# A Simple Mechanism for Beyond-Pairwise Correlations in Integrate-and-Fire Neurons

- David A. Leen
^{1}and - Eric Shea-Brown
^{1, 2, 3, 4}Email author

**5**:17

https://doi.org/10.1186/s13408-015-0030-9

© Leen and Shea-Brown 2015

**Received: **31 October 2014

**Accepted: **23 July 2015

**Published: **1 September 2015

## Abstract

The collective dynamics of neural populations are often characterized in terms of *correlations* in the spike activity of different neurons. We have developed an understanding of the circuit mechanisms that lead to correlations among cell pairs, but little is known about what determines the population firing statistics among larger groups of cells. Here, we examine this question for a simple, but ubiquitous, circuit feature: common fluctuating input arriving to spiking neurons of integrate-and-fire type. We show that this leads to strong beyond-pairwise correlations—that is, correlations that cannot be captured by maximum entropy models that extrapolate from pairwise statistics—as for earlier work with discrete threshold crossing (dichotomous Gaussian) models. Moreover, we find that the same is true for another widely used, doubly stochastic model of neural spiking, the linear–nonlinear cascade. We demonstrate the strong connection between the collective dynamics produced by integrate-and-fire and dichotomous Gaussian models, and show that the latter is a surprisingly accurate model of the former. Our conclusion is that beyond-pairwise correlations can be both broadly expected and possible to describe by simplified (and tractable) statistical models.

## Keywords

## 1 Introduction

Interest in the collective dynamics of neural populations is rapidly increasing, as new recording technologies yield views into neural activity on larger and larger scales, and new statistical analyses yield potential consequences for the neural code [4, 5, 9, 20, 28, 34]. A fundamental question that arises as we seek to quantify these population dynamics is the statistical *order* of correlations among spiking activity in different neurons. That is, can the co-dependence of spike events in a set of neurons be described by an (overlapping) set of correlations among pairs of neurons, or are there irreducible higher-order dependencies as well? Recent studies show that purely pairwise statistical models are successful in capturing the spike outputs of neural populations under some stimulus conditions [22, 27, 28]. At the same time, different stimuli or different (or larger) populations can produce beyond-pairwise correlations [9, 16, 18, 31, 33]. In these studies, and in the present paper, beyond-pairwise correlations are defined by comparing with a pairwise maximum entropy (PME) model of spike trains: that is, a statistical model built with minimal assumptions about collective spiking beyond the rates of spiking in single cells and correlations in the spikes from cell pairs.

Despite these rich empirical findings, we are only beginning to understand what *dynamical* features of neural circuits determine whether or not they will produce substantial beyond-pairwise *statistical* correlations. Recent work has suggested that one of these mechanisms is common—or correlated—input fluctuations arriving simultaneously at multiple neurons [1, 2, 10, 14, 23]; importantly, this is a feature that occurs in many neural circuits found in biology [3, 24, 32]. In particular, [1, 14] showed that common, Gaussian input fluctuations, when “dichotomized” so that inputs over a given threshold produce spikes, give rise to strong beyond-pairwise correlations in the spike output of large populations of cells. This is an interesting result, as a step function thresholding mechanism produces beyond-pairwise correlations in spike outputs starting with purely pairwise (Gaussian) inputs.

A natural question is whether more realistic, dynamical mechanisms of spike generation—beyond “static” step function transformations—will also serve to produce strong beyond-pairwise correlations based on common input processes. In this paper, we show that the answer is yes, and connect several widely used models of neural spiking to explain why. In particular we show that, in contrast to the PME, the dichotomous Gaussian model gives a highly accurate description of the complete correlation structure of an integrate-and-fire population with common inputs.

## 2 Results

### 2.1 An Exponential Integrate-and-Fire Population with Common Inputs

*N*exponential integrate-and-fire (EIF) neurons [6, 8]. Each cell’s membrane voltage \(V_{i}\), \(i=1,\ldots,N\), evolves according to

Each cell’s input current \(I_{i}(t)\) has a constant (DC) level *γ*, and a white noise term with amplitude *σ*. The noise term has two components. The first is the common input \(\xi_{c}(t)\), which is shared among all neurons. The second is an independent white noise \(\xi_{i}(t)\); the relative amplitudes are scaled so that the inputs to different cells are correlated with (Pearson’s) correlation coefficient *λ* (as in, e.g., [7, 13, 26], cf. [3]).

