Fractured columnar small-world functional network organization in volumes of L2/3 of mouse auditory cortex

Abstract

The sensory cortices of the brain exhibit large-scale functional topographic organization, such as the tonotopic organization of the primary auditory cortex (A1) according to sound frequency. However, at the level of individual neurons, layer 2/3 (L2/3) A1 appears functionally heterogeneous. To identify if there exists a higher-order functional organization of meso-scale neuronal networks within L2/3 that bridges order and disorder, we used in vivo two-photon calcium imaging of pyramidal neurons to identify networks in three-dimensional volumes of L2/3 A1 in awake mice. Using tonal stimuli, we found diverse receptive fields with measurable colocalization of similarly tuned neurons across depth but less so across L2/3 sublayers. These results indicate a fractured microcolumnar organization with a column radius of ∼50 µm, with a more random organization of the receptive field over larger radii. We further characterized the functional networks formed within L2/3 by analyzing the spatial distribution of signal correlations (SCs). Networks show evidence of Rentian scaling in physical space, suggesting effective spatial embedding of subnetworks. Indeed, functional networks have characteristics of small-world topology, implying that there are clusters of functionally similar neurons with sparse connections between differently tuned neurons. These results indicate that underlying the regularity of the tonotopic map on large scales in L2/3 is significant tuning diversity arranged in a hybrid organization with microcolumnar structures and efficient network topologies.

Significance Statement.

While there is a wealth of information about the response of single neurons to sensory stimuli, it is unclear how large populations of neurons encode information. We investigate functional connectivity and organization of large populations of neurons (∼1,000) in L2/3 of the auditory cortex. We find that neurons in L2/3 of the auditory cortex encode stimuli in functional networks with microcolumnar structure and small-world topology. We quantify how efficiently these complex networks are embedded in physical space utilizing several novel analytical techniques that highlight higher order (i.e. more than pairwise) cooperativity in the network. These techniques are scalable to larger datasets; thus, we lay the analytical groundwork for future acquisition methods that enlarge the imaging volumes accessible with single-cell resolution.

Introduction

On a broad scale, neurons in sensory cortices are organized in functional maps in which neurons with similar stimulus selectivity are located close to each other. This organization is largely inherited from the peripheral receptors, e.g. the cochlea or retina, and in the auditory cortex forms a large-scale map of frequency preference, or tonotopy. Thus, tonal sound stimuli yield activity in topographically localized regions. In the primary auditory cortex (A1) a large-scale tonotopic organization is present along a cardinal, mostly rostro-caudal, direction of A1 (1–3). However, while this smooth gradient exists on the coarse scale, neighboring A1 neurons in the superficial layers (L2/3) can show quite different frequency preferences, while neighboring neurons in the thalamocortical recipient layers (L4) are more similar to each other (1, 4–10). Behaviorally relevant sound stimuli are encoded by small populations of cells in L2/3 (11–18) suggesting that even though L2/3 seems functionally heterogeneous, it might contain embedded networks.

Functional networks can be constructed from activity relationships such as signal correlations (SCs), which quantify how neurons respond to stimuli similarly. Calculating the spatial pattern of pairwise correlations shows that neighboring neurons show high correlations (5, 6, 19, 20) consistent with a high degree of local connectivity (21, 22), and Granger–Causal relationships in small networks (11, 23). The all-to-all pairwise relationships are far too dense to map out the most meaningful cell–cell couplings, so constraints such as thresholds are imposed to infer the topology of functional networks (24–27). Moreover, a large-scale orderly spatial functional organization is a salient feature of sensory cortices, and wiring constraints might influence the network topology. For example, systems that exhibit Rentian scaling are cost-efficiently embedded in physical space (28–30), meaning that functionally similar network nodes are mostly colocalized. Beyond the physical embedding of the network is the topological organization of the network itself. A common observation of many functional networks is the feature of small-world network organization, characterized by a high clustering coefficient and low mean path length (31, 32). Here, we investigated the 3D functional network topology of A1 L2/3 using chronic in vivo two-photon imaging in awake mice.

We presented tonal stimuli while imaging ∼1,000 neurons simultaneously to extract and analyze the topology of the functional networks within a cortical volume (∼370 × 370 × 100 µm). We found that volumes of L2/3 of the auditory cortex have a large diversity of tuning in relatively small volumes, yet still showed coarse spatial organization of similarly tuned neurons. Neurons were organized into microcolumns where neuronal tuning was similar with respect to depth but became dissimilar with respect to lateral distance away from the core of the columns. We constructed functional networks and found that functionally linked neurons were clustered in space, exhibited Rentian scaling, and that the functional network topology showed characteristics of small-world network organization. These two lines of evidence suggest that L2/3 auditory cortex neuronal networks fit characteristics of a cost-efficient organization, where functionally similar neurons are embedded in microcolumns while local tuning diversity is maintained, possibly to facilitate rapid plasticity and integration of information across the auditory cortex.

Results

We investigated the activity of large populations of neurons in awake mice by imaging cortical volumes ∼370 × 370 × 100 μm utilizing a Physik Instruments P-725 PIFOC Objective Scanner (Fig. 1A, left). Each dataset contained five imaging planes spaced 20 μm apart, each imaged at ∼4.3 Hz (Fig. 1A, right). We imaged seven mice ages P126–189 (159 ± 23 mean ± std) at the time of imaging to collect nine volumetric fields of view (FOV), each containing between 449 and 1,432 responding neurons (837 ± 369 mean ± std) for 7,532 total responding neurons recorded.

