Home | About | Journals | Submit | Contact Us | Français |

**|**HHS Author Manuscripts**|**PMC2678680

Formats

Article sections

Authors

Related links

J Neurosci. Author manuscript; available in PMC 2009 October 15.

Published in final edited form as:

PMCID: PMC2678680

NIHMSID: NIHMS109801

Jonathon Shlens,^{1,}^{3} Greg D. Field,^{1} Jeffrey L. Gauthier,^{1} Martin Greschner,^{1} Alexander Sher,^{2} Alan M. Litke,^{2} and E.J. Chichilnisky^{1}

The publisher's final edited version of this article is available free at J Neurosci

See other articles in PMC that cite the published article.

Synchronized firing among neurons has been proposed to constitute an elementary aspect of the neural code in sensory and motor systems. However, it remains unclear how synchronized firing affects the large-scale patterns of activity and redundancy of visual signals in a complete population of neurons. We recorded simultaneously from hundreds of retinal ganglion cells in primate retina, and examined synchronized firing in completely sampled populations of ~50–100 ON-parasol cells, which form a major projection to the magnocellular layers of the lateral geniculate nucleus. Synchronized firing in pairs of cells was a subset of a much larger pattern of activity that exhibited local, isotropic spatial properties. However, a simple model based solely on interactions between adjacent cells reproduced 99% of the spatial structure and scale of synchronized firing. No more than 20% of the variability in firing of an individual cell was predictable from the activity of its neighbors. These results held both for spontaneous firing and in the presence of independent visual modulation of the firing of each cell. In sum, large-scale synchronized firing in the entire population of ON-parasol cells appears to reflect simple neighbor interactions, rather than a unique visual signal or a highly redundant coding scheme.

Much of what is known about neural coding arises from studies of single neurons. However, in many neural circuits, the concerted activity of multiple cells cannot be predicted based on recordings from single cells, because the firing of different cells is not statistically independent. For example, retinal ganglion cells (RGCs), which transmit all visual information from the eye to the brain, exhibit strong *synchronized firing* – a tendency to fire nearly simultaneously more frequently than expected by chance (Arnett, 1978; Arnett and Spraker, 1981; Mastronarde, 1983a; Mastronarde, 1983b; Mastronarde, 1983c; Mastronarde, 1989; Meister et al., 1995; DeVries, 1999). Numerous studies based on recordings from pairs of nearby RGCs reveal that synchronized firing reflects a combination of common synaptic inputs and gap junction coupling (Mastronarde, 1983a; Mastronarde, 1983b; Mastronarde, 1983c; Mastronarde, 1989; Dacey and Brace, 1992; Jacoby et al., 1996; Stafford and Dacey, 1997; Brivanlou et al., 1998; Hu and Bloomfield, 2003; Hidaka et al., 2004; Ishikane et al., 2005; Amthor et al., 2005; Ackert et al., 2006; Trong and Rieke, 2008). Theoretical and empirical studies suggest that synchronized firing in the retina could play an important role in processing and transmitting information (Meister, 1996; Nirenberg et al., 2001; Schneidman et al., 2006; Pillow et al., 2008), as it may in other sensory, memory and motor systems (Laurent, 2002; Harris, 2005; Usrey and Reid, 1999; Narayanan et al., 2005).

However, to understand how synchronized firing affects the neural code of the retina, it is necessary to know how entire collections of RGCs – rather than just pairs – fire in concert. Recent work suggests that synchronized firing in local clusters of 7–10 RGCs can be understood on the basis of pairwise interactions (Schneidman et al., 2006; Shlens et al., 2006). But almost nothing is known about concerted activity in larger collections of RGCs, leaving open two central questions about how synchronized firing affects the neural code of the retina.

First, it is unclear whether large, distinctive patterns of synchronized firing could be produced by spatially extended sources of common input, potentially providing a unique visual signal to the brain (Schnitzer and Meister, 2003). In the mammalian retina, a diverse collection of wide-field amacrine cell types provide extensive lateral connections to RGCs (Dacey and Brace, 1992; Stafford and Dacey, 1997; MacNeil and Masland, 1998). These cells have been proposed to underlie visual functions such as the segregation of object and background motion (Masland, 2001; Baccus and Meister, 2002; Olveczky et al., 2003; Baccus, 2007), by simultaneously influencing activity in large collections of RGCs. The spatial extent of amacrine cell connectivity suggests that their impact may be difficult to observe in the small, local groups of RGCs previously examined.

Second, it is unclear whether the pairwise interactions in a complete population of cells may combine to dominate the activity of any individual cell (Schneidman et al., 2006; Nirenberg and Victor, 2007; Bethge and Berens, 2007; Tang et al., 2008). Extrapolations based on recordings from up to 10 cells have suggested that that the activity of each RGC could be strongly dictated by the surrounding population activity even if individual pairwise interactions are weak, implying high network redundancy (Schneidman et al., 2006). This prediction has never been tested in a population large enough or complete enough that the hypothesized redundancy could be observed.

Using large-scale electrophysiological recordings in the primate retina (Litke et al., 2004; Frechette et al., 2005), we probe the spatial scale and structure of synchronized firing in complete populations of ~50–100 ON-parasol RGCs, which uniformly sample visual space and form a major projection to the lateral geniculate nucleus. We report that patterns of synchronized firing in this population are significantly larger than previously appreciated. However, we also find that network activity can be summarized accurately on the basis of pairwise interactions between neighboring cells (Shlens et al., 2006; Schneidman et al., 2006), and therefore that no additional widespread interactions are implicated. Based on pairwise interactions, we show that no more than 20% of the activity each ON-parasol cell is accounted for by the activity of the network, indicating that although synchrony is powerful, the resulting signals are not massively redundant. These findings hold in the presence of visual stimulation, indicating that the observed structure of synchronized firing remains relevant during visual signaling.

Preparation and recording methods are described elsewhere (Litke et al., 2004; Frechette et al., 2005; Chichilnisky and Kalmar, 2002). Briefly, eyes were obtained from 12 deeply and terminally anesthetized macaque monkeys (*Macaca mulatta*) used by other experimenters in accordance with institutional guidelines for the care and use of animals. Immediately after enucleation, the anterior portion of the eye and vitreous were removed in room light and the eye cup was stored in darkness for at least 20 minutes prior to dissection, in a bicarbonate buffered Ames’ solution (Sigma; St. Louis, MO) bubbled with 95% O_{2}, 5% CO_{2} at 32–34 degrees C, pH 7.4. Under infrared illumination pieces of peripheral retina 3–5 mm in diameter, isolated from the retinal pigment epithelium, were placed flat against a planar array of 512 extracellular microelectrodes, covering an area of approximately 1,800 × 900 µ*m*. Data were obtained from 15–120 minute periods of recording. The preparation was perfused with Ames’ solution.

