Minimum and Maximum Entropy Distributions for Binary Systems with Known Means and Pairwise Correlations

Abstract

Maximum entropy models are increasingly being used to describe the collective activity of neural populations with measured mean neural activities and pairwise correlations, but the full space of probability distributions consistent with these constraints has not been explored. We provide upper and lower bounds on the entropy for the minimum entropy distribution over arbitrarily large collections of binary units with any fixed set of mean values and pairwise correlations. We also construct specific low-entropy distributions for several relevant cases. Surprisingly, the minimum entropy solution has entropy scaling logarithmically with system size for any set of first- and second-order statistics consistent with arbitrarily large systems. We further demonstrate that some sets of these low-order statistics can only be realized by small systems. Our results show how only small amounts of randomness are needed to mimic low-order statistical properties of highly entropic distributions, and we discuss some applications for engineered and biological information transmission systems.

1. Introduction

Maximum entropy models are central to the study of physical systems in thermal equilibrium [1], and they have recently been found to model protein folding [2,3], antibody diversity [4], and neural population activity [5,6,7,8,9] quite well (see [10] for a different finding). In part due to this success, these types of models have also been used to infer functional connectivity in complex neural circuits [11,12,13] and to model collective phenomena of systems of organisms, such as flock behavior [14].

This broad application of maximum entropy models is perhaps surprising since the usual physical arguments involving ergodicity or equality among energetically accessible states are not obviously applicable for such systems, though maximum entropy models have been justified in terms of imposing no structure beyond what is explicitly measured [5,15]. With this approach, the choice of measured constraints fully specifies the corresponding maximum entropy model. More generally, choosing a set of constraints restricts the set of consistent probability distributions—the maximum entropy solution being only one of all possible consistent probability distributions. If the space of distributions were sufficiently constrained by observations so that only a small number of very similar models were consistent with the data, then agreement between the maximum entropy model and the data would be an unavoidable consequence of the constraints rather than a consequence of the unique suitability of the maximum entropy model for the dataset in question.

In the field of systems neuroscience, understanding the range of allowed entropies for given constraints is an area of active interest. For example, there has been controversy [5,16,17,18,19] over the notion that small pairwise correlations can conspire to constrain the behavior of large neural ensembles, which has led to speculation about the possibility that groups of ∼200 neurons or more might employ a form of error correction when representing sensory stimuli [5]. Recent work [20] has extended the number of simultaneously recorded neurons modeled using pairwise statistics up to 120 neurons, pressing closer to this predicted error correcting limit. This controversy is in part a reflection of the fact that pairwise models do not always allow accurate extrapolation from small populations to large ensembles [16,17], pointing to the need for exact solutions in the important case of large neural systems. Another recent paper [21] has also examined specific classes of biologically-plausible neural models whose entropy grows linearly with system size. Intriguingly, these authors point out that entropy can be subextensive, at least for one special distribution that is not well matched to most neural data (coincidentally, that special case was originally studied nearly 150 years ago [22] when entropy was a new concept, though not in the present context). Understanding the range of possible scaling properties of the entropy in a more general setting is of particular importance to neuroscience because of its interpretation as a measure of the amount of information communicable by a neural system to groups of downstream neurons.

Previous authors have studied the large scale behavior of these systems with maximum entropy models expanded to second- [5], third- [17], and fourth-order [19]. Here we use non-perturbative methods to derive rigorous upper and lower bounds on the entropy of the minimum entropy distribution for any fixed sets of means and pairwise correlations possible for arbitrarily large systems (Equations (1)–(9); Figure 1 and Figure 2). We also derive lower bounds on the maximum entropy distribution (Equation (8)) and construct explicit low and high entropy models (Equations (28) and (29)) for a broad array of cases including the full range of possible uniform first- and second-order constraints realizable for large systems. Interestingly, we find that entropy differences between models with the same first- and second-order statistics can be nearly as large as is possible between any two arbitrary distributions over the same number of binary variables, provided that the solutions do not run up against the boundary of the space of allowed constraint values (see Section 2.1 and Figure 3 for a simple illustration of this phenomenon). This boundary is structured in such a way that some ranges of values for low order statistics are only satisfiable by systems below some critical size. Thus, for cases away from the boundary, entropy is only weakly constrained by these statistics, and the success of maximum entropy models in biology [2,3,4,5,6,7,8,9,11,14], when it occurs for large enough systems [17], can represent a real triumph of the maximum entropy approach.

