Modeling Higher-Order Correlations within Cortical Microcolumns
[['Electrophysiology is rapidly moving towards high density recording techniques capable of capturing the simultaneous activity of large populations of neurons .', 'This raises the challenge of understanding how networks encode and process information in ways that go beyond tuning properties or feedforward receptive field models .', 'Modeling the distribution of states in a network provides a way to discover communication patterns between neurons or functional groupings such as cell assemblies which may exhibit a more direct relation to stimulus or behavioral variables .', 'The Ising model , originally developed in the 1920s to describe magnetic interactions [1] , has been used to statistically characterize electrophysiological data , particularly in the retina [2] , and more recently for cortical recordings [3] , [4] .', 'This model treats spikes from a population of neurons binned in time as binary vectors and captures dependencies between cells with the maximum entropy distribution for pairwise dependencies .', 'This has been shown to provide a good model for small groups of cells in the retina [5] , though it is unable to capture dependencies higher than second-order .', 'In this work , we apply maximum entropy models to neural population recordings from the visual cortex .', 'Cortical networks have proven more challenging to model than the retina: The magnitude and importance of pairwise correlations between cortical cells is controversial [6] , [7] and higher-order correlations , i . e . correlations which cannot be captured by a pair-wise maximum entropy model , play a more important role [8]–[10] .', 'One of the challenges with current recording technologies is that we can record simultaneously only a tiny fraction of the cells that make up a cortical circuit .', "Sparse sampling together with the complexity of the circuit mean that the majority of a cell's input will be from cells outside the recorded population .", 'In adult cat visual cortex , direct synaptic connections have been reported to occur between 11%–30% of nearby pairs of excitatory neurons in layer IV [11] , while a larger fraction of cell pairs show “polysynaptic” couplings [12] , defined by a broad peak in the cross-correlation between two cells .', 'This type of coupling can be due to common inputs ( either from a different cortical area or lateral connections ) or a chain of monosynaptic connections .', 'A combination of these is believed to give rise to most of the statistical interactions between recorded pairs of cells .', 'The Ising model , which assumes only pairwise couplings , is well suited to model direct ( and symmetric ) synaptic coupling , but cannot capture interactions involving more than two cells .', 'We propose a new approach , that addresses both incomplete sampling and common inputs from other cell assemblies , by extending the Ising model with a layer of hidden units or latent variables .', 'The resulting model is a semi-Restricted Boltzmann Machine ( sRBM ) , which combines pairwise connections between visible units with an additional set of connections to hidden units .', 'Estimating the parameters of energy-based models , to which Ising models and Boltzmann machines belong , is computationally hard because these models cannot be normalized in closed form .', 'For both Ising models and Boltzmann machines with hidden units , the normalization constant is intractable to compute , consisting of a sum over the exponential number of states of the system .', 'This makes exact maximum likelihood estimation impossible for all but the smallest systems and necessitates approximate or computationally expensive estimation methods .', 'In this work , we use Minimum Probability Flow ( MPF [13] , [14] , in the context of neural decoding see [4] , [15] ) to estimate parameters efficiently without computing the intractable partition function .', 'It provides a straightforward way to estimate the parameters of Ising models and Boltzmann machines for high-dimensional data .', 'Another challenge in using energy-based models is the evaluation of their likelihood after fitting to the data , which is again made difficult due to the partition function .', 'To compute probabilities and compare the likelihood of different models , annealed importance sampling ( AIS ) [16] was used to estimate the partition function .', 'Combining these two methods for model estimation and evaluation , we show that with hidden units , Boltzmann machines can capture the distribution of states in a microcolumn of cat visual cortex significantly better than an Ising model without hidden units .', 'The higher-order structure discovered by the model is spatially organized and specific to cortical layers , indicating that common input or recurrent connectivity within individual layers of a microcolumn are the dominant source of correlations .', 'Applied to spatiotemporal patterns of activity , the model captures temporal structure in addition to dependencies across different cells , allowing us to predict spiking activity based on the history of the network .'], ['We estimated Ising , RBM and sRBM models for populations of cortical cells simultaneously recorded across all cortical layers in a microcolumn of cat V1 in response to long , continuous natural movies presented at a frame rate of 150 Hz .', 'Code for the model estimation is available for download at http://github . com/ursk/srbm .', 'Fig . 