Mohammad Mirzadeh

Minimum energy desynchronizing control for coupled neurons

We employ optimal control theory to design an event-based, minimum energy, desynchronizing control stimulus for a network of pathologically synchronized, heterogeneously coupled neurons. This works by optimally driving the neurons to their phaseless sets, switching the control off, and letting the phases of the neurons randomize under intrinsic background noise. An event-based minimum energy input may be clinically desirable for deep brain stimulation treatment of neurological diseases, like Parkinson’s disease. The event-based nature of the input results in its administration only when it is necessary, which, in general, amounts to fewer applications, and hence, less charge transfer to and from the tissue. The minimum energy nature of the input may also help prolong battery life for implanted stimulus generators. For the example considered, it is shown that the proposed control causes a considerable amount of randomization in the timing of each neuron’s next spike, leading to desynchronization for the network.

A second-order discretization of the nonlinear Poisson-Boltzmann equation over irregular geometries using non-graded adaptive Cartesian grids

In this paper we present a finite difference scheme for the discretization of the nonlinear Poisson-Boltzmann (PB) equation over irregular domains that is second-order accurate. The interface is represented by a zero level set of a signed distance function using Octree data structure, allowing a natural and systematic approach to generate non-graded adaptive grids. Such a method guaranties computational efficiency by ensuring that the finest level of grid is located near the interface. The nonlinear PB equation is discretized using finite difference method and several numerical experiments are carried which indicate the second-order accuracy of method. Finally the method is used to study the supercapacitor behaviour of porous electrodes.

A conservative discretization of the Poisson-Nernst-Planck equations on adaptive Cartesian grids

In this paper we present a novel hybrid finite-difference/finite-volume method for the numerical solution of the nonlinear Poisson-Nernst-Planck (PNP) equations on irregular domains. The method is described in two spatial dimensions but can be extended to three dimensional problems as well. The boundary of the irregular domain is represented implicitly via the zero level set of a signed distance function and quadtree data structures are used to systematically generate adaptive grids needed to accurately capture the electric double layer near the boundary. To handle the nonlinearity in the PNP equations efficiently, a semi-implicit time integration method is utilized. An important feature of our method is that total number of ions in the system is conserved by carefully imposing the boundary conditions, by utilizing a conservative discretization of the diffusive and, more importantly, the nonlinear migrative flux term. Several numerical experiments are conducted which illustrate that the presented method is first-order accurate in time and second-order accurate in space. Moreover, these tests explicitly indicate that the algorithm is also conservative. Finally we illustrate the applicability of our method in the study of the charging dynamics of porous supercapacitors.

Minimum energy spike randomization for neurons

With inspiration from Arthur Winfrees idea of randomizing the phase of an oscillator by driving its state to a set in which the phase is not defined, i.e., the phaseless set, we employ a Hamilton-Jacobi-Bellman approach to design a minimum energy control law that effectively randomizes the next spiking time for a two-dimensional conductance-based model of noisy oscillatory neurons. The control is initially designed for the deterministic system through the numerical solution of the Hamilton-Jacobi-Bellman partial differential equation for the cost-to-go function, from which the minimum energy stimulus can be found; then its performance is investigated in the presence of noise. It is shown that such control causes a considerable amount of randomization in the timing of the neurons next spike.