Our results also have relevance for engineered information transmission systems. We show that empirically measured first-, second-, and even third-order statistics are essentially inconsequential for testing coding optimality in a broad class of such systems, whereas the existence of other statistical properties, such as finite exchangeability [23], do guarantee information transmission near channel capacity [24,25], the maximum possible information rate given the properties of the information channel. A better understanding of minimum entropy distributions subject to constraints is also important for minimal state space realization [26,27]—a form of optimal model selection based on an interpretation of Occam’s Razor complementary to that of Jaynes [15]. Intuitively, maximum entropy models impose no structure beyond that needed to fit the measured properties of a system, whereas minimum entropy models require the fewest “moving parts” in order to fit the data. In addition, our results have implications for computer science as algorithms for generating binary random variables with constrained statistics and low entropy have found many applications (e.g., [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]).

Problem Setup

To make these ideas concrete, consider an abstract description of a neural ensemble consisting of N spiking neurons. In any given time bin, each neuron i has binary state $s_i$ denoting whether it is currently firing an action potential ($s_i=1$) or not ($s_i=0$). The state of the full network is represented by $\overrightarrow{s}=(s_1,\dots ,s_N)\in {\{0,1\}}^N$. Let $p(\overrightarrow{s})$ be the probability of state $\overrightarrow{s}$ so that the distribution over all $2^N$ states of the system is represented by $\mathbf{p}\in {[0,1]}^{2^N}$, ${\sum }_{\overrightarrow{s}}p(\overrightarrow{s})=1$. Although we will use this neural framework throughout the paper, note that all of our results will hold for any type of system consisting of binary variables.

In neural studies using maximum entropy models, electrophysiologists typically measure the time-averaged firing rates ${\mu }_i=〈s_i〉$ and pairwise event rates ${\nu }_{ij}=〈s_i{s}_j〉$ and fit the maximum entropy model consistent with these constraints, yielding a Boltzmann distribution for an Ising spin glass [44]. This “inverse” problem of inferring the interaction and magnetic field terms in an Ising spin glass Hamiltonian that produce the measured means and correlations is nontrivial, but there has been progress [19,45,46,47,48,49,50]. The maximum entropy distribution is not the only one consistent with these observed statistics, however. In fact, there are typically many such distributions, and we will refer to the complete set of these as the solution space for a given set of constraints. Little is known about the minimum entropy permitted for a particular solution space.

Our question is: Given a set of observed mean firing rates and pairwise correlations between neurons, what are the possible entropies for the system? We will denote the maximum (minimum) entropy compatible with a given set of imposed correlations up to order n by $S_n$ (${\tilde{S}}_n$). The maximum entropy framework [5] provides a hierarchical representation of neural activity: as increasingly higher order correlations are measured, the corresponding model entropy $S_n$ is reduced (or remains the same) until it reaches a lower limit. Here we introduce a complementary, minimum entropy framework: as higher order correlations are specified, the corresponding model entropy ${\tilde{S}}_n$ is increased (or unchanged) until all correlations are known. The range of possible entropies for any given set of constraints is the gap ($S_n-{\tilde{S}}_n$) between these two model entropies, and our primary concern is whether this gap is greatly reduced for a given set of observed first- or second-order statistics. We find that, for many cases, the gap grows linearly with the system size N, up to a logarithmic correction.

2. Results

We prove the following bounds on the minimum and maximum entropies for fixed sets of values of first and second order statistics, $\left\{{\mu }_i\right\}=\left\{〈s_i〉\right\}$ and $\left\{{\nu }_{ij}\right\}=\left\{〈s_i{s}_j〉\right\}$, respectively. All entropies are given in bits.

For the minimum entropy:

$${log}_2\left(\frac{N}{1+(N-1)\overline{\alpha }}\right)\le {\tilde{S}}_2\le {log}_2\left(1+\frac{N(N+1)}{2}\right),$$

(1)

where $\overline{\alpha }$ is the average of ${\alpha }_{ij}={(4{\nu }_{ij}-2{\mu }_i-2{\mu }_j+1)}^2$ over all $i,j\in \{1,\dots ,N\}$, $i\ne j$. Perhaps surprisingly, the scaling behavior of the minimum entropy does not depend on the details of the sets of constraint values—for large systems the entropy floor does not contain tall peaks or deep valleys as one varies $\left\{{\mu }_i\right\}$, $\left\{{\nu }_{ij}\right\}$, or N.

