Neurological Disorders

Derivation 1: Amyloid-Beta Aggregation Kinetics

The Smoluchowski coagulation equation models amyloid aggregation. Let $c_k(t)$ be the concentration of aggregates of size $k$:

$$\frac{dc_k}{dt} = \frac{1}{2}\sum_{i+j=k} K_{ij} c_i c_j - c_k \sum_{j=1}^{\infty} K_{kj} c_j + J_k$$

where $K_{ij}$ is the aggregation kernel and $J_k$ accounts for production and clearance. For a simplified two-compartment model (monomers $m$ and plaques $P$):

$$\frac{dm}{dt} = \alpha - \beta m - \gamma m^2, \quad \frac{dP}{dt} = \gamma m^2 - \delta P$$

where $\alpha$ is production rate, $\beta$ is monomer clearance, $\gamma$ is aggregation rate, and $\delta$ is plaque clearance. The steady state plaque burden is $P^* = \gamma (\alpha/\beta)^2 / \delta$. Anti-amyloid therapies target different parameters: secretase inhibitors reduce $\alpha$, immunotherapy increases $\delta$.

Derivation 2: Basal Ganglia Rate Model of Parkinsonism

The Albin-DeLong model describes basal ganglia dysfunction through altered firing rates. In the normal state, the firing rate of the GPi (output nucleus) is:

$$r_{\text{GPi}} = r_0 - w_D \cdot r_{\text{Str}_D} + w_{\text{STN}} \cdot r_{\text{STN}}$$

where $r_{\text{Str}_D}$ is the direct pathway striatal output and $r_{\text{STN}}$ is the subthalamic nucleus rate. Dopamine depletion (D = 0 vs. normal D = 1) affects both pathways:

$$r_{\text{Str}_D} = f(D \cdot I_{\text{ctx}}), \quad r_{\text{Str}_I} = f((2-D) \cdot I_{\text{ctx}})$$

D1 receptors in the direct pathway are excitatory ($r \propto D$), while D2 receptors in the indirect pathway are inhibitory ($r \propto 2-D$). With dopamine depletion, the direct pathway weakens (less movement facilitation) and the indirect pathway strengthens (more movement suppression), producing the hypokinetic symptoms of PD. The resulting excessive GPi output inhibits thalamocortical circuits.

Derivation 3: Epileptor Model and Seizure Bifurcations

The Epileptor (Jirsa et al., 2014) is a minimal model capturing seizure onset, evolution, and termination. The fast variables describe seizure dynamics:

$$\dot{x}_1 = y_1 - f_1(x_1, x_2) - z + I_1$$

$$\dot{y}_1 = 1 - 5x_1^2 - y_1$$

The slow variable $z$ (representing metabolic processes) controls transitions:

$$\dot{z} = \frac{1}{\tau_0}\left(4(x_1 - x_0) - z\right)$$

Seizure onset occurs via a saddle-node bifurcation when $z$ decreases below a critical value, and termination occurs via a homoclinic bifurcation when $z$increases. The parameter $x_0$ controls the seizure threshold: more negative values make seizures less likely. This model predicts stereotyped seizure dynamics and has been used to create patient-specific virtual brain models for surgical planning.

Derivation 4: Spreading Depolarization in Stroke

Ischemic stroke triggers spreading depolarization (SD) waves — slow ($\sim$3 mm/min) waves of neuronal depolarization that expand the infarct. The reaction-diffusion model:

$$\frac{\partial [K^+]_o}{\partial t} = D_K \nabla^2 [K^+]_o + f([K^+]_o) - g([K^+]_o)$$

where $[K^+]_o$ is extracellular potassium, $D_K \approx 3 \times 10^{-6}$ cm$^2$/s is the diffusion coefficient, $f$ is the release function (positive feedback), and$g$ is the pump/uptake function. The wave speed is approximately:

$$v \approx 2\sqrt{D_K \cdot f'(K_0^+)} \approx 3 \text{ mm/min}$$

Each SD wave in ischemic tissue causes further energy depletion, expanding the penumbra into infarct. The number of SD waves correlates with final infarct size. Therapeutic targets include NMDA receptor blockers (reducing $f$) and Na$^+$/K$^+$-ATPase enhancers (increasing $g$).

Derivation 5: Network Degeneration Spreading Model

Neurodegenerative diseases spread along neural connections. The network spreading model (Raj et al., 2012) describes atrophy propagation on the connectome:

$$\frac{d\mathbf{x}}{dt} = -\beta \mathbf{L} \mathbf{x}$$

where $\mathbf{x}$ is the vector of regional atrophy, $\mathbf{L}$ is the graph Laplacian of the structural connectome, and $\beta$ is the spreading rate. The solution:

$$\mathbf{x}(t) = e^{-\beta \mathbf{L} t} \mathbf{x}_0 = \sum_k e^{-\beta \lambda_k t} (\mathbf{u}_k^T \mathbf{x}_0) \mathbf{u}_k$$

Starting from a seed region $\mathbf{x}_0$, atrophy spreads preferentially along highly connected pathways. The eigenmodes of $\mathbf{L}$ predict spatial patterns of degeneration: low eigenvalue modes (affecting connected regions) dominate at early stages, while high eigenvalue modes emerge later. This model successfully predicts the progression patterns of Alzheimer's (from entorhinal cortex) and Parkinson's (from substantia nigra).