Multiscale Stochastic Neuron Modeling : with applications in deep brain stimulation

In recent years deep brain stimulation (DBS) has seen success in curing adverseeffects of several diseases, among those Parkinson. Current method for treatmentuses implanted electrodes of the brain which stimulate neurons via potential fields.The precise mechanism with which DBS works is still being researched. To this end amodel allowing for seamless coupling of DBS-signals and neuron behavior will aid intesting and further development of the existing DBS-signals. We simulate the ionic channels in the neuronal membrane as well as the synapticchannels in the dendrites. The scheme has been implemented using URDME, aMATLAB research code, where a C-code solver is available. A neuron tree is loadedthrough the TREES toolbox from which a connectivity matrix can be formulated. Foreach time step the propagation of the electric potential in the neuronal membrane iscalculated in MATLAB using a Crank-Nicholson scheme. The membrane current isthen calculated and (through Livelink for COMSOL) it is sent to a time dependentPDE-solver which calculates the extracellular potential created by a action potential. Convergence of the interspike interval (ISI) as the time step decreases is shown, aswell as when the space discretization of a neuronal structure is refined. A majority ofthe computational time is spent evaluating the stochastic simulation of ion-channels,and computes the solution of a reference test in ~80s, compared to the ODE modelwhich takes ~30s. Due to the highly parallel nature of the stochastic solver this timecould be decreased. We also show that the stochastic model of a neuron has a different threshold currentfor a potential spike compared to the deterministic model, a systematic study is doneto find the threshold gradient for the stochastic case. Further, the propagationthrough a chain of neurons is simulated where the obtained potential field is realistic