The voltage on each electrode was digitized at 20 kHz and stored for off-line analysis. Details of recording methods and spike sorting are given elsewhere (Litke et al., 2004). Briefly, spikes were identified using a threshold of four times the voltage SD. For each spike, the waveform of the spike and the simultaneous waveforms on 6 adjacent electrodes were extracted. Three to five waveform features were identified using principal components analysis. A mixture of Gaussians model was fit to the distribution of features using expectation maximization (Duda et al., 2001). The number of clusters and initial conditions for the model was determined automatically using an adapted watershed transformation (Castleman, 1996; Roerdink and Meijster, 2001) . All clusters were visually inspected and when necessary, a mixture of Gaussians model was fitted using manually selected initial conditions. Clusters with a large number of refractory period violations (more than 10% estimated contamination based on refractory period violations), or spike rates below 1 Hz, were excluded from further analysis.

An optically reduced stimulus from a gamma-corrected cathode ray tube computer display refreshing at 120 Hz was focused on the photoreceptor outer segments. The mean intensity was adjusted to a low photopic light level by including neutral density filters in the light path. The mean photon absorption rate for the long (middle, short) wavelength sensitive cones was approximately equal to the rate that would have been caused by a spatially uniform monochromatic light of wavelength 561 (530, 430) nanometers and intensity 4300 (4200, 2400) photons/µ*m*^{2}/sec, incident on the photoreceptors. For the collection of ON-parasol cells shown in Figure 1a–b, the mean firing rate during exposure to a steady, spatially uniform display at this light level was 11.8 ± 3.6 Hz and 22.4 ± 4.8 Hz for each preparation respectively. Across all preparations examined firing rates varied from roughly 5–20 Hz.

Islands of large-scale synchronized firing in populations of primate ON-parasol RGCs across four preparations (a–d). Each oval represents the 0.9, 1.0, 1.0 and 1.0 SD outline of the Gaussian fit to the receptive field in each preparation, respectively. **...**

Spatio-temporal receptive fields were measured using a dynamic white noise stimulus in which the intensity of each display phosophor at each pixel location was selected randomly and independently over space and time from a binary distribution. RMS stimulus contrast was 96%, stimulus duration was 30 minutes. The pixel size (60 µ*m* in 10 preparations, 96 µ*m* and 120 µ*m* in two others) was selected to accurately capture the spatial structure of parasol cell receptive fields. For each RGC, the spike-triggered average stimulus was computed; this summarizes how the cell integrates visual inputs over space and time (Marmarelis and Naka, 1972; Chichilnisky, 2001). An elliptical 2-dimensional Gaussian function was fitted to the the spatial profile; outlines in Figure 1 represent the boundary for these fits at the SD values specified in the figure caption. The averaged major and minor axes of the 1 SD contour of each parasol cell was ~ 200µ*m*.

The following method was used to calculate the fraction of spikes in each cell that were synchronized with those of any of its immediate neighbors in the mosaic, over and above the chance expectation. For a given cell, consider the observed probability *p* with which the cell fired simultaneously (±5 ms) with any of its immediate neighbors. Also consider the probability *q* that this would occur if the neurons fired independently, obtained by multiplying the respective firing probabilities of the cells. The probability that the cell participates in unexpected synchronized firing is thus *p* – *q*. Dividing by the total firing probability of the cell *r* yields the fraction of spikes that were part of an unexpected firing pattern, (*p*–*q*)/*r*. Across 20 cells in each of four preparations, this value was 31±5%, 22±4%, 15±2% and 14±4%, respectively (mean ± SD).

The invariant spatial properties of synchronized firing were quantified by measuring the degree of directional preference in the strength of synchrony. For each cell of interest in the mosaic interior, a polygon was constructed in which the angle and magnitude of each point measures the angle and strength of synchrony with neighboring neurons. The angle and magnitude of these points were uncorrelated (*R ^{2}* = 0.02,

Several statistics as well as the formulation of the pairwise-adjacent model required defining adjacent cells in the ON-parasol mosaic. A non-parametric technique was used to define adjacency based on receptive field center locations. Each location in space was associated with the nearest receptive field center. This definition delineates a collection of boundaries between territories associated with neighboring cells, known as a Voronoi tessellation (Voronoi, 1907; Okabe et al., 2000). Pairs of cells with shared boundaries are connected with a line in Figure 4a.

The Voronoi tessellation misidentifies adjacency if a hole exists in the mosaic due to an unrecorded cell (e.g. Figure 1a, bottom left). Likewise, the tessellation misidentifies adjacency between cells along the edge of the mosaic. To deal with the latter problem, a secondary stipulation was added that adjacency requires a shared edge within the convex hull of the collection of receptive field centers.

Maximum entropy methods are used in statistical inference to identify an unknown distribution given several constraints which are insufficient to fully specify the answer. The method involves selecting the distribution with the greatest *entropy* consistent with the constraints,

(1)

where *H*[*P*] = –Σ_{x} *P*(*x*)log_{2} P(*x*) is the entropy of a firing pattern distribution P(*x*) and {λ_{i}} are Lagrange multipliers enforcing the constraints (Jaynes, 1957a; Jaynes, 1957b; Cover and Thomas, 1991; Amari, 2001). In our application the specified constraints are the firing rates expressed as probabilities and the joint firing patterns *P*(*x*_{i},*x*_{j}) for pairs of neurons *x*_{i}, *x*_{j} that are adjacent. The Lagrange multipliers enforcing each pairwise-adjacent constraint specify the sensitivity of the maximum entropy solution to slight changes to the strength of synchrony (Figure 4a, red lines). Note that a model with pairwise-adjacent constraints where a spike (or no spike) labeled with a +1 (or −1) is identical to a nearest-neighbor Ising model (Schneidman et al., 2006; Tang et al., 2008). The interaction terms of this model (Figure 4a) do not directly reflect the strength of synchronized firing between a pair of neurons: for example, interaction terms can propagate correlation to explain observed synchronized firing between non-adjacent cells (Shlens et al., 2006).

A dual form of this problem is to recognize that computing the maximum entropy distribution is equivalent to computing the maximum likelihood estimate of the Lagrange multipliers {λ_{i}} given the data (Csiszar, 1975; Berger et al., 1996). Calculating the maximum likelihood estimate of the parameters is a special case of estimating the parameters of a discrete Markov random field (Jordan, 1998) or equivalently, a Boltzmann machine (Hinton and Sejnowski, 1986). This problem is recognized to be log-concave everywhere, meaning that a unique global maximum exists and can be found with techniques from convex optimization (Press et al., 1988; Malouf, 2002; Berger et al., 1996; Malouf, 2002; Darroch and Ratcliff, 1972), simulated annealing (Hinton and Sejnowski, 1986; Hertz et al., 1991), or approximations to the full likelihood (Perez, 1998; Hinton, 2002). In the case of large models, estimating model parameters is an area of active research (X et al., 2006).