1a ) shows an example frame from one of the movies .', 'Model parameters were estimated using MPF with an regularization penalty on the model parameters to prevent overfitting .', 'To compute and compare likelihoods , the models were normalized using AIS .', 'Here we present data from two animals , one with 22 single units ( B4 ) , another with 36 units ( T6 ) , as well as a multiunit recording with 26 units ( B4M ) .', 'Fig . 1b ) shows spiking data from session B4 in 20 ms bins , with black squares indicating a spike in a bin .', 'Spatiotemporal datasets were constructed by concatenating spikes from consecutive time bins .', 'Pairs of cells show weak positive correlations , shown in Fig . 1c ) , and noise correlations computed from 60 repetitions of a 30s stimulus are similarly small and positive .', 'For all recordings , the population was verified to be visually responsive and the majority of cells were orientation selective simple or complex cells .', 'As recordings were performed from a single cortical column , receptive fields shared the same retinotopic location and have similar orientation selectivity , differing mostly in size , spatial frequency and phase selectivity .', 'See [17] for a receptive field analysis performed on the same raw data .', 'The estimated model parameters for the three different types of models ( Ising , RBM and sRBM ) are shown in Fig . 2 for session B4 .', 'The sparseness penalty , chosen to optimize likelihood on a validation dataset , results in many of the parameters being zero .', 'For the Ising model', '( a ) we show the coupling as a matrix plot , with lines indicating anatomical layer boundaries .', 'The diagonal contains the bias terms , which are negative since all cells are off the majority of the time .', 'The matrix has many small positive weights that encourage positive pairwise correlations .', 'In', '( b ) we show the hidden units of the RBM as individual bar plots , with the bars representing connection strengths to visible units .', 'The topmost bar corresponds to the hidden bias of the unit , and hidden units are ordered from highest to lowest variance .', 'The units are highly selective in connectivity: The first unit almost exclusively connects to cells in the deep ( granular and subgranular ) cortical layers .', 'The second unit captures correlations between cells in the superficial ( supergranular ) layers .', 'The correlations are of high order , with 10 and more cells receiving input from a hidden unit .', 'The remaining units connect fewer cells , but still tend to be location-specific .', 'Only the hidden units that have non-zero couplings are shown .', 'Additional hidden units are turned off by the sparseness penalty , which was chosen to maximize likelihood on the cross-validation dataset .', 'The interpretation of hidden units is quite similar to the pairwise terms of the Ising model: positive coupling to a group of visible units encourages these units to become active simultaneously , as the energy of the system is lowered if both the hidden unit and the cells it connects to are active .', 'Thus the hidden units become switched on when cells they connect to are firing ( activation of hidden units not shown ) .', 'The sRBM combines both pairwise and hidden connections and hence is visualized with a pairwise coupling matrix and bar plots for hidden units .', 'With the larger number of parameters , the best model is even more sparse in the number of nonzero parameters .', 'The remaining pairwise terms predominantly encode negative interactions , and much of the positive coupling has been explained away by the hidden units .', 'These give rise to strong positive couplings within either superficial ( II/III ) or intermediate ( IV ) and deep ( V/VI ) layers , which explain the majority of structure in the data .', 'The more succinct explanation for dependencies between recorded neurons is via connections to shared hidden units , rather than direct couplings between visible units .', 'The RBM and sRBM in this comparison were both estimated with 22 hidden units , but we show only units that did not turn off entirely due to the sparseness penalty .', 'In this example , a sparseness penalty of was found to be optimal for all three models .', 'In order to ascertain to what degree the stimulus driven component of activity accounts for the learned higher-order correlations , we augmented the above models with a dynamic bias term that consists of the log of the average instantaneous firing probability of each cell over repeated presentations of the same stimulus .', 'In the case that all trained parameters were zero , this model would assign a firing probability to all neurons identical to that in the peri-stimulus time histogram ( PSTH ) .', 'In Fig . 