We emphasize that the bounds in Equation (1) are valid for arbitrary sets of mean firing rates and pairwise correlations, but we will often focus on the special class of distributions with uniform constraints:

$$\begin{array}{cc}\hfill {\mu }_i& =\mu ,\mathrm{for}\mathrm{all}i=1,\dots ,N\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\nu }_{ij}& =\nu ,\mathrm{for}\mathrm{all}i\ne j.\hfill \end{array}$$

(3)

The allowed values for $\mu$ and $\nu$ that can be achieved for arbitrarily large systems (see Appendix A) are

$${\mu }^2\le \nu \le \mu .$$

(4)

For uniform constraints, Equation (1) reduces to

$${log}_2\left(\frac{N}{1+(N-1)\alpha }\right)\le {\tilde{S}}_2\le {log}_2\left(1+\frac{N(N+1)}{2}\right),$$

(5)

where $\alpha (\mu ,\nu )={(4(\nu -\mu )+1)}^2$. In most cases, the lower bound in Equation (1) asymptotes to a constant, but in the special case for which $\mu$ and $\nu$ have values consistent with the global maximum entropy solution ($\mu =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ and $\nu =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$), we can give the higher bound:

$${log}_2\left(N\right)\le {\tilde{S}}_2\le {log}_2\left(N\right)+2.$$

(6)

For the maximum entropy, it is well known that:

$$S_2\le N,$$

(7)

which is valid for any set of values for $\left\{{\mu }_i\right\}$ and $\left\{{\nu }_{ij}\right\}$. Additionally, for any set of uniform constraints that can be achieved by arbitrarily large systems (Equation (4)) other than $\nu =\mu$, which corresponds to the case in which all N neurons are perfectly correlated, we can derive upper and lower bounds on the maximum entropy that each scale linearly with the system size:

$$w+xN\le S_2\le N,$$

(8)

where $w=w(\mu ,\nu )\ge 0$ and $x=x(\mu ,\nu )>0$ are independent of N.

An important class of probability distributions are the exchangeable distributions [23], which are distributions over multiple variables that are symmetric under any permutation of those variables. For binary systems, exchangeable distributions have the property that the probability of a sequence of ones and zeros is only a function of the number of ones in the binary string. We have constructed a family of exchangeable distributions, with entropy ${\tilde{S}}_2^{exch}$, that we conjecture to be minimum entropy exchangeable solutions. The entropy of our exchangeable constructions scale linearly with N:

$$C_{1}N-\mathcal{O}\left({log}_{2}N\right)\le {\tilde{S}}_2^{exch}\le C_{2}N,$$

(9)

where $C_1=C_1(\mu ,\nu )$ and $C_2=C_2(\mu ,\nu )$ do not depend on N. We have computationally confirmed that this is indeed a minimum entropy exchangeable solution for $N\le 200$.

Figure 1 illustrates the scaling behavior of these various bounds for uniform constraints in two parameter regimes of interest. Figure 2 shows how the entropy depends on the level of correlation ($\nu$) for the maximum entropy solution ($S_2$), the minimum entropy exchangeable solution(${\tilde{S}}_2^{exch}$), and a low entropy solution (${\tilde{S}}_2^{con}$), for a particular value of mean activity ($\mu =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$) at each of two system sizes, $N=5$ and $N=30$.

2.1. Limits on System Growth

Physicists are often faced with the problem of having to determine some set of experimental predictions (e.g., mean values and pairwise correlations of spins in a magnet) for some model system defined by a given Hamiltonian, which specifies the energies associated with any state of the system. Typically, one is interested in the limiting behavior as the system size tends to infinity. In this canonical situation, no matter how the various terms in the Hamiltonian are defined as the system grows, the existence of a well-defined Hamiltonian guarantees that there exists a (maximum entropy) solution for the probability distribution for any system size.

However, here we are studying the inverse problem of deducing the underlying model based on measured low-order statistics [19,45,46,47,48,49]. In particular, we are interested in the minimum and maximum entropy models consistent with a given set of specified means and pairwise correlations. Clearly, both of these types of (potentially degenerate) models must exist whenever there exists at least one distribution consistent with the specified statistics, but as we now show, some sets of constraints can only be realized for small systems.