Fig. 1.

Imaging volumes of auditory cortex neurons to determine functional properties. A) Volumes were acquired via the use of an objective mover (Physik Instrumente P-725.4CD PIFOC®) during 2p imaging of neurons expressing GCaMP6s. Data was collected from five distinct z-planes where neurons were manually identified. B) Example tone-evoked responses in one neuron. C) Left: Tone-evoked responses for all stimuli in the same example neuron. Individual trials are shown as thin gray lines, and the average time course is overlaid as the thick black line. Right: the tuning curve produced by averaging the tone-evoked response across trials for each stimulus. D) Example imaged volume with neurons colored according to BF (N = 1,432 total responding neurons). The histogram of best frequencies for this field of view is shown on the right. E) IQR of best frequencies in each field of view (median IQR = 2.5 octaves, N = 9).

Volumes of L2/3 auditory cortex are diversely tuned with localized tuning organization

To assess the functional properties of volumes of L2/3 auditory cortex, we presented a range of tones and evaluated the tone-evoked responses of each neuron (Fig. 1B). The average response magnitude across all trial repetitions was used to produce a tuning curve for each neuron and determine the frequency that produced the maximal response (best frequency, BF) (Fig. 1C). Some neurons did not have a clear preference to a particular frequency and were deemed untuned. The BF was heterogeneous within each FOV (Fig. 1D). To quantitatively assess the extent of tuning diversity, we computed the interquartile range (IQR) of the BF of all tuned neurons in each FOV (Fig. 1E). We found an average interquartile range of best frequency (IQR_BF) of 2 ± 1 octaves (mean ± std) indicating diversely tuned populations of neurons as expected for L2/3 of A1 (4–6).

We next assessed the spatial organization of tuned neurons by our established method (6, 19). In short, the tuning diversity is computed in local neighborhoods around each neuron, and then the size of that local neighborhood is increased to assess the spatial extent of the tuning diversity. We adapted this method to three dimensions by increasing the size of a column rather than a 2D circle (Fig. 2A). We began by increasing the radius of a column with height fixed at the size of the volumetric FOV (100 μm). We found that tuning diversity increased as pairwise radial distance increased (Fig. 2B and C; left. All datasets shown in Fig. S3A). We next fixed the radius of the column at 50 μm and parametrically increased the height (Fig. 2B and C; right). We also assessed this height dependency in columns with fixed radii of 25, 75, and 100 μm (Fig. S3B). We detected no change in tuning diversity with respect to the depth of the imaged volume. In conjunction, these results indicate that similarly tuned neurons in L2/3 of the auditory cortex tend to organize in vertical columns of relatively small radii (<100 μm), and tuning diversity rapidly increases in the dimension orthogonal to cortical depth.

Fig. 2.

Tuning heterogeneity increases with radial pairwise distance. A) Localization of tuning similarity is assessed for one neuron by a vertical column of varying radius. This process is repeated for every neuron. B) Local tuning diversity for one example FOV (plots for all nine datasets are shown in Fig. S3). Left: IQR_BF as a function of column radius, averaged across all neurons. Shaded gray error bars represent 95% confidence intervals. The horizontal dashed line represents the IQR_BF of the entire FOV. Vertical dashed lines indicate radii highlighted in panel C. Right: IQR_BF as a function of column height, averaged across all neurons. C) Local tuning diversity in each FOV. Left: IQR_BF of three column widths for each imaging dataset. The mean and standard deviation of IQR_BF across all neurons are shown for each column width. Right: Conventions as in the left panel but for variable column height at a fixed radius of 50 µm.

Tuned neurons have high-order, nonrandom spatial organization

We have shown that there is some spatial organization of functional properties in L2/3 of the auditory cortex with second-order (pairwise) measures. However, higher-order metrics such as topological data analysis (33) are needed to capture cooperation involving larger groups of neurons. Minkowski functionals (MFs) (34–36) provide a characterization of a point pattern that captures high-order correlations among all points simultaneously rather than just pairwise distances. MFs are computed by centering a circle on each point, functionally expanding the radius of that circle, and then calculating measures (such as area) on the topological patterns as a function of radius (Fig. 3A). This method creates a “fingerprint” for each point pattern defined by neuronal locations. From the two-dimensional patterns, circumference, area, and connectivity (the Euler number) are calculated as a function of radius (see Methods section for details). We first investigated if tuned neurons are organized in a spatial pattern that is distinct from the spatial arrangement of randomly selected neurons in the same FOV (Fig. 3A; bottom left). The differential MF is produced by subtracting from the MF of tuned neuron locations the average MF across 10 random MFs (Fig. 3A; bottom right).

Fig. 3.

MFs suggest higher-order spatial organization. A) Using MFs to analyze spatial locations by expanding a circle centered on each point. Three example radii are shown to illustrate increasing overlap with increasing radius. Bottom Left: The fractional Minkowski functional of area for the example point pattern (black line). Thin gray lines represent the MF from each random surrogate point pattern. Bottom right: The differential Minkowski functional produced by subtracting the fractional MF from the average of all randomizations. B) The differential MFs for flattened volumes of each dataset. Each color represents a different FOV. The MFs for area, perimeter, and Euler’s characteristics are shown left to right. C) Differential MFs from each individual z-plane in all nine datasets where color indicates z-plane depth. Portions of these lines that are solid represent interaction distances with a significant difference from random surrogates.