2d ) the couplings for the Ising model with stimulus terms are shown .', 'As the pairwise couplings now only capture additional structure beyond correlations explained by the stimulus , they tend to be weaker than in the Ising model without stimulus terms .', 'In particular the bias terms on the diagonal are almost completely explained away by the dynamic bias .', 'The same reasoning applies to the RBM with PSTH terms , which is shown in e ) .', 'Although the couplings are weaker than for the pure RBM , the basic structure remains , with the first two hidden units explaining correlations within superficial and deep groups of cells , respectively .', 'This shows that the learned coupling structure can not be explained purely from higher-order stimulus correlations and receptive field overlap .', 'Even when stimulus-induced correlations are fully accounted for , the correlation structure captured by the RBM remains similar and higher-order correlations are the dominant driver of correlated firing .', 'For a quantitative comparison between models , we computed normalized likelihoods using Annealed Importance Sampling ( AIS ) to estimate the partition function .', 'For each model , we generated 500 samples through a chain of annealing steps .', 'To ensure convergence of the chain , we use a series of chains varying the number of annealing steps and verify that the estimate of the partition function stabilizes to within at least 0 . 02 bits/s ( see Fig . S1 ) .', 'For models of size 20 and smaller we furthermore computed the partition function exactly to verify the AIS estimate .', 'Fig . 3a ) shows a comparison of excess log likelihood for the three different models and on all three datasets .', ', which we define as the gain in likelihood over the independent firing rate model , is computed in units of bits/spike for the full population .', 'Both higher-order models outperform the Ising model in fitting the datasets significantly .', 'Error bars are standard deviation computed from 10 models with different random subsets of the data used for learning and validation , and different random seeds for the parameters and the AIS sampling runs .', 'Fig . 3b ) shows the excess log likelihood for the models with stimulus terms .', 'Due to the additional computational complexity , these models were only estimated for the small B4 data set .', 'The left of the two bar plots shows that including the stimulus information through the PSTH greatly increases the likelihood , even the PSTH only model without coupling terms outperforms the Ising and RBM models by about 0 . 7 bits/s .', 'Including coupling terms still increases the likelihood , which is particularly visible on the right bar plot which shows the log likelihood gain relative to the PSTH model .', 'Including higher-order coupling terms still provides a significant gain over the pairwise model , confirming that there are higher-order correlation in the data beyond those induced by the stimulus .', 'Each of the models was estimated for a range of sparseness parameters bracketing the optimal using 4-fold cross-validation on a holdout set , and the results are shown for the optimal choice of for each model .', 'Additional insight into the relative performance of the models can be gained by comparing model probabilities to empirical probabilities for the various types of patterns .', 'Fig . 4 shows scatter plots of model probabilities under the different models against pattern frequencies in the data .', 'Patterns with a single active cell , two simultaneously active cells , etc . are distinguished by different symbols .', 'As expected from the positive correlations , the independent model ( yellow ) shown in', 'a ) consistently overestimates the probabilities of cells being active individually , so these patterns fall above the identity line , while all other patterns are underestimated .', 'For comparison , the Ising model is shown in the same plot ( blue ) , and does significantly better , indicated by the points moving closer to the identity line .', 'It still tends to fail in a similar way though , with many of the triplet patterns being underestimated as the model cannot capture triplet correlations .', 'In', 'b ) , this model is directly compared against the RBM ( green ) .', 'Except for very rare patterns , most points are now very close to the identity line , as the model can fully capture higher-order dependencies .', 'Hidden units describe global dependencies that greatly increase the frequency of high order patterns compared to individually active cells .', 'The 5% and 95% confidence intervals for the counting noise expected in the empirical frequency of states are shown as dashed lines .', 'The solid line is the identity .', 'Inserts in both models show the distribution of synchrony , , where is the number of cells simultaneously firing in one time bin .', 'This metric has been used for example in [18] to show how pairwise models fail to capture higher-order dependencies .', 'In the case of the T6 