Identification of Pattern Completion Neurons in Neuronal Ensembles Using Probabilistic Graphical Models

Abstract

Neuronal ensembles are groups of neurons with coordinated activity that could represent sensory, motor, or cognitive states. The study of how neuronal ensembles are built, recalled, and involved in the guiding of complex behaviors has been limited by the lack of experimental and analytical tools to reliably identify and manipulate neurons that have the ability to activate entire ensembles. Such pattern completion neurons have also been proposed as key elements of artificial and biological neural networks. Indeed, the relevance of pattern completion neurons is highlighted by growing evidence that targeting them can activate neuronal ensembles and trigger behavior. As a method to reliably detect pattern completion neurons, we use conditional random fields (CRFs), a type of probabilistic graphical model. We apply CRFs to identify pattern completion neurons in ensembles in experiments using in vivo two-photon calcium imaging from primary visual cortex of male mice and confirm the CRFs predictions with two-photon optogenetics. To test the broader applicability of CRFs we also analyze publicly available calcium imaging data (Allen Institute Brain Observatory dataset) and demonstrate that CRFs can reliably identify neurons that predict specific features of visual stimuli. Finally, to explore the scalability of CRFs we apply them to in silico network simulations and show that CRFs-identified pattern completion neurons have increased functional connectivity. These results demonstrate the potential of CRFs to characterize and selectively manipulate neural circuits.

SIGNIFICANCE STATEMENT We describe a graph theory method to identify and optically manipulate neurons with pattern completion capability in mouse cortical circuits. Using calcium imaging and two-photon optogenetics in vivo we confirm that key neurons identified by this method can recall entire neuronal ensembles. This method could be broadly applied to manipulate neuronal ensemble activity to trigger behavior or for therapeutic applications in brain prostheses.

Introduction

Following Lorente de Nó's (1938) proposal that cortical circuits are designed to sustain recurrent reverberating chains of activity, Hebb (1949) postulated that recurrent neuronal activity could generate cell assemblies, functional multineuronal units, as building blocks within neuronal microcircuits. In addition, Hebb speculated that strong recurrent connections between a given set of neurons would increase the probability to reactivate a whole group if only one of the elements was activated. This property, known as pattern completion, represents an efficient mechanism to retrieve information allowing the reactivation of previously stored information triggered by a few neurons (Marr, 1971; Hopfield, 1982; Rolls and Treves, 1994). Acknowledging this tradition, we use the term “pattern completion” to define neurons whose activation can recall an entire ensemble (Carrillo-Reid et al., 2016, 2019).

A neuronal ensemble is a group of neurons with coordinated activity that represents an experimental or behavioral condition (Carrillo-Reid and Yuste, 2020a,b). Advances in two-photon calcium imaging and two-photon optogenetics have made possible the simultaneous reading and writing of cortical ensemble activity with single-cell resolution in awake animals (Rickgauer et al., 2014; Packer et al., 2015). Coactivating sets of neurons can artificially imprint neuronal ensembles in vivo (Carrillo-Reid et al., 2016). Neuronal ensembles can also be recalled by targeting individual neurons with two-photon holographic optogenetics, demonstrating pattern completion and causality between neuronal ensemble activity and learned behaviors (Carrillo-Reid et al., 2019; Marshel et al., 2019; Robinson et al., 2020). The identification of pattern completion neurons (p.c. neurons) could enable the systematic manipulation of neuronal ensembles.

Graph theory has been used in neuroscience to describe the structural and functional organization of entire brains (Bullmore and Sporns, 2009). Graphs are usually constructed with nodes representing brain regions (Bettencourt et al., 2007) and edges representing information flow (Iturria-Medina et al., 2008). Many studies have constructed graphs for functional analysis of brain data from fMRI, EEG, and electrode arrays, taking brain regions (Achard and Bullmore, 2007; Fair et al., 2008; Hagmann et al., 2008), voxels (Eguíluz et al., 2005; van den Heuvel et al., 2008; Zuo et al., 2012) or electrode position (Downes et al., 2012) as nodes, and cross-correlation, mutual information, or Granger causality as edges (Fair et al., 2008; Bullmore and Sporns, 2009; Micheloyannis et al., 2009; Wang et al., 2010; Khazaee et al., 2015). Graphs are usually associated with a restricted set of parameters that describe the weight and direction of edges obtained by pairwise metrics, underscoring network interactions. To understand the role of pattern completion neurons in awake animals it is necessary to generate graphical models that capture the interaction between neurons and subsequently identify and manipulate individually neurons with pattern completion capability that could have a potential role orchestrating network activity (Carrillo-Reid and Yuste, 2020a).

Neuronal ensembles form a network structure that can be naturally characterized with probabilistic graphical models, where nodes and edges are biologically meaningful, representing neurons and their functional connections, respectively. Here, we used conditional random fields (Koller and Friedman, 2009), a combination of graph theory and probabilistic modeling that uses graphs to simplify and express the conditional interaction among a collection of variables. We applied CRFs to existing data from a recent work that used two-photon optogenetics to investigate pattern completion properties of neurons from mouse primary visual cortex in vivo (Carrillo-Reid et al., 2016), and we demonstrate that the parameters obtained from CRFs can predict neurons with the ability to recall artificially imprinted ensembles. To test the broad applicability of CRFs, we also analyzed publicly open calcium imaging data (Allen Institute Brain Observatory dataset) and demonstrated that our approach can identify neurons that reliably predict different features of visual stimuli and could potentially be targeted to recall percepts. Finally, to test the scalability of CRFs we used in silico network simulations and demonstrated that few neurons from Hopfield-style networks have pattern completion capability and increased functional connectivity.

Materials and Methods

Animals and surgery

All experimental procedures were conducted in accordance with the National Institutes of Health and Columbia University Institutional Animal Care and Use Committee and have been described previously (Carrillo-Reid et al., 2016, 2019). Briefly, simultaneous two-photon imaging and two-photon optogenetic experiments were performed on C57BL/6 male mice. Virus AAV1-syn-GCaMP6s-WPRE-SV40 and AVVdj-CaMKIIa-C1V1(E162T)-TS-P2A-mCherry-WPRE were injected simultaneously into layer 2/3 of left primary visual cortex (2.5 mm lateral and 0.3 mm anterior from the λ, 200 μm from pia). After 3 weeks mice were anesthetized with isoflurane (1–2%) and a titanium head plate was attached to the skull using dental cement. Dexamethasone sodium phosphate (2 mg/kg) and enrofloxacin (4.47 mg/kg) were administered subcutaneously. Carprofen (5 mg/kg) was administered intraperitoneally. After surgery animals received carprofen injections for 2 d as postoperative pain medication. A reinforced thinned skull window for chronic imaging (2 mm in diameter) was made above the injection site using a dental drill. A 3 mm circular glass coverslip was placed and sealed using a cyanoacrylate adhesive (Drew et al., 2010). Imaging experiments were performed 7–28 d after head plate fixation. During recording sessions mice were awake (head fixed) and could move freely on a circular treadmill.