A drawback of these methods is that most optimization techniques require an explicit calculation of the probability distribution, which in turn requires normalizing across all possible firing patterns. In the case of synchronized firing, the firing pattern distribution across *n* neurons consists of a set of binary random variables *x* = (*x*_{1}, *x*_{2}, …, *x _{n}*) with over 2

Because estimating the model parameters is complex, validation of the results was performed in two ways. First, the convergence of the procedure was examined by checking its ability to reproduce the desired constraints. In the case of the pairwise-adjacent model this amounts to matching the firing rates of individual neurons and the degree of synchronized firing between adjacent neurons. The trained model accurately matched the observed firing rate and strength of synchrony (*R ^{2}* > 0.99). The remaining error may reflect the convergence criterion selected and statistical variability between the training and testing data sets. Second, the overall ability of the learning procedure to correctly infer the parameters of a truly pairwise-adjacent distribution was tested. 30 minutes worth of samples were generated from a pairwise-adjacent model, and a new pairwise-adjacent model was estimated on these simulated data samples. The newly trained model recovered an unbiased estimate of the parameters {λ

Models were compared using likelihood analysis. Details of this analysis are given elsewhere (Shlens et al., 2006). Briefly, the likelihood of a data set under a given model is the probability of having observed that data set under the assumption that the model is correct. For each of the statistics examined (Figure 3), the number of observations was counted, producing a histogram *c* = {*c*_{i}}, where *m* = Σ* _{i}c_{i}* is the total number of measurements. This produces an estimate of the probability of each value of the statistic,

The likelihood shrinks multiplicatively with increasing measurements (larger *m*). A quantity that is invariant to the number of measurements is the average likelihood The Kullback-Leibler divergence from information theory (*D*_{KL}) bears a simple relationship to the average likelihood:

in the limit of infinite measurements (*m* → ∞). Estimating the KL divergence in a finite data is nontrivial and subject to severe biases, especially in high-dimensional data. To address this issue, techniques for estimating entropy efficiently were used (Paninski, 2003; Krichevsky and Trofimov, 1981; Nemenman et al., 2002; Orlitsky et al., 2004).

The selection of time scale (or bin size) for defining synchronized firing could influence the results. Ideally, the choice of bin size should match the width of the cross-correlation peak (~10 ms; Figure 2b in (Shlens et al., 2006)), and encompass a significant fraction of the refractory period, in order to accurately reflect the strength of correlation while avoiding statistical dependencies associated with refractoriness. Selecting too small of a bin size could produce a poor estimate of correlation strength due to limited counting statistics. Selecting too large a bin size could mask synchrony by averaging over times outside the window of synchronization.

Subsampling analysis reveals that 512 electrode recordings capture scale of synchronized firing. Complete mosaic of neurons in four preparations (including Figure 1a–b; see letter labels) is systematically enlarged from center. Values reported **...**

A twofold change in the bin size did not significantly affect the results. For the preparation of Figure 3a (top panels), using bin sizes of 5 ms and 20 ms, the pairwise-adjacent model accounted for, respectively, (99.2%, 99.4%, 99.5%) and (99.0%, 98.2%, 98.6%) of the departures from independence in the three statistics examined, similar to the results obtained with a 10 ms bin size (see insets in Figure 3a). The fraction of firing entropy of an individual RGC predictable from its neighbors was also little affected by bin size. For the retina in Figure 4a, using bin sizes of 5 ms and 20 ms, 7.3%±0.9% and 5.1%±1.1% of firing entropy respectively was predictable from the firing of neighboring cells. Similar results were obtained in the presence of visual stimulation.

The pairwise-adjacent model failed to explain ~1–2% of the deviations from statistical independence (Figure 3). Several potential sources of the deviations were examined.

First, when the waveforms of spikes in different neurons overlap in time, the recorded (summed) voltage waveform differs from the waveform produced by either cell alone, potentially causing spikes from both cells to be missed in the spike sorting procedure and leading to an underestimate of synchronized firing events. To test for this possibility, the following control procedure was performed. 30 minutes of data was sampled from a pairwise-adjacent model fitted to recorded data. Then artifacts were introduced that mimicked the hypothetical spike sorting bias. In a first test, roughly 5% of pairwise synchronous events between adjacent neurons were removed. In a second test, 5% of triplet synchronous events between adjacent neurons were removed. A new pairwise-adjacent model was then fitted to and compared with the altered data sets. In both manipulations, no systematic biases were found in the three statistics of interest (Figure 3) and the model accounted for 99.3, 99.6 and 99.6% of the failures of statistical independence. Thus, artifacts associated with the spike sorting procedure are unlikely to entirely account for the observed failures of the model.

Second, to assess whether the failures were attributable to non-stationarity or finite counting statistics, the accuracy of the model was compared to that of a second firing pattern distribution (*P*_{emp}) which consists of the firing probabilities estimated directly from an independent period of recording from the same neurons. The accuracy of *P*_{emp} as a model for *P*_{obs} reveals the reproducibility of the experiment. *P*_{emp} always accounted for at least 99.5% of the deviations from statistical independence. Thus, the failures of the model cannot be explained entirely by counting statistics or non-stationarity of the recordings.

Third, the possibility of systematic deviations between the model and data were examined in more detail (Figure 3, bottom rows). In 11 of the 12 preparations examined, the model did not systematically overpredict or under-predict any of the three statistics examined. In one preparation (Figure 1a), the model over-predicted the frequency of large firing patterns but not small ones, and the over-prediction grew systematically with size of the firing pattern. Thus, although the results were not entirely consistent across data sets, systematic deviations could not explain the failures of the model in all data sets.

Finally, a potential source of the discrepancy between data and model is an implicit assumption that was made about the temporal structure of the data: that the sequential time samples from the recording are independent observations. This assumption does not take into account interactions over time in the firing of cells. A notable temporal interaction is action potential refractoriness, which reduces the probability that a neuron will generate a spike for a period of several milliseconds after the occurrence of a spike. For temporal interactions to have a significant effect on the results, the distribution of firing statistics must depend substantially on the pattern of activity in preceding time samples. To test this, time samples were divided into two groups: those for which the preceding time sample contained more than the median number of neurons firing, and those for which the preceding time sample contained fewer than the median. Each of the three statistics was then examined separately for the two groups of firing patterns. The effect of the preceding time sample was large. For example, in the preparation in Figure 1b, the Kullback-Leibler divergence between the full distribution of numbers of neurons firing and the group with high and low preceding firing was 1.74e-4 and 1.30e-4 bits/neuron respectively. These values represent 2.4% and 3.2% of the magnitude of the departures from statistical independence in the data, respectively. Similar departures were observed for the other statistics examined. Thus, the effect of temporal interactions on the measured firing pattern statistics was potentially large enough to account for the observed failures of the model.