data set with 36 cells shown here , the Ising model and RBM both provide a good fit to the distribution of synchrony in the observed data .', 'Note that any error in estimating the partition function of the models would lead to a vertical offset of all points .', 'Thus visually checking the alignment of the data cloud around the identity line provides a visual verification that there are no catastrophic errors in the estimation of the partition function .', 'Unfortunately we cannot use this alignment as a shortcut to compute the partition function without sampling , e . g . by defining such that the all zeros state has the correct frequency , as this assumes a perfect model fit .', 'For instance , regularization tends to reduce model probabilities of the most frequent states , so this estimate of would systematically overestimate the likelihood of regularized models .', 'We note , however , that for higher-order models with no regularization this estimate does indeed agree well with the AIS estimate .', 'The same models can be used to capture spatiotemporal patterns by treating previous time steps as additional cells .', 'Consecutive network states binned at 6 . 7 ms were concatenated in blocks of up to 13 time steps , for a total network dimensionality of 130 with 10 cells .', 'These models were cross-validated and the sparseness parameters optimized in the same way as for the instantaneous model .', 'This allows us to learn kernels that describe the temporal structure of interactions between cells .', 'In Fig . 5 we compare the relative performance of spatiotemporal Ising and higher-order models as a function of the number of time steps included in the model .', 'To create the datasets , we picked a subset of 10 cells with the highest firing rates from the B4 dataset ( 4 cells from subgranular , 2 from granuar and 4 from supergranular layers ) and concatenated blocks of up to 13 subsequent data vectors .', 'This way models of any dimensionality divisible by 10 can be estimated .', 'The number of parameters of the RBM and Ising model were kept the same by fixing the number of hidden units in the RBM to be equal to the number of visible units , the sRBM was also estimated with a square weight matrix for the hidden layer .', 'As before , the higher-order models consistently outperform the Ising model .', 'The likelihood per spike increases with the network size for all models , as additional information from network interactions leads to an improvement in the predictive power of the model .', 'The curve for the Ising model levels off after a dimensionality of about 30 is reached , as higher-order structure that is not well captured by pairwise coupling becomes increasingly important .', 'However , the likelihood of higher-order models continues to increase through the entire experimental range .', 'The insert in the figure shows the entropy of the models , normalized by the data dimensionality by dividing by the number of frames and neurons .', 'The entropy was computed as where the expectation of the energy was estimated by initializing 100 , 000 samples using the holdout data set , and then running 2000 steps of Gibbs sampling .', 'Due to temporal dependencies additional frames carry less entropy , but we do not reach the point of extensivity where the additional entropy per frame reaches a constant value .', 'As the RBM is better able to explain additional structure in new frames , the additional entropy for new frames is much less than for the Ising model .', 'A similar observation has been made in [5] , where Ising and higher-order models for 100 retinal ganglion cells were compared to models for 10 time steps of 10 cells .', 'It is noteworthy that temporal dependencies are similar to dependencies between different cells , in that there are strong higher-order correlations not well described by pairwise couplings .', 'These dependencies extend surprisingly far across time ( at least 87 ms , corresponding to the largest models estimated here ) and are of such a form that including pairwise couplings to these states does not increase the likelihood of the model .', 'This has implications e . g . for GLMs that are typically estimated with linear spike coupling kernels which will likely miss these interactions .', 'To predict spiking based on the network history , we can compute the conditional distribution of single units given the state of the rest of the network .', 'This is illustrated for a network with 15 time steps for a dimensionality of 150 .', 'This model is not included in the above likelihood comparison , as the AIS normalization becomes very expensive for this model size .', 'Fig . 