You’re basically asking: if the cortex is a pile of coupled oscillators, which knobs get nudged when genes push the system toward “twitchy” (seizure-prone) and “moody” (cycle-prone)? Short answer: the knobs that raise gain, slow recovery, and increase synchrony.

1) Single-cell: Izhikevich-style neuron

I’ll assume you meant the Izhikevich model:

\dot v = 0.04v^2+5v+140 - u + I,\quad \dot u = a(bv-u),\quad \text{if } v\ge 30\text{ mV, then } v\leftarrow c,\; u\leftarrow u+d.

Instability shows up as parameter shifts that make spikes cheaper and recovery weaker:

• ↑I (drive) or less negative c (reset) → easier re-spiking, lower rheobase. Think persistent Na or Cav1.2 up, M-current down.
• ↓a (slower recovery) and/or ↓d (smaller after-spike kick to u) → weaker adaptation, more tendency to tonic spiking/bursting. Epileptiform patterns emerge as the system approaches Hopf/SNIC thresholds.
• ↑b (stronger v→u coupling) can generate low-threshold spiking and subthreshold oscillations; depending on operating point it promotes rhythmicity.

Translation: channelopathies and GABA deficits tilt these four dials toward “faster to fire, slower to settle,” which is what neurons do when they’re auditioning for a seizure.

2) Cortical column: Wilson–Cowan / neural-mass

For an E/I mass model

\tau_E \dot E = -E + F(w_{EE}E - w_{EI}I + P_E),\quad \tau_I \dot I = -I + F(w_{IE}E - w_{II}I + P_I),

instability is a classic loop-gain story:

• ↑wEE, ↓wEI, ↓wII, ↑P_E, steeper F’ push the fixed point through a Hopf bifurcation into oscillations; keep pushing and you get larger, slower, spike-wave–like cycles and eventually depolarization block regimes.
• τE/τI mismatch shifts frequency bands (gamma→beta/theta) and widens the oscillatory region. Many seizure models are literally “move parameters until the Hopf line is crossed.”
• JR/Jansen–Rit columns show the same: low input or high gain give paroxysmal rhythms; coupling columns strengthens path to hypersynchrony.