Visual stimulation

Visual stimuli were generated using MATLAB Psychophysics Toolbox and displayed on an LCD monitor positioned 15 cm from the right eye at 45° to the long axis of the animal. Population activity corresponding to two-photon stimulation of targeted neurons in layer 2/3 of visual cortex was recorded with the monitor displaying a gray screen with mean luminescence similar to drifting gratings. The imaging setup and the objective were completely enclosed with blackout fabric and a black electrical tape. Visual stimuli consisted of full-field sine wave drifting gratings (100% contrast, 0.035 cycles/°, 2 cycles/s) drifting in two orthogonal directions presented for 4 s, followed by 6 s of mean luminescence (Carrillo-Reid et al., 2016, 2019).

Simultaneous two-photon calcium imaging and photostimulation

Two-photon imaging and optogenetic photostimulation were performed with two different femtosecond-pulsed lasers attached to a commercial microscope. An imaging laser (λ = 940 nm) was used to excite a genetically encoded calcium indicator (GCaMP6s), and a photostimulation laser (λ = 1064 nm) was used to excite a red shifted opsin (C1V1), which preferentially responds to longer wavelengths (Packer et al., 2012).

The two laser beams on the sample were individually controlled by two independent sets of galvanometric scanning mirrors. The imaged field of view was ∼240 × 240 μm (25× numerical aperture 1.05, XLPlan N objective), comprising 60–100 neurons.

We adjusted the power and duration of photostimulation so that the amplitude of calcium transients evoked by C1V1 activation mimic the amplitude of calcium transients evoked by visual stimulation with drifting gratings. Single-cell photostimulation was performed with a spiral pattern delivered from the center of the cell to the boundaries of the soma as has been previously published (Carrillo-Reid et al., 2016).

Image processing

Image processing was performed with ImageJ (version 1.42q, National Institutes of Health) and custom-made programs written in MATLAB as previously described (Carrillo-Reid et al., 2016, 2019). Acquired images were processed to correct motion artifacts using TurboReg. Neuronal contours were automatically identified using independent component analysis and image segmentation (Mukamel et al., 2009). Calcium transients were computed as changes in fluorescence: (F_i–F_o)/F_o, where F_i denotes the fluorescence intensity at any frame, and F_o denotes the basal fluorescence of each neuron. Spikes were inferred from the first-time derivative of calcium signals with a threshold of 3 SDs above noise level. We constructed an N × F binary matrix, where N denotes the number of active neurons, and F represents the total number of frames for each movie.

Population vectors

We constructed multidimensional population vectors that represent the simultaneous activation of different neurons. Peaks of synchronous activity describe population vectors (Carrillo-Reid et al., 2008). Only high-activity frames were used. An activity threshold was determined by generating 1000 shuffled raster plots and comparing the distribution of the random peaks against the peaks of synchrony observed in the real data (Shmiel et al., 2006). We tested the significance of population vectors against the null hypothesis that the synchronous firing of neuronal pools is given by a random process (Shmiel et al., 2006; Carrillo-Reid et al., 2015a). Such population vectors can be used to describe the network activity as a function of time (Schreiber et al., 2003; Stopfer et al., 2003; Brown et al., 2005; Sasaki et al., 2007; Carrillo-Reid et al., 2008). The number of dimensions for each experiment is given by the total number of active cells (N). Neuronal ensembles are defined by the concomitant firing of neuronal groups at different times. The similarity between two population vectors was defined by their normalized inner product, which represents the cosine of the angle between such vectors. A similarity index close to 1 indicates that both vectors point to a close direction in an N-dimensional space, therefore the neurons that define each vector are almost the same. A similarity index of 0 denotes that the pair of vectors are orthogonal, so the neurons that define each vector are different (Carrillo-Reid et al., 2015a,b; Carrillo-Reid and Yuste, 2020b).

Allen Institute Brain Observatory dataset

To demonstrate the general applicability of our approach we analyzed a publicly available dataset from the Allen Brain Observatory (http://observatory.brain-map.org/visualcoding) along with the software development kit (SDK) for extracting fluorescence and (F_i–F_o)/F_o (http://alleninstitute.github.io/AllenSDK/) by the Allen Institute of Brain Science (de Vries et al., 2020). Spikes were detected by first low-pass filtering the (F_i–F_o)/F_o traces, then a threshold of 5 SDs above noise level on the first derivative of filtered (F_i–F_o)/F_o. The experiments identifiers s used are 511507650, 511509529, 511510650, 511510670, 511510718, and 511510855.

Conditional random fields

We constructed CRFs as previously published (Tang et al., 2016), using indicator feature vectors $x=\left[x^1,x^2,...,x^M\right],$ where $x^m\in {\mathcal{X}}^N$, for each edge and node, and target binary population activity vectors $y=\left[y^1,y^2,...,y^M\right],$ where $y^m\in {\mathcal{Y}}^N$ for $N$ neurons and $M$ samples (time points). The ${\mathcal{X}}^N$ and ${\mathcal{Y}}^N$ describe an N-dimensional space where the dimensionality describes the total number of active neurons. For each sample, the conditional probability can be expressed as follows:

$$p\left(y^m\text{|}{x}^m;\theta \right)=\frac{\exp \left(〈\varphi \left(x^m,y^m\right),\theta 〉\right)}{Z\left(x^m;\theta \right)},$$

where $\varphi$ is a vector of the distribution expressed in log-linear form, $\theta$ is a vector of parameters with parameters for the log-linear distribution, and $Z$ is the partition function as follows:

$$Z\left(x^m;\theta \right)={\displaystyle \sum }_{y\in \mathcal{Y}}\exp \left(〈\varphi \left(x^m,y\right),\theta 〉\right).$$

The conditional probability can be factored over a graph structure $G=\left(V,\mathcal{A}\right)$, where $V$ is the collection of nodes representing observation variables and target variables, and $\mathcal{A}$ is the collection of subsets of $V$. Based on the graph structure, the model parameters can be written separately for nodes and edges as $\varphi =\{{\varphi }_V,{\varphi }_A\}$ and $\theta =\left\{{\theta }_V,{\theta }_A\right\}\in {\mathrm{\Re }}^N$. Given binary $x$ and $y$, node parameters ${\varphi }_V$ and ${\theta }_V$ include two sets of distribution and the parameters corresponding to the node state 0 and 1, whereas edge parameters ${\varphi }_A$ and ${\theta }_A$ include four sets of distribution and the parameters, corresponding to the edge states 00, 01, 10, and 11 (see Fig. 2). The conditional dependencies can be then written as follows:

$$p\left(Y\text{|}X;\theta \right)=\frac{\text{exp}({{\displaystyle \sum }}_{i\in V}{\theta }_i{\varphi }_i\left(X,Y_i\right)+{{\displaystyle \sum }}_{\alpha \in \mathcal{A}}{\theta }_{\alpha }{\varphi }_{\alpha }\left(X,Y_{\alpha }\right))}{Z\left(X;\theta \right)}.$$ (1)

Figure 2.

