Improved Mathematical Models of Parkinson's Disease with Hopf Bifurcation and Huntington's Disease with Chaos

Résumé

Abstract Studying mathematical models for Parkinson's disease and Huntington's disease by using delay differential equations is important to clarify the importance of synchronization between different basal ganglia loops. We used the delay circuit RLC (resistor, inductor, capacitor) model to illustrate excitation and inhibition of the motor cortex from the direct pathway and the indirect pathway in the basal ganglia. A term has been added to the mathematical model without time delay in the case of the hyperdirect pathway. It is proposed to add a non-linear term to adjust the synchronization. We studied Hopf bifurcation conditions for the proposed models. The desynchronization of response times between the direct pathway and the indirect pathway leads to different symptoms of Parkinson's disease. Tremor occurs when the time delay in the dopaminergic pathway is increased by the indirect pathway. The simulation confirmed that tremor occurs and the motor cortex is in an inhibited state. If the time delay in the dopaminergic pathway is increased by the direct pathway, this leads to a critical increase in the activity of the motor cortex. The activity of the motor cortex is regulated by the hyperdirect pathway. The simulation showed bradykinesia occurs when we switch from one movement to another that is less exciting for the motor cortex. A decrease of GABA in the striatum or delayed excitation of the substantia nigra from the subthalamus may be a major cause of Parkinson's disease. An increase in the response time delay in one of the pathways results in the chaotic movement characteristic of Huntington's disease.

Citer ce document

Accès au document

Ce document est hébergé sur son dépôt institutionnel d'origine.

Auteur(s)

Statistiques

Consultations : 1

Téléchargements : 0