6a ) shows the learned weights of 18 randomly chosen nonzero hidden units for a spatiotemporal RBM model with 150 hidden units .', 'Each subplot corresponds to one hidden unit , which connects to 10 neurons ( vertical axis ) across 15 time steps or 100 ms ( horizontal axis ) .', 'Some units specialize in spatial coupling across different cells at a constant time lag .', 'As the model has no explicit notion of time , the time lag of these spatial couplings is not unique and the model learns multiple copies of the same spatial pattern .', 'Thus while there are 55 nonzero hidden units , the number of unique patterns is much smaller so that the effective representation is quite sparse .', 'The remaining units describe smooth , long-range temporal dependencies , typically for small groups of cells .', 'Both of these subpopulations capture higher-order structure connecting many neurons that cannot be well approximated with pairwise couplings .', 'By conditioning the probability of one cell at one time bin on the state of the remaining network , we can compute how much information about a cell is captured by the model over a naive prediction based on the firing rate of the cell .', 'This conditional likelihood for each cell is plotted in Fig . 6b ) in a similar way to excess log likelihood for the entire population in Fig . 5 , except in units of bits per second rather than bits per spike .', 'While the result here reflects our previous observation that Boltzmann machines with hidden units outperform Ising models , we note that the conditional probabilities are easily normalized in closed form since they describe a one-dimensional state space .', 'Thus we can ensure that the likelihood gain holds independent of the estimation of and is not due to systematic errors in sampling from the high-dimensional models .', 'Fig . 6c ) provides a more intuitive look at the prediction .', 'For 1 s of data from one cell , where 5 spikes occur , we show the conditional firing probabilities for the three models given 100 ms of history of itself and the other cells .', 'Qualitatively , the models perform well in predicting spiking probabilities , suggesting it might compare favorably to prediction based on GLMs or Ising models [19] .'], ['While there has been a resurgence of interest in Ising-like maximum entropy models for describing neural data , progress has been hampered mainly by two problems .', 'First , estimation of energy based models is difficult since these models cannot be normalized in closed form .', 'Evaluating the likelihood of the model thus requires approximations or a numerical integral over the exponential number of states of the model , making maximum likelihood estimation computationally intractable .', 'Even the pairwise Ising model is typically intractable to estimate , and various approximations are required to overcome this problem .', 'Second , the number of model parameters to be estimated grows very rapidly with neural population size .', 'If correlations up to order are considered , the number of parameters is proportional to .', 'In general , fully describing the distribution over states requires a number of parameters which is exponential in the number of neurons .', 'This can be dealt with by cutting off dependencies at some low order , by estimating only a small number of higher-order coupling terms , or by imposing some specific form on the dependencies .', 'We attempted to address both of these problems here .', 'Parameter estimation was made tractable using MPF , and latent variables were shown to be an effective way of capturing high order dependencies .', 'This addresses several shortcomings that have been identified with the Ising model .', 'As argued in [20] , models with direct ( pairwise ) couplings are not well suited to model data recorded from cortical networks .', 'Since only a tiny fraction of the neurons making up the circuit are recorded , most input is likely to be common input to many of the recorded cells rather than direct synapses between them .', 'While this work compares Generalized Linear Models ( GLMs ) such as models of the retina [21] and for LGN [22] to linear dynamical systems ( LDS ) models , the argument applies equally for the models presented here .', 'Another shortcoming of the Ising model and some previous extensions is that the number of parameters to be estimated does not scale favorably with the dimensionality of the network .', 'The number of pairwise coupling terms in GLM and Ising models scales with the square of the number of neurons , so with the amounts of data typically collected in electrophysiological experiments it is only possible to identify the parameters for small networks with a few tens of cells .', 'This problem is aggravated by including higher-order couplings: for example the number of third order coupling parameters scales with the cube of the data dimensionality .', 'Therefore attempting to estimate these coupling parameters directly is a daunting task that usually requires approximations and strong regularization .', 'Early attempts at modeling higher-order structure side-stepped these technical issues by focussing on structure in very small networks .', 'Ohiorhenuan noted that Ising models fail to explain structure in cat visual cortex [8] and was able to model triplet correlations [9] by considering very small populations of no more than 6 neurons .', 'Similarly , Yu et al . [3] , [10] show that over the scale of adjacent cortical columns of anesthetized cat visual cortex , small