Classification of visual stimuli from calcium imaging data using CRFs. A, In the experimental setup, simultaneous two-photon imaging and two-photon optogenetics were performed in layer 2/3 of primary visual cortex in head-fixed freely moving mice. B, Graphical representation of CRFs. Circles represent neurons, squares represent added nodes depicting visual stimuli, shaded nodes (x) represent observed data, and white nodes (y) represent true states of the neurons and are connected by edges that indicate their mutual dependencies. Node potentials are defined over the two possible states of each node, and edge potentials are defined over the four possible states of each existing edge, depending on the state of adjacent nodes. C, Representation of graphical structure with added stimulus nodes. Each stimulus introduces an additional node to the graph. D, Ratio of log likelihood predicting horizontal (top, red) and vertical (bottom, blue) drifting gratings calculated by CRF models. Colored stripes indicate visual stimuli. Scale bar, 10 s. E, The accuracy of real models is higher compared with shuffled models (n = 8 mice, 16 models; p < 0.0001 (****); Mann–Whitney test).

This model is a generalized version of Ising models, which have been previously applied to model neuronal networks (Schaub and Schultz, 2012). The log likelihood of each observation can be then written as follows:

$$\ell \left(\theta ;X^m,Y^m\right)=〈\varphi \left(X^m,Y^m\right),\theta 〉-\log Z\left(X^m\right).$$ (2)

To reduce the complexity of the model, we first learn a sparse graph structure that represents variable dependencies, then we construct a CRF on the learned structure. Given the inferred binary spikes from raw imaging data, we constructed CRF models in the following two steps: (1) structure learning and (2) parameter learning.

For structure learning, we learned a sparse graph structure $G=\left(V,\mathcal{A}\right)$ using ${\ell }_1$-regularized neighborhood-based logistic regression for each node r (Ravikumar et al., 2010) as follows:

$$\underset{{\theta }_{\backslash r}}{\min }\left\{\ell \left({\theta }^s;x\right)+{\lambda }_s{\Vert {\theta }_{\backslash r}^s\Vert }_1\right\},$$

where

$$\ell \left({\theta }^s;x\right)=-\frac{\mathrm{1}}{n}{\displaystyle \sum }_{i=1}^n\log \frac{\exp \left(\mathrm{2}{x}_r{{\displaystyle \sum }}_{t\in V\backslash r}{\theta }_{rt}^s{x}_t\right)}{\exp \left(\mathrm{2}{x}_r{{\displaystyle \sum }}_{t\in V\backslash r}{\theta }_{rt}^s{x}_t\right)+\mathrm{1}}$$

and

$${\theta }_{\backslash r}^s=\left\{{\theta }_{ru}^s,u\in V\backslash r\right\},$$

where $\backslash r$ denotes except r, and ${\theta }^s$ is a vector of regression parameters for structure learning. Here, ${\lambda }_s$ is a regularization parameter that controls the sparsity (or conversely, the density) of the constructed graph structure. This is essentially a logistic regression of variable $X_r$ on the other variables $X_{\backslash r}$, with ${\ell }_1$-regularization. The regression coefficients thus represent the neighborhood structure and the sign pattern. We implemented ${\ell }_1$ regularization using a highly efficient elastic-net algorithm solving using penalized maximum likelihood as follows:

$$\underset{{\beta }_0,\beta }{\text{min}}\frac{1}{N}{\displaystyle \sum }_{i=1}^N{w}_{i}l\left(y_i,{\beta }_0+{\beta }^T{x}_i\right)+{\lambda }_s\left[\frac{\left(1-\alpha \right)}{2}{\Vert \beta \Vert }_2^2+\alpha {\Vert \beta \Vert }_1\right],$$

where α = 1, and thus is equivalent to a standard LASSO (least absolute shrinkage and selection operator). We constructed a large number of structures (100+) across a broad range of ${\lambda }_s$, from which we stochastically selected a number of structures for subsequent parameter learning. We ensured proper bracketing by including the largest and smallest ${\lambda }_s$ structure in this priming set. In future iterations of parameter learning, the selected sets of structures were selected using the log likelihood of learned models as feedback. The final graph structure was obtained by thresholding the edge potentials with a given density preference $d$. Edges with potential values within the top $d$ quantile were kept as the final structure. It is worth noting that although $d$ could bias the result, varying $d$ does not lead to density values that differ much. This is because of the sparsity induced by the ${\ell }_1$ regularizer.

For parameter learning, we aim to learn the edge and node potential parameters $=\left\{{\theta }_V,{\theta }_A\right\}$. Based on the learned structure, we used the Bethe approximation to approximate the partition function and iterative Frank–Wolfe methods to perform parameter estimation by maximizing the log likelihood of the observations with a quadratic regularizer (Tang et al., 2016) as follows:

$$\ell \left(\theta ;X,Y\right)={\displaystyle \sum }_{m=1}^M\ell \left(\theta ;X^m,Y^m\right)-\frac{{\lambda }_p}{2}{\Vert \theta \Vert }^2.$$

Here, ${\lambda }_p$ is a regularization that controls the learned parameters and helps prevent overfitting. Cross-validation was done to find the best ${\lambda }_s$, $d$, and ${\lambda }_p$ via held-out model likelihood. We varied ${\lambda }_s$ with six values between 0.002 and 0.5, d with six values between 0.25 and 0.3, and ${\lambda }_p$ with five values between 10 and 10 000, all sampled uniformly. To obtain the best model parameters, 90% data were used for training, and 10% data were withheld. Cross-validation was done using all possible combinations of the above parameters and calculating the likelihood of the withheld data. Then, the best model parameters were determined by selecting the parameter set with a locally maximum likelihood in the parameter space. This empirically results in models with connectivity density of ∼10 to 40%, corresponding to the previously reported connectivity probability in layer 2/3 of V1 (Yoshimura et al., 2005; Ko et al., 2011).

