Optimal Control of Models of Neural Population Dynamics

神经系统在数学上被描述为受控的动态系统。它们受到外部刺激输入（“控制”）的影响，这可以是自然的——通过突触连接——也可以是人工的——通过外部电或磁刺激。数学方法可以帮助理解神经系统对刺激反应的潜在机制。在这里，我们将使用非线性最优控制理论（OCT）的方法来研究有效控制信号的设计，并研究它们的性质，其应用有两个方面：在“合成”场景中，人们可以研究如何有效地操纵神经系统（例如调节大脑活动）。在“分析”场景中，人们可以根据优化原则（例如由进化强制执行）研究神经动力系统的自然设计。控制信号的效率是用代价函数来衡量的，代价函数将输入的强度与接近目标状态进行交换。在这个项目中，我们使用庞特里亚金原理来数值计算成本梯度。然后迭代逼近最优控制。传统的最优控制方法仅限于可以定义与时间相关的精确目标状态的设置。相比之下，任何试图强制执行振荡、网络动态同步或锁定的场景，无论控制开始时的阶段或振荡的确切形状如何，都需要对标准方法进行扩展。在这里，我们想研究如何使OCT方法适应这种情况。为此，我们提出了新的成本函数，并研究了它们在单神经群体、网络基序和大型全脑模型中的适用性和实用性。我们将重点关注两种不同复杂程度的神经质量模型：相对简单的Wilson-Cowan模型，它描述了周期性耦合的兴奋性和抑制性神经元群体激活的时间演化，以及来自随机连接的兴奋性和抑制性自适应指数积分和火神经元网络的平均场模型（“平均场AdEx模型”）。最优控制信号的计算需要对这些模型的状态演化进行数值模拟，为此我们使用了Neurolib，这是一个开源的Python软件框架，用于建模神经动力学。我们将提供额外的算法作为Neurolib的开源扩展，这将使我们能够计算最优控制信号或上述（和其他）模型，用于确定性和噪声系统，以及单个神经种群及其网络的模型。该软件将以模块化的方式实现，这样它可以很容易地调整到特定场景的要求，以促进未来对神经系统刺激的研究。

Neural systems are described mathematically as controlled dynamical systems. They are affected by external stimulating inputs ("control"), which can be natural - via synaptic connections - or artificial - via external electric or magnetic stimulation. The mathematical approach can help to understand the underlying mechanisms responsible for the reaction of neural systems to stimulation. Here we will study the design of efficient control signals using methods from nonlinear optimal control theory (OCT) and investigate their properties, with applications being twofold: In "synthetic" scenarios, one can study how to manipulate neural systems efficiently (e.g. modulate brain activity). In "analytic" scenarios, one can study the natural design of neural dynamical systems in terms of optimization principles (e.g. enforced by evolution). The efficiency of a control signal is measured in terms of a cost functional that trades the strength of the input against the closeness to a target state. In this project, we use Pontryagin’s Principle to numerically compute a cost gradient. We then approach the optimal control iteratively. The conventional optimal control approach is limited to settings where an exact time-dependent target state can be defined. By contrast, any scenario where one would try to enforce oscillations, synchrony of the network dynamics, or locking, irrespective of the phases at control onset or of the exact shape of the oscillations, requires an extension of the standard approach. Here we want to study how to adapt the OCT approach to such settings. To this end, we suggest novel cost functionals and study their applicability and usefulness in models of single neural populations, network motifs, and large whole-brain models. We will focus on two neural mass models of different levels of complexity: the comparatively simple Wilson-Cowan model, which describes the time-evolution of the activation of recurrently coupled excitatory and inhibitory populations of neurons, and a mean-field model that is derived from a network of randomly connected excitatory and inhibitory adaptive exponential integrate-and-fire neurons (“mean-field AdEx model”). The computation of optimal control signals requires numerical simulations of the state evolution of these models, for which we use Neurolib, an open-source Python software framework for modeling neural dynamics. We will provide additional algorithms as an open-source extension to Neurolib, which will enable us to compute optimal control signals or the aforementioned (and other) models, for deterministic and noisy systems, and for models of single neural populations and networks thereof. The software will be implemented in a modular fashion, such that it can easily be adjusted to the requirements of specific scenarios to facilitate future studies of stimulation of neural systems.