A Simple Example

To illustrate this point, consider the following example, which is arguably the simplest case exhibiting a cap on system size. At least one system consisting of N neurons can be constructed to satisfy the uniform set of constraints: $\mu =0.1$ and $\nu =0.0094<{\mu }^2=0.01$, provided that $2\le N\le 150$, but no solution is possible for $N>150$. To prove this, we first observe that any set of uniform constraints that admits at least one solution must admit at least one exchangeable solution (see Appendix A and Appendix D). Armed with this fact, we can derive closed-form solutions for upper and lower bounds on the minimum value for $\nu$ consistent with N and $\mu$ (Appendix Equation (A25)), as well as the actual minimum value of $\nu$ for any given value of N, as depicted in Figure 3.

Thus, this system defined by its low-level statistics cannot be grown indefinitely, not because the entropy decreases beyond some system size, which is impossible (e.g., Theorem 2.2.1 of [25]), but rather, because no solution of any kind is possible for these particular constraints for ensembles of more than 150 neurons. We believe that this relatively straightforward example captures the essence of the more subtle phenomenon described by Schneidman and colleagues [5], who pointed out that a naive extrapolation of the maximum entropy model fit to their retinal data to larger system sizes results in a conundrum at around $N\approx 200$ neurons.

For our simple example, it was straightforward to decide what rules to follow when growing the system—both the means and pairwise correlations were fixed and perfectly uniform across the network at every stage. In practice, real neural activities and most other types of data exhibit some variation in their mean activities and correlations across the measured population, so in order to extrapolate to larger systems, one must decide how to model the distribution of statistical values involving the added neurons.

Fortunately, despite this complication, we have been able to derive upper and lower bounds on the minimum entropy for arbitrarily large N. In Appendix C, we derive a lower bound on the maximum entropy for the special case of uniform constraints achievable for arbitrarily large systems, but such a bound for the more general case would depend on the details of how the statistics of the system change with N.

Finally, we mention that this toy model can be thought of as an example of a “frustrated” system [51], in that traveling in a closed loop of an odd number of neurons involves an odd number of anticorrelated pairs. By “anticorrelated”, we mean that the probability of simultaneous firing of a pair of neurons is less than chance, $\nu <{\mu }^2$, but note that $\nu$ is always non-negative due to our convention of labeling active and inactive units with zeros and ones, respectively. However, frustrated systems are not typically defined by their correlational structure, but rather by their Hamiltonians, and there are often many Hamiltonians that are consistent with a given set of observed sets of constraints, so there does not seem to be a one-to-one relationship between these two notions of frustration. In the more general setting of nonuniform constraints, the relationship between frustrated systems as defined by their correlational structure and those that cannot be grown arbitrarily large is much more complex.

2.2. Bounds on Minimum Entropy

Entropy is a strictly concave function of the probabilities and therefore has a unique maximum that can be identified using standard methods [52], at least for systems that possess the right symmetries or that are sufficiently small. In Section 2.3, we will show that the maximum entropy $S_2$ for many systems with specified means and pairwise correlations scales linearly with N (Equation (8), Figure 1). We obtain bounds on minimum entropy by exploiting the geometry of the entropy function.

2.2.1. Upper Bound on the Minimum Entropy

We prove below that the minimum entropy distribution exists at a vertex of the allowed space of probabilities, where most states have probability zero [53]. Our challenge then is to determine in which vertex a minimum resides. The entropy function is nonlinear, precluding obvious approaches from linear programming, and the dimensionality of the probability space grows exponentially with N, making exhaustive search and gradient descent techniques intractable for $N\gtrsim 5$. Fortunately, we can compute a lower (upper) bound ${\tilde{S}}_2^{lo}$ (${\tilde{S}}_2^{hi}$) on the entropy of the minimum entropy solution for all N (Figure 1), and we have constructed two families of explicit solutions with low entropies (${\tilde{S}}_2^{con}$ and ${\tilde{S}}_2^{con2}$; Figure 1 and Figure 2) for a broad parameter regime covering all allowed values for $\mu$ and $\nu$ in the case of uniform constraints that can be achieved by arbitrarily large systems (see Equation (A24), Figure A2 in the Appendix).

