Bahman Tahayori

Modeling extracellular electrical stimulation: I. Derivation and interpretation of neurite equations

Neuroprosthetic devices, such as cochlear and retinal implants, work by directly stimulating neurons with extracellular electrodes. This is commonly modeled using the cable equation with an applied extracellular voltage. In this paper a framework for modeling extracellular electrical stimulation is presented. To this end, a cylindrical neurite with confined extracellular space in the subthreshold regime is modeled in three-dimensional space. Through cylindrical harmonic expansion of Laplaces equation, we derive the spatio-temporal equations governing different modes of stimulation, referred to as longitudinal and transverse modes, under types of boundary conditions. The longitudinal mode is described by the well-known cable equation, however, the transverse modes are described by a novel ordinary differential equation. For the longitudinal mode, we find that different electrotonic length constants apply under the two different boundary conditions. Equations connecting current density to voltage boundary conditions are derived that are used to calculate the trans-impedance of the neurite-plus-thin-extracellular-sheath. A detailed explanation on depolarization mechanisms and the dominant current pathway under different modes of stimulation is provided. The analytic results derived here enable the estimation of a neurites membrane potential under extracellular stimulation, hence bypassing the heavy computational cost of using numerical methods.

Modelling extracellular electrical stimulation: III. Derivation and interpretation of neural tissue equations

OBJECTIVE A common approach in modelling extracellular electrical stimulation is to represent neural tissue by a volume conductor when calculating the activating function as the driving term in a cable equation for the membrane potential. This approach ignores the cellular composition of tissue, including the neurites and their combined effect on the extracellular potential. This has a number of undesirable consequences. First, the two natural and equally valid choices of boundary conditions for the cable equation (i.e. using either voltage or current) lead to two mutually inconsistent predictions of the membrane potential. Second, the spatio-temporal distribution of the extracellular potential can be strongly affected by the combined cellular composition of the tissue. In this paper, we develop a mean field volume conductor theory to overcome these shortcomings of available models. APPROACH This method connects the microscopic properties of the constituent fibres to the macroscopic electrical properties of the tissue by introducing an admittivity kernel for the neural tissue that is non-local, non-instantaneous and anisotropic. This generalizes the usual tissue conductivity. A class of bidomain models that is mathematically equivalent to this class of self-consistent volume conductor models is also presented. The bidomain models are computationally convenient for simulating the activation map of neural tissue using numerical methods such as finite element analysis. MAIN RESULTS The theory is first developed for tissue composed of identical, parallel fibres and then extended to general neural tissues composed of mixtures of neurites with different and arbitrary orientations, arrangements and properties. Equations describing the extracellular and membrane potential for the longitudinal and transverse modes of stimulation are derived. SIGNIFICANCE The theory complements our earlier work, which developed extensions to cable theory for the micro-scale equations of neural stimulation that apply to individual fibres. The modelling framework provides a number of advantages over other approaches currently adopted in the literature and, therefore, can be used to accurately estimate the membrane potential generated by extracellular electrical stimulation.

Modeling extracellular electrical stimulation: II. Computational validation and numerical results

The validity of approximate equations describing the membrane potential under extracellular electrical stimulation (Meffin et al 2012 J. Neural Eng. 9 065005) is investigated through finite element analysis in this paper. To this end, the finite element method is used to simulate a cylindrical neurite under extracellular stimulation. Laplaces equations with appropriate boundary conditions are solved numerically in three dimensions and the results are compared to the approximate analytic solutions. Simulation results are in agreement with the approximate analytic expressions for longitudinal and transverse modes of stimulation. The range of validity of the equations describing the membrane potential for different values of stimulation and neurite parameters are presented as well. The results indicate that the analytic approach can be used to model extracellular electrical stimulation for realistic physiological parameters with a high level of accuracy.

Modelling extracellular electrical stimulation: IV. Effect of the cellular composition of neural tissue on its spatio-temporal filtering properties

OBJECTIVE The objective of this paper is to present a concrete application of the cellular composite model for calculating the membrane potential, described in an accompanying paper. APPROACH A composite model that is used to determine the membrane potential for both longitudinal and transverse modes of stimulation is demonstrated. MAIN RESULTS Two extreme limits of the model, near-field and far-field for an electrode close to or distant from a neuron, respectively, are derived in this paper. Results for typical neural tissue are compared using the composite, near-field and far-field models as well as the standard isotropic volume conductor model. The self-consistency of the composite model, its spatial profile response and the extracellular potential time behaviour are presented. The magnitudes of the longitudinal and transverse components for different values of electrode-neurite separations are compared. SIGNIFICANCE The unique features of the composite model and its simplified versions can be used to accurately estimate the spatio-temporal response of neural tissue to extracellular electrical stimulation.

Rabi resonance in spin systems: Theory and experiment

The response of a magnetic resonance spin system is predicted and experimentally verified for the particular case of a continuous wave amplitude modulated radiofrequency excitation. The experimental results demonstrate phenomena not previously observed in magnetic resonance systems, including a secondary resonance condition when the amplitude of the excitation equals the modulation frequency. This secondary resonance produces a relatively large steady state magnetisation with Fourier components at harmonics of the modulation frequency. Experiments are in excellent agreement with the theoretical prediction derived from the Bloch equations, which provides a sound theoretical framework for future developments in NMR spectroscopy and imaging.