Given that the fractional MF of any measure decreases from one toward zero as a function of radius, the rate at which it decreases indicates the extent of spatial clustering or dispersion of the point pattern. Therefore, by comparing a MF to its randomly sampled MFs we can draw conclusions about spatial clustering based on whether it decreases faster or slower than random surrogates. Therefore, in the differential MF, negative values indicate spatial clustering with respect to the rest of the field of view, positive values indicate spatial dispersion with respect to the field of view, and equivalency to zero (within confidence intervals of random samples) indicates spatial randomness within the field of view.

We used MFs to assess whether the spatial organization of tuned neurons exhibits spatial randomness or if there is an underlying structure to the observed salt-and-pepper organization of the tuned neurons. We started by collapsing the z-dimension in each volume onto a single plane to assess columnar organization and radial deviations from random point patterns drawn from tuned neurons. We found that each FOV had small-scale (short distance) deviations from random in the fractional MFs (Fig. 3B). There was no generalizable clustering or dispersion within these small-scale deviations from random, as some datasets were more clustered than random while others were more dispersed. However, within a single dataset, all three functionals (area, perimeter, and Euler characteristic) followed the same trend of clustering or dispersion. These results show that the arrangement of tuned neurons has high-order spatial order at small distances (Fig. 3B).

Collapsing the z-dimension gives a description of spatial topologies orthogonal to cortical depth but ignores the physiological limit of neuronal proximity. To further assess the local columnar organization, we next looked at each z-plane separately, given that individual z-planes retain the physiological limitation of how closely neurons can be embedded in the x–y dimension. When considering each z-plane separately, we again find higher-order organization of tuned neurons at short interaction lengths typically approaching random organization on average at radii of ∼50 µm, but variability was high (Fig. 3C). There was not a generalized trend of clustering or dispersion with respect to depth, as each z-plane seemed to have different deviations from random in most datasets. However, the small-scale deviations from random were evident in all datasets. These results are an independent validation of our finding from Fig. 2 that despite the seemingly random nature of salt-and-pepper tuning within L2/3 A1, there exists a nonrandom organization of tuned neurons on a small scale.

Functional networks show colocalization of similar receptive fields

We showed that tuned neurons have nonrandom spatial organization and that the BF of neurons exhibits columnar organization in volumes of L2/3 of the auditory cortex. However, the BF metric reduces receptive fields to a single value and requires the distinction between tuned and untuned neurons. This distinction might obscure salient organizational properties. We thus sought to investigate if neurons were functionally organized when all response properties were considered. Signal correlations (SCs) measure how similarly two neurons respond to stimuli, providing insight into functional connectivity (Fig. 4A). SCs of the FOV have a wide distribution with a slight bias toward positive values (Fig. 4B), indicating functional similarity among the populations of neurons.

Fig. 4.

Functional network construction and spatial relationships. A) SCs between neurons were assessed based on the similarity of their tuning curves. Tuning curves from an example high correlation neuronal pair and low correlation neuronal pair are shown on the left and right, respectively. Example correlations from 20 neuronal pairs are shown as a portion of the whole correlation matrix from one FOV. B) Signal correlation cumulative distribution functions from all FOVs. Dashed lines indicate the distribution mean from each FOV. C) Cartoon illustrating functional network construction. The full network on the left can be pruned according to a threshold value or according to an MST algorithm. D) Probability distributions of distances between neurons that hold connections in the functional network. Left: Distribution of the distances to the nearest nodes that share a connection in the functional network. Right: Distribution of average distances to all connected neighbors for each node. Dashed lines indicate the probability distribution from all pairwise distances in the FOV.

Utilizing functional relationships, we can identify the underlying network architecture of the population and its physical embedding in the cortical volume. The all-to-all pairwise relationships are far too dense to map out any meaningful topology, so constraints such as thresholds must be imposed to identify the most salient functional relationships (24–27). However, different thresholds can lead to vastly different underlying topologies (37), and therefore, such networks need to be assessed at a range of thresholds (38).

Another method for pruning the network to the most salient and efficient connections is through a minimum spanning tree (MST) algorithm. The benefits of utilizing an MST algorithm instead of a thresholded network approach are that no arbitrary threshold needs to be assigned and that the MST network includes every neuron within the population, even if it is not strongly correlated to any other neuron. MSTs have been used to describe network topology in transportation and communication networks (39, 40) and the spatial structure of brain responses using electroencephalogram (EEG), functional magnetic resonance imaging (fMRI), and magnetoencephalography (MEG) data (25, 41, 42). Changes in MSTs have been used to describe developmental changes in network topology (43) and in children with autism spectrum disorder (43, 44). The limitation of MSTs is that they create a sparsely connected acyclic network, meaning they lack clustering and potentially omit several strongly correlated links. In light of this, we utilize both thresholded functional networks and MST functional networks.

In thresholded functional networks, the threshold value must be chosen such that the graph is not too sparse but not too redundantly connected (Fig. 4C). The sparsity of a network is quantified by the average degree across all neurons in the network, where a neuron's degree is its number of functional connections. Utilizing this metric, we thresholded graphs such that the average degree was >2 but less than the square of the number of neurons, which has been previously reported as a region of intermediate sparsity (45). We report metrics as an average over the several graphs in this degree range unless stated otherwise. Functional networks were also constructed by the MST algorithm (Fig. 4C; right).