Action potentials were recorded extracellularly from retinal ganglion cells (RGCs) in isolated macaque monkey retina perfused with physiological saline solution, using an array of 512 recording electrodes (Litke et al., 2004; Frechette et al., 2005). The receptive field of each RGC was identified using reverse correlation with a white noise stimulus (Chichilnisky, 2001). Cells were segregated into distinct functional classes according to their receptive field characteristics. ON-parasol cells, which project to the magnocellular layers of the lateral geniculate nucleus, were identified by their distinctive receptive field size, density, and light response kinetics (Chichilnisky and Kalmar, 2002; Field et al., 2007). As expected from previous work, the receptive fields of ON-parasol cells formed a regular lattice, or mosaic, uniformly sampling the region of retina recorded (Figure 1) (DeVries and Baylor, 1997; Chichilnisky and Kalmar, 2002; Frechette et al., 2005). The completeness of the recorded mosaics indicated that nearly every ON-parasol cell in this 4 × 8 degree region of retina was recorded in each of 12 preparations (*n* = 104, 48, 56, 56, 58, 66, 68, 75, 81, 66, 65 and 54 neurons). Subsets of one of these data sets were used in prior work (Shlens et al., 2006).

To examine multi-neuron synchronized firing, the activity of the entire population of ON-parasol cells was recorded in the presence of spatially uniform, steady photopic illumination. The prevalence of synchronized firing was determined by calculating the fraction of spikes in each cell that were synchronized with those of any of its immediate neighbors in the mosaic, over and above chance expectation (see Methods). Across four preparations this value ranged from 14% to 31%. Thus, a substantial fraction of the spikes produced by any given ON-parasol cell reflect synchronized firing.

To examine the spatial structure of multi-neuron synchronized firing, movies were generated in which each frame represented the activity of all cells in a 10 ms time bin. The receptive field of each cell was filled if the cell fired a spike (Supplementary Movie 1). Synchronized firing was examined at the 10 ms time scale because this time scale captured much of the auto-correlation within a cell and cross-correlation between cells (42% and 83% of variance, respectively; see Methods for expanded discussion of time scale) (Shlens et al., 2006). Several snapshots of activity are shown for two recordings in Figure 1a–b; these data were selected to emphasize synchronized firing. Synchronized firing occured at spatial scales substantially larger than the distance between nearest neighbors in the mosaic. Specifically, synchronized firing events including up to 44% of the population, and up to 26 spatially contiguous neurons, were observed in the four preparations examined. In one preparation (Figure 1a), firing patterns of this magnitude were observed every 30 minutes. If the cells fired independently, such patterns would occur only once every ~6,000 years. Therefore, the observed patterns of activity clearly deviate from the predictions of independence, and significantly exceed the scale of multi-neuron synchronized firing previously reported in the retina (Schnitzer and Meister, 2003; Shlens et al., 2006; Schneidman et al., 2006).

Some regularities in the patterns of synchronized firing are apparent from visual inspection (Figure 1a–b, Supplementary Movie 1). First, as expected from previous work (Shlens et al., 2006), synchronized firing was spatially *localized*: active cells tended to be clustered in a local region, the size of which covaried with the average receptive field size of the ON-parasol cells recorded. Second, synchronized firing was spatially *isotropic*: the strength of synchrony was approximately invariant to the orientation of the line joining a cell pair, exhibiting deviations from circular symmetry no larger than 7% (see Methods; data not shown). Third, synchrony was approximately *translation invariant*: its spatial extent was approximately independent of the location of the recorded cells: no distinct boundaries or stereotyped motifs appeared in recorded firing patterns (e.g. (Ikegaya et al., 2004)), and each cell was equally likely to exhibit synchronized firing with each of its neighbors. These features of synchronized firing were observed consistently in all preparations examined.

The above findings suggest that synchronized firing in ON-parasol cells may be summarized parsimoniously, provided that the measurements encompass a sufficiently large region of retina. The sufficiency of the recording was tested by examining three statistics of synchronized firing. The first statistic was the fraction of neurons firing in a given time sample. For example, in the bottom panel of Figure 1b, a total of 13 neurons fired (20%). The second statistic was the number of firing neurons in spatially contiguous groups consisting of more than one neuron (see Methods). For example, in the bottom panel of Figure 1b, two contiguous firing groups were observed, sizes 10 and 3. The third statistic was the fraction of pairs of adjacent firing cells. For example, in the bottom panel of Figure 1b, there were 18 pairs of adjacent firing cells (11%). (Note that a more complete approach to assess synchronized firing would be to estimate the frequency of all 2^{n} possible firing patterns in *n* neurons (Shlens et al., 2006), which is impractical when *n* is in the range of 50–100 as in the present data).

If the present recordings were sufficient to capture the full extent of synchronized firing, then the values of these statistics should reach asymptotes with a collection of cells no larger than the total number of cells recorded. To test this prediction, these statistics were calculated using spatially contiguous subsets of recorded cells in four preparations (Figure 2). For small groups of cells similar to those examined in previous work (Schneidman et al., 2006; Shlens et al., 2006) (e.g. 7–10 neurons), the three statistics either exhibited large variance or assumed mean values significantly different from the values observed with the entire population (gray bar). However, these statistics approached asymptotic values, with gradually shrinking variance, with about 40 cells.

Large synchronized firing patterns appeared not to represent singular events, but instead samples from a continuum, as shown by the distributions of the three statistics in Figure 3a (top row, grey bars). These distributions also reveal the spatial extent of synchronized firing. For example, in the recording of Figure 1a, for over 99.9% of the recorded patterns, no more than 25 neurons (40%), and no more than 15 spatially contiguous neurons (23%), fired synchronously. Finally, the three statistics may be used to summarize quantitatively the departures from statistical independence in the population (Figure 3a; top row, dashed curves; Supplementary Movie 4). The large observed deviations reflect the prominence of synchronized firing in the population.

Can the observed patterns of synchronized firing be explained with a simple model? One possibility is that they can be understood entirely on the basis of measured interactions between pairs of cells (Schneidman et al., 2006; Shlens et al., 2006; Tang et al., 2008), or (most simply) between pairs of adjacent cells in the mosaic (Shlens et al., 2006). This hypothesis was tested using the observed firing pattern statistics (Figure 3, top row).

In order to generate predictions of population responses based purely on pairwise and adjacent interactions, the *maximum entropy* methodology developed in recent work was used (Schneidman et al., 2003; Schneidman et al., 2006; Shlens et al., 2006). To introduce the approach, let *x*_{i} {1,0} indicate whether neuron *i* fires (or not) during a single time bin, and let *P*_{obs}(*x _{i}*) indicate the probability that it fires (or not). Similarly, let

To test the model, simulated firing patterns were generated and compared to the data. Qualitatively, the accuracy of the model predictions may be assessed visually by comparing movies generated from the model and data (Supplementary Movie 1 and Supplementary Movie 3). Quantitatively, the firing patterns produced by the model closely matched all three measured statistics of the data (Figure 3a, top row, solid curves). Strikingly, the model correctly predicted the frequency of spatially contiguous islands of up to at least 30 firing neurons, even though model parameters were constrained only by the firing probabilities of pairs of adjacent cells. Across 4 preparations the observed frequency of firing patterns matched that predicted from the fitted pairwiseadjacent model (Figure 3a, bottom row). Large events encompassing > 40% of neurons firing or > 15 contiguous neurons firing (Figure 3a, bottom row, grey box) occurred infrequently and were subject to large counting variability. In spite of this variability, the pairwise-adjacent model produced an unbiased prediction of each firing pattern across preparations.