subnetworks of 10 cells are better characterized with a dichotomized Gaussian model than the pairwise maximum entropy distribution .', 'While the dichotomized Gaussian [23] is estimated only from pairwise statistics , it carries higher-order correlations that can be interpreted as common Gaussian inputs [24] .', 'However these correlations are implicit in the structure of the model and not directly estimated from the data as with the RBM , so it is not clear that the model would perform as well on different datasets .', 'Given that higher-order correlations are important to include in statistical models of neural activity , the question turns to how these models can be estimated for larger data sets .', 'In this section , we focus on two approaches that are complementary to our model using hidden units .', 'The increasing role of higher-order correlations in larger networks was first observed in [25] , where Ising models were fit via MCMC methods to the same 40 cell retina dataset that was analyzed in terms of subsets of 10 cells in [2] .', 'This point is further emphasized by Schneidman and Ganmor in [5] , who caution that trying to model small subsets ( 10 cells ) of a larger network to infer properties of the full network may lead to incorrect conclusions , and show that for retinal networks higher-order correlations start to dominate only once a certain network size is reached .', 'Therefore they address the same question as the present paper , i . e . how to capture order correlations without the accompanying growth in the number of free parameters in a larger network .', 'In their proposed Reliable Interaction Model ( RIM ) , they exploit the sparseness of the neural firing patterns to argue that it is possible to explicitly include third , forth and higher-order terms in the distribution , as most higher-order coupling terms will be zero .', 'Therefore the true distribution can be well approximated from a small number of these terms , which can be calculated using a simple recursive scheme .', 'In practice , the main caveat is that only patterns that appear in the data many times are used to calculate the coupling terms .', "While the model by construction assigns correct relative probabilities to observed patterns , the probability assigned to unobserved patterns is unconstrained , and most of the RIM's probability mass may thus be assigned to states which never occur in the data .", 'The second alternative to the RBM with hidden units is to include additional low-dimensional constraints in an Ising model .', 'In the “K-pairwise” model [18] , [26] , in addition to constraining pairwise correlations , a term is introduced to constrain the probability of neurons being active simultaneously .', 'This adds very little model complexity , but significantly improves the model of the data .', 'This is shown , for example , by computing conditional predictions in a similar fashion to that shown in Fig . 6c ) , where the K-pairwise model for a population of 100 retinal ganglion cells has an 80% correlation with the repeated trial PSTH .', 'In contrast to the RBM , however , this model is not structured in a way that can be easily interpreted in terms of functional connectivity .', 'To estimate these models for large numbers ( ) of neurons , the authors leverage the sampling algorithm described in [27] , an -regularized histogram Monte Carlo method .', 'In addition to proposing a faster ( though slower than MPF ) parameter estimation method for this class of models , Tkačik and colleagues address the difficulty in sampling from the model and computing the partition function .', 'In our experiments the overall limiting factor is the Gibbs sampler in the AIS partition function estimation .', 'Tkačik et al . use a more efficient sampling algorithm ( Wang-Landau ) to compute partition functions and entropy of their models .', 'As an even simpler approach to the partition function problem , they suggest that it can be obtained in closed form if the empirical probability of at least one pattern in the data is accurately known .', 'A case in point is the all zeros pattern that is typically frequent for recordings with sparsely firing neurons .', 'Unfortunately , this approach is limited in that it assumes that the probability the model assigns to the state is identical to the empirical probability of the state .', 'In the case that the model has not been perfectly fit to the data , or in the case that the data does not belong to the model class , this will lead to an incorrect estimate of the partition function .', 'Since the activity we are modeling is in response to a specific stimulus , one may rightfully question whether the observed higher-order correlations in neural activity are simply due to higher-order structure contained in the stimulus , as opposed to being an emergent property of cortical networks .', 'In an attempt to tease apart the contribution of the stimulus , we included a nonparametric PSTH term in the model .', 'However , this can capture arbitrarily complex stimulus transformations using the trial-averaged