Our goal is to minimize the entropy S as a function of the probabilities $p_i$ where

$$S\left(\mathbf{p}\right)=-\sum _{i=1}^{n_s}{p}_i{log}_2{p}_i,$$

(10)

$n_s$ is the number of states, the $p_i$ satisfy a set of $n_c$ independent linear constraints, and $p_i\ge 0$ for all i. For the main problem we consider, $n_s=2^N$. The number of constraints—normalization, mean firing rates, and pairwise correlations—grows quadratically with N:

$$\begin{array}{c}\hfill n_c=1+\frac{N(N+1)}{2}.\end{array}$$

(11)

The space of normalized probability distributions $\mathcal{P}=\{p:{\sum }_{i=1}^{n_s}{p}_i=1,p_i\ge 0\}$ is the standard simplex in $n_s-1$ dimensions. Each additional linear constraint on the probabilities introduces a hyperplane in this space. If the constraints are consistent and independent, then the intersection of these hyperplanes defines a $d=n_s-n_c$ affine space, which we call $\mathcal{C}$. All solutions are constrained to the intersection between $\mathcal{P}$ and $\mathcal{C}$, and this solution space is a convex polytope of dimension ≤d, which we refer to as $\mathcal{R}$. A point within a convex polytope can always be expressed as a linear combination of its vertices; therefore, if $\left\{{\mathbf{v}}_i\right\}$ are the vertices of $\mathcal{R}$, we may express

$$\mathbf{p}=\sum _i^{n_v}{a}_i{\mathbf{v}}_i,$$

(12)

where $n_v$ is the total number of vertices and ${\sum }_i^{n_v}{a}_i=1$.

Using the concavity of the entropy function, we will now show that the minimum entropy for a space of probabilities S is attained on one (or possibly more) of the vertices of that space. Moreover, these vertices correspond to probability distributions with small support—specifically a support size no greater than $n_c$ (see Appendix B). This means that the global minimum will occur at the (possibly degenerate) vertex that has the lowest entropy, which we denote as ${\mathbf{v}}_{*}$:

$$\begin{array}{cc}\hfill S\left(\mathbf{p}\right)& =S\left(\sum _i^{n_v}{a}_i{\mathbf{v}}_i\right)\hfill \\ & \ge \sum _i^{v_s}{a}_{i}S\left({\mathbf{v}}_i\right)\hfill \\ & \ge \left(\sum _i^{v_s}{a}_i\right)S\left({\mathbf{v}}_{*}\right)\hfill \\ & =S\left({\mathbf{v}}_{*}\right).\hfill \end{array}$$

(13)

It follows that ${\tilde{S}}_2=S\left({\mathbf{v}}_{*}\right)$.

Moreover, if a distribution satisfying the constraints exists, then there is one with at most $n_c$ nonzero $p_i$ (e.g., from arguments as in [41]). Together, these two facts imply that there are minimum entropy distributions with a maximum of $n_c$ nonzero $p_i$. This means that even though the state space may grow exponentially with N, the support of the minimum entropy solution for fixed means and pairwise correlations will only scale quadratically with N.

The maximum entropy possible for a given support size occurs when the probability distribution is evenly distributed across the support and the entropy is equal to the logarithm of the number of states. This allows us to give an upper bound on the minimum entropy as

$$\begin{array}{cc}\hfill {\tilde{S}}_2\le {\tilde{S}}_2^{hi}& ={log}_2\left(n_c\right)\hfill \\ & ={log}_2\left(1+\frac{N(N+1)}{2}\right)\hfill \\ & \approx 2{log}_2\left(N\right),N\gg 1.\hfill \end{array}$$

(14)

Note that this bound is quite general: as long as the constraints are independent and consistent this result holds regardless of the specific values of the $\left\{{\mu }_i\right\}$ and $\left\{{\nu }_{ij}\right\}$.

2.2.2. Lower Bound on the Minimum Entropy

We can also use the concavity of the entropy function to derive a lower bound on the entropy ${\tilde{S}}_2^{lo}$ as in Equation (1):

$${\tilde{S}}_2(N,\left\{{\mu }_i\right\},\left\{{\nu }_{ij}\right\})\ge {\tilde{S}}_2^{lo}={log}_2\left(\frac{N^2}{N+{\sum }_{i\ne j}{\alpha }_{ij}}\right),$$

(15)

where ${\tilde{S}}_2(N,\left\{{\mu }_i\right\},\left\{{\nu }_{ij}\right\})$ is the minimum entropy given a network of size N with constraint values $\left\{{\mu }_i\right\}$ and $\left\{{\nu }_{ij}\right\}$, and the sum is taken over all $i,j\in \{1,\dots ,N\}$, $i\ne j$ and ${\alpha }_{ij}={(4{\nu }_{ij}-2{\mu }_i-2{\mu }_j+1)}^2$. Taken together with the upper bound, this fully characterizes the scaling behavior of the entropy floor as a function of N.