Node strength

As 0 and 1 on graph nodes represent the nonactive and active state of neurons, the edge potential term 11 represents the coactive state of connected neurons, whereas 01 and 10 represent antiactivate, and 00 represent the state that both nodes (neurons) are nonactive. Because the nodes are mostly inactive during recording, the 00 term is the strongest term, whereas only the 11 term captures neuronal ensemble coactivation. Therefore, we defined the node strength as the sum of the 11 term of edge potentials from all connecting edges as follows:

$$s\left(i\right)={\displaystyle \sum }_{j=1}^{N_E\left(i\right)}{\varphi }_{11}\left(i,j\right).$$

Here, $N_E\left(i\right)$ denotes the number of connecting edges for node $i$. The defined node strength reflects the importance of a given cell in coactivating other cells.

Shuffling method

To generate shuffled models, we first randomized the spike raster matrices while preserving the activity per cell and per frame. Then, we trained CRF models using the shuffled spike matrices, with the cross-validated ${\lambda }_s$, $d$, and ${\lambda }_p$ from the real model. This procedure was repeated 100 times. The random level of node strengths was determined by averaging node strengths from shuffled models.

Identifying putative pattern completion neurons

To find the most influential neurons for each condition, we iterated through all the neurons and identified their contribution in predicting the stimulus conditions in the population. To this end, for the $i^{\text{th}}$ neuron in the population, we compared its activity or inactivity in all $M$ frames for each model. With the two resulting population vectors in the $m^{\text{th}}$ frame among all samples, we calculated the log probability of them coming from the trained CRF model as follows:

$$p_{i,1}^m=p\left(y^m\text{|}{x}_{\backslash i}^m,x_i^m=1;\theta \right)$$

$$p_{i,0}^m=p\left(y^m\text{|}{x}_{\backslash i}^m,x_i^m=0;\theta \right).$$

Then, we computed the log likelihood ratio as follows:

$${\ell }_{i,1-0}=\left\{\log p_{i,1}^m-\log p_{i,0}^m\right\},m=1,...,M$$

and calculated the standard receiver operating characteristic (ROC) curve with the ground truth as the timing of each presented visual stimuli. The prediction ability of all nodes for all presented stimuli is then represented by an area under curve (AUC) matrix $A$, where $A_{i,d}$ represents the AUC value of node $i$ predicting stimulus $d$. Additionally, we calculated the node strength $S=\left\{s_i\right\}$ of each neuron in the CRF model.

We then computed prediction AUC $A^r$ and node strength $S^r$ of each node r from 100 CRF models trained on shuffled data. The final ensemble for stimulus $d$ is defined as follows: $\left\{i\right|A_{i,s}>\text{mean}\left(A_s^r\right)+\text{std}\left(A_s^r\right),S_i>\text{mean}\left(S^r\right)+\text{std}\left(S^r\right)\}$.

Neural network simulation to study pattern completion

The simulated neural network was generated using 10 Hopfield nets embedded in a larger matrix. Each Hopfield network was generated synchronously and deterministically (Gerstner et al., 2014). The binary state S of neuron i in the following time step (t + 1) given j inputs and weight w_ij was described by the following:

$$S_i\left(t+1\right)={\displaystyle \sum }_j{w}_{ij}{S}_j\left(t\right).$$

Every neuron was fully connected according to the weight matrix w, defined by the following equation:

$$w_{ij}=\frac{1}{N}{\displaystyle \sum }_{u}en{s}_i^{u}en{s}_j^u,$$

where ens_i^u is the state of neuron i in the u^th ensemble. Only one ensemble was active at any given time. External stimuli were produced by initializing the network to a noisy ensemble state (20% noise), and thereafter the network was permitted to iterate for 5 time steps to simulate the pattern completion of the ensemble. In total, the matrix contained 810 neurons and 100 ensembles. The probability of any given neuron being active during an ensemble external to its respective Hopfield net was fixed at 2.5%.

Data analysis

CRF models were trained using the Columbia University Yeti and Habanero shared High Performance Computing Cluster (see Figs. 2–6). Benchmarking and simulation modeling were performed on a custom-built computer using an Intel Core i9-9900K processor and NVIDIA Titan RTX (see Figs. 7, 8). MATLAB R2016a, R2019b, and R2020a (MathWorks) were used for data analysis.

Figure 6.

CRF efficacy to model pattern completion neurons from cortical ensembles. A, Cosine similarity from putative pattern completion neurons extracted from CRFs that have been down sampled or up sampled (left) or randomly sampled (right). Black triangle indicates the original ensemble size. Note that removing or adding neurons decreases the ability to predict visual stimuli. B, AUC values of down-sampled or up-sampled putative pattern completion neurons (left) or randomly sampled neurons (right). C, ROC curves of down-sampled or up-sampled neurons (left) or randomly sampled neurons (right; n = 8 mice, 28 ensembles; Wilcoxon rank sum test).

Figure 7.

Pattern completion neurons in simulated networks. A, Raster-plot of simulated data sorted by Hopfield networks (100 ensembles from 10 Hopfield nets). B, Probability distribution of simulated network. The network's response to trained stimuli was comparable to the level of activity observed in high-activity frames in our in vivo datasets (dashed line). Population activity was defined as the percentage of neurons active during any given frame. C, The distribution of cosine similarities comparing the weights of the simulated network and CRF model for each neuron. Dash line denotes shuffled data. D, Top-ranked neurons according to their score in both simulated network and CRFs. Correlation coefficient and linear regression of scores calculated from the simulated network and CRF model (ρ = 0.8592; r² = 0.738). E, Comparison between putative pattern completion neurons identified in simulated networks and CRFs for different set sizes. F, Pattern completion neurons from each ensemble have higher scores than nonpattern completion neurons (n = 100 ensembles from 10 Hopfield networks; p.c. neurons score = 0.7891 ± 0.0452; non-p.c. neurons score = 0.4794 ± 0.0399; p < 0.0001 (****); mean ± SD; Mann–Whitney test).

Figure 8.

Modeling CRFs with different parameters. A, Run time of structural learning across small (n = 64), medium (n = 500), and large (n = 1000) datasets of increasing size. Each run generated 100 structures using λ_s values between 1e-5 and 0.5. With datasets of 1000 neurons and 10 000 population vectors (prescreened for the inclusion of only high-activity frames), run times did not surpass 5 h. B, Mean run time of parameter learning for individual models using small (n = 64), medium (n = 500), and large (n = 1000) datasets of increasing size. With datasets of 1000 neurons and 10,000 population vectors (prescreened for the inclusion of only high-activity frames), run times did not surpass 10 min per model. Approximately one-half of this run time comprised block-coordinate Frank–Wolfe iterations. The remainder of this run time can be attributed to the initiation of each model and frequent reporting of progress.

We didn't use any statistical method to predetermine sample size. Sample sizes are similar to those reported previously (Carrillo-Reid et al., 2016, 2019). Statistical tests were done in GraphPad Prism 5. Statistical details of each specific test can be found in the article and the figure legends.

