Analysis of Graph Invariants in Functional Neocortical Circuitry Reveals Generalized Features Common to Three Areas of Sensory Cortex
Transmission and processing of information in the brain is in large part determined by the connectivity between neurons 1 ., The neocortical microcircuit hypothesis states that the neocortex is composed of repeated elements of a generalized circuit that are tweaked for specialization in each area 2 ., Supporting this hypothesis , local synaptic connectivity in the neocortex is non-random and is at least partly determined by neuron location and class 2–12 ., These rules imply that there is a probabilistic or partially stereotyped wiring diagram ., The extent to which these rules generalize across the neocortex , however , is unclear ., Analysis of neocortical microcircuit spiking activity in different brain regions has revealed common dynamical features 12–15 , suggesting that circuits may share similarities between regions ., In this study , we use the spatiotemporal correlations of firing activity between neurons to generate functional wiring diagrams 15–19 ., Modeling studies have shown a clear relationship between connectivity and neural firing 15 , 20–24 ., This suggests that we can gain insight into the underlying structure and organization of cortical circuitry by analyzing the emergent dynamics of large populations of neocortical neurons ., Here we employed high speed two-photon calcium imaging 25 to densely sample the spiking activity of up to 1126 neurons within a 1 . 1 mm diameter field of view , spanning multiple columns and layers in three different areas of the sensory neocortex ., We then applied post-processing algorithms to detect spatiotemporal relationships between spiking neurons and modeled this activity as wiring diagrams , or graphs 15 ., Graph theory is a useful technique to quantify network dynamics , and has been increasingly applied in the neural context to understand brain connectivity patterns 21 , 26 , 27 ., One potential approach to identify invariant features of functional wiring diagrams within and between areas of cortex is to isolate graph isomorphisms ., For example , the unlabeled graphs and are isomorphic when any two nodes and of are connected in if and only if that connection exists in ., However , such an analysis currently remains intractable in graphs of sizes analyzed here , as the best known algorithm runs in polynomial time 28 ., Perhaps more importantly , the organizational features of connectivity that have been described to date reflect probabilistic , rather than deterministic microcircuit architectures 5 , 8 , 9 , making it unlikely that connectivity patterns in the brain are formally isomorphic ., In order to test the postulate that the organization of functional circuitry generalizes across the neocortex , we instead applied functions that are invariant to labeling of the nodes of the graph ., In other words , if A is the adjacency matrix describing graph , we wanted to describe the function such that , where is the × permutation matrix 29 ., In the context of our study , we aimed to identify features of a neuronal circuit wiring diagram that are invariant to the particular identities of the neurons ., Thus , we characterized each neuron only by the connections it had with other neurons ., While neurons and activation patterns between animals and regions may vary in their individual details , these abstract , global characteristics of circuit structure stay constant , even following the relabeling of the neurons ., By investigating label-independent features , called graph invariants , we hoped to disregard features of the functional circuit that may be susceptible to over-fitting , and focus on features that are stable across slices and areas of the neocortex ., Many graph invariants have been previously described , such as maximum degree and MAXCUT value 29 ., Some particularly useful invariants include the graph eigenvalues and eigenvectors 29 , 30 ., We apply these analyses to functional wiring diagrams generated from imaging data from three sensory neocortical areas to test the validity of a functional analogue to a generalized circuit architecture of the neocortex ., All procedures were performed in accordance and approved by the Institutional Animal Care and Use Committee at the University of Chicago ., To foster reproducibility and fast development of future work based upon these results , we have published functional graph analysis tools under an open source , GPLv3 license , available here: https://github . com/ssgrn/GraphInvariantsNeocortex ., All statistical analyses were performed with MATLAB ( MathWorks ) ., Unless otherwise noted , data are presented as mean ± SD ., All values in the text are in reference to the Pearson correlation computed with the command corrcoef ., For nonparametric distribution comparison between the three sensory areas , the Kruskal-Wallis test ( KW-test ) was implemented via the kruskalwallis function ., The nonparametric Komolgorov-Smirnov test ( KS-test ) , noted at use , was used to compare fitted distributions to data ., The Komolgorov-Smirnov test were implemented using the command kstest2 ., For tests of significance , was used as the cutoff ., Algebraic connectivity and eigenvector centrality were computed using the MIT Toolbox for Network Analysis ( http://strategic . mit . edu/downloads . php ? page=matlab_networks ) ., Graph figures were generated using the open source Python graph visualization tool NetworkX ( http://networkx . github . io/ ) ., Circular variance was computed with the MATLAB Toolbox for Circular Statistics 32 ., We compared our data to two null models: random topologies and -nearest neighbors topologies ., Each random topology was formed by preserving the locations of neurons in a corresponding functional topology and then assigning a 0 . 