To derive this lower bound, note that the concavity of the entropy function allows us to write

$$S\left(\mathbf{p}\right)\ge -{log}_2{\parallel \mathbf{p}\parallel }_2^2.$$

(16)

Using this relation to find a lower bound on $S\left(\mathbf{p}\right)$ requires an upper bound on ${\parallel \mathbf{p}\parallel }_2^2$ provided by the Frobenius norm of the correlation matrix $C\equiv \langle \overrightarrow{s}{\overrightarrow{s}}^T\rangle$ (where the states are defined to take vaues in ${\{-1,1\}}^N$ rather than ${\{0,1\}}^N$)

$${\parallel \mathbf{p}\parallel }_2^2\le \frac{{\parallel C\parallel }_F^2}{N^2}.$$

(17)

In this case, ${\parallel C\parallel }_F^2$ is a simple function of the $\left\{{\mu }_i\right\}$, $\left\{{\nu }_{ij}\right\}$:

$${\parallel C\parallel }_F^2=N+\sum _{i\ne j}{(4{\nu }_{ij}-2{\mu }_i-2{\mu }_j+1)}^2.$$

(18)

Using Equations (17) and (18) in Equation (16) gives us our result (see Appendix H for further details).

Typically, this bound asymptotes to a constant as the number of terms included in the sum over ${\alpha }_{ij}$ scales with the number of pairs ($\mathcal{O}(N^2)$), but in certain conditions this bound can grow with the system size. For example, in the case of uniform constraints, this reduces to

$${\tilde{S}}_2^{lo}={log}_2\left(\frac{N}{1+(N-1)\alpha }\right),$$

(19)

where $\alpha (\mu ,\nu )={(4(\nu -\mu )+1)}^2$.

In the special case

$$\nu =\mu -\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.,$$

(20)

$\alpha$ vanishes allowing the bound in Equation (15) to scale logarithmically with N.

In the large N limit for uniform constraints, we know $\mu \ge \nu \ge {\mu }^2$ (see Appendix A), therefore the only values of $\mu$ and $\nu$ satisfying Equation (20) are

$$\mu =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,\nu ={\mu }^2=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right..$$

(21)

Although here the lower bound grows logarithmically with N, rather than remaining constant, for many large systems this difference is insignificant compared with the linear dependence $S_0=N$ of the maximum entropy solution (i.e., N fair i.i.d. Bernoulli random variables). In other words, the gap between the minimum and maximum possible entropies consistent with the measured mean activities and pairwise correlations grows linearly in these cases (up to a logarithmic correction), which is as large as the gap for the space of all possible distributions for binary systems of the same size N with the same mean activities but without restriction on the correlations.

2.3. Bounds on Maximum Entropy

An upper bound on the maximum entropy is well-known and easy to state. For a given number of allowed states $n_s$, the maximum possible entropy occurs when the probability distribution is equally distributed over all allowed states (i.e., the microcanonical ensemble), and the entropy is equal to

$$\begin{array}{cc}\hfill S_2\le S_2^{hi}& ={log}_2\left(n_s\right)\hfill \\ & =N.\hfill \end{array}$$

(22)

Lower bounds on the maximum entropy are more difficult to obtain (see Appendix C for further discussion on this point.). However, by restricting ourselves to uniform constraints achievable by arbitrarily large systems (Equations (2) and (3)), we can construct a distribution that provides a useful lower bound on the maximum possible entropy consistent with these constraints:

$$S_2\ge S_2^{con}=w+xN,$$

(23)

where

$$\begin{array}{cc}\hfill w=& -\beta {log}_2\beta -(1-\beta ){log}_2(1-\beta ),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill x=& (1-\beta )\left[-\eta {log}_2\eta -(1-\eta ){log}_2(1-\eta )\right]\hfill \end{array}$$

(25)

and

$$\begin{array}{cc}\hfill \beta & =\frac{\nu -{\mu }^2}{1+\nu -2\mu },\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \eta & =\frac{\mu -\nu }{1-\mu }.\hfill \end{array}$$

(27)

It is straightforward to verify that w and x are nonnegative constants for all allowed values of $\mu$ and $\nu$ that can be achieved for arbitrarily large systems. Importantly, x is nonzero provided $\nu <\mu$ (i.e., $\beta \ne 1\cap \eta \ne \{0,1\}$), so the entropy of the system will grow linearly with N for any set of uniform constraints achievable for arbitrarily large systems except for the case in which all neurons are perfectly correlated.