To quantify the spatial layout of these functional networks, we assessed the distribution of the network in cortical space. For each neuron in the functional network, we determined which other neuron in the network is its nearest physical neighbor. We then compared the distribution of these nearest-neighbor distances in the functional networks to the distribution of all pairs of neurons (Fig. 4D; left). Since network sparsity can influence this measure, we split the thresholded networks into three groups of increasing sparsity based on average degree. We found that for the most densely connected networks (low threshold), most nodes are connected to a neighbor that is in very close proximity (<100 µm). As the threshold is increased and networks become more sparse, the nearest-neighbor distance increases but is still strongly biased toward small distances. MST networks also maintain a bias toward close nearest neighbors but have a long-tailed distribution toward long distances. Given that the MST network connects all possible nodes at least once, this suggests that there are several neurons with weak correlations to the rest of the population that are dispersed in cortical space.

This highlights a key difference between the thresholded network analysis and MST analysis, where MST networks of N neurons are sparsely connected with only N + 1 connections and include every neuron, even if they only possess weak correlations. As expected, the MST nearest-neighbor distribution looks most similar to the high threshold distribution because a higher threshold causes increased connection sparsity, making it less likely for all functionally connected neurons to be in very close proximity (i.e. increased sparsity leads to increased average path length). Next, we compute a similar measure by averaging the physical distance of all connections for each node (Fig. 4D; right). We find that all distributions shift rightward toward larger distances compared to the closest neighbor distribution, with a peak at 125–150 µm reflecting the average distance of a node's neighbors in the functional network. The MST network maintains the strongest bias of small-distance connections. This is likely due to the high sparsity of the MST network, meaning that most salient functional connections are colocalized in close proximity in cortical space. The average neighbor distance distributions closely reflecting the shape of all pairwise distances indicate that our functional networks are not selectively subsampling from neurons that are in close proximity but rather include populations spanning the whole FOV.

Functional networks exhibit properties of Rentian scaling

Since spatial functional organization is a salient feature of sensory cortices, it may be informative to assess how functional networks are embedded in physical space. By combining the topological organization of the functional network with the physical location of connections in the cortex, we can assess if there are spatial clusters of functionally similar neurons and the efficiency of how they are embedded. As alluded to earlier, the limited physical space available for the brain motivates the idea that neurons are functionally organized in space to optimize wiring costs but still maintain topological complexity and integration of information (46). Rentian scaling is a topo-physical property that uses the topological and physical organization of networks to assess whether the network is cost-efficiently embedded in physical space. A network that exhibits scale-invariant topo-physical scaling is said to exhibit Rentian scaling and has been shown to exist in several functional neuronal networks (28–30). Systems that exhibit Rentian scaling are said to be cost-efficiently embedded in physical space, meaning that functionally similar neurons are mostly colocalized.

We assessed whether our functional networks exhibited Rentian scaling by using the 3D Rentian scaling function from the Brain Connectivity Toolbox (38). The network is partitioned into random cubes many times, then for each cube, the number of nodes contained (N) is counted, and the number of connections crossing the cube barrier (E) is counted (Fig. 5A). If the scaling relationship between the number of edges and the number of nodes from all boxes is well-described by a power law, it indicates Rentian scaling of the network. To quantitatively assess Rentian scaling, these values are plotted in log–log space where the linear relationship is assessed to determine power law fit (Fig. 5B). We found that thresholded functional networks had Rent’s exponents of 0.98 ± 0.05 (Fig. 5C; top left) and were better fit by a power law than an exponential (Fig. 5C; bottom left) suggesting that these networks exhibit Rentian scaling. Similarly, the MST functional networks had power law fit exponents of 0.98 ± 0.02 (Fig. 5C; top right) and were much better fit by a power law than exponential (Fig. 5C; bottom right). These results indicate that the physical location and density of connections between functionally similar neurons exhibit Rentian scaling. In order to assess the cost-efficiency of the network embedding, we next use a rewiring randomization on each network where each node's connections are randomized many times while the overall degree distribution is maintained. We found that for most networks, Rentian exponents increased in the rewired network, indicating a degradation in cost-efficiency of the physical embedding (Fig. 5D). However, several networks had reduced Rentian exponents, indicating that the random rewiring improved the efficiency of the embedded network. This may indicate that the functional networks we have constructed from similarities in tonal responses represent just one of many subnetworks within the cortex with varying levels of cost-efficient embedding.

Fig. 5.

Functional networks exhibit Rentian scaling. A) Rentian scaling is assessed by embedding the functional network in physical space, randomly placing many boxes, and then counting the edges through the box and nodes contained within it. One example box is shown for an FOV. B) Example plot to assess the scaling relationship between the number of edges (E) and number of nodes (N) from each randomly placed box. C) Best-fit parameters assessing Rentian scaling in each functional network. Top: Power law fit exponents for thresholded networks and MST networks in each FOV. Error bars represent standard deviation across different correlation thresholds in thresholded networks. Bottom: R² values from each network to assess power law fit against exponential fit. Color corresponds to the dataset in the top graph, where multiple points represent different correlation thresholds. D) Change in best-fit exponents when each connection in the functional network is randomly rewired for thresholded networks (left) and MST networks (right). Error bars represent standard deviation across different correlation thresholds in thresholded networks.