Data availability

Step-by-step implementation of the code and associated notations can be found at the following: https://github.com/hanshuting/graph_ensemble for the code used to implement CRF models on a cluster and https://github.com/darikoneil/Identification-of-Pattern-Completion-Neurons-in-Neuronal-Ensembles-using-Probabilistic-Graphical-Mod for the code used to run simulated data and CRF models on a single computer.

Results

Neuronal ensembles as multidimensional population vectors

As neuroscience moves from the study of individual neurons to neural ensembles as the substrate of brain functions, it is necessary to develop methods to identify neurons that are able to recall entire ensembles (Carrillo-Reid et al., 2017). The development of optical techniques has allowed the simultaneous recording and manipulation of neurons with single-cell precision (Rickgauer et al., 2014; Packer et al., 2015). However, studies performing simultaneous recordings usually perform pair correlations or the average of all recorded neurons as a measurement of network activity. As neuronal ensembles are related to specific functions that can be repeated at different times, they can be optimally defined by binary multidimensional population vectors (see above, Materials and Methods) where each vector represents the activity of all observed neurons at a given time window, and the dimensionality of the vectors is defined by the number of observed neurons (Fig. 1A). In this context, neuronal ensembles could be understood as clusters of population vectors that point in a similar direction in a high-dimensional space (Fig. 1B). The advantage of conceptualizing neuronal ensembles as population vectors, instead of pairwise correlations between neurons or averaged activity, is that large datasets could be visualized as changes in population vectors in real time, whereas pairwise correlations of neurons are dependent on the recording time length (Carrillo-Reid and Yuste, 2020b). Thus, we built CRFs using population vectors that take into account all the recorded neurons simultaneously, allowing the comparison of network activity under different experimental conditions to highlight neurons with pattern completion capability that could be used to orchestrate entire functional networks.

Figure 1.

Characterization of neuronal ensembles as multidimensional vectors. A, Population activity represented as population vectors. The activity from the entire population at each time point (t) is represented by a binary vector (v1 to vt). B, Schematic representation of population vectors in an N-dimensional space, where n denotes the total number of observed neurons. Each point in a multidimensional space represents a population vector defined by a coactive group of neurons (left). Dashed arrows indicate the dimensionality of the data (d1 to dn). For clarity only population vectors v1 and v2 are depicted by solid arrows. A neuronal ensemble could be defined by a cluster of population vectors (right).

Building CRF models from population responses to oriented stimuli

Cortical neurons from layer 2/3 in primary visual cortex are organized as neuronal ensembles, where each ensemble depicts a given orientation of visual stimuli (Miller et al., 2014; Carrillo-Reid et al., 2015b). To investigate the ability of CRF models to identify groups of neurons that could recall the representation of a specific orientation of visual stimuli, we analyzed population responses to drifting gratings from layer 2/3 neurons of primary visual cortex in awake head-fixed mice from previously published data (Carrillo-Reid et al., 2016; Fig. 2A).

CRFs model the conditional distribution p(y|x) of a network, where x represents observations and y represents labels associated with a graphical structure (Sutton and McCallum, 2012). As no assumptions are made on x, CRFs can describe the conditional distribution with complex dependencies in observation variables associated with a graphical structure that is used to constrain the interdependencies between labels (Fig. 2B). CRFs have been successfully applied in diverse areas of machine learning that involve multidimensional data such as analysis of texts (Peng et al., 2011), bioinformatics (Sato and Sakakibara, 2005; Liu et al., 2006), computer vision (Brecht et al., 2004; Sminchisescu et al., 2006), and natural language processing (Lafferty et al., 2001; Choi et al., 2005), but their application to identify neurons with pattern completion capability that could allow the study of the properties and mechanisms of neuronal ensemble formation is lacking.

Population vectors representing the coordinated activity of neuronal groups were inferred from calcium imaging recordings and used as training data. We defined activity events from each neuron as nodes in an undirected graph, where each node can have two values, 0 corresponding to nonactivity and 1 corresponding to activity. In this way, nodes interact with each other through connecting edges, which have four possible configurations, 00, 01, 10, and 11, depending on the values of the two nodes adjacent to the edge. The two values associated with nodes and the four values associated with edges are characterized by a set of parameters called node potentials $({\varphi }_0,{\varphi }_1)$ and edge potentials $({\varphi }_{00},{\varphi }_{01},{\varphi }_{10},{\varphi }_{11})$ respectively (Fig. 2B). These parameters, also known as potential functions, reflect the scores of individual values on each node and edge. Using part of the observation data (e.g., training data), we estimated the model's structure (e.g., graph connectivity) and parameters (e.g., the potentials) via maximum likelihood (see above, Materials and Methods). We then performed cross-validation on held-out data to identify the optimal level of sparsity and regularization (see above, Materials and Methods). Once the model was learned from the data, the normalized product of the corresponding nodes and edge potentials describe a probability distribution that gives a scalar score when a given neuronal population becomes active. This is a normalized probability over all possible network states (e.g., all binary configurations of the nodes) and is estimated by maximizing likelihood on past observed population vectors (see above, Materials and Methods).

To integrate information about the external stimulus along with the observed neuronal data, we added an additional node for each type of stimulus that was presented to the animal. This node was set to 1 when the corresponding stimulus was on and 0 when the stimulus was off (Fig. 2B). In this way, CRFs modeled the conditional probability of network states given the observations for different visual stimuli. Therefore, by treating visual stimuli as added nodes and comparing the output likelihood of observing each stimulus, CRFs were able to model different visual stimuli from observed data. The nodes directly connected to the added nodes generate a specific model for each experimental condition (Fig. 2C). Given two different visual stimuli (horizontal or vertical drifting gratings; y_s1 and y_s2, respectively), the logarithm of probability corresponding to observing each stimulus is defined by $p_{\text{s}1}=p\left(x^m\text{|}{y}^m,y_{\text{s}1}=1,y_{\text{s}2}=0\right)$ and $p_{\text{s}2}=p\left(x^m\text{|}{y}^m,y_{\text{s}1}=0,y_{\text{s}2}=1\right)$. Thus, assuming that the a priori probability of visual stimuli is uniform, the higher probability value through comparing the log-likelihood ratio $\log \left(p_{\text{s}1}\right)-\log \left(p_{\text{s}2}\right)$ can be used to classify each stimuli (Fig. 2D). To quantify the classification performance of CRFs, we calculated the accuracy percentage in experimental data and compared it with shuffled models in which node potentials and edge potentials from the real model were randomly shuffled (Fig. 2E). The accuracy of real models was significantly higher than shuffled models (n = 8 mice; real model accuracy = 79.82 ± 8.34%; shuffled model accuracy = 61.21 ± 22.55%; mean ± SD; p < 0.0001; Mann–Whitney test) demonstrating that CRFs can model different orientations of drifting gratings from calcium imaging population recordings.