5 probability of forming a directed edge between every neuron in the field of view ., Each -nearest neighbors topology was formed by preserving the locations of neurons in a corresponding functional topology and then forming a directed edge from neuron A to neuron B if neuron B was one of -nearest neighbors of neuron A . In all analyses , we used ., To determine whether A1 , S1 , and V1 functional circuit wiring diagrams exhibited invariant features , we monitored neuronal activity in 43 slices from each region of the mouse neocortex ( 11 of A1 , 21 of S1 , and 11 of V1 ) using high speed multi-photon calcium imaging 15 , 25 , 31 ., Spontaneous circuit activity requires intact excitatory amino acid transmission 15 , 33 , sufficient oxygenation 34 and corresponds to UP states within single neurons which comprise the functional circuit 15 , 35 ., Previous reports have found that spontaneous activity delineates all of the possible multi-neuronal patterns within a sampled population and that a sensory input activates only a subset of these patterns 14 , 36 ., By monitoring spontaneous activity in the imaged field of view , we hoped to maximize the number of pairwise correlations within the imaged populations ., We imaged the flow of activity through large populations of neurons ( A1: 595±101 cells , S1: 704±157 cells , V1: 734±129 cells ) at the mesoscale in a two-dimensional circular imaging plane with a diameter of 1 . 1 mm that comprised multiple layers and columns with single-cell resolution ( Figure 1A ) ., We confirmed activity was not biased to any one lamina and that our sampling was uniform across our field of view , since the amount of activity observed across all circuit events did not differ between layers ( , KW-test; see Methods for explanation of laminar identification ) ., Because temporal resolution of multi-photon microscopy is compromised at these spatial scales , we used the heuristically optimized path scan technique 25 ( Figure 1B ) , which allowed us to achieve fast frame rates ( frame duration 86±17 . 7 ms ) that did not differ between regions ( , KW-test ) ., We deconvolved calcium fluorescence changes of each detected neuron into spike trains ( Figure 1C ) 31 and generated rasters of spiking activity for the entire imaged population of neurons ( Figure 1D ) ., All regions of the sensory neocortex showed a common capacity for emergent , multi-neuronal patterned activity , characterized by discrete periods ( >500 ms ) of correlated action potential generation within subsets of neurons ., Circuit events were separated by periods of quiescence and we refer to these distinct , clustered epochs of spontaneous action potentials as individual circuit events ., The start and finish of a circuit event was easily resolvable because the field of view was either quiescent , corresponding to a DOWN state in a single neuron , or was active , corresponding to a UP state in a single neuron 35 ., One circuit event lasted 1203±456 ms in A1 , 1568±885 ms in S1 , and 1342±698 ms in V1 ., We imaged 82 total circuit events in A1 , 268 total events in S1 , and 104 total events in V1 ., Using this data , we generated graphical abstractions , or circuit topologies , corresponding to functional activity over all circuit events observed in a single field of view ., Neurons were represented as nodes in each graph ., Edges between nodes were directional and formed according to the following rule: neuron A was considered functionally connected to neuron B if neuron B fired in the subsequent frame ( Figure 1E ) ., These edges were then weighted according to how many times this single frame lagged correlation occurred , normalized to the number of events in that field of view ., Thus , stronger edge weights indicated reliable , correlated spiking , whereas weaker edge weights indicated unreliable , weakly correlated spiking ( Figure 1E ) ., The resultant graphs contained a large number of edges ( median: 3 . 4×104 functional connections , range: 4 . 2×105 functional connections ) ., Note that although a functional relationship between neurons increases the probability of them having a synaptic connection 18 , 33 , a linear relationship between each functional edge and a synaptic connection does not exist 16 ., Rather , given our method of inference , the functional connectivity measure captured the flow of activity through the network during a circuit event ., There is an ongoing debate on whether the cortical column , which is oriented perpendicular to pia , regulates and shapes the flow of information in sensory cortices 47 ., Coronal slices allowed us to image activity patterns with near simultaneity across all lamina ., Using this data , we assessed directional flow in functional graphs by computing the angle and distance between the source and destination of directed functional connections relative to the orientation of pia ., Flow maps are plots that capture direction of circuit flow with points scattered at a radius and angle about the origin ., represents the distance of the functional connection from the source to the sink , and represents