ABSTRACT
Using delay differential equations to study mathematical models of Parkinson's disease and Huntington's disease is important to show how important it is for synchronization between basal ganglia loops to work together. We used the delay circuit RLC (resistor, inductor, capacitor) model to show how the direct pathway and the indirect pathway in the basal ganglia excite and inhibit the motor cortex, respectively. 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 appears when the response time in the indirect pathway increases at rest. The simulation confirmed that tremor occurs and the motor cortex is in an inhibited state. The direct pathway can increase the time delay in the dopaminergic pathway, which significantly increases the activity of the motor cortex. The hyperdirect pathway regulates the activity of the motor cortex. 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.
Subject(s)
Huntington Disease , Motor Cortex , Parkinson Disease , Huntington Disease/physiopathology , Huntington Disease/metabolism , Humans , Parkinson Disease/physiopathology , Motor Cortex/physiopathology , Nonlinear Dynamics , Basal Ganglia/physiopathology , Models, Neurological , Models, Theoretical , Computer Simulation , Tremor/physiopathologyABSTRACT
The Van der Pol equation is the mathematical model of a second-order ordinary differential equation with cubic nonlinearity. Several studies have been adding time delay to the Van der Pol model. In this paper, the differential equation of the Van der Pol model and the RLC (resistor-inductor-capacitor) circuit are deduced as a delay differential equation. The Van der Pol delay model contains two delays, which allows the re-use of its applications in the suggested equation. The Taylor series was used to deduce ordinary differential equations from the delay differential equations in the case of small delays. Also, the model for Parkinson's disease modification is described as the Van der Pol model. A numerical simulation of the delay differential equations has been done to show the different cases that the delay differential equations can express using the MATLAB program.