pathology-cognitive

Legend 1: Boolean Network Transition Functions and Threshold Equations

This model defines how individual molecular events ($e_i$) activate the seven key pathological pathways ($P_i$) using logical operators (AND $\land$, OR $\lor$).

Pathway Transition Functions ($\Phi$):

• P1 (ACE2 / TLR engagement): $\Phi_1: P_1 \leftrightarrow e_3 \lor e_4$

• P2 (Neuropilin-1 entry): $\Phi_2: P_2 \leftrightarrow e_5$

• P3 (Spike-driven amyloidogenesis): $\Phi_3: P_3 \leftrightarrow e_1 \land e_2$

• P4 (Inflammasome / NLRP3): $\Phi_4: P_4 \leftrightarrow e_6 \land e_7$

• P5 (SAMHD1 inactivation): $\Phi_5: P_5 \leftrightarrow (e_8 \lor e_9) \land e_{10}$

• P6 (cGAS–STING interferonopathy): $\Phi_6: P_6 \leftrightarrow e_{11} \land e_{12}$

• P7 (Immunological engram fixation): $\Phi_7: P_7 \leftrightarrow e_{13} \land e_{14}$

Systemic Worsening Index & Synergistic Collapse:

To evaluate systemic deterioration rather than isolated pathway activation, the Systemic Worsening Index is defined as the sum of all active pathways:

$$W(P) = \sum_{i=1}^{7} P_i$$

The critical threshold ($W_{critical}$) representing a synergistic collapse into Alzheimer's disease-like pathology is met only when at least four of the seven pathways are active ($k \ge 4$), with the strict requirement that the essential gatekeeper pathways, P5 and P6, are concurrently activated.

$$W_{critical} = 1 \iff W(P) \ge 4 \land P_5 = 1 \land P_6 = 1$$