2.4. Low-Entropy Solutions

In addition to the bounds we derived for the minimum entropy, we can also construct probability distributions between these minimum entropy boundaries for distributions with uniform constraints. These solutions provide concrete examples of distributions that achieve the scaling behavior of the bounds we have derived for the minimum entropy. We include these so that the reader may gain a better intuition for what low entropy models look like in practice. We remark that these models are not intended as an improvement over maximum entropy models for any particular biological system. This would be an interesting direction for future work. Nonetheless, they are of practical importance to other fields as we discuss further in Section 2.6.

Each of our low entropy constructions, ${\tilde{S}}_2^{con}$ and ${\tilde{S}}_2^{con2}$, has an entropy that grows logarithmically with N (see Appendix E, Appendix F and Appendix G, Equations (1)–(6)):

$$\begin{array}{cc}\hfill {\tilde{S}}_2^{con}& =⌈{log}_2\left(N\right)+1⌉\hfill \\ & \le {log}_2\left(N\right)+2,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\tilde{S}}_2^{con2}& \le {log}_2\left({⌈N⌉}_p({\lceil N\rceil }_p-1)\right)-1+{log}_2\left(3\right)\hfill \\ & \le {log}_2\left(N(2N-1)\right)+{log}_2\left(3\right),\hfill \end{array}$$

(29)

where $\lceil .\rceil$ is the ceiling function and ${\lceil .\rceil }_p$ represents the smallest prime at least as large as its argument. Thus, there is always a solution whose entropy grows no faster than logarithmically with the size of the system, for any observed levels of mean activity and pairwise correlation.

As illustrated in Figure 1a, for large binary systems with uniform first- and second-order statistics matched to typical values of many neural populations, which have low firing rates and correlations slightly above chance ([5,6,7,8,9,11,16]; $\mu =0.1$, $\nu =0.011$), the range of possible entropies grows almost linearly with N, despite the highly symmetric constraints imposed (Equations (2) and (3)).

Consider the special case of first- and second-order constraints (Equation (21)) that correspond to the unconstrained global maximum entropy distribution. For these highly symmetric constraints, both our upper and lower bounds on the minimum entropy grow logarithmically with N, rather than just the upper bound as we found for the neural regime (Equation (6); Figure 1a). In fact, one can construct [21,22] an explicit solution (Equation (28); Figure 1b and Figure 2a,d,e,h) that matches the mean, pairwise correlations, and triplet-wise correlations of the global maximum entropy solution whose entropy ${\tilde{S}}_2^{con}$ is never more than two bits above our lower bound (Equation (15)) for all N. Clearly then, these constraints alone do not guarantee a level of independence of the neural activities commensurate with the maximum entropy distribution. By varying the relative probabilities of states in this explicit construction we can make it satisfy a much wider range of $\mu$ and $\nu$ values than previously considered, covering most of the allowed region (see Appendix F and Appendix G) while still remaining a distribution whose entropy grows logarithmically with N.

2.5. Minimum Entropy for Exchangeable Distributions

We consider the exchangeable class of distributions as an example of distributions whose entropy must scale linearly with the size of the system unlike the global entropy minimum which we have shown scales logarithmically. If one has a principled reason to believe some system should be described by an exchangeable distribution, the constraints themselves are sufficient to drastically narrow the allowed range of entropies although the gap between the exchangeable minimum and the maximum will still scale linearly with the size of the system except in special cases. This result is perhaps unsurprising as the restriction to exchangeable distributions is equivalent to imposing a set of additional constraints (e.g., $p\left(100\right)=p\left(010\right)=p\left(001\right)$, for $N=3$) that is exponential in the size of the system.