Identification of neurons with pattern completion properties

After the successful generation of CRF models that describe different experimental conditions (vertical or horizontal drifting-gratings; Fig. 2D) we investigated whether the parameters obtained from such models could be used to identify neurons with putative pattern completion properties. To identify pattern completion neurons we set the activity of each neuron to 1 or 0 and compared the output logarithm probability difference (probability ratio) $\log p_{i,1}^m-\log p_{i,0}^m=\log p\left(y^m\text{|}{x}_{/i}^m,x_i^m=1\right)-\log p\left(y^m\text{|}{x}_{/i}^m,x_i^m=0\right)$ using the inferred CRF models (Fig. 3A). In other words, we tested how good each neuron is to predict each model. We calculated single neuron performance by binarizing the logarithm probability difference (Fig. 3B) and calculated the area under the ROC curve (AUC). Because pattern completion neurons are likely to have strong functional connectivity, we used the node strength obtained from CRF models that represents the summation of edge potentials (${\varphi }_{11}$ terms) from all connecting edges for each node in the graph. In this way, strongly connected neurons have high node strength, whereas weakly connected neurons have low node strength (Fig. 3C). Finally, we defined putative pattern completion neurons as the ones that can be used to predict each experimental condition with higher performance (high AUC) and have high node strength (Fig. 3D). It is important to highlight that some neurons could take part in several ensembles simultaneously (Carrillo-Reid et al., 2015b). Such neurons that respond to different experimental conditions have high functional connectivity but are not able to recall a specific ensemble (Carrillo-Reid et al., 2019), therefore the use of AUC is fundamental to distinguish neurons with high functional connectivity that can recall a specific ensemble. Our analysis shows that neurons with putative pattern completion capability have significantly higher AUC compared with nonpattern completion (non-p.c.) neurons that have high node strength [Fig. 3E; n = 14 mice; AUC p.c. neurons = 0.8063 ± 0.1277; AUC non-p.c. neurons = 0.2520 ± 0.1649; mean ± SD; p < 0.0001; Mann–Whitney test]. Noticeably, not all the neurons with high node strength are able to predict the given stimuli.

Figure 3.

Identification of putative pattern completion neurons using CRFs. A, Schematic representation of putative pattern completion neurons extraction from CRF models with added nodes for two different visual stimuli. The activity of the $i^{\text{th}}$ neuron is set to 1 or 0 at each frame, and the log likelihood $p_{i,1}^m$ and $p_{i,0}^m$ of modified population vectors is calculated. B, Log-likelihood inference and prediction for cortical putative pattern completion neurons representing horizontal (top, red) or vertical (bottom, blue) drifting gratings. C, Graphical representation of node strength. D, Putative pattern completion neurons are identified by high AUC and high node strength (top right quadrant). Confidence levels were defined from CRF models of shuffled data (gray bars). E, AUC of putative pattern completion neurons with high node strength is higher than nonpattern completion neurons (n = 14 mice, 28 models; p < 0.0001 (****); Mann–Whitney test).

Optogenetic confirmation of pattern completion prediction using CRF models

We then tested the CRF prediction of putative pattern completion neurons using previously published data, where artificially imprinted cortical ensembles were created by the repetitive activation of a group of neurons with two-photon optogenetics and recalled by activating specific members of the ensemble, a property known as pattern completion (Carrillo-Reid et al., 2016). To do so, we used the structural and classification parameters of CRFs learned from simultaneous two-photon imaging and two-photon optogenetic experiments with single-cell resolution before and after the imprinting protocol and demonstrated that pattern completion neurons increased their connectivity after the imprinting protocol (Fig. 4A). To clearly visualize the change in graph connectivity induced by the imprinting protocol, we constructed isomorphic graphs from the CRF models and arranged them in a circular visualization (Fig. 4B). After the artificial ensemble was imprinted, neurons with pattern completion capability that were able to recall an entire ensemble showed significantly increased AUC (Fig. 4C; n = 6 mice; AUC before = 0.4570 ± 0.1256; AUC after = 0.8568 ± 0.0952; mean ± SD; p = 0.0313; Wilcoxon matched-pairs signed-rank test). These analyses demonstrate that structural and classification parameters inferred with CRFs can be used not only to identify pattern completion neurons but also to study the mechanisms involved in the reconfiguration and recalling of neuronal ensembles from optogenetically perturbed microcircuits.

Figure 4.

Confirming pattern completion with two-photon optogenetics. A, Graphical models obtained using CRFs from simultaneous two-photon imaging and two-photon optogenetic stimulation of a neuron with pattern completion capability before (left) and after (right) two-photon optogenetic ensemble imprinting. Ensemble connections are highlighted with blue lines. Scale bar, 50 μm. B, Circular graphs of CRF models before (pre-) and after (post-) ensemble imprinting. Connections between ensemble neurons are shown in blue. Red represents photo-stimulated neuron (arrow). C, Node strength and AUC values showed network changes of neurons with pattern completion capability. The photo-stimulated neuron is represented in red before (left) and after (right) ensemble imprinting. Confidence levels calculated from random data are depicted by gray bars.

Applicability of CRF models to publicly available datasets

One of the challenges for the identification of putative pattern completion neurons consists in the applicability of the methods to data obtained in different laboratories under diverse experimental conditions. To explore the general applicability of our method, we analyzed publicly open datasets (Allen Brain Observatory), which contain data from layer 2/3 of primary visual cortex consisting in several visual stimuli types with different experimental settings. We generated CRF models for each of four orientations of drifting gratings at a temporal frequency (TF) of 1 and compared the performance of the model for the same orientation but with different TFs. For each model, neurons with high AUCs and high node strength showed concomitant responses to the corresponding orientation (Fig. 5A). Putative pattern completion neurons were also able to define specific neuronal ensembles for each orientation (Fig. 5B). The ROC curves comparing the performance of the model with TF = 1 against data obtained with the same orientation but with different TFs showed that our approach could also be used to detect changes in specific features of visual stimuli in layer 2/3 of primary visual cortex as different TFs showed a degraded classification performance (Fig. 5C,D; classification AUC: AUC_TF=1 = 0.79 ± 0.15, AUC_TF=2 = 0.71 ± 0.21, AUC_TF=4 = 0.69 ± 0.17, AUC_TF=8 = 0.62 ± 0.17, AUC_TF=15 = 0.56 ± 0.14; p_1,4 < 0.05, p_1,8 < 0.01, p_1,15 < 0.001; n = 5 animals, 20 ensembles; mean ± SD; Wilcoxon rank sum test). Therefore, our approach was able to find neurons with putative pattern completion properties, from a publicly open dataset, that are tuned to a particular orientation of drifting gratings for a given TF (Fig. 5E), suggesting that CRFs could be used to recall percepts with varied features.

