The Kalman filter is a fundamental filtering algorithm that fuses noisy sensory data, a previous state estimate, and a dynamics model to produce a principled estimate of the current state. It assumes, and is optimal for, linear models and white Gaussian noise. Due to its relative simplicity and general effectiveness, the Kalman filter is widely used in engineering applications. Since many sensory problems the brain faces are, at their core, filtering problems, it is possible that the brain possesses neural circuitry that implements equivalent computations to the Kalman filter. The standard approach to Kalman filtering requires complex matrix computations that are unlikely to be directly implementable in neural circuits. In this paper, we show that a gradient-descent approximation to the Kalman filter requires only local computations with variance weighted prediction errors. Moreover, we show that it is possible under the same scheme to adaptively learn the dynamics model with a learning rule that corresponds directly to Hebbian plasticity. We demonstrate the performance of our method on a simple Kalman filtering task, and propose a neural implementation of the required equations.
翻译:Kalman 过滤器是一种基本的过滤算法,它将噪音感官数据、先前的状态估计和动态模型结合起来,以得出对当前状态的有原则的估计。它假定线性模型和白色高斯噪音,并且是最佳的。由于它的相对简单性和一般效果,Kalman 过滤器在工程应用中被广泛使用。由于大脑面临的许多感官问题,就其核心而言,是过滤问题,因此大脑可能拥有神经电路,与Kalman 过滤器进行等量的计算。Kalman 过滤的标准方法需要复杂的矩阵计算,而这种计算在神经电路中是不可能直接执行的。在本文中,我们表明对Kalman 过滤器的梯度-白度近似只需要局部计算,而有不同的加权预测错误。此外,我们表明,在同一计划下,有可能以与Hebbian 塑料直接相适应的学习规则来适应性地学习动态模型。我们的方法在简单的Kalman 过滤任务上的表现,并提议执行所需的神经等式。