical description

The model in its version with two potassium compartments is formulated as a system of five ordinary differential equations that describe the evolution of:

• two extracellular potassium concentrations – one close to the neurons ($[K]_{o1}$) and one in a distant compartment ($[K]_{o2}$),
• the intracellular sodium concentration ($[Na]_{i}$),
• the population‑averaged membrane polarisation ($V$),
• and the synaptic resource ($x$).

The firing rate $\nu$ of the neuronal population is a sigmoidal function of $V$, and the sodium‑potassium pump current $I_{pump}$ depends on both potassium and sodium concentrations. The full set of equations is given below:

${d[K]_{o1} \over dt}={[K]_{o2}-[K]_{o1} \over \tau _{K1}}+\delta _{K}~\nu (t)-2\beta I_{pump}(t)$

${d[K]_{o2} \over dt}={[K]_{bath}-[K]_{o2} \over \tau _{K2}}+{[K]_{o1}-[K]_{o2} \over \tau _{K1}}$

${d[Na]_{i} \over dt}={{[Na]_{i}}^{0}-[Na]_{i} \over \tau _{Na}}+\delta _{Na}\nu (t)-3I_{pump}(t)$

${dV \over dt}={u\left(t\right)-V \over \tau _{m}}$

${dx \over dt}={1-x \over \tau _{x}}-\delta _{x}x~\nu (t)$

where the total input current $u(t)$ includes potassium‑mediated, synaptic, and noise contributions:

$u(t)=g_{K}(V_{K}(t)-V_{K}^{0})+G_{syn}(x-0.5)~\nu (t)+\sigma ~\eta (t)$

The potassium reversal potential $V_{K}$ and the pump current $I_{pump}$ are given by:

$V_{K}(t)=26.6~\ln([K]_{o2}/130)$

$I_{pump}(t)=\rho /\left(1+\exp(3.5-[K]_{o1})/(1+\exp((25-[Na]_{o1})/3)\right)$

and the firing rate is:

$\nu (t)=100\left(2/\left(1+\exp((6-V)/10)\right)-1\right)$

To capture single‑neuron behaviour, a quadratic integrate and fire neuron is added, receiving the same input $u(t)$ as the population:

${dU \over dt}={g_{U} \over C_{U}}(U-U_{1})(U-U_{2})+{g_{L} \over C_{U}}u(t),~~~~~if~~~U>U^{th}~~then~~U=U_{reset}$

This system exhibits rich bifurcation behaviour, allowing the transition between interictal and ictal states to be studied as a function of ionic concentrations and model parameters.

Simulation parameters

The parameters used to produce the figure above are:

$\tau _{K1}=25s,~\tau _{K2}=250s,~\tau _{Na}=20s,~\tau _{x}=2s,~\tau _{m}=0.01s,$ $~\delta _{K}=0.04~mM,~\delta _{Na}=0.03~mM,~\delta _{x}=0.01,~\rho =0.8,~\beta =10,~\sigma =8,$ $~G_{syn}/g_{L}=2.5~mV\cdot s,~g_{K}/g_{L}=0.5,~g_{L}=5~nS,$ $[K]_{o}^{0}=3~mM,~[K]_{bath}=8.5~mM,~[Na]_{i}^{0}=10~mM,$ $g_{U}=0.4,~C_{U}=0.2,~U^{th}=25,~U_{reset}=-50,~U_{1}=-60,~U_{2}=-40,~U_{0}=-70$

Implementation

The original implementation of the model is written in Python and is available from the online database ModelDB