Figure 5.

Applicability of CRFs to public calcium imaging data. A, Calcium transients of representative pattern completion neurons identified from CRF models for different orientations of drifting gratings. Gray stripes indicate visual stimuli for the corresponding orientation of drifting gratings for each neuronal ensemble. B, Temporal course of neuronal ensemble activation. Each neuronal ensemble was defined by all putative pattern completion neurons with high AUC and high node strength for each orientation of drifting gratings. Each row shows the activation of a specific neuronal ensemble (black lines). Scale bar, 200 frames. C, ROC curves of neuronal ensembles predicting different temporal frequencies (TF: 1, 2, 4, 8, and 15 Hz). Dashed line represents random classification performance. D, AUC for different TFs (classification AUC: TF 1 = 0.7906 ± 0.0301, TF 2 = 0.7158 ± 0.0407, TF 4 = 0.7009 ± 0.0323, TF 8 = 0.6212 ± 0.0335, TF 15 = 0.5642 ± 0.0288; p_1,2 = 0.2277 ns; p_1,4 = 0.0423*; p_1,8 = 0.001**; p_1,15 = 4.2583e-05***). Note that prediction performance of CRF models with TF = 1 decreases with data with different TFs. E, Preferred orientation selectivity of neuronal ensembles for TF = 1 Hz. The radius of each circle depicts AUC values from 0 (center) to 1 (border). Dotted inner circles represent random performance (AUC = 0.5; n = 6 mice, 24 ensembles; Wilcoxon rank sum test).

Quantifying CRF prediction performance for ensembles

We next investigated whether the neurons highlighted by CRFs were optimal for predicting visual stimuli. To do so, we randomly shuffled population vectors containing all putative pattern completion neurons from cortical ensembles by adding or removing elements from the group and examined the stimulus prediction performance. The similarity function (Fig. 6A, left), prediction performance (Fig. 6B, left) and ROC curves (Fig. 6C, left) of population vectors formed with detected cortical ensembles had a maximum value when the neurons depicted by CRFs were unchanged (original neurons similarity 0.2147 ± 0.0206; prediction AUC 0.8109 ± 0.0242; mean ± SEM). These results showed that population vectors formed with neurons identified by CRFs represent an efficient population to predict external visual stimuli. This raises the question of whether such population vectors are a specific nonrandom subgroup. To answer this, we randomly sampled a subset from all the population of neurons ranging from 10 to 190% of the total size of neuronal ensembles. Indeed, we observed that the similarity function (Fig. 6A, right), prediction performance (Fig. 6B, right) and ROC curves (Fig. 6C, right) from random groups of neurons was significantly lower than those of CRF models (best similarity 0.1000 ± 0.0101, AUC 0.5045 ± 0.0071; mean ± SEM), indicating that neurons identified with CRF parameters achieved a classification performance significantly better than random sets of neurons. These results suggest that targeting all the neurons that were active at different trials of a given experimental condition could reduce the accuracy of recalled ensembles.

Modeling CRFs with different parameters using simulated neural networks

To test the performance of CRFs in modeling the pattern completion properties of simulated neural networks, we embedded 10 Hopfield networks within a larger network of weaker connectivity (Fig. 7A; see above, Materials and Methods). To achieve biologically plausible population vectors, we constructed our network so that the mean of neuronal activity following stimuli presentation matched the mean of neuronal activity of high-activity frames in our in vivo datasets (Fig. 7B). To replicate a plausible level of underdetermination, we modeled this simulated network at eight values of the structure-learning regularization term, which is a regularization parameter that controls the sparsity of the graphical structure λ_s (7.9767e-05-0.1793), and two values of the parameter-learning regularization term, which controls the learned parameters and helps in preventing overfitting, λ_p (10, 10,000). First, we assessed how accurately our model could describe the functional connectivity of our simulated network. To accomplish this, we used cosine similarity to measure the synaptic weight matrix of the simulated network with the edge potentials of our model. We found a striking similarity between the synaptic weights of each neuron and respective edge potentials in the CRFs (cosine similarity = 0.9615 ± 0.0017; mean ± SEM), which was not observed in a shuffled model (cosine similarity = 0.0052; Fig. 7C). Thereafter, we investigated whether using the real connectivity of the simulated network or the learned connectivity of the model would identify the same pattern completion neurons. To compare performance using one metric, we averaged the scores for prediction performance (AUC) and normalized node strength. We found the correlation between real and modeled ensembles to be very strong (r² = 0.7380, ρ = 0.8592; Fig. 7D). To determine whether both methods would identify the same neurons for targeted stimulation, we compared sets of neurons selected by each method using the Jaccard index. We found that both methods targeted similar groups of pattern completion neurons, and this relationship was maintained across variations in set size (Fig. 7E). Our analysis of these targeted groups confirmed that a handful of neurons in each ensemble possessed pattern completion capability demonstrated by significantly higher scores between pattern completion neurons and nonpattern completion neurons (Fig. 7F; n = 100 ensembles; p.c. neurons score = 0.7891 ± 0.0452; non-p.c. neurons score = 0.4794 ± 0.0399; p < 0.0001; mean ± SD; Mann–Whitney test), supporting the hypothesis that neurons with pattern completion capability have increased functional connectivity (Carrillo-Reid et al., 2016, 2019).

Discussion

Graph theory analysis of functional connectivity in cortical microcircuits

In this study, we use a computational method, CRFs, to identify pattern completion neurons in calcium imaging recordings of mouse visual cortex in vivo. As opposed to traditional descriptive approaches for neuronal ensemble identification and network analyses, based on correlations between pairs of neurons, machine learning methods represent an empirically grounded new approach to create models that aim to capture the functional structure of neuronal circuits while providing structural information about the network properties of individual neurons within a population.