Correlation networks exhibit weak small-world propensity

Beyond the physical embedding of the network is the topological organization of the network itself. A common observation of many functional networks is the feature of small-world network topology. Small-world network organization is characterized by a high clustering coefficient and low mean path length (31, 32). Small-world networks represent a middle ground between perfectly ordered and completely random networks and are indicative of a network with optimal information transmission (47).

We aimed to characterize the topology of the functional networks by assessing if they exhibit small-world organization. If the functional organization of L2/3 organization was completely random, there would be no evident tonotopic organization, and functionally similar neurons would be randomly dispersed. On the other extreme, a completely regular (lattice) network would represent a population of neurons with perfect tonotopy and, therefore, little to no tuning diversity, where each receptive field is sharply tuned to one specific stimulus. In both extremes, the wiring cost of the network is inefficient because either there are many redundant connections or there is a lack of integration of information. Small-world networks represent a cost-efficient middle ground between these two extremes (Fig. 6A). The small-world propensity can be computed by comparing the clustering coefficient and mean path length of the real data to graphs generated from a Watts–Strogatz model (47, 48). To assess small-world propensity, we compared the correlation graph to a regular lattice graph characterized by a high clustering coefficient and a random graph characterized by low mean path length. We found that our functional networks had relatively low clustering coefficients (Fig. 6B, left) and midrange mean path lengths (Fig. 6B, right). Utilizing the clustering coefficient and mean path length measurements from the functional networks and regular and random Watts–Strogatz surrogates, we can next calculate the small-world propensity (47). We find that the thresholded functional networks have a small-world propensity between 0.4 and 0.5 (N = 9) (Fig. 6C), which indicates a weak small-world topological organization (49). Given that MST graphs lack clustering by definition since they are trees, we did not perform small-world analysis on the MST functional networks.

Fig. 6.

Small-world network topology. A) Schematic showing that the connectivity of small-world networks exists in a middle ground between regular networks and completely random networks. B) Left: Clustering coefficient of functional networks in (from left to right) experimental datasets, regular surrogates, and random surrogates. Error bars represent standard deviation across network thresholds. Right: Mean path length of functional networks and corresponding regular and random surrogates. C) Distribution of small-world propensity values computed at each network threshold. Each line is a separate dataset.

The evidence of small-world organization in thresholded functional networks indicates that there are clusters of functionally similar neurons in conjunction with important long-range connections to provide short path lengths between all neurons. This may imply that the tonotopic gradient in L2/3 of the auditory cortex could be designed such that there is organization according to frequency (represented by highly clustered groups of similarly tuned neurons) for low-level stimuli, yet also wired according to small-world topology to encode more complex stimuli that recruit many tonotopic regions together (via tonotopic “outliers” scattered throughout the gradient). Our observation of relatively weak small-world topology complements other findings of order on small scales: the characterization of collectively active neuronal groups based on their local neighborhood tuning similarity (Fig. 2C), short nearest-neighbor distances within functional networks (Fig. 4D), and higher-order organization, especially on short length scales as determined with MFs (Fig. 3).

Noise correlation networks mirror networks based on SCs

We further analyzed the functional organization of these volumes by computing noise correlations (NCs) and carrying out the same analyses as for SCs. NCs measure the trial-to-trial covariability of neuronal pairs as a proxy of direct anatomical connections between neurons or shared sources of activity perturbations. Using these pairwise NCs, we constructed functional networks to assess topology and spatial organization. Pairwise NCs were overall lower in magnitude than SCs (Fig. S4A). Nearest neighbors in the NC functional networks were much closer in proximity when compared to SC functional networks, while the overall average neighbor distance was similar (Fig. S4B). This indicates that neurons with the most similar trial-to-trial response fluctuations are highly colocalized, suggesting a direct anatomical connection indicative of a columnar structure. While NC functional networks exhibited Rentian exponents that were best described by a power law compared to an exponential (Fig. S4C), there was no clear consensus on the change in these exponents when networks were randomly rewired (Fig. S4D). Furthermore, we found NC networks had network topology closer to random than SC networks, as measured by clustering coefficient, mean path length, and small-world propensity (Fig. S4E and F). This network topology closer to random in NC networks suggests that trial-to-trial response fluctuations do not organize in a well-defined network topology and that “spontaneous activity” exists in a mostly random network structure, while network activations during sound processing cause a small-world network topology to emerge enabling efficient stimulus processing.

Discussion

By imaging thousands of neurons in 3D, we reveal functional organization in the local populations of neurons in L2/3 of the primary auditory cortex (A1), which are heterogeneously tuned and have population-wide response variability. We find that functionally similar neurons tend to organize into coarse microcolumnar arrangements with column radii of ∼50 µm. Minkowski functional analysis confirms this arrangement to be nonrandom on small scales. We further show that the well-known salt-and-pepper tuning characteristics of L2/3 neurons are arranged in a small-world topology with Rentian scaling. This indicates that functionally similar neurons are cost-efficiently embedded in interconnected, colocalized clusters with sparse functional connections between spatially distant auditory cortex neurons. While the columnar cortical organization has extensively been studied in the cortex (50, 51), this microcolumnar organization within the highly recurrent L2/3 is novel, especially as we find it on small scales of only 100 µm in depth in each imaged volume. Thus, while the surface of L2/3 seemed to be organized in a salt-and-pepper fashion, we show that there is an underlying structure. Our results complement findings based on histological observations of the microcolumnar organization of A1 (52–54). Indeed, imaging in rodents is consistent with the presence of microcolumns in mouse A1 on the scale of <100 µm (55). Similarly, imaging in the mouse visual cortex shows spatial clustering of similarly tuned neurons (56).

