An official website of the United States government
Abstract Burst suppression is a phenomenon in which the electroencephalogram (EEG) of a pharmacologically sedated patient alternates between higher frequency and amplitude bursting to lower frequency and amplitude suppression. The level of sedation can be quantified by the burst suppression ratio (BSR), which is defined as the amount of time that an EEG is suppressed over the amount of time measured. Maintaining a stable BSR in patients is an important clinical problem, which has led to an interest in the development of BSR-based closed-loop pharmacological control systems. Methods to address the problem typically involve pharmacokinetic (PK) modeling that describes the dynamics of drug infusion in the body as well as signal processing methods for estimating burst suppression from EEG measurements. In this regard, simulations, physiological modeling, and control design can play a key role in producing a solution. This paper seeks to add to prior work by incorporating a Schnider PK model with the Wilson–Cowan neural mass model to use as a basis for robust control design of biophysical burst suppression dynamics. We add to this framework actuator modeling, real-time burst suppression estimation, and uncertainty modeling in both the patient's physical characteristics and neurological phenomena to form a closed-loop system. Using the Ziegler–Nichols tuning method for proportional-integral-derivative (PID) control, we were able to control this system at multiple BSR set points over a simulation time of 18 h in both nominal, patient varied with noise added and with reduced performance due to BSR bounding when including patient variation and noise as well as neurological uncertainty.
more »
« less
Iterated Crank–Nicolson Runge–Kutta Methods and Their Application to Wilson–Cowan Equations and Electroencephalography Simulations
The Wilson–Cowan model has been widely applied for the simulation of electroencephalography (EEG) waves associated with neural activities in the brain. The Runge–Kutta (RK) method is commonly used to numerically solve the Wilson–Cowan equations. In this paper, we focus on enhancing the accuracy of the numerical method by proposing a strategy to construct a class of fourth-order RK methods using a generalized iterated Crank–Nicolson procedure, where the RK coefficients depend on a free parameter c2. When c2 is set to 0.5, our method becomes a special case of the classical fourth-order RK method. We apply the proposed methods to solve the Wilson–Cowan equations with two and three neuron populations, modeling EEG epileptic dynamics. Our simulations demonstrate that when c2 is set to 0.4, the proposed RK4-04 method yields smaller errors compared to those obtained using the classical fourth-order RK method. This is particularly visible when the spectral radius of the connection matrix or the excitation-inhibition coupling coefficient is relatively large.
more »
« less
- PAR ID:
- 10631911
- Publisher / Repository:
- Foundations
- Date Published:
- Journal Name:
- Foundations
- Volume:
- 4
- Issue:
- 4
- ISSN:
- 2673-9321
- Page Range / eLocation ID:
- 673 to 689
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
ABSTRACT This paper presents high-order Runge–Kutta (RK) discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry. The scheme can preserve a general polytropic equilibrium state and achieve total energy conservation up to machine precision with carefully designed spatial and temporal discretizations. To achieve the well-balanced property, the numerical solutions are decomposed into equilibrium and fluctuation components that are treated differently in the source term approximation. One non-trivial challenge encountered in the procedure is the complexity of the equilibrium state, which is governed by the Lane–Emden equation. For total energy conservation, we present second- and third-order RK time discretization, where different source term approximations are introduced in each stage of the RK method to ensure the conservation of total energy. A carefully designed slope limiter for spherical symmetry is also introduced to eliminate oscillations near discontinuities while maintaining the well-balanced and total-energy-conserving properties. Extensive numerical examples – including a toy model of stellar core collapse with a phenomenological equation of state that results in core bounce and shock formation – are provided to demonstrate the desired properties of the proposed methods, including the well-balanced property, high-order accuracy, shock-capturing capability, and total energy conservation.more » « less
-
null (Ed.)Solving phase equations for systems with high degrees of nonlinearities is cumbersome. However, in the case of two coupled canonical oscillators, that is, a reduced model of translated Wilson–Cowan neuronal dynamics, under slowly varying amplitude and rotating wave approximations, we suggested a convenient way to find their average relative phase evolution. This approach enabled us to find an explicit solution for the average relative phase of the two coupled canonical oscillators based on the original neuronal model parameters, and importantly, to find their phase-locking constraint. This methodology is straightforward to implement in any Wilson–Cowan-type coupled oscillators with applications in gradient frequency neural networks (GFNNs).more » « less
-
Abstract In this work we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker–Planck equation and the Keller–Segel equations. The two proposed schemes are first-order accurate in time, explicitly solvable, and second-order and fourth-order accurate in space, which are obtained via finite difference implementation of the classical continuous finite element method. The fully discrete schemes are proved to be positivity preserving and energy dissipative: the second-order scheme can achieve so unconditionally while the fourth-order scheme only requires a mild time step and mesh size constraint. In particular, the fourth-order scheme is the first high order spatial discretization that can achieve both positivity and energy decay properties, which is suitable for long time simulation and to obtain accurate steady state solutions.more » « less
-
In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh 2 and the second order TVD-RK scheme needs $$ \tau \le \rho {h}^{\frac{4}{3}}$$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h .more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/foundations4040042