the angle between the source and the sink ., We measured the amount of angular clustering of activity flow in sensory areas by computing the circular variance of functional connections ., The clustering of points at a particular angle indicates stereotypy of functional flow across events in a neighborhood of the functional topology ., We calculated the amount of angular clustering by computing the circular variance of the set of points ., Circular variance is defined as:The value of the circular variance varies from 0 to 1; the lower the value , the tighter the clustering of points about a single mean angle ., In functional circuit topologies from all three areas of the sensory neocortex , flow covered the entire angular space , regardless of the pairwise distance , or radius , spanned by the functional connection ( Figure 5A ) ., We found that the spread of circular variance increased for functional connections which spanned the largest distances , most likely due boundaries imposed by pia , internal capsule , or field of view ( Figure 5B ) ., Thus we did not find a canonical circuit flow in spontaneous cortical activity regardless of sensory area ., The highly distributed nature of functional topologies suggested that large fields of view are necessary to fully capture invariant features of functional topology ., We sought to confirm this hypothesis by examining the spatial dependency of connectedness in functional topologies ., Connectedness in the context of an imaged field of view can be described as an aperture problem: large interlinked networks look like disjoint groups of interacting cells if viewed only in small parts , while viewing the entire network at once reveals one giant component ., For efficient computation in our graph invariant framework , we examined this problem in the following way: disjoint modules of network activity could be characterized as a weakly connected functional topology with a small algebraic connectivity ., We explored how algebraic connectivity of the functional topology was modulated by two variables: minimum weight and field of view size ., Because edge weight corresponds to the reliability of an observation of a spike correlation , thresholding minimum weight in a functional topology pruned its weaker edges ., We defined field of view size as the maximum pairwise distance between any two neurons investigated ., Together , these variables represented spatial and sampling bias during experiments ., We found that the algebraic connectivity of functional topologies followed similar trajectories in all three sensory areas: smaller fields of view and the exclusion of the weakest functional connections resulted in weakly connected graphs ( Figure 6A ) ., Taken together , these data suggest that one must employ large fields of view and low edge weight thresholds to capture an independent functional circuit ., Interestingly , we found a field of view size in each sensory area at which the algebraic connectivity seemed to reach capacity or asymptote; above this distance , larger fields of view did not result in significantly increased connectivity ., This finding suggested that a subsample size less than 1 . 1 square mm would capture a complete functional circuit topology ., To further understand the interplay between experimental field of view and the topology of the functional circuits , we specified a general model of Field of View ( FOV ) Error , or how well a functional topology is captured as a function of field of view size ( Figure 6B ) ., FOV error varies with the distribution of functional connections inherent to each neocortical region ( Figure 2 , right column ) ., Formally , let denote the existence of a functional connection between neurons and , and denote the pairwise distance between and ., Let be a pairwise distance ., Then , We computed the average FOV error over all pairwise combinations of neurons in all sensory areas as a function of ., To achieve less than 10 percent FOV error , we found that must be at least 676 microns in A1 , 660 microns in S1 , and 583 microns in V1 ( Figure 6C ) ., This corresponds to a minimum of 430 neurons in A1 , 510 neurons in S1 , and 478 neurons in V1 by computing a cumulative distribution of neuronal density based on the probability distributions of pairwise distances in our fields of view ( Figure 6D ) ., In contrast , we found less than 10 percent FOV error was achieved with just 93 microns in -nearest neighbors topologies , and 884 microns ( almost the entire imaging field of view ) in topologies with a uniform random spatial distribution of functional connectivity ( Figure 6C ) ., In the random graphs , error dropped linearly as field of view size was increased ( =\u200a0 . 