Centre for Neural Circuits and Behaviour, University of Oxford, Oxford, United Kingdom.
Institute of Science and Technology Austria, Klosterneuburg, Austria.
PLoS Comput Biol. 2022 Aug 15;18(8):e1010365. doi: 10.1371/journal.pcbi.1010365. eCollection 2022 Aug.
Neuronal networks encode information through patterns of activity that define the networks' function. The neurons' activity relies on specific connectivity structures, yet the link between structure and function is not fully understood. Here, we tackle this structure-function problem with a new conceptual approach. Instead of manipulating the connectivity directly, we focus on upper triangular matrices, which represent the network dynamics in a given orthonormal basis obtained by the Schur decomposition. This abstraction allows us to independently manipulate the eigenspectrum and feedforward structures of a connectivity matrix. Using this method, we describe a diverse repertoire of non-normal transient amplification, and to complement the analysis of the dynamical regimes, we quantify the geometry of output trajectories through the effective rank of both the eigenvector and the dynamics matrices. Counter-intuitively, we find that shrinking the eigenspectrum's imaginary distribution leads to highly amplifying regimes in linear and long-lasting dynamics in nonlinear networks. We also find a trade-off between amplification and dimensionality of neuronal dynamics, i.e., trajectories in neuronal state-space. Networks that can amplify a large number of orthogonal initial conditions produce neuronal trajectories that lie in the same subspace of the neuronal state-space. Finally, we examine networks of excitatory and inhibitory neurons. We find that the strength of global inhibition is directly linked with the amplitude of amplification, such that weakening inhibitory weights also decreases amplification, and that the eigenspectrum's imaginary distribution grows with an increase in the ratio between excitatory-to-inhibitory and excitatory-to-excitatory connectivity strengths. Consequently, the strength of global inhibition reveals itself as a strong signature for amplification and a potential control mechanism to switch dynamical regimes. Our results shed a light on how biological networks, i.e., networks constrained by Dale's law, may be optimised for specific dynamical regimes.
神经网络通过定义网络功能的活动模式来编码信息。神经元的活动依赖于特定的连接结构，但结构和功能之间的联系还没有完全被理解。在这里，我们采用一种新的概念方法来解决这个结构-功能问题。我们不是直接操纵连接，而是专注于上三角矩阵，它在由 Schur 分解得到的给定规范正交基中表示网络动力学。这种抽象允许我们独立地操纵连接矩阵的特征谱和前馈结构。使用这种方法，我们描述了广泛的非正规瞬态放大谱，并为了补充动态状态的分析，我们通过特征向量和动力学矩阵的有效秩来量化输出轨迹的几何形状。反直觉的是，我们发现缩小特征谱的虚部分布会导致在线性和非线性网络中产生高度放大的线性和持久动力学状态。我们还发现了放大和神经元动力学维度之间的权衡，即神经元状态空间中的轨迹。能够放大大量正交初始条件的网络产生的神经元轨迹位于神经元状态空间的同一子空间中。最后，我们研究了兴奋性和抑制性神经元网络。我们发现全局抑制的强度与放大的幅度直接相关，即减弱抑制权重也会降低放大，并且特征谱的虚部分布随兴奋性到抑制性和兴奋性到兴奋性连接强度比的增加而增长。因此，全局抑制的强度本身就是放大的一个强有力的特征，也是切换动态状态的潜在控制机制。我们的结果揭示了生物网络（即受戴尔定律约束的网络）如何针对特定的动态状态进行优化。