To summarize these findings, the similarity of the distributions of the three statistics for the model (*P _{model}*) and the data (

The accurate fit of the pairwise-adjacent model made possible an analysis of the influence of shared signals on the activity of each cell. A natural measure of shared signals is the *information* that the activity of the surrounding population contains about the activity of the reference cell. Specifically, the information about the activity of a reference cell, *y*, contained in the activity of its neighbors, *x*_{1},…,*x _{n}*, is measured by the reduction in uncertainty about the activity of the reference cell obtained from knowledge of the neighbor activity

First, the information that the activity of a single neighboring cell contained about the activity of a reference cell was estimated, i.e. *I*(*y*; *x*_{1}). For five example cells, this quantity was small, reflecting weak pairwise coupling (Figure 4b, left-most points). Next, the analysis was repeated for increasing numbers of neighbors, accumulated in order of distance, up to *n* = 15 cells. For cells in Figure 4b, the amount of information in the surrounding activity increased with the number of neighbors considered, reaching an asymptotic maximum of 0.06 ±0.01 bits with 15 cells. Across all reference cells in this data set (excluding cells with fewer than 6 recorded neighbors, to avoid edge effects), the total information about the activity of the reference cell contained in its 15 nearest neighbors was 0.056±0.0087 bits. In three other preparations this value was, 0.094±0.025 bits, 0.08±0.011 bits and 0.039±0.01 bits (Figure 4c). Just 6 immediate neighbors provided the large majority (90%±7.4%) of the information about the firing of the reference cell (Figure 4c).

To assess the relative strength of network interactions, the information *I*(*y*; *x*_{1}) was compared with the entropy of the firing of the reference cell, *H*(*y*). For the preparation in Figure 4a, this comparison is plotted with red points in Figure 4d. Note that the variation in entropy across reference cells solely reflects the variation in firing rates of the individual cells. A best fit line (not shown) indicates that the information was roughly 7.0±1.1% of the entropy of the firing of the reference cell (see Methods for analysis across bin sizes). Across three other preparations, the network activity accounted for 18.9±3.2%, 11.2±1.7% and 6.5±1.5% of the entropy. In summary, no more than 20% of the uncertainty in the firing of an individual cell was accounted for by the activity of the surrounding population (Figure 4d, dashed line).

The above results refer only to synchronized firing in the spontaneous activity of RGCs. However, the relevance of synchronized firing for retinal function ultimately depends on its role in visual signaling to the brain, and it is possible that the influence of the network on an individual cell changes when RGC firing is modulated by a visual stimulus (Figure 1c–d, Supplementary Movie 2). To test this possibility, the pairwise adjacent model was fitted to data obtained in the presence of a white noise stimulus, in which the intensity and color of each pixel of the display varied randomly and independently over time. The stimulus was strong enough to drive firing in the cells: it was used to extract the receptive fields shown in Figure 1. However, the pixels were small enough that they generated no more than 8.4% additional synchronized firing in adjacent cells (as determined by shuffle correction (Perkel et al., 1967; Palm et al., 1988)). Thus, this stimulus provided a test of whether visually modulated activity that is approximately independent in each cell alters the relative influence of the network.

Figure 3b and Figure 4e shows the results from 12 retinas. The pairwise-adjacent model accurately predicted the frequency of firing patterns in the retina from Figure 1d (Figure 3b, top row) as well as 11 other retinas (Figure 3b, bottom row). Importantly, the pairwise-adjacent model predicted the frequency of large-scale firing events (> 15 contiguous neurons; Figure 3b, bottom row, grey box) and accounted for 98.0–99.8% of the deviations from statistical independence. For the same four preparations in Figure 4d, the activity in the neighbors accounted for 6.3±1.0%, 18.6±3.3%, 9.5±1.5% and 5.1±1.4% of the activity of the reference cell, respectively (Figure 4e). Thus, as the visual function of RGCs was engaged by independent stimulus modulations, the pairwise-adjacent model remained accurate and the relative influence of shared and unique signals in each cell remained roughly fixed.

The present findings reveal the large-scale structure of multi-neuron synchronized firing in a population of RGCs that transmits a complete visual representation to the brain. ON-parasol cells fired synchronized spikes in large, spatially contiguous groups, and 15–30% of spikes in each cell were associated with multineuron synchronized firing not expected by chance. However, a simple model, based on pairwise interactions between immediate neighbors in the population, accounted for 99% of the spatial scale and structure of synchronized firing. These neighbor interactions accurately predicted the frequencies of the largest patterns of electrical activity measured. Despite strong synchronized firing, the activity of any given cell was only weakly redundant: no more than 20% of the firing of each cell was predictable from the activity of its neighbors. These findings were virtually unchanged by independent visual stimulation of the cells. In summary, synchronized firing in a complete population of ON-parasol cells appears to reflect simple neighbor interactions, rather than a unique collective visual signal or a highly redundant coding scheme.

A previous finding in larval tiger salamander retina indicated that RGCs often fire in large groups, raising the possibility that synchronized firing patterns represent a unique, elementary symbol in the neural code of the retina (Schnitzer and Meister, 2003). However, because that study lacked information about the spatial structure and cell type organization of synchronized firing, it left open the question of what the unique symbol could be. Other studies have also examined multi-neuron synchronized firing, but at a scale significantly smaller than the patterns of activity in the neural population, providing an incomplete picture (Schneidman et al., 2006; Shlens et al., 2006).

The present work characterizes the spatial “footprint” of synchronized firing in one cell type at a sufficiently large scale to capture all the apparent structure. The data did not reveal singular and unique synchronized firing events in the ON-parasol population, nor any particular tendency toward large-scale synchronization. Instead, the data suggest that large-scale synchronized firing events form part a continuum of patterns of collective electrical activity produced by neighbor interactions.

Although synchronized firing was strong and structured, the activity of each ON-parasol cell was only weakly redundant with the activity of surrounding ON-parasol cells. This finding relates to two predictions from previous studies.