Optimal stimulus current waveshape for a hodgkin-huxley model neuron

Traditionally, rectangular Lilly-type current pulses have been employed to electrically stimulate a neuron. In this paper, we utilize a least squares optimisation approach to assess the optimality of rectangular pulses in the context of electrical current stimulation. To this end, an appropriate cost function to minimise the total charge delivered to a neuron while keeping the waveshape sufficiently smooth, is developed and applied to a Hodgkin-Huxley ionic model of the neural action potential. Cubic spline parameters were utilized to find the optimal stimulation profile for a fixed peak current. Simulation results demonstrate that the optimal stimulation profile for a specified single neuron is a non-rectangular pulse whose shape depends upon the maximum allowable current as well as the stimulus duration.

Revisiting the Bloch equation through averaging

A novel approach to finding an approximate analytic solution to the Bloch equation is developed in this paper. The method is based on time scaling and averaging of the Bloch equation after transformation to a rotating frame of reference. In order to accomplish the scaling, a novel time scaled magnetisation vector is introduced. The resultant time scaled system is subsequently approximated through averaging, a technique that to the best of our knowledge, has not previously been applied in the nuclear magnetic resonance context. Our proposed method of approximating the solution to the Bloch equation is valid for continuous wave excitation as well as the traditional pulse excitation with an arbitrary envelope, making this a widely applicable technique unlike previously proposed methods. Comparison of the approximate analytic solution and simulation results clearly indicates that the error is negligible when the field inhomogeneities are small compared to the excitation field amplitude. Extremum seeking techniques may be applied to determine the optimal excitation, given the form of the approximate solution. This result is applicable to a range of research areas including nuclear magnetic resonance, magnetic resonance imaging and optical resonance problems.

Magnetic resonance described in the excitation dependent rotating frame of reference

An excitation dependent rotating frame of reference to observe the magnetic resonance phenomenon is introduced in this paper that, to the best of our knowledge, has not been used previously in the nuclear magnetic resonance context. The mathematical framework for this new rotating frame of reference is presented based on time scaling the Bloch equation after transformation to the classical rotating frame of reference whose transverse plane is rotating at the Larmor frequency. To this end, the Bloch equation is rewritten in terms of a magnetisation vector observed from the excitation dependent rotating frame of reference. The resultant Bloch equation is referred to as the time scaled Bloch equation. In the excitation dependent rotating frame of reference whose coordinates are rotating at the instantaneous Rabi frequency the observed magnetisation vector is a much slower signal than the true magnetisation in the rotating frame of reference. As a result the ordinary differential equation solvers have the ability to solve the time scaled version of the Bloch equation with a larger step size resulting in a smaller number of samples for solving the equation to a desired level of accuracy. The simulation results for different types of excitation are presented in this paper. This method may be used in true Bloch simulators in order to reduce the simulation time or increase the accuracy of the numerical solution. Moreover, the time scaled Bloch equation may be employed to determine the optimal excitation pattern in magnetic resonance imaging as well as designing pulses with better slice selectivity which is an active area of research in this field.

Challenging the optimality of the pulse excitation in magnetic resonance imaging

Abstract The design of excitation signals for Magnetic Resonance Imaging (MRI) is cast as an optimal control problem. An appropriate cost criterion, the Signal Contrast Efficiency (SCE), is developed. It is to be optimised subject to dynamics expressed by the Bloch equation. The solution to the optimisation problem is potentially useful for all forms of MRI including structural and functional imaging. Here, we demonstrate that signals other than pulse excitations, which are ubiquitous in MRI, can provide adequate excitation, thus challenging the optimality and ubiquity of pulsed signals. A class of on-resonance piece-wise continuous amplitude modulated signals is introduced. It is shown that despite the bilinear nature of the Bloch equations, the optimisation problem is largely analytically tractable for this class of signals, using Galerkin approximation methods. Simulations demonstrate that this class of signals may provide an attractive alternative to pulsed excitation signals for MRI.

Revisiting excitation pattern design for magnetic resonance imaging through optimisation of the signal contrast efficiency

Abstract The design of excitation signals for Magnetic Resonance Imaging (MRI) is cast as an optimal control problem. Here, we demonstrate that signals other than pulse excitations, which are ubiquitous in MRI, can provide adequate excitation, thus challenging the optimality and ubiquity of pulsed signals. A class of on-resonance piecewise continuous amplitude modulated signals is introduced. It is shown that despite the bilinear nature of the Bloch equations, the spins system response is largely analytically tractable for this class of signals, using Galerkin approximation methods. To challenge the optimality of the pulse excitation, an appropriate cost criterion, the Signal Contrast Efficiency (SCE), is developed. It is to be optimised subject to dynamics expressed by the Bloch equations. To solve the problem the Bloch equation is transferred to the excitation dependent rotating frame of reference. The numerical solutions to the problem for different tissue types show that for a short period of time, pulse excitations provide the maximum signal contrast. However, the problem should be solved for longer periods of time which may result in a different answer than a pulse. For this purpose, the approximate analytic solution which is derived based on averaging the Bloch equation in the excitation dependent rotating frame of reference will be used to find the optimal excitation pattern. The solution to the optimisation problem is potentially useful for all forms of MRI including structural and functional imaging. The objective of this paper is to show that while classically transient response of pulses have been monitored so far, the optimal excitation pattern may be the steady state response of a non-pulse excitation.