The layers of A1 transform incoming information from the thalamus (57, 58), with sensory inputs arriving in layer 4 (L4), which are then relayed to L2/3 via divergent and heterogeneous projections (22). Therefore, the functional network organization described here is not only due to feed-forward inputs from L4 but is also shaped by recurrent intralaminar connections within L2/3. The excitatory connections from L4 to L2/3 originate from relatively small areas (22, 59), yet there are abundant intralaminar connections within L2/3 (22, 60, 61). Therefore, common input from L4 could dominate local nearby correlations while intralaminar L2/3 connections contribute to the fractured overall organization.

The functional networks we assess in this work mathematically connect neurons that have similar receptive fields. We cannot assert that any functional connections represent anatomical connections between neurons but rather that neurons have similar inputs. In the case of L2/3 neurons, these are inputs from thalamorecipient L4. Given that our sound stimuli did not span the entirety of all possible stimuli, our assessment of Rentian scaling and cost-efficiency of how networks are embedded is specific to the utilized stimulus set and how it is encoded in this subpopulation of A1. It remains to be seen if other stimuli activate populations of neurons that are more cost-efficiently embedded in the cortex, or it could be that the cost-efficiency we observe is the best that the cortex can achieve given limited physical space to encode stimuli of unending potential complexity.

We showed that L2/3 functional networks exhibited Rentian scaling indicative of efficient physical embedding; however, random rewiring of the network improved the efficiency in some cases. We assessed networks that represent how this stimulus set is encoded in a subpopulation of neurons. These may represent one set of subnetworks from countless subnetworks embedded in this cortical space to encode the vast array of complex stimuli.

Our finding that the networks approach randomness on the length scales imaged poses important questions that the next generation of large field-of-view single neuron resolution imaging systems will be able to address: We know from widefield imaging and electrophysiological mapping showing tonotopic maps that a functional network drawn from all A1 neurons would show re-emergence of order on larger length scales. Thus, our data indicate the existence of a “valley of randomness” at intermediate-length scales surrounded by distinct small-scale and large-scale organizations. We propose that this intermixing on intermediate length scales allows the A1 to retain small-world characteristics for the A1 as a whole, with sparse connections between frequency bands that retain Rentian scaling and small-world characteristics in order to support optimal encoding of sound stimuli.

Methods

Experimental methods

All procedures were approved by the University of Maryland Institutional Animal Care and Use Committee. Mice: We used 7 mice ages P126–189 (159 ± 23 mean ± std) at the time of imaging. Thy1-GCaMP6s × CBA F1 mice were used for 2P calcium imaging of cortical neurons. We crossed Thy1-GCaMP6s (JAX: 024275) mice with CBA/CaJ mice (JAX: 000654), which are known for their exceptional hearing (62), to create Thy1-GCaMP6s × CBA F1 mice that retain good hearing as adults (19, 63).

Surgery: Mice were prepared as described previously (11, 19, 64). In brief, two hours before surgery in order to prevent infection and cortical edema, mice were injected with dexamethasone (5 mg/kg). For surgery, mice were anesthetized with isoflurane (3–4% for induction and 1.5–2% for maintenance) and injected with dexamethasone and atropine (0.1 mg/kg) at the beginning of the surgery. The body temperature was maintained at 38 °C. The hair on the scalp was removed via plucking and a hair removal agent (Nair). After disinfecting the with alternating swabs of betadine and 70% ethanol, the skin overlying the skull and temporal muscle was removed, and the temporal bone was exposed by resecting the temporal muscle. The headpost was attached to the skull with a combination of cyanoacrylate (Vetbond) and dental acrylic (C&B Metabond). We removed a ∼3 mm circular section of the skull above A1 and implanted a cranial window. The cranial window consisted of the coverslips: two 3 mm circular glass coverslips and one 5 mm circular coverslip. The edges of the window were filled with a clear silicone elastomer (Kwik-Sil) and the window was then affixed in place with dental acrylic. We coated the dental acrylic and headpost in iron oxide to prevent optical reflections. Postoperatively, mice received injections of meloxicam (0.5 mg/kg) and were allowed to recover for at least a week before imaging.

In vivo two-photon imaging: Imaging was performed in animals as described previously (11, 19, 64). After recovery from surgery mice were acclimatized to the microscope. We then commenced imaging. We used widefield imaging to determine the A1 location by its characteristic rostro-caudal tonotopic axis (64). All two-photon imaging was performed on a microscope (Bergamo II series, B248, Thorlabs) using a pulsed femtosecond Ti:Sapphire 2P laser (Vision S, Coherent, tuned to 940 nm) using ThorImage and ThorSync software. We imaged a ∼370 µm by 370 µm field of view at a 30 Hz frame rate. Volume scanning was achieved using a Physik Instrument P-725.4CD PIFOC Objective mover. We scanned a volume with 100 µm depth with six z-planes spaced 20 µm apart and one flyback z-plane, resulting in a sampling rate of ∼4.3 Hz for each z-plane. Due to slow flyback effects, we only analyzed data from five z-planes to avoid double-counting cells. Figure S1A shows the frame timing with respect to the piezo location and each z-plane, whereas Fig. S1B shows how frame triggers intersect with trial stimulus triggers.

