All experimental procedures were approved by the Institutional Animal Care and Use Committee (IACUC) at the University of California, Santa Barbara and were conducted in accordance with the guidelines of the US Department of Health and Human Services. The animals used in the study were adult, triple transgenic mice of the genotype TITL-GCaMP6s (Ai94, Madisen et al., 2015) X Emx1-Cre (Jackson Labs #005628) X ROSA:LNL:tTA (Jackson Labs #011008) to express the calcium indicator GCaMP6s in cortical excitatory neurons. The mice were housed in a 12-h reverse light/dark cycle and had ad libitum access to food and water. A total of 10 recordings (5 spontaneous and 5 stimulated) were obtained from 8 mice, with some mice contributing multiple recordings on different days. At least one week before experiments, mice were surgically implanted with a 5mm optical glass coverslip over the right posterior cortex and a stainless steel headplate, both adhered with cyanoacrylate glue (Oasis Medical) and dental cement (Parkell Metabond), as previously described (Schneider et al., 2025). After recovery, intrinsic signals optical imaging (ISOI) (Kalatsky and Stryker, 2003) was performed to measure cortical retinotopic maps used to delineate visual area boundaries, also as described previously (Yu et al., 2022; Smith et al., 2017). These were used to validate craniotomy targeting and to register the two-photon fields of view to visual areas via vascular landmarks.

The calcium imaging datasets were recorded at the University of California, Santa Barbara, using a custom-built two-photon mesoscope (Diesel2p) (Yu et al., 2021). The mesoscope allows for imaging large fields of view (herein, 3 mm × 3-4 mm) at resonant speeds and cellular resolution. The imaging was performed at a median frame rate of 7.5 Hz and 1.46 microns per pixel. Fields of view were positioned to cover primary visual cortex (V1) and several higher visual areas (HVAs) at once, at depths of 150–200 μm to capture L2/3 cortical neurons. Imaging sessions lasted approximately 45 minutes, during which the mice were either in a spontaneous state (no visual stimulus) or were presented with drifting grating patches as visual stimuli. The visual stimuli were presented on a 90 cm curved monitor placed 14.5 cm from the mouse's left eye and tilted 10 degrees downward to cover approximately 150 degrees of visual angle in azimuth and 70 degrees in elevation. Drifting grating patches were presented sequentially as 20-degree-wide squares tiling the visual field area above. Each had a spatial frequency of either 0.04 or 0.1 cpd and a temporal frequency of 1 or 4 Hz, and were presented in 8 different orientations (0, 45, 90, 135, 180, 225, 270, and 315 degrees). The parameters of the patches in each sequence were randomly shuffled across successive presentations to enforce an even stimulus distribution. Finally, a projective correction was applied to screen stimuli to account for the near placement of a flat screen and the spherical eye of the mouse (Smith, 2012); with the correction applied, images strictly maintain the intended geometry and location across the visual field relative to the eye center. Visible emission from the screen was blocked from the imaging objective using a custom opaque plastic cone placed between the headplate and objective. Spontaneous recordings were obtained in the absence of any visual stimuli, with the mice either in total darkness or with the same screen set to a static middle gray image. Further details of the recordings used in this study are provided in the Supplementary material.

The raw calcium imaging data were first preprocessed using the Suite2p pipeline (Pachitariu et al., 2016), which includes motion correction and alignment using the registration module. The motion correction step in Suite2p involved aligning the frames of the imaging data to correct for any movement artifacts. The cell detection and extraction of the fluorescence traces were performed using the EXTRACT algorithm (Inan et al., 2017; Dinç et al., 2021), which identifies regions of interest (ROIs) corresponding to individual neurons and extracts their fluorescence signals, while correcting for any neuropil contamination. The extracted fluorescence traces were then deconvolved to estimate the underlying spiking rate using the deep-learning-based CASCADE algorithm (Rupprecht et al., 2021). Finally, to obtain the discretized binary states of each neuron, the deconvolved traces were fitted using a Hidden Markov Model (HMM) to identify the most probable state (Active: 1 or Quiet: 0) of each neuron at each time point (see Figure 1). This approach allows us to avoid setting an arbitrary threshold for each neuron and provides a probabilistic framework for identifying discrete states from the deconvolved calcium traces (Diana et al., 2019; Etter et al., 2020). The binarized states were then used for the correlation and information-theoretic analyses.

To measure the interdependencies between the neural activity traces recorded from the neurons in the visual cortex, we build upon mutual information (MI). MI quantifies the mutual dependence between two variables by computing the amount of information obtained about one random variable through observing another random variable. For two discrete random vectors U and V, the mutual information of U given V is defined as:

where u and v are values of the vector random variables U and V, respectively; p(u, v) is the joint probability distribution function; and p(u) and p(v) are the marginal probability distribution functions.

