Feedback through graph motifs relates structure and function in complex networks

Abstract

In physics, biology and engineering, network systems abound. How does the connectivity of a network system combine with the behavior of its individual components to determine its collective function? We approach this question for networks with linear time-invariant dynamics by relating internal network feedbacks to the statistical prevalence of connectivity motifs, a set of surprisingly simple and local statistics of connectivity. This results in a reduced order model of the network input-output dynamics in terms of motifs structures. As an example, the new formulation dramatically simplifies the classic Erdős-Rényi graph, reducing the overall network behavior to one proportional feedback wrapped around the dynamics of a single node. For general networks, higherorder motifs systematically provide further layers and types of feedback to regulate the network response. Thus, the local connectivity shapes temporal and spectral processing by the network as a whole, and we show how this enables robust, yet tunable, functionality such as extending the time constant with which networks remember past signals. The theory also extends to networks composed from heterogeneous nodes with distinct dynamics and connectivity, and patterned input to (and readout from) subsets of nodes. These statistical descriptions provide a powerful theoretical framework to understand the functionality of real-world network systems, as we illustrate with examples including the mouse brain connectome.

Recurrent interactions can explain the variance in single trial responses

“…The imaging data used throughout this manuscript was described in a previous study [4]. Briefly, the activity of L2/3 excitatory neurons expressing GCaMP6s in mouse visual cortex (73-347 neurons; n = 8 animals; 21 distinct fields of view; Fig 1A) were imaged using two photon laser scanning microscopy (25)(26)(27)(28)(29)(30)(31)(32)(33) Hz; [4,33]). Mice were awake and allowed to freely run on a linear treadmill while viewing drifting square-wave gratings presented in 12 directions in pseudo-random order interleaved with mean-luminance matched grey screen; each trial was a 5-minute block of stimulus presentation in this format.…”

“…In visual cortex, visually tuned neurons with similar stimulus selectivity are more likely to be synaptically interconnected [25] and these connections manifest as specific motifs [24] which coordinate synaptic integration [26]. Moreover, populations of neurons have been shown to exhibit higherorder state correlations [23,27], and topological network features have been shown to shape spike propagation and information transfer [26,28,29]. However, the relative role of local synaptic connectivity, as compared to stimulus-related input or more global variables such as locomotion, in the generation of a sensory representation, particularly in real time, remains unclear.…”

Cyclic transitions between higher order motifs underlie sustained activity in asynchronous sparse recurrent networks

“…390 The necessity of higher-order patterns for stable activity has strong implications for 391 neural coding. Previous work has already demonstrated that correlations enhance 392 coding, with triplet correlations having an advantage over pairwise 393 [5,16,18,[21][22][23][52][53][54]. The neural code must rest upon a foundation of stable 394 propagation of spikes, which we have shown in turn rests on higher-order motifs and 395 coordinated integration.…”

Connectivity inference from neural recording data: Challenges, mathematical bases and research directions

“…These abstract metrics facilitate comparison of the structure of di-verse neural populations. For example, Hu et al (2016) proposed a method to relate the network statistics of connectivity of linear point processes or Hawkes models (see section 5.1.6) to its function.…”