Sound stimuli: Sounds were presented as described previously (11, 19, 64). We used a free-field electrostatic speaker (ES1, TDT Inc) driven by an ED1 speaker driver. The speaker was calibrated by recording a 70 dB 4–64 kHz white noise with a calibrated microphone to find the speakers' transfer function. We then used the inverse of the transfer function to equalize the output of the speaker, resulting in a flat Freq/dB curve. We tested this calibration using pure tones at 70 dB and ensured that the presented sound level was <5 dB from the target. During 2P imaging sessions, we presented 1-s sinusoidally amplitude-modulated tones (3 to 48 kHz, 0.5 octave spacing, and 5 Hz full-depth modulation) at 70 dB. Each unique stimulus (nine frequencies) was presented 20 times for a total of 180 trials. In addition to the tone, each trial contained 1 s prestimulus silence. Trials were separated by a variable 6–10 s intertrial interval. On average, the tuning (BF) of neurons in the imaging FOVs was in the 8–16 kHz band.

Data analysis

Two-photon image analysis

Two-photon image sequences were processed using methods previously described (19, 65). Image sequences were corrected for x–y drifts and movement artifacts using discrete Fourier transform registration (66) implemented in MATLAB (Mathworks). Neurons were identified manually from an average image across all motion-corrected images. Ring-like regions of interest (ROI) boundaries were programmatically drawn based on the method described by Chen et al. (67). Overlapping ROI pixels (due to closely juxtaposed neurons) were excluded from the analysis. For each selected neuron, a raw fluorescence signal over time (F_soma) was extracted by averaging across pixels from the ROI overlying the soma. Neuropil (NP) correction was performed on the raw fluorescence of all soma ROIs (F_soma) (68). In short, the NP ROI was drawn based on the outer boundary of the soma ROI and extended from one pixel beyond the soma ROI outer boundary to 15 μm, excluding any pixels assigned to neighboring somata. The resulting fluorescence intensity (F) used for analysis was calculated as F = F_soma − (α×F_NP), where we use a default value of α = 0.7 to reduce fluorescence contamination from the NP (68). The NP-corrected fluorescence for each neuron was then converted to a relative fluorescence amplitude (ΔF/F₀), where ΔF = (F − F₀). F₀ was estimated using a sliding window approach that calculated the average fluorescence of points less than the 50th percentile in a 10-s sliding window, as similarly reported in previous studies (69, 70).

We tested the sensitivity of NP subtraction by investigating how fluorescent traces and pairwise correlations were altered at different NP subtraction coefficients. First, to investigate how individual neuronal traces are altered, we computed the self-similarity of ΔF/F traces at each level of NP subtraction. NP subtraction was performed at 11 different values (α = 0 to α = 1 at 0.1 intervals), and the correlation between ΔF/F traces from the same neuron at each α value was computed. ΔF/F traces were minimally altered with respect to the percentage of NP subtracted except at large differences (i.e. differences of 70% or greater) (Fig. S2A). Within the range of 60–90% NP subtracted, there was nearly no difference in ΔF/F traces as indicated by neuronal self-correlations centered on unity (Fig. S2A). Next, we aimed to investigate how pairwise correlations, as measured by similarities in tuning curves, were altered with respect to NP percent subtraction. We found that NP subtraction decreases pairwise correlations as expected, though the effect was minimal (Fig. S2B). The difference in SCs on a single neuron basis from 70 to 0% NP subtracted was −0.07± 0.17 (mean ±std), indicating a very small average decrease even at a large difference in NP subtracted. The difference in SCs is significantly smaller within the range of 60–90% subtraction (Fig. S2B). These results indicate that the amount of NP signal subtracted has an expected effect of lowering pairwise correlations, though the effect is very minimal.

Neuronal receptive fields and tuning

Neuronal tuning and responsiveness were assessed using methods described previously (19). Neurons in which the ΔF/F₀ signal was significantly modulated by sound presentation (and deemed “responsive”) were defined by ANOVA (P < 0.01) across baseline (prestimulus) and all sound presentation periods. Neuron receptive fields (tuning curves) were determined as the average ΔF/F₀ response during the stimulus presentation for each frequency across all stimulus repetitions. BF for each neuron was determined as the stimulus frequency that elicited the highest average ΔF/F₀ response (peak of the tuning curve). To determine if a neuron was tuned or untuned, we computed a z-score from the tuning curve to find peaks that deviated significantly from the mean of the tuning curve. To isolate the most distinctly tuned neurons, tuned neurons were defined as the upper 30% quantile of the max z-score distribution from all responding neurons. The tuned/untuned distinction was only utilized for analyses of BF, whereas all correlation and functional networks made use of all responding neurons, regardless of how selectively tuned they were.

Minkowski functionals

MFs are computed by centering a circle on each cell center of interested (tuned cells) and then functionally expanding the radius of that circle (Fig. 3A). The radial expansion of the circles eventually leads to overlapping regions between circles and the formation of topological “objects.” The topological pattern is characterized by calculating measures (such as area) on the objects that emerge as a function of radius. MFs can describe patterns in n-dimensional space by using n + 1 measures. For example, in a two-dimensional pattern, the topology is characterized using the area measure A(r), perimeter measure P(r), and Euler characteristic E(r). The area measure and perimeter measure are the total area and total circumference of the resulting objects, excluding the multiplicity of overlapping regions. The Euler characteristic sometimes referred to as the connectivity measure, is defined as the number of objects minus the number of holes in the resulting pattern. Each measure is typically converted to its fractional form (a(r), p(r), and e(r)) where it is divided by the total measure, including the multiplicity of overlapping regions.