We quantify the population output by binning spikes with temporal resolution \(\Delta t = 10\mbox{ ms}\) (see Fig. 1). (On rare occasions (<0.4 % of the bins; see Fig. 1, caption) multiple spikes from the same neuron can occur in the same bin. These are considered as a single spike.) The spike *firing rate* is quantified by *μ*, the probability of a spike occurring in a bin for a given neuron. *Pairwise correlation* in the simultaneous spiking of neurons *i*, *j* is quantified by the correlation coefficient \(\rho =\operatorname{Cov}(n_{i},n_{j})/\mathrm{Var}\), where \(n_{i}\), \(n_{j}\) are the \(\{0,1\}\) spike events for the cells and Var is their (identical) variance \(\mu (1-\mu)\).

### 2.2 Emergence of Strong Beyond-Pairwise Correlations in EIF Populations

*k*out of the

*N*cells fire simultaneously (as in, e.g., [1, 2, 14, 16]). Figure 2(a) illustrates these distributions. The question we ask is: Do beyond-pairwise correlations play an important role in determining the population-wide spike-count distribution?

*μ*for each neuron and pairwise spike correlation

*ρ*for each pair of neurons, while making minimal further assumptions on the joint probability distribution [9, 16, 18, 31, 33], cf. [15, 30]. For a population of

*N*neurons with identical means

*μ*and pairwise correlations

*ρ*, as for our simple circuit model, the PME model gives a distribution of population spike counts,

*Z*is a normalization factor and parameters

*α*and

*β*are adjusted numerically [14]. (Specifically, we use the function fminunc to find parameters

*α*and

*β*which minimize the negative likelihood of spike counts

*k*from simulations of the EIF model, under the model \(P_{\mathrm{PME}}(k)\).)

Figure 2(a) demonstrates that, for small populations, the corresponding PME and EIF distributions are similar. However, for populations larger than about \(N=30\) neurons, strong differences emerge. This difference in population spike-count distributions demonstrates that the EIF model produces beyond-pairwise correlations that strongly impact the structure of population firing. This is because the moments of the population spike-count distribution at a given order are determined by the moments—and hence correlations—of spikes among sets of cells of up to that order. Because the PME and EIF models have matched first- and second-order moments but different population spike-count distributions, they must differ in their beyond-pairwise correlations.

*ρ*and mean firing rates

*μ*. Additionally, as in [14] (cf. [17]), the Jensen–Shannon divergence grows with increasing population size

*N*. Moreover, the divergence increases with increasing pairwise correlation and decreasing mean firing rate.

### 2.3 A Linear–Nonlinear Cascade Model That Approximates EIF Spike Activity and Produces Beyond-Pairwise Correlations

We next study the impact of common input on beyond-pairwise correlations in a widely used point process model of neural spiking. This is the linear–nonlinear cascade model, where each neuron fires as a (doubly stochastic) inhomogeneous Poisson process. We use a specific linear–nonlinear cascade model that is fit to EIF dynamics. This both establishes that the common input mechanism is sufficient to drive beyond-pairwise correlations in the cascade model, and develops a semi-analytic theory for the population statistics in the EIF system.

*F*[19]:

*N*neurons produces spikes independently. Thus, the probability of

*k*cells firing simultaneously is

*A*and static nonlinearity

*F*described above. We note that [25] derive a related expression for a different definition of synchronous output for a neural population.

This said, the LNL model does not produce a perfect fit to the EIF outputs, the most obvious problem being the overestimation of the zero spike probabilities, which in the \(N=100\) case are overestimated by almost 100 % (the tail probabilities are also underestimated). Notably, the LNL fits become almost perfect for lower correlations i.e. \(\rho= 0.05\) (data not shown). This suggests the discrepancies are due to failures of the LNL approximation for large fluctuations in the instantaneous spiking rates \(r(t)\) (see Fig. 3(b)); these fluctuations are smaller at lower correlation values, which lead to smaller signal currents in the LNL formulation. While further work would be required to trace the precise origin of this discrepancy, we conjecture that one factor is the lack of a refractory period in the LNL model, which will impact firing statistics most strongly during and after fluctuations to high instantaneous rates.

