Applied Mathemathics seminar

Motivated by issues related to treating certain neurological diseases such

as Parkinson's disease by a method called electrical deep brain stimulation,

we consider applying optimal control methods to both mathematical models of neurons and in vitro neurons. Patients suffering from Parkinson's disease experience involuntary tremors that typically affect the distal portion of their upper limbs. It has been hypothesized that these tremors are associated with simultaneous spiking of a cluster of neurons in the thalamus and basal ganglia regions of the brain. In a healthy situation, the periodic ring of neurons is not synchronized, but they can engage in a pathological synchrony and all fire at the same time which results in release of strong action potentials that trigger the downstream muscles with periodic shocks, manifested as tremors.

 

In this talk, we investigate the control of different neuronal systems

using methods of optimal control. The neuronal systems considered range

from simple one-dimensional phase models to multi-dimensional conductance-based models, both on a single neuron level and on a population level. The optimal control methods considered produce event-based, continuous-time, typically bounded input stimuli that can optimally achieve the desired control objective. The optimality criterion considered is minimum energy. The control objectives of interest are the interspike interval for single neurons and desynchrony for populations of neurons. The applicability of the interspike interval controller is shown in practice by testing it on single in vitro pyramidal neurons in the CA1 region of rat hippocampus.