One study of RGCs in larval salamander retina predicted, based on an extrapolation from small groups of cells, that interactions among all RGCs in a small region of the retina could produce highly ordered or “freezing” behavior (Schneidman et al., 2006), a phenomenon not observed in the present work. However, two important differences in the studies bear mention. First, the previous study was performed using stimuli with substantial spatial redundancy, which would be expected to elevate synchronized firing (for discussion, see (Nirenberg and Victor, 2007; Bethge and Berens, 2007; Roudi et al., in press)). Second, the extrapolation from the data to the complete RGC population was based on the approximation that correlated activity between pairs of cells is independent of distance within the recorded region. This approximation implies that the total interaction strength would grow as the square of the number of cells. In the present work, examining an essentially complete collection of cells uniformly tiling a large region of retina, correlated activity declines systematically with distance (Shlens et al., 2006) and is captured by local interactions between neighboring cells. Thus the total interaction strength scales linearly with the number of cells, and the prediction of what could occur in a fully connected local population (Schneidman et al., 2006) remains untested.

In a similar vein, a second study of larval salamander RGCs revealed substantial (up to ~40%) redundancy in the visual signals carried by pairs of RGCs, particularly in cell pairs with highly overlapping receptive fields (Puchalla et al., 2005). In principle, this high pairwise redundancy could produce even higher redundancy in a full population of such cells. By contrast, the present results show that the information carried about the activity of a ON-parasol cell by all the surrounding ON-parasol cells is no more than ~20%. This value provides a bound on the visual signal redundancy in the entire ON-parasol population (Equation 11 in (Schneidman et al., 2003)). However, it does not provide a measure of redundancy across all RGC populations in a local region of retina.

A simple hypothesis may reconcile the present and previous findings: firing and visual signals are largely non-redundant among RGCs of a single type, which exhibit limited receptive field overlap, but highly redundant across all RGCs of different types in a small region of retina, which exhibit substantial receptive field overlap. This would be predicted if the dominant source of noise in the retina were photoreceptor noise (see below), because RGCs with overlapping receptive fields presumably sample from the same photoreceptors.

It is worth noting that different RGC types project to distinct targets in the brain and probably subserve distinct visual functions (Berson, 2008). Therefore, high redundancy between different cell types in a local region may reveal little about the functional impact of synchronized firing, whereas low redundancy within a cell type over an extended region may be relevant for understanding downstream computations.

Previous studies have revealed that synchronized firing in RGCs of several species arises from a combination of gap junction coupling and common synaptic input (Dacey and Brace, 1992; Jacoby et al., 1996; Stafford and Dacey, 1997; Hu and Bloomfield, 2003; Hidaka et al., 2004; Mastronarde, 1989; Brivanlou et al., 1998). In principle, either of these mechanisms could mediate synchronized firing observed here. Recent work using paired intracellular recordings indicates that gap junction coupling (potentially through amacrine cells (Dacey and Brace, 1992)) and correlated synaptic input contribute to synchronized firing among ONparasol cells, although the latter dominates (Trong and Rieke, 2008). Although the present work provides no additional insight into mechanism, it does suggest that the mechanisms underlying synchronized firing in pairs of neighboring ON-parasol cells may suffice to explain the patterns of activity in the entire network. One possibility is that a single noise source, namely photoreceptor noise, dominates both firing variability in individual RGCs and shared variability in neighboring RGCs ((Trong and Rieke, 2008); F. Rieke, personal communication).

It remains possible that novel stimuli could activate distinct retinal circuits that would create more complex patterns of RGC activity than would be produced by neighbor interactions. For example, recent work has shown that looming stimuli (Ishikane et al., 2005) and reversing moving stimuli (Schwartz et al., 2007) generate unique patterns of synchronized firing in frog and salamander RGCs, respectively. Whether such effects are present in the ON-parasol population remains to be seen.

The present results highlight the value of statistical approaches to probe the effect of synchronized firing on visual signals. Although the maximum entropy approach has the benefit of relying on few assumptions, its utility may be limited to examining simultaneous activity because of the large data sets required to probe firing dynamics and stimulus dependencies. An alternative approach is a parametric model of light response that includes interactions in the spiking activity of adjacent cells and also provides an explicit, optimal decoding algorithm for the neural signals (Pillow et al., 2008). This model may be useful for probing the temporal structure of synchronized firing and its role in visual signaling.

The regularity of the observed firing patterns made it possible to summarize their structure and scale with a few statistics and a simple model. Previous findings suggest that such a summary of network interactions may prove more difficult in cortical circuits (Yoshimura et al., 2005; Yoshimura and Callaway, 2005; Callaway, 1998), where the anatomical and functional organization may not be as simple or as precisely registered as in the retina. However, it remains possible that the functional organization of at least some central circuits is simple, even though their anatomical arrangement is complicated by factors such as multiple overlaid maps of sensory variables (e.g. primary visual cortex (Ohki et al., 2005; Ohki et al., 2006)). Furthermore, increasing knowledge about cell type organization in the brain (Callaway, 2005; Gobel et al., 2007; Mitchell et al., 2007) might reveal a more regular structure in patterned firing within each defined neural population.

Click here to view.^{(18K, pdf)}

Click here to view.^{(10M, mov)}

Click here to view.^{(9.9M, mov)}

Click here to view.^{(8.9M, mov)}

Click here to view.^{(9.7M, mov)}

We thank D. Petrusca, W. Dabrowski, A. Grillo, M. Grivich, P. Grybos, P. Hottowy, and S. Kachiguine for technical development; F. Rieke, M.J. Black and L. Paninski for discussions, E. Simoncelli for comments on the manuscript, E. Callaway, H. Fox, M. Taffe and K. Osborn for providing access to retinas, C. Hulse for technical assistance, and S. Barry for machining. This research was supported in part by the National Science Foundation through TeraGrid resources provided by the San Diego Supercomputer Center. This work was supported by NSF IGERT training grant DGE-0333451 and Miller Institute for Basic Research in Science (JS), DAAD and DFG (MG), NIH NRSA and Chapman Foundation (JLG), Helen Hay Whitney Foundation (GDF), Burroughs Wellcome Fund Career Award at Scientific Interface (AS), NSF grant PHY-0417175 (AML) and NIH grant EY017736 (EJC).