9995 ) ., Thus , it appears that large FOVs result in fewer errors about underlying functional topology , and that the field of view error is lessened by skew in the likelihood of a connection toward shorter distances ., All regions of the sensory neocortex showed a common capacity for spontaneous circuit activations that emerged from the underlying local synaptic connectivity 15 ., Using the statistical dependencies of spiking between pairs of neurons , we generated directed and weighted functional graphs ., This approach revealed a scaling relationship between A1 and S1 15 , but was unable to delineate exactly what graph features were common to both regions ., In this study , we conducted an analysis of graph invariance in functional circuit topologies generated from three regions of sensory neocortex in order to extend the graph theoretic approach toward delineating generalized rules of connectivity ., The graph invariant framework allowed us to examine how circuits are similar , by considering how graph properties independent of neuronal labeling are consistent between areas ., This represents a top-down approach which extracted global features of functional connectivity from large , dense sampling of neuronal activity in the neocortex ., This analysis revealed multiple graph invariants that are consistent across sensory areas ., The structure of neocortical functional topologies were well-characterized by non-random connectivity that was not merely dependent on spatial proximity , despite the fact that the probability of functional connection peaked proximally ., In all areas , distal connections were required to achieve connected graphs , reminiscent of the daisy arrangement of dense local and patchy distal neocortical connections suggested by neuronal anatomy 2 , 48 ., We found that functional topologies of all areas were connected , and the degree of connectivity was statistically indistinguishable between areas ., Moreover , functional connections were structured even within a local circuit of the functional topology ., We found that eigenvector centrality , a measure of influence in local flow , is log-normally distributed in all sensory areas , and is highly correlated with out-degree , and weakly correlated with in-degree ., The size of functional topology does not scale with the number of neurons in the field of view , revealing that circuit activity is comprised of structured activations of subsets of neurons ., Local circuit flow comprehensively covers angular space regardless of spatial scale , which is inconsistent with a canonical flow of spontaneous activity ., Finally , our analysis revealed that given a large imaged field of view , a minimal numerical sample size was necessary to minimize the error of falsely characterizing two neurons as being independent ., In summary , the invariant features revealed by this study suggest the existence of a generalized functional circuit throughout the sensory neocortex , strengthening the argument that the neocortical microcircuit hypothesis should be framed as probabilistic rules of connectivity and organization ., This is not to say that label-dependent features do not play a role in mediating the structure of functional topology ., For example , although connectivity is strongly biased towards spatial proximity between neurons , the -nearest neighbors rule and random topologies poorly recapitulated functional topologies in the data ., This indicates that other connectivity rules that are not simply dependent on spatial proximity , such as those based on cell types 11 , 49 , likely play an important role ., As another example , we found that the distribution of eigenvector centrality , which strongly correlates with out-degree in all areas , is highest in V1 , and that the ratio of the number of open sequences to closed sequences , which stays constant as a function of path length in all areas , is highest in V1 ., These analyses suggest that V1 may be more feedforward than A1 and S1 , a result consistent with previous studies 37 , 42 ., The translation of the eigenvector centrality distribution seen in Figure 4A may represent a tweaking of a generalized rule ( fitting to a log-normal distribution ) to optimize the circuit for a particular function ( feedforwardness ) ., In general , it is possible that the specialization of the circuit to the overall function of the cortical area is label-dependent , or dependent on emergent properties of cell phenotypes ., However , despite the fact that label dependent rules of connectivity are likely present , by investigating global features of functional circuit topology that are invariant to the details of individual neurons , we are able to reveal abstract structural rules present in functional wiring in a computationally efficient manner ., We emphasize that our functional approach does not necessarily identify causal connectivity , but rather pairwise correlative dynamics 50 ., However , we also note that there is a relationship between structure and function 18 ., This relationship is likely enhanced in this study as the high sampling density employed here should dramatically increase the likelihood that a correlation could reflect a causal connection , since the likelihood of a synaptic connection increases with spatial proximity 8 ., We consider the slice preparation to be an isolated system that allows us to study the local connectivity that defines cortical microcircuitry and remove the potentially conflating influence of long modulatory and long afferent inputs ., This approach allowed us to maximize the imaged field of view and the corresponding numerical sample of neurons ., In addition , coronal slices allowed us to examine the potential influence of laminar boundaries on functional circuitry ., We found that a field of view of approximately 640 µm is necessary to correctly establish functional dependence between two neurons in the sensory neocortex ., This field of view results from having a minimal numerical sampling while having sufficient distal functional connections that are necessary to generate a connected graph ., The necessity of distal functional connections that extend beyond layers and columns may indicate that functional circuits