Here, we compute the mutual information between binarized and deconvolved calcium traces. The HMM-based binarization helps to mitigate the effects of noise and variability in the calcium imaging data, and has been useful in the identification of neuronal assemblies (Diana et al., 2019) and decoding behavior (Etter et al., 2020).

To quantify the correlation length in the neural activity, we compute the pairwise Pearson correlation coefficient between the binarized spiking activity of all pairs of neurons in each dataset. The correlation coefficients are binned according to the inter-neuron distance, and the average correlation coefficient is then computed for each of the logarithmically spaced distance bins. The correlation length λ is then estimated by fitting an exponential decay of the average correlation coefficient with distance:

Here, C(d) is the average correlation coefficient at distance d, C₀ is the initial correlation coefficient at the minimum distance, C_∞ is the asymptotic correlation coefficient at large distances, which accounts for the baseline correlation in mesoscale calcium imaging data at large distances. The correlation length λ provides a characteristic length scale for the decay of correlations in space. The parameters, C₀, C_∞ and λ in Equation 5 were estimated by a non-linear least squares fitting algorithm using the SciPy library in Python (Virtanen et al., 2020). To visualize the decay of correlations with distance, we also defined the normalized average correlation coefficient:

which allows for consistent comparison of the decay of correlations across different datasets and conditions.

The Z-scored Synergy and Redundancy values decay slowly with distance, with fitted λ beyond the field of view. To obtain more robust estimates of the spatial extent of information decay in the neural activity, we estimate the effective information length λ_eff as the normalized area under the curve of the fitted exponential decay function:

where d₀ = 100μm is the minimum distance and d_max = 1500μm is the maximum distance in the field of view such that sufficient pairs of neurons are available.

Within each dataset, we compute the pairwise synergistic and redundant interactions between neurons based on the normalized PID components. We formalize these interactions as networks, where the nodes are neurons and the edge weights correspond to the normalized synergistic (or redundant) information between the neurons. We then obtain the associated k-nearest neighbor (kNN) graph, whereby we retain only the top k strongest connections for each neuron. This results in a sparsified network representation that highlights the most relevant interactions while reducing noise and spurious connections.

To carry out our analysis using partial network decomposition, we consider the two unweighted networks (synergistic and redundant) and a combined network that contains the union of the edge sets from the synergistic and redundant layers (see Figure 3). This combined representation allows us to analyse the interplay between synergistic and redundant interactions and their contributions to the overall information propagation in the neural system. Using partial network decomposition (Luppi et al., 2024), we identify complementary, shared, and unique shortest paths between pairs of neurons across the synergistic and redundant layers (see Figure 3). This analysis provides insights into how the different types of information interactions contribute to the overall connectivity and information flow in the network across different spatial scales.

Briefly, the framework considers shortest paths between pairs of nodes as a measure of efficiency. Suppose we have a combined network with two layers, A and B, representing synergistic and redundant interactions, respectively. For each pair of nodes (i, j), we identify the shortest paths in each layer separately, denoted as d_A(i, j) and d_B(i, j). We then consider the shortest path on the combined network, which includes the union of the edges from both layers, and we denote it as d_A∪B(i, j). The partial network decomposition then classifies the shortest paths into three categories:

By analyzing the distribution of complementary, shared, and unique paths, we can characterize how the contribution of synergistic and redundant interactions varies over paths of different lengths in the network.