- Ackert J, Wu S, Lee J, Abrams J, Hu E, Perlman I, Bloomfield S. Light-induced changes in spike synchronization between coupled ON direction selective ganglion cells in the mammalian retina. J Neurosci. 2006;26:4206–4215. [PubMed]
- Amari S. Information geometry on hierarchy of probability distributions. IEEE Transactions on Information Theory. 2001;47:1701–1711.
- Amthor F, Tootle J, Grzywacz N. Stimulus-dependent correlated firing in directionally selective retinal ganglion cells. Vis Neurosci. 2005;22:769–787. [PubMed]
- Arnett D. Statistical dependence between neighboring retinal ganglion cells in goldfish. Exp Brain Res. 1978;32:49–53. [PubMed]
- Arnett D, Spraker T. Cross-correlation analysis of the maintained discharge of rabbit retinal ganglion cells. J Physiol. 1981;317:29–47. [PubMed]
- Baccus S. Timing and computation in inner retinal circuitry. Annu Rev Physiol. 2007;69:271–290. [PubMed]
- Baccus S, Meister M. Fast and slow contrast adaptation in retinal circuitry. Neuron. 2002;36:909–919. [PubMed]
- Berger A, Della Pietra S, Della Pietra C. A maximum entropy approach to natural language processing. Computational Linguistics. 1996;22:1–36.
- Berson D. Retinal ganglion cell types and their central projections. In: Masland R, editor. The Senses: A Comprehensive Reference. Oxford, UK: Elsevier; 2008.
- Bethge M, Berens P. Near-maximum entropy models for binary neural representations of natural images. Adv. Neural Information Processing Systems. 2007;20:1–8.
- Brivanlou I, Warland D, Meister M. Mechanisms of concerted firing among retinal ganglion cells. Neuron. 1998;20:527–539. [PubMed]
- Callaway E. Local circuits in primary visual cortex of the macaque monkey. Annu Rev Neurosci. 1998;21:47–74. [PubMed]
- Callaway E. A molecular and genetic arsenal for systems neuroscience. Trends Neurosci. 2005;28:196–201. [PubMed]
- Castleman KR. Digital image processing. Englewood Cliffs, N.J: Prentice Hall; 1996.
- Chichilnisky E. A simple white noise analysis of neuronal light responses. Network: Computation in Neural Systems. 2001;12:199–213. [PubMed]
- Chichilnisky E, Kalmar R. Functional asymmetries in ON and OFF ganglion cells of primate retina. J Neurosci. 2002;22:2737–2747. [PubMed]
- Cover T, Thomas J. Elements of information theory. Wiley, New York: Wiley series in telecommunications; 1991. Vol.
- Csiszar I. I divergence geometry of probability distributoins and minimization problems. Annals of Probability. 1975;3:146–158.
- Dacey D, Brace S. A coupled network for parasol but not midget ganglion cells in the primate retina. Vis Neurosci. 1992;9:279–290. [PubMed]
- Darroch J, Ratcliff D. Generalized iterative scaling for log-linear models. Ann Math Stats. 1972;43:1470–1480.
- DeVries S. Correlated firing in rabbit retinal ganglion cells. J Neurophysiol. 1999;81:908–920. [PubMed]
- DeVries S, Baylor D. Mosaic arrangement of ganglion cell receptive fields in rabbit retina. J Neurophysiol. 1997;78:2048–2060. [PubMed]
- Duda R, Hart P, Stork D. Pattern classification. New York: Wiley & Sons; 2001.
- Field G, Sher A, Gauthier J, Greschner M, Shlens J, Litke A, Chichilnisky E. Spatial properties and functional organization of small bistratified ganglion cells in primate retina. J Neurosci. 2007;27:13261–13272. [PubMed]
- Frechette E, Sher A, Grivich M, Petrusca D, Litke A, Chichilnisky E. Fidelity of the ensemble code for visual motion in primate retina. J Neurophysiol. 2005;94:119–135. [PubMed]
- Gamerman D, Lopes HF. Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition (Texts in Statistical Science Series) Chapman & Hall/CRC; 2006.
- Gobel W, Kampa B, Helmchen F. Imaging cellular network dynamics in three dimensions using fast 3D laser scanning. Nat Methods. 2007;4:73–79. [PubMed]
- Harris K. Neural signatures of cell assembly organization. Nat Rev Neurosci. 2005;6:399–407. [PubMed]
- Hertz J, Krogh A, Palmer RG. Introduction to the theory of neural computation. v. 1. Redwood City, Calif: Addison-Wesley Pub. Co; 1991. Vol. Santa Fe Institute studies in the sciences of complexity.Lecture notes.
- Hidaka S, Akahori Y, Kurosawa Y. Dendrodendritic electrical synapses between mammalian retinal ganglion cells. J Neurosci. 2004;24:10553–10567. [PubMed]
- Hinton G. Training products of experts by minimizing contrastive divergence. Neural Comput. 2002;14:1771–1800. [PubMed]
- Hinton G, Sejnowski T. Learning and relearning in boltzmann machines. In: Rumelhart D, McClelland J, editors. Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 1. Cambridge, MA: MIT Press; 1986. pp. 45–76. Foundations.
- Hu E, Bloomfield S. Gap junctional coupling underlies the short-latency spike synchrony of retinal alpha ganglion cells. J Neurosci. 2003;23:6768–6777. [PubMed]
- Ikegaya Y, Aaron G, Cossart R, Aronov D, Lampl I, Ferster D, Yuste R. Synfire chains and cortical songs: temporal modules of cortical activity. Science. 2004;304:559–564. [PubMed]
- Ishikane H, Gangi M, Honda S, Tachibana M. Synchronized retinal oscillations encode essential information for escape behavior in frogs. Nat Neurosci. 2005;8:1087–1095. [PubMed]
- Jacoby R, Stafford D, Kouyama N, Marshak D. Synaptic inputs to ON parasol ganglion cells in the primate retina. J Neurosci. 1996;16:8041–8056. [PubMed]
- Jaynes E. Information theory and statistical mechanics. Phys Rev. 1957a;106:620–630.
- Jaynes E. Information theory and statistical mechanics II. Phys Rev. 1957b;108:171–190.
- Jordan MI. Learning in Graphical Models (Adaptive Computation and Machine Learning) The MIT Press; 1998.
- Kennel M, Shlens J, Abarbanel H, Chichilnisky E. Estimating entropy rates with bayesian confidence intervals. Neural Comput. 2005;17:1531–1576. [PubMed]
- Krichevsky R, Trofimov V. The performance of universal coding. IEEE Trans. Inf. Theo. 1981;28:199–207.
- Laurent G. Olfactory network dynamics and the coding of multidimensional signals. Nat Rev Neurosci. 2002;3:884–895. [PubMed]
- Litke A, Bezayiff N, Chichilnisky E, Cunningham W, Dabrowski W, Grillo A, Grivich M, Grybos P, Hot-towy P, Kachiguine S, Kalmar R, Mathieson K, Petrusca D, Rahman M, Sher A. What does the eye tell the brain? development of a system for the large scale recording of retinal output activity. IEEE Trans. Nucl. Sci. 2004:1434–1440.
- MacNeil M, Masland R. Extreme diversity among amacrine cells: implications for function. Neuron. 1998;20:971–982. [PubMed]
- Malouf R. A comparison of algorithms for maximum entropy parameter estimation. Stockholm, Sweden. Proceedings of the Sixth Conference on Natural Language Learning (CoNLL-2002); 2002. pp. 49–55. Morgan Kauffman.
