Modeling and Detection of Critical Slowing Down in Epileptic Dynamics

TL;DR

A multi-stable slow-fast system to capture critical slowing down in epileptic dynamics is introduced, and regions of attraction for stable states are constructed, shedding light on how dynamic bifurcations drive pathological oscillations.

Abstract

Epilepsy is a common neurological disorder characterized by abrupt seizures. Although seizures may appear random, they are often preceded by early warning signs in neural signals, notably, critical slowing down, a phenomenon in which the system's recovery rate from perturbations declines when it approaches a critical point. Detecting these markers could enable preventive therapies. This paper introduces a multi-stable slow-fast system to capture critical slowing down in epileptic dynamics. We construct regions of attraction for stable states, shedding light on how dynamic bifurcations drive pathological oscillations. We derive the recovery rate after perturbations to formalize critical slowing down. A novel algorithm for detecting precursors to ictal transitions is presented, along with a proof-of-concept event-based feedback control strategy to prevent impending pathological oscillations. Numerical studies are conducted to validate our theoretical findings.

Figures (8)

• Figure 1: Illustration of critical slowing down
• Figure 2: Bifurcation diagram of system \ref{['old']} illustrating the emergence and disappearance of equilibria and limit cycles across parameter regimes.
• Figure 3: Region of attraction of steady state $\mathcal{E}_1$. If $\sigma(0)<c_1$, the system will always converge to $\mathcal{E}_1$ for any $x(0)$ and $y(0)$. For $c_1<\sigma(0)<c_2$, the convergence occurs within the cylindrical region. When $\sigma(0)>c_2$, the system will always converge to undesired oscillations.
• Figure 4: Perturbations at $t=40$ drive $\sigma$ beyond $-0.7$. Despite initial convergence towards the steady state, $x$ and $y$ quickly transition to pathological oscillations after $\sigma$ passes the critical point $0$. Parameters: $a=b=1, \omega=2, c_1 =-0.9, c_2=-0.7, c_3=0.5$, and $\varepsilon=0.1$.
• Figure 5: Comparison of convergence speeds for different initial $\sigma(0)$ under the same perturbation $(0,0)$ to $x$ and $y$. Parameters: $a=b=1$, $\omega = 5$, $c_1=-0.9, c_2=-0.7, c_3=0.5$, and $\varepsilon=0.01$.

Theorems & Definitions (8)

• Lemma 1
• Theorem 1
• proof
• Example 1
• Example 2
• Theorem 2
• proof
• Example 3