In the past decades, graph theory has been applied to characterize the structure and function of neuronal networks (Achard and Bullmore, 2007; Bettencourt et al., 2007; Fair et al., 2008; Hagmann et al., 2008; Iturria-Medina et al., 2008; Supekar et al., 2008; Yu et al., 2008; Downes et al., 2012; Zuo et al., 2012; Cunningham and Yu, 2014; Chiang et al., 2016). Most of these studies used recordings across multiple brain regions (Achard and Bullmore, 2007; Fair et al., 2008; Zuo et al., 2012; Hinne et al., 2013; Chiang et al., 2016), and only a few focused on the general network properties of cortical circuits with recordings from single neurons (Bonifazi et al., 2009; Stetter et al., 2012; Gururangan et al., 2014; Yatsenko et al., 2015). The majority of methods applied to infer network properties in brain slices (Mao et al., 2001; Cossart et al., 2003; Ikegaya et al., 2004; Stetter et al., 2012; Gururangan et al., 2014) or in vivo (Yatsenko et al., 2015) operate on the correlation matrix and aim to recover the functional dependencies between observed pairs of neurons. However, these methods are model-free and therefore incapable of describing the overall network dynamics based on the probability distribution of neuronal ensembles. Our method provides an alternative by directly modeling the statistical dependencies among all the neurons. The generation of a graphical model of the circuit provides a direct link to the functional connectivity structure of the circuit, enabling targeted manipulations of components, as opposed to dimensionality reduction or encoding methods.

CRFs graphical models identify pattern completion neurons in neuronal ensembles

Cortical ensembles represent coactive neuronal populations that could represent the computational unit of brain functions (Mao et al., 2001; Bonifazi et al., 2009; Miller et al., 2014; Carrillo-Reid et al., 2015a, 2016, 2019). The structural and functional organization of brain microcircuits could be characterized by these modular properties. For this reason, it is necessary to identify pattern completion neurons in these cortical modules that can efficiently recall different brain states (Gururangan et al., 2014; Carrillo-Reid et al., 2015a).

The computational difficulty in constructing CRFs lies in recovering the global normalizer (the partition function) and the corresponding gradients. With an arbitrary graph structure, this problem is often intractable. But recent advances that combine Bethe free energy approximation and Frank–Wolfe methods for inference and learning model parameters allow fast and a relatively accurate construction of cyclic CRFs (Tang et al., 2016). Thus, CRFs could be applied to datasets with thousands of interconnected neurons. However, for datasets with more neurons (and therefore more random variables and larger networks), CRFs (like most machine learning approaches) would require an increasingly large number of samples (population vectors) in the training dataset.

Advantages and limitations of CRFs compared with other classification algorithms

Compared with fully generative models such as Markov random fields and Bayesian networks that make assumptions on the dependencies among all the observed variables from the model, CRFs only model the hidden system states dependent on observed features. Because no independence assumptions are made among observed variables, CRFs avoid potential errors introduced by unobserved common inputs. Additionally, given the finite number of network states described by population activity, the conditional distribution is enough for making predictions and for identifying neurons in each state. Compared with other discriminative finite-state models such as the maximum entropy Markov model (MEMM), CRFs use global normalizers to overcome the local bias in MEMM, induced by local normalizers, and achieve higher accuracy in diverse applications (Lafferty et al., 2001). Also, compared with encoders that adjust a series of constants associated with each neuron, CRFs can be used to explore changes in functional connectivity patterns (as we demonstrate here) and their relation to learning or behavioral states. Therefore, CRFs appear promising for modeling neuronal ensemble functional connectivity and for identifying members that could be easily manipulated by two-photon optogenetics.

The overall activity of multiple cells in a given time window can be understood as a multidimensional array of population vectors where vectors pointing to a similar location in space can be considered as a group. We previously showed that population vectors defining a group (i.e., a neuronal ensemble) can be extracted from multidimensional arrays by performing singular value decomposition (SVD; Carrillo-Reid et al., 2015a). Although SVD can identify cortical ensembles reliably, it lacks a structured graphical model that allows the systematic study of changes in network properties and the identification of neurons with pattern completion capability. One limitation for the current CRF learning algorithm is the computation of a sparse graph structure, which is less prone to overfitting of the data. Certainly, because of some modeling assumptions, computational considerations and the finiteness of training data, the learned graphical structure and parameters may not be the globally best probabilistic model of the data. However, it is recovered efficiently rather than requiring an exhaustive and unrealistic exploration of all possible structures and parameter combinations. Additionally, approximations during the parameter learning step can sometimes compromise the global optimality guarantees.

Applicability of CRFs to broader datasets

To assess the scalability of CRFs, we also performed benchmarking tests using sample artificial networks of 64, 500, and 1000 neurons with up to 10,000 frames of activity. We found that 100 λ_s runs completed in <270 min of CPU time when using the largest dataset (Fig. 8A). Parameter estimation was trivial in comparison, requiring just 600 s of CPU time per model at the largest dataset (Fig. 8B). Both structural learning and parameter estimation can be easily parallelized by neighborhood (structural learning) and by model (parameter estimation), leading to highly efficient wall clock run times. Furthermore, given the sequential execution of structural learning and parameter estimation, it is also possible to substitute more efficient methods of structural learning, such as Pearson's correlation, to further prune wall clock run times. Given the efficient wall clock run times, it is feasible for future research to consider expanding our approach to incorporate multiframe ensembles and other temporal interactions or use continuous (instead of binarized) neural data by replacing the linear programming solver used during parameter estimation. Finally, given the efficiency of parameter estimation, the development of equally efficient structural learning methods will allow for the analysis of streaming network data.

Identification of pattern completion neurons for optogenetic targeting

One key advantage of CRFs is the ability to model the circuit explicitly in a manner than can be used for targeted manipulation. Indeed, the results from the present work (Fig. 4) and those from our previous publication (Carrillo-Reid et al., 2019), demonstrate that neurons with high node strength and AUC values are particularly prone to be pattern completion neurons. As one practical example of an application of this method, nontargeted electrical stimulation of visual cortex has been used for decades as an attempt to provide useful visual sensations to patients who have lost the functionality of their eyes (Brindley and Lewin, 1986). To improve prostheses, one could in principle train patients using devices with a large number of electrodes (Shepherd et al., 2013). Our results suggest that after a given network has been imprinted (Carrillo-Reid et al., 2016), the identification of neurons with pattern completion capability could be used to reduce the number of active points that require stimulation and pinpoint them with accuracy. The further development of network models based on population activity that can predict a given set of features embedded in visual stimuli will be crucial for the efficient manipulation of cortical ensembles.

Finally, it has been shown that the connectivity of diverse systems described by graphs with complex topologies follow a scale-free power law distribution (Barabasi and Albert, 1999). Similarly, cortical ensembles described by CRFs could be characterized by a subset of neurons with strong synaptic connections. The existence of neurons with pattern completion capability has been demonstrated in previous studies where perturbing the activity of single neurons was able to change the overall network dynamics (Hagmann et al., 2008; Bonifazi et al., 2009; Carrillo-Reid et al., 2016).

In closing, our results suggest that the structural and predictive parameters defined by CRF models could be used in the design of closed-loop experiments with single-cell resolution to investigate the role of a specific subpopulation of neurons in a given cortical microcircuit during different behavioral events (Carrillo-Reid et al., 2019).