response to predict the response to a new repetition of the same stimulus .', 'As an “oracle model” , it does not only capture the part of the response that could be attributed to a feed-forward receptive field , but also captures contextual modulation effects mediated by surrounding columns and feedback from higher brain areas , essentially making it “too good” as a stimulus model .', 'The RBM and Ising models are then relegated to merely explain the trial to trial variability in our experiments .', 'Not including stimulus terms and finding the best model to explain the correlations present in the data , irrespective of whether they are due to stimulus or correlated variability , seems to be an equally valid approach to discover functional connectivity in the population .', 'GLMs [21] can be used to model each cell conditioned on the rest of the population .', 'While mostly used for stimulus response models including stimulus terms , they are easily extended with terms for cross-spike coupling , which capture interactions between cells .', 'GLMs have been successfully augmented with latent variables [20] , for instance to model the effect of common noisy inputs on synchronized firing at fast timescales [28] .', 'A major limitation of GLMs is that current implementations can only be estimated efficiently if they are linear in the stimulus features and network coupling terms , so they are not easily generalized to higher-order interactions .', 'Two approaches have been used to overcome this limitation for stimulus terms .', 'The GLM can be extended with additional nonlinearities , preserving convexity on subproblems [22] .', 'Alternatively , the stimulus terms can be packaged into nonlinear features which are computed in preprocessing and usually come with the penalty of a large increase in the dimensionality of the problem [29] .', 'However , we are not aware of any work applying either of these ideas to spike history rather than stimulus terms .', 'Another noteworthy drawback of GLMs is that instantaneous coupling terms cannot be included [20] , so instantaneous correlations cannot be modeled
['Introduction', 'Results', 'Discussion', 'Materials and Methods']
['We statistically characterize the population spiking activity obtained from simultaneous recordings of neurons across all layers of a cortical microcolumn .', 'Three types of models are compared: an Ising model which captures pairwise correlations between units , a Restricted Boltzmann Machine ( RBM ) which allows for modeling of higher-order correlations , and a semi-Restricted Boltzmann Machine which is a combination of Ising and RBM models .', 'Model parameters were estimated in a fast and efficient manner using minimum probability flow , and log likelihoods were compared using annealed importance sampling .', 'The higher-order models reveal localized activity patterns which reflect the laminar organization of neurons within a cortical column .', 'The higher-order models also outperformed the Ising model in log-likelihood: On populations of 20 cells , the RBM had 10% higher log-likelihood ( relative to an independent model ) than a pairwise model , increasing to 45% gain in a larger network with 100 spatiotemporal elements , consisting of 10 neurons over 10 time steps .', 'We further removed the need to model stimulus-induced correlations by incorporating a peri-stimulus time histogram term , in which case the higher order models continued to perform best .', 'These results demonstrate the importance of higher-order interactions to describe the structure of correlated activity in cortical networks .', 'Boltzmann Machines with hidden units provide a succinct and effective way to capture these dependencies without increasing the difficulty of model estimation and evaluation .']
['Communication between neurons underlies all perception and cognition .', "Hence , to understand how the brain's sensory systems such as the visual cortex work , we need to model how neurons encode and communicate information about the world .", 'To this end , we simultaneously recorded the activity of many neurons in a cortical column , a fundamental building block of information processing in the brain .', 'This allows us to discover statistical structure in their activity , a first step to uncovering communication pathways and coding principles .', 'To capture the statistical structure of firing patterns , we fit models that assign a probability to each observed pattern .', 'Fitting probability distributions is generally difficult because the model probabilities of all possible states have to sum to one , and enumerating all possible states in a large system is not possible .', 'Making use of recent advances in parameter estimation , we are able to fit models and test the quality of the fit to the data .', 'The resulting model parameters can be interpreted as the effective connectivity between groups of cells , thus revealing patterns of interaction between neurons in a cortical circuit .']
['computational neuroscience', 'neuroscience', 'biology and life sciences', 'computational biology']