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Analyzing the functional connectivity of the brain is an enormous challenge, as deciphering functional connectivity requires knowledge of functional responses and connections. One promising strategy is analyzing the spatial pattern of activity correlations across cell populations. In the primary auditory cortex (A1), cells respond to different sound features. On the large scale, there exists a tonotopic map, which is fractured at the small scale, raising the question of whether functional connections are spatially ordered or disordered. To test whether functional connectivity on a local and a global scale is also disordered, we first designed a robust statistical model to estimate parameters and test for the significance of the estimated correlation maps. We developed an inference method that allows efficient model fitting and statistical testing to project the correlation maps to 2D space. We then performed in vivo two-photon calcium imaging in layer 2/3 of A1 with pure tones (PT) or a combination of two tones (TT; harmonically related or not). We found that the spatial patterns of signal correlations (SCs) depend on the type of sound stimuli that were presented. The functional 2D maps of PT-driven SCs are more restricted to local neurons than TT signal correlations which showed more global textures. 2D SC patterns for harmonic stimuli showed spatially distinct relationships. TT SCs revealed spatially precise functional connectivity between harmonically related neurons. Thus, even though the frequency preference of neighboring neurons in A1 is functionally diverse, the functional connection pattern of these neurons is functionally precise and harmonically related. 

Extracting directional connectivity in a neuronal ensemble from spiking observations is a key challenge in understanding the circuit mechanisms of brain function. Existing methods proceed in two stages, by first estimating the latent processes that govern spiking, followed by characterizing connectivity using said estimates. As such, the extracted networks in the second stage are highly sensitive to the accuracy of the estimates in the first stage. In this work, we introduce a framework to directly extract Granger causal links from spiking observations, without requiring intermediate time-domain estimation, by explicitly modeling the endogenous and exogenous latent processes that underlie spiking activity. Our proposed method integrates several techniques such as point processes, state-space modeling and Pólya-Gamma augmentation. We demonstrate the utility of our proposed approach using simulated data and application to real data from the rat brain during sleep.