We first investigate the presence of long-range correlations in the neural activity recorded from the visual cortex of awake mice. In Figure 4A (left), we plot the average Pearson correlation coefficients between the binarized spiking activity of two neurons as a function of the distance between the neurons (binned), for both spontaneous and visually stimulated conditions. The results show that the average correlations decay with distance, but remain significantly above zero even at distances of several millimeters, indicating the presence of long-range correlations in the neural activity. In Supplementary Materials, we show how the shape of the correlation distribution changes with distance. The fitted correlation functions (Equation 5) are also shown in Figure 4A (left). From each fit, we obtain values for the initial correlation C₀, correlation length λ and asymptotic correlation C_∞. The normalized correlations (Equation 6) shown in Figure 4A (right) highlight the increased correlation length observed during visual stimulation. Figure 4B shows that the initial correlation C₀ is not significantly different (Mean difference = 0.002, p = 0.0871 and Hedge's g = 0.91) among the datasets in the two conditions, yet the estimated correlation lengths are significantly larger (Mean difference = 603.21μm, p = 0.0055 and Hedge's g = 2.25) during visual stimulation compared to spontaneous activity, suggesting that visual input enhances long-range correlations in the visual cortex. An increased correlation length is a signature of a system operating closer to criticality (Stanley, 1971). It must be noted that although an increase in correlations is expected due to the visual stimulus for short distances (≈200 − 300μm) (Marshel et al., 2011), we observe increased correlations up to ≈900μm. This indicates a distinct state of the cortical network that supports longer-range correlations rather than short-range stimulus-related co-activations. However, other factors, such as changes in attentional states or arousal levels during visual stimulation, may also contribute to the observed increase in correlation length (Vinck et al., 2015). Therefore, further investigation is needed to fully understand the biological mechanisms that modulate these long-range correlations. Furthermore, we observe that the fitted correlation functions fit the data well for large distances, but tend to overestimate correlations at short distances (see Figure 4A). This may be due to the presence of local heterogeneities and anti-correlations among the neurons in the field of view (between 100 − 200μm), which are not captured by the exponential decay model. Future work could explore more complex correlation functions to better capture the full range of correlation behaviors observed in the data.

Next, we apply PID and NuMIT to decompose the time-delayed mutual information (TDMI) between pairs of neurons into Z-scored redundant and synergistic components. The TDMI was estimated at a time delay of τ = 1 time step. The average Z-scored synergy and redundancy as a function of distance between pairs of neurons are shown in Figure 5. Both redundancy (Figure 5A, left) and synergy (Figure 5B, left) exhibit a decay with distance, but average initial redundancy (Z-score = 1.24 ± 0.06) is generally higher than synergy (Z-score = 0.23 ± 0.021), and the decay is slower for redundancy (λ_eff = 1278 ± 11μm) compared to synergy (λ_eff = 1230 ± 41μm). This suggests that redundant information is more prevalent and more spatially extended in the visual cortex. These findings are consistent with previous studies that have reported that sensory regions of the brain exhibit high levels of redundancy, which may serve to enhance the robustness and reliability of sensory processing (Luppi et al., 2022). The normalized Z-score plots on the right highlight the distinct spatial profiles of synergy and redundancy in the two contexts (see Figure 5A, B, right). While the spatial decay of redundancy is similar in both spontaneous and visually stimulated conditions, synergy exhibits a slower decay during visual stimulation compared to spontaneous activity. This indicates the enhanced role of synergistic interactions in supporting long-range correlations during visual processing.

The fits also allow us to obtain effective information lengths for synergy and redundancy across all datasets, as shown in Figure 5C. We observe that while the redundancy information lengths do not significantly change (Mean difference = 8.33μm, p = 0.150 and Hedge's g = 0.61) between spontaneous and stimulated datasets, the synergy information lengths are significantly larger (Mean difference = 56.67μm, p = 0.023 and Hedge's g = 1.08) during visual stimulation compared to spontaneous activity. This suggests that visual stimulation enhances the spatial extent of synergistic interactions in the visual cortex, which may facilitate more efficient information processing and integration across different regions, whereas stimulation has little effect on the spatial extent of redundant interactions. Overall, these results highlight the unique role of synergistic interactions in mediating the increase of the spatial extent of long-range correlations, a signature of criticality.

To further investigate the interplay between synergistic and redundant interactions in the visual cortex, we apply partial network decomposition to combined networks, where neurons are nodes and the unweighted edges of the sparsified redundancy and synergy layers are constructed from the corresponding normalized components.

We then compute the complementary, shared, and unique paths that contribute to the propagation of information in the combined network representing the neurons of the visual cortex and their different interactions.

The proportion of complementary, shared, and unique paths as a function of path length is shown for both the spontaneous and stimulated conditions in Figure 6. We observe that the proportion of complementary paths increases with path length, indicating that synergistic and redundant interactions work together to facilitate efficient information processing across longer distances. The proportion of unique paths decreases with path length, and there are very few shared paths across all path lengths. These findings suggest that synergistic and redundant interactions complement each other over longer paths while maintaining a unique presence over shorter paths.

During stimulation, we observe a significant increase in the proportion of long (path length ≥4) complementary paths (Mean difference = 0.105, p = 0.0008) and a decrease in the proportion of long unique paths in the redundancy layer (Mean difference = −0.09, p = 0.0005), compared to spontaneous activity. This suggests that enhanced cooperative interactions between synergy and redundancy networks are observed at the expense of unique paths in the redundancy network during visual stimulation. These results highlight the dynamic nature of information interactions in the brain and their modulation by sensory input.