### 2.4 The Dichotomized Gaussian (DG) Model Gives an Excellent Description of the EIF Population Activity

So far we have studied the emergence of beyond-pairwise correlations in two spiking neuron models—the EIF model, described in terms of a stochastic differential equation, and the LNL model, which is a continuous-time reduction of the EIF to a doubly stochastic point process. Next, we show how these results connect to earlier findings for a more general and abstracted statistical model. This is the Dichotomous Gaussian (DG) model, which has been shown analytically to produce beyond-pairwise correlations and to describe empirical data from neural populations [1, 2, 14, 33].

In the DG framework, spikes either occur or fail to occur independently and discretely in each time bin. Specifically, at each time *N* neurons receive a correlated Gaussian input variable with mean *γ* and correlation *λ*. Each neuron applies a step nonlinearity (Heaviside function) to its inputs, spiking only if its input is positive. Input parameters *γ* and *λ* are chosen to match two target firing statistics: the spike rate *μ* and the correlation coefficient *ρ*.

*ρ*and firing rate

*μ*, the rest of their population statistics match almost exactly over the full range of population sizes, for firing rates \(\mu =0.1\) and a variety of correlation values

*ρ*. Panel b(ii) shows that the match degrades somewhat for higher firing rates.

*k*are. In prior work [14] it was shown that this statistic grows linearly (i.e., extensively) with population size

*N*for the DG model, and the figure shows that the same holds for the EIF and LIF models. This growth stands, as first noted by [14], in marked contrast to the heat capacity for the PME model, which saturates at a population of approximately \(N = 30\) neurons.

We next develop the mathematical connection between the DG and the EIF models, via our description of the LNL model above.

*i*th neuron, \(Z_{i} = \gamma+ \sqrt{1-\lambda} T_{i} + \sqrt{\lambda} c\). Here, \(T_{i}\) is a Gaussian random variable (with unit variance) which is independent for each neuron (the independent input),

*c*is a Gaussian random variable that is common input to all neurons (the common input), and

*γ*is a constant term giving the mean input. The probability of a spike is given by a step function applied to the input. For a given realization of the common input

*c*, this is \(P(Z_{i} > 0 | c)\). We can again define a “

*L*” function similar to that in Eq. (4):

*k*is similar to Eq. (5):

*λ*.

*c*variable we obtain

Thus, after the transformation the only difference between the LNL and DG models is the functions \(L(c)\) vs. \(\tilde{L}( f(c))\). Figure 6(b) shows that these functions largely agree over about 2 standard deviations of the Gaussian pdf of values of the common input signal *c*.^{1} This reveals why the LNL and DG—and, by extension, the EIF—models all produce such similar population-level outputs, including their higher-order structure.

## 3 Conclusion

We have shown that Exponential-Integrate-and-Fire (EIF) neurons receiving common input give rise to strong beyond-pairwise correlations—that is, distributions of population spike counts that cannot be described by a pairwise maximum entropy (PME) approach. Moreover, the population output that results can be predicted from a linear–nonlinear (LNL) cascade model, which forms a tractable reduction of the EIF neuron. Beyond giving an explicit formula for the EIF population spike-count distribution, our findings for the LNL-cascade model demonstrate that common input will drive beyond-pairwise correlations in a widely used class of point process models.

Finally, we show that there is a surprisingly exact connection between the population dynamics of the EIF- and LNL-cascade models and that of the (apparently) simpler Dichotomized Gaussian (DG) model of [1, 14]. The success of the DG model in capturing EIF population statistics is significant for two reasons. First, it suggests one reason why this abstracted model has been able to capture the population output recorded from spiking neurons [33]. Second, because the DG model is a special case of a Bernoulli generalized linear model (see the appendix), our finding indicates that this very broad class of statistical models may be able to capture the higher-order population activity in neural data. A key feature of these models would be the inclusion of common fluctuations in the spike probabilities of cells in each time bin (cf. [11]); such models can also be extended to include spike history-dependent terms. This would then capture an effect missing here: temporal correlations in spike trains (e.g., refractory effects).

## Declarations

### Acknowledgements

The authors thank Liam Paninski for helpful insights. This work was supported by the Burroughs Wellcome Fund Scientific Interfaces Program and NSF grant CAREER DMS-1056125. ESB gratefully acknowledges a Simons Fellowship in Mathematics, and wishes to thank the Allen Institute founders, Paul G. Allen and Jody Allen, for their vision, encouragement and support.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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