We compared the MF of each response pattern to 10 MFs calculated on random samples from the cell locations within the field of view using the same number of points as in the actual response pattern. This serves as a control for bias in the possible cell locations in the field of view as well as bias due to the number of points in the pattern of interest. A mean random MF and confidence intervals are generated from the random samples and subtracted from the actual MF to produce what we refer to as the differential Minkowski functional. We analyze MFs for each pattern as a function of radius from r = 0 to r = R, where R is half of the maximum distance between two neurons in a field of view. This allows every circle within a pattern to interact. We, therefore, describe the extent of clustering, dispersion, and randomness at a range of cortical distances from tens to hundreds of microns. Portions of the Minkowski functional are deemed significantly different from random if they are at least two standard deviations different than the random surrogates.

Pairwise correlations

Pairwise correlations were computed using methods as described previously (19). To assess the functional similarity of neurons, we computed SCs for each neuronal pair (Fig. 4A) by calculating the correlation coefficient of their tuning curves utilizing MATLAB's “corrcoef” function, which performs the following:

$$\rho (A,B)={\displaystyle \frac{1}{N-1}{\displaystyle \sum _{i=1}^N}\left({\displaystyle \frac{A_i-{\mu }_A}{{\sigma }_A}}\right)\left({\displaystyle \frac{B_i-{\mu }_B}{{\sigma }_B}}\right)}$$

where A and B represent the tuning curve of each neuron, each with N elements, where µ and σ represent the mean and standard deviation, respectively. NCs were obtained for each cell pair by finding the covariance of trial-to-trial response fluctuations about the mean. This was done by calculating a frequency response area (FRA) for each trial block and calculating its difference from the true FRA (mean across trials). The “corrcoef” function was used on pairs of trial block fluctuation FRAs. The correlation coefficient values from each trial block were averaged to arrive at one noise correlation value for each neuronal pair.

Functional networks construction and quantification

We construct functional networks from pairwise correlations in the neuronal population data. However, not every pairwise correlation within the rich neuronal population data is entirely meaningful. Therefore, when constructing functional networks, some connections need to be pruned to assess any meaningful topology within the network, which we accomplish in two separate applications via (i) thresholding or (ii) a topological constraint (MST). The problem with (i) is that a choice of correlation threshold can have a drastic effect on the observations. As a result, in this study, we chose to analyze all our thresholded correlation networks at a range of thresholds. A full range of thresholds from max correlation to minimum correlation would include many graphs with very few connections or far too many redundant connections. Therefore, we sampled many threshold values within a region of intermediate sparsity such that the average degree was >2 but less than the square of the number of neurons (45). We then reported each metric as a mean across all thresholds with associated error and found very little difference between these networks unless stated otherwise. For the MST method, the network is pruned such that the overall weight of the graph is minimized. In a response pattern of N nodes, the MST picks N–1 edges such that every node within the graph is connected while minimizing the total edge weight of the tree and without creating any loops within the graph. The edge weight is defined as one minus the value of the pairwise correlation. This allows the weight-minimizing algorithm to pick out the most interconnected neurons. In both methods, the degree of each neuron is the number of connections to it in the functional network. Thresholding correlations instead of MST allows for interesting topologies to appear (cyclical regions) and allows computations of clustering coefficients, whereas MST is an acyclic graph by definition. However, thresholding correlations will ignore any weak correlations completely and omit neurons, whereas MST benefits by including every neuron with at least one connection, even if it has a weak correlation.

Clustering coefficient values and Rentian scaling assessment were performed with MATLAB code from the Brain Connectivity Toolbox (38). Goodness-of-fit statistics for power law versus exponential (Fig. 5C) was assessed using MATLAB's “fit” function. Small-world propensity was calculated as previously described (47). Regular and random surrogate networks used for small-world propensity were calculated using the “WattsStrogatz” function in MATLAB to produce networks with the same number of nodes and average degree but with rewiring probabilities at both extremes.

The minimum spanning tree

To further quantify the topology of population responses, we apply methods from graph theory. With any given response pattern, we treat each of the neuronal locations as a node. Each node has a pairwise relationship called an edge, and each edge has a weight that can represent any quantitative relationship between the two nodes. In this work, we investigate functional connectivity utilizing SCs as the edge weight to infer similar feed-forward input to neurons. The edge weight is defined as one minus the value of the pairwise correlation. This allows the weight-minimizing algorithm to pick out the neurons with the most similar inputs. In a response pattern of N nodes, the MST picks N–1 edges such that every node within the graph is connected while minimizing the total edge weight of the tree and without creating any loops within the graph (Fig. 4B).

Supplementary Material

Supplementary material is available at PNAS Nexus online.

Funding

This was supported by National Institutes of Health U19 NS107464 (P.O.K. and W.L.) and National Institutes of Health R01DC017785 (P.O.K.).

Author Contributions

Z.B., W.L., and P.O.K. conceived the study. K.S.S. and Z.B. performed experiments. Z.B. analyzed data. Z.B., W.L., and P.O.K. wrote the manuscript.

Data Availability

All data will be available at the JHU Research Data Repository (https://archive.data.jhu.edu) upon publication.