Nicola Wilten, Tripp Bryan, Scott Matthew
Department of Applied Mathematics, University of Waterloo Waterloo, ON, Canada.
Department of Systems Design Engineering, University of WaterlooWaterloo, ON, Canada; Center for Theoretical Neuroscience, University of WaterlooWaterloo, ON, Canada.
Front Comput Neurosci. 2016 Feb 29;10:15. doi: 10.3389/fncom.2016.00015. eCollection 2016.
A fundamental question in computational neuroscience is how to connect a network of spiking neurons to produce desired macroscopic or mean field dynamics. One possible approach is through the Neural Engineering Framework (NEF). The NEF approach requires quantities called decoders which are solved through an optimization problem requiring large matrix inversion. Here, we show how a decoder can be obtained analytically for type I and certain type II firing rates as a function of the heterogeneity of its associated neuron. These decoders generate approximants for functions that converge to the desired function in mean-squared error like 1/N, where N is the number of neurons in the network. We refer to these decoders as scale-invariant decoders due to their structure. These decoders generate weights for a network of neurons through the NEF formula for weights. These weights force the spiking network to have arbitrary and prescribed mean field dynamics. The weights generated with scale-invariant decoders all lie on low dimensional hypersurfaces asymptotically. We demonstrate the applicability of these scale-invariant decoders and weight surfaces by constructing networks of spiking theta neurons that replicate the dynamics of various well known dynamical systems such as the neural integrator, Van der Pol system and the Lorenz system. As these decoders are analytically determined and non-unique, the weights are also analytically determined and non-unique. We discuss the implications for measured weights of neuronal networks.
计算神经科学中的一个基本问题是如何连接脉冲神经元网络以产生所需的宏观或平均场动力学。一种可能的方法是通过神经工程框架（NEF）。NEF方法需要所谓的解码器，这些解码器通过一个需要大型矩阵求逆的优化问题来求解。在这里，我们展示了如何针对I型和某些II型发放率，根据其相关神经元的异质性解析地获得解码器。这些解码器生成函数的近似值，这些近似值在均方误差上以1/N的形式收敛到所需函数，其中N是网络中的神经元数量。由于其结构，我们将这些解码器称为尺度不变解码器。这些解码器通过NEF权重公式为神经元网络生成权重。这些权重迫使脉冲网络具有任意和规定的平均场动力学。用尺度不变解码器生成的权重渐近地都位于低维超曲面上。我们通过构建脉冲θ神经元网络来证明这些尺度不变解码器和权重曲面的适用性，这些网络复制了各种著名动力系统的动力学，如神经积分器、范德波尔系统和洛伦兹系统。由于这些解码器是解析确定的且不唯一，权重也是解析确定的且不唯一。我们讨论了对神经网络测量权重的影响。