- Marmarelis P, Naka K. White-noise analysis of a neuron chain: an application of the wiener theory. Science. 1972;175:1276–1278. [PubMed]
- Masland R. The fundamental plan of the retina. Nat Neurosci. 2001;4:877–886. [PubMed]
- Mastronarde D. Correlated firing of cat retinal ganglion cells. i. spontaneously active inputs to x-and y-cells. J Neurophysiol. 1983a;49:303–324. [PubMed]
- Mastronarde D. Correlated firing of cat retinal ganglion cells. II. responses of x-and y-cells to single quantal events. J Neurophysiol. 1983b;49:325–349. [PubMed]
- Mastronarde D. Interactions between ganglion cells in cat retina. J Neurophysiol. 1983c;49:350–365. [PubMed]
- Mastronarde D. Correlated firing of retinal ganglion cells. Trends Neurosci. 1989;12:75–80. [PubMed]
- Meister M. Multineuronal codes in retinal signaling. Proc Natl Acad Sci USA. 1996;93:609–614. [PubMed]
- Meister M, Lagnado L, Baylor D. Concerted signaling by retinal ganglion cells. Science. 1995;270:1207–1210. [PubMed]
- Mitchell J, Sundberg K, Reynolds J. Differential attention-dependent response modulation across cell classes in macaque visual area v4. Neuron. 2007;55:131–141. [PubMed]
- Narayanan N, Kimchi E, Laubach M. Redundancy and synergy of neuronal ensembles in motor cortex. J Neurosci. 2005;25:4207–4216. [PubMed]
- Nemenman I, Shafee F, Bialek W. Advances in neural information processsing systems. Vol. 14. Cambridge, MA: MIT Press; 2002. Entropy and inference, revisited.
- Nirenberg S, Carcieri S, Jacobs A, Latham P. Retinal ganglion cells act largely as independent encoders. Nature. 2001;411:698–701. [PubMed]
- Nirenberg S, Victor J. Analyzing the activity of large populations of neurons: how tractable is the problem? Curr Opin Neurobiol. 2007 [PMC free article] [PubMed]
- Ohki K, Chung S, Ch’ng Y, Kara P, Reid R. Functional imaging with cellular resolution reveals precise micro-architecture in visual cortex. Nature. 2005;433:597–603. [PubMed]
- Ohki K, Chung S, Kara P, Hubener M, Bonhoeffer T, Reid R. Highly ordered arrangement of single neurons in orientation pinwheels. Nature. 2006;442:925–928. [PubMed]
- Okabe A, Boots B, Sugihara K, Chiu SN. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley: (Wiley Series in Probability and Statistics); 2000.
- Olveczky B, Baccus S, Meister M. Segregation of object and background motion in the retina. Nature. 2003;423:401–408. [PubMed]
- Orlitsky A, Santhanam N, Zhang J. Universal compression of memoryless sources over unknown alphabets. IEEE Trans. Inf. Theo. 2004;50:1469–1481.
- Palm G, Aertsen A, Gerstein G. On the significance of correlations among neuronal spike trains. Biol Cybern. 1988;59:1–11. [PubMed]
- Paninski L. Convergence properties of three spike-triggered analysis techniques. Network. 2003;14:437–464. [PubMed]
- Perez P. Markov random fields and images. CWI Quarterly. 1998;11:413–437.
- Perkel D, Gerstein G, Moore G. Neuronal spike trains and stochastic point processes. II. simultaneous spike trains. Biophys J. 1967;7:419–440. [PubMed]
- Pillow J, Shlens J, Paninski L, Sher A, Litke A, Chichilnisky E, Simoncelli E. Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature. 2008;454:995–999. [PMC free article] [PubMed]
- Press W, Flannery B, Teukolsky S, Vetterling W. Numerical recipes in C. Cambridge, England: Cambridge University Press; 1988.
- Puchalla J, Schneidman E, Harris R, Berry M. Redundancy in the population code of the retina. Neuron. 2005;46:493–504. [PubMed]
- Roerdink J, Meijster A. The watershed transform: Definitions, algorithms and parallelization strategies. Fundamenta Informaticae. 2001;41:187–228.
- Roudi Y, Nirenberg S, Latham P. Pairwise maximum entropy models for studying large biological systems: when they can work and when they can not. PLos Computational Biology. in press. [PMC free article] [PubMed]
- Schneidman E, Berry M, Segev R, Bialek W. Weak pairwise correlations imply strongly correlated network states in a neural population. Nature. 2006;440:1007–1012. [PMC free article] [PubMed]
- Schneidman E, Bialek W, Berry M. Synergy, redundancy, and independence in population codes. J Neurosci. 2003;23:11539–11553. [PubMed]
- Schneidman E, Still S, Berry M, Bialek W. Network information and connected correlations. Phys Rev Lett. 2003;91:238701. [PubMed]
- Schnitzer M, Meister M. Multineuronal firing patterns in the signal from eye to brain. Neuron. 2003;37:499–511. [PubMed]
- Schwartz G, Taylor S, Fisher C, Harris R, Berry Mn Synchronized firing among retinal ganglion cells signals motion reversal. Neuron. 2007;55:958–969. [PMC free article] [PubMed]
- Shlens J, Field G, Gauthier J, Grivich M, Petrusca D, Sher A, Litke A, Chichilnisky E. The structure of multi-neuron firing patterns in primate retina. J Neurosci. 2006;26:8254–8266. [PubMed]
- Stafford D, Dacey D. Physiology of the a1 amacrine: a spiking, axon-bearing interneuron of the macaque monkey retina. Vis Neurosci. 1997;14:507–522. [PubMed]
- Swendsen R, Wang J. Nonuniversal critical dynamics in monte carlo simulations. Physical Review Letters. 1987;58:86–88. [PubMed]
- Tang A, Jackson D, Hobbs J, Chen W, Smith J, Patel H, Prieto A, Petrusca D, Grivich M, Sher A, Hottowy P, Dabrowski W, Litke A, Beggs J. A maximum entropy model applied to spatial and temporal correlations from cortical networks in vitro. J Neurosci. 2008;28:505–518. [PubMed]
- Trong P, Rieke F. Origin of correlated activity between parasol retinal ganglion cells. Nat Neurosci. 2008 [PMC free article] [PubMed]
- Usrey W, Reid R. Synchronous activity in the visual system. Annu Rev Physiol. 1999;61:435–456. [PubMed]
- Voronoi G. Nouvelles applications des parameètres continus à la theéorie des formes quadratiques. Journal fuür die Reine und Angewandte Mathematik. 1907;133:97–178.
- X L, S R, D H, Black M. European Conference on Computer Vision. Vol. II. Springer; 2006. Efficient belief propagation with learned higher-order markov random fields; pp. 269–282.
- Yoshimura Y, Callaway E. Fine-scale specificity of cortical networks depends on inhibitory cell type and connectivity. Nat Neurosci. 2005;8:1552–1559. [PubMed]
- Yoshimura Y, Dantzker J, Callaway E. Excitatory cortical neurons form fine-scale functional networks. Nature. 2005;433:868–873. [PubMed]

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's Canada Institute for Scientific and Technical Information in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |