Observers for Canonic Models of Neural Oscillators
Observers for Canonic Models of Neural Oscillators
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DOI：
10.1051/mmnp/20105206
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发表时间：
2010-01-01
影响因子：
2.2
通讯作者：
van Leeuwen, C.
中科院分区：
文献类型：
--
作者：
Fairhurst, D.;Tyukin, I.;van Leeuwen, C.
我们考虑一类定义为耦合的非线性常微分方程组的非线性振子的状态和参数估计问题。可观测变量仅限于状态向量和输入信号的几个分量。这类系统描述了一组控制神经膜诱发电位动力学的经典模型，包括Hodgkin-Huxley、Hindmarsh-Rose、Fitzhugh-Nagumo和Morris-LeCar模型。我们在观测器设计的经典框架内考虑了这些模型的状态和参数重构问题。这个框架为非线性微分方程组的状态和参数重构问题提供了计算上有效的解决方案，只要这些方程是所谓的自适应观测器规范形式。我们证明，尽管典型的神经振荡器是局部可观测的，但它们并不是自适应正则观测器形式。此外，我们还证明了不存在参数无关的微分同态，使得这些模型的原始方程可以转化为自适应正则观测器形式。然而，我们证明了对于Hindmarsh-Rose和Fitzhugh-Nagumo模型，参数相关的坐标变换可以将这些系统表示为自适应观测器规范形式。这允许以指数收敛速度重建未知状态和参数值，至少部分地并且直到(双)线性变换。为了避免仅部分重构的问题，同时又能处理未知参数以非线性方式进入系统的更一般的非线性模型，我们提出了一种新的状态和参数重构方法。该方法结合了标准李亚普诺夫设计的优点和基于正不变性和小增益定理的更灵活的设计和分析技术。我们表明，这种灵活性允许克服在这个问题中出现的病态和非唯一性问题。通过简单的数值算例说明了该方法的有效性。
We consider the problem of state and parameter estimation for a class of nonlinear oscillators defined as a system of coupled nonlinear ordinary differential equations. Observable variables are limited to a few components of state vector and an input signal. This class of systems describes a set of canonic models governing the dynamics of evoked potential in neural membranes, including Hodgkin-Huxley, Hindmarsh-Rose, FitzHugh-Nagumo, and Morris-Lecar models. We consider the problem of state and parameter reconstruction for these models within the classical framework of observer design. This framework offers computationally-efficient solutions to the problem of state and parameter reconstruction of a system of nonlinear differential equations, provided that these equations are in the so-called adaptive observer canonic form. We show that despite typical neural oscillators being locally observable they are not in the adaptive canonic observer form. Furthermore, we show that no parameter-independent diffeomorphism exists such that the original equations of these models can be transformed into the adaptive canonic observer form. We demonstrate, however, that for the class of Hindmarsh-Rose and FitzHugh-Nagumo models, parameter-dependent coordinate transformations can be used to render these systems into the adaptive observer canonical form. This allows reconstruction, at least partially and up to a (bi)linear transformation, of unknown state and parameter values with exponential rate of convergence. In order to avoid the problem of only partial reconstruction and at the same time to be able to deal with more general nonlinear models in which the unknown parameters enter the system nonlinearly, we present a new method for state and parameter reconstruction for these systems. The method combines advantages of standard Lyapunov-based design with more flexible design and analysis techniques based on the notions of positive invariance and small-gain theorems. We show that this flexibility allows to overcome ill-conditioning and non-uniqueness issues arising in this problem. Effectiveness of our method is illustrated with simple numerical examples.