represent information from multiple octaves in A1 51 whiskers in S1 52 , or a natural visual scene in V1 53 ., Our data are consistent with anatomical studies that have revealed a patchy , distributed axonal structure which has been postulated to limit signal redundancy while enabling the potential for integration of information within local populations of neurons 48 , 54 ., For these hypotheses to be properly evaluated , future work toward understanding the role of connectivity in cortical dynamics and behavior will require a combination of research at the in-vitro and in-vivo level ., Interestingly , we found that the connectedness of the topology depended not only on the size of the field of view , but also on whether the most unreliable connections were considered ., In a previous study employing a network model , we similarly found that weak connections were necessary to recapitulate experimentally observed circuit dynamics 15 ., In this study , functional topologies became sparse and modular as minimum thresholds on weight were increased , likely because fewer functional connections were reliable ., When only the most reliable functional connections were considered , the topologies were sparsely connected regardless of sensory area ., By investigating invariant metrics without setting thresholds on how reliably active the neurons were , we did not bias ourselves to only investigating the most reliable connections ., Such a bias may lead to subsampling errors , exactly parallel to the problems that arise from using small fields of view ., Since circuit topologies become highly connected with the inclusion of weak functional connections , weak connections may be necessary to provide a large dynamic range similar to a previous study of mouse V1 37 , 42 ., These data and analyses suggest that the generalized features of functional circuitry identified in this study maximize the capacity of this system to represent the sensory environment .
Correlations in local neocortical spiking activity can provide insight into the underlying organization of cortical microcircuitry ., However , identifying structure in patterned multi-neuronal spiking remains a daunting task due to the high dimensionality of the activity ., Using two-photon imaging , we monitored spontaneous circuit dynamics in large , densely sampled neuronal populations within slices of mouse primary auditory , somatosensory , and visual cortex ., Using the lagged correlation of spiking activity between neurons , we generated functional wiring diagrams to gain insight into the underlying neocortical circuitry ., By establishing the presence of graph invariants , which are label-independent characteristics common to all circuit topologies , our study revealed organizational features that generalized across functionally distinct cortical regions ., Regardless of sensory area , random and -nearest neighbors null graphs failed to capture the structure of experimentally derived functional circuitry ., These null models indicated that despite a bias in the data towards spatially proximal functional connections , functional circuit structure is best described by non-random and occasionally distal connections ., Eigenvector centrality , which quantifies the importance of a neuron in the temporal flow of circuit activity , was highly related to feedforwardness in all functional circuits ., The number of nodes participating in a functional circuit did not scale with the number of neurons imaged regardless of sensory area , indicating that circuit size is not tied to the sampling of neocortex ., Local circuit flow comprehensively covered angular space regardless of the spatial scale that we tested , demonstrating that circuitry itself does not bias activity flow toward pia ., Finally , analysis revealed that a minimal numerical sample size of neurons was necessary to capture at least 90 percent of functional circuit topology ., These data and analyses indicated that functional circuitry exhibited rules of organization which generalized across three areas of sensory neocortex .
Information in the brain is represented and processed by populations of interconnected neurons ., However , there is a lack of a clear understanding of the structure and organization of circuit wiring , particularly at the mesoscale which spans multiple columns and layers ., In this study , we sought to evaluate whether functional circuit architecture generalizes across the neocortex , testing the existence of a functional analogue to the neocortical microcircuit hypothesis ., We analyzed the correlational structure of spontaneous circuit activations in primary auditory , somatosensory , and visual neocortex to generate functional topologies ., In these graphs , neurons were represented as nodes , and time-lagged firing between neurons were directed edges ., Edge weights reflected how many times the lagged firing occurred and was synonymous to the strength of the functional connection between two neurons ., The presence of label-independent features , identified by investigating functional circuit topologies under a graph invariant framework , suggest that functionally distinct areas of the neocortex carry features of a generalized functional cortical circuit ., Furthermore , our analyses show that the simultaneous recording of large sections of cortical circuitry is necessary to recognize these features and avoid undersampling errors .
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