Theoretical framework and bifurcation analysis for deep recurrent neural networks inferred from neural measurements

在理论神经科学中，我们构建神经电路的数学模型，以深入了解实验测量背后的计算原理和动力学机制。深度学习方面的最新进展可能有助于在某种程度上自动化这一过程：我们可以直接在数据上训练递归神经网络(RNN)，以复制、预测或自由生成观察到的记录，而不是明确地建立一个可以基于理论家的见解来解释数据的模型。虽然这是一种有希望的方法，我们的团队在过去几年做出了很大贡献，但它也带来了新的挑战：由于此类模型是由算法设计的，而不是我们自己设计的，因此对它们的计算和动力学特性的分析通常更具挑战性。此外，当前的RNN训练算法通常假设底层系统是静态的(参数随时间变化不变)，这一假设在神经科学中经常会被违反，特别是在需要任何形式的学习或自适应行为的任务中。允许RNN(或任何动态系统)中的参数自适应变化，反过来将导致系统动力学在某些临界点发生突变，即所谓的分叉。分支对于解释神经系统和精神病学中的许多观察结果都是非常重要的。目前的建议解决了这些挑战：在WP1中，我们基于非线性动力系统和分支理论的概念，为通过深度学习从实验数据中推断出的一类特殊的RNN建立了一个数学框架。该框架将能够打开“黑匣子”，并彻底表征经验推断的RNN在不同参数范围内的动力学行为，从而推导出训练系统本身的重要计算和功能性质。在WP2中，我们将扩展现有的RNN训练算法，以允许参数随时间或实验试验的变化。这将使，连同WP1中计划的数学进步，能够在实验数据中彻底表征分叉(以及特定类型的分叉)。在WP3中，我们将使用WP1和WP2中开发的数学概念和算法来解决关于啮齿动物规则学习的一个长期存在(但仍未解决)的假设：通过分析直接基于神经记录训练的RNN，我们将测试在学习新行为规则期间经验观察到的神经种群活动的突然变化是否确实是由于分支。我们还将检查哪种类型的分叉可能是所观察到的神经活动变化的基础，因为这具有重要的功能含义。这里发展的方法和概念将适用于许多科学和工程领域，远远超出神经科学。

In theoretical neuroscience, we construct mathematical models of neural circuits to gain insight into the computational principles and dynamical mechanisms underlying experimental measurements. Recent advances in deep learning may help to automatize this process to some degree: Rather than explicitly building a model which could explain the data based on the theoretician’s insights, we may train recurrent neural networks (RNNs) directly on data to reproduce, predict, or freely generate the observed recordings. While this is a promising approach to which our group has substantially contributed in the past years, it also comes with new challenges: Since such models were designed by an algorithm and not by ourselves, the analysis of their computational and dynamical properties is generally more demanding. Moreover, current RNN training algorithms usually assume that the underlying system is stationary (with parameters constant across time), an assumption that will often be violated in neuroscience, especially in tasks that require any form of learning or adaptive behavior. Allowing for adaptive parameter changes in RNNs (or any dynamical system), in turn, will lead to abrupt transitions in system dynamics at some critical points, so-called bifurcations. Bifurcations are hugely important both from a theoretical perspective as well as for explaining many observations in neural systems and psychiatry.The present proposal addresses these challenges: In WP1, we develop a mathematical framework for a special class of RNNs inferred from experimental data by deep learning, based on concepts from nonlinear dynamical systems and bifurcation theory. This framework will enable to open the “black box” and thoroughly characterize the dynamical behavior of empirically inferred RNNs within different parameter regimes, and thereby to deduce important computational and functional properties of the trained-on system itself. In WP2, we will extend existing RNN training algorithms to allow for parameter changes across time or experimental trials. This will enable, together with the planned mathematical advances in WP1, to thoroughly characterize bifurcations (and the specific type of bifurcation) in experimental data. In WP3 we will use the mathematical concepts and algorithms developed in WP1 and WP2 to address a long-standing (but still unresolved) hypothesis about rule learning in rodents: Through analysis of RNNs directly trained on neural recordings, we will test whether the empirically observed abrupt changes in neural population activity during learning of a new behavioral rule are indeed due to bifurcations. We will also check which type of bifurcation may underlie the observed neural activity changes, as this has important functional implications. The methods and concepts evolved here will be applicable in many scientific and engineering areas, far beyond neuroscience.