While a direct computational solution to the general problem of finding the minimum entropy solution becomes intractable for $N\gtrsim 5$, the situation for the exchangeable case is considerably different. In this case, the high level of symmetry imposed means that there are only $n_s=N+1$ states (one for each number of active neurons) and $n_c=3$ constraints (one for normalization, mean, and pairwise firing). This makes the problem of searching for the minimum entropy solution at each vertex of the space computationally tractable up into the hundreds of neurons.

Whereas the global lower bound scales logarithmically, our computation illustrates that the exchangeable case scales with N as seen in Figure 1. The large gap between ${\tilde{S}}_2^{exch}$ and ${\tilde{S}}_2$ demonstrates that a distribution can dramatically reduce its entropy if it is allowed to violate sufficiently strong symmetries present in the constraints. This is reminiscent of other examples of symmetry-breaking in physics for which a system finds an equilibrium that breaks symmetries present in the physical laws. However, here the situation can be seen as reversed: Observed first and second order statistics satisfy a symmetry that is not present in the underlying model.

2.6. Implications for Communication and Computer Science

These results are not only important for understanding the validity of maximum entropy models in neuroscience, but they also have consequences in other fields that rely on information entropy. We now examine consequences of our results for engineered communication systems. Specifically, consider a device such as a digital camera that exploits compressed sensing [54,55] to reduce the dimensionality of its image representations. A compressed sensing scheme might involve taking inner products between the vector of raw pixel values and a set of random vectors, followed by a digitizing step to output N-bit strings. Theorems exist for expected information rates of compressed sensing systems, but we are unaware of any that do not depend on some knowledge about the input signal, such as its sparse structure [56,57]. Without such knowledge, it would be desirable to know which empirically measured output statistics could determine whether such a camera is utilizing as much of the N bits of channel capacity as possible for each photograph.

As we have shown, even if the mean of each bit is $\mu =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, and the second- and third-order correlations are at chance level ($\nu =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$; $〈s_i{s}_j{s}_k〉=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.$, for all sets of distinct $\{i,j,k\}$), consistent with the maximum entropy distribution, it is possible that the Shannon mutual information shared by the original pixel values and the compressed signal is only on the order of ${log}_2\left(N\right)$ bits, well below the channel capacity (N bits) of this (noiseless) output stream. We emphasize that, in such a system, the transmitted information is limited not by corruption due to noise, which can be neglected for many applications involving digital electronic devices, but instead by the nature of the second- and higher-order correlations in the output.

Thus, measuring pairwise or even triplet-wise correlations between all bit pairs and triplets is insufficient to provide a useful floor on the information rate, no matter what values are empirically observed. However, knowing the extent to which other statistical properties are obeyed can yield strong guarantees of system performance. In particular, exchangeability is one such constraint. Figure 1 illustrates the near linear behavior of the lower bound on information (${\tilde{S}}_2^{exch}$) for distributions obeying exchangeability, in both the neural regime (cyan curve, panel (a)) and the regime relevant for our engineering example (cyan curve, panel (b)). We find experimentally that any exchangeable distribution has as much entropy as the maximum entropy solution, up to terms of order ${log}_2\left(N\right)$ (see Appendicies).

This result has potential applications in the field of symbolic dynamics and computational mechanics, which study the consequences of viewing a complex system through a finite state measuring device [27,58]. If we view each of the various models presented here as a time series of binary measurements from a system, our results indicate that bitstreams with identical mean and pairwise statistics can have profoundly different scaling as a function of the number of measurements (N), indicating radically different complexity. It would be interesting to explore whether the models presented here appear differently when viewed through the $ϵ$-machine framework [27].

In computer science, it is sometimes possible to construct efficient deterministic algorithms from randomized ones by utilizing low entropy distributions. One common technique is to replace the independent binary random variables used in a randomized algorithm with those satisfying only pairwise independence [59]. In many cases, such a randomized algorithm can be shown to succeed even if the original independent random bits are replaced by pairwise independent ones having significantly less entropy. In particular, efficient derandomization can be accomplished in these instances by finding pairwise independent distributions with small sample spaces. Several such designs are known and use tools from finite fields and linear codes [33,34,60,61,62], combinatorial block designs [32,63], Hadamard matrix theory [42,64], and linear programming [41], among others [22]. Our construction here of two families of low entropy distributions fit to specified mean activities and pairwise statistics adds to this literature.

3. Discussion

Ideas and approaches from statistical mechanics are finding many applications in systems neuroscience [65,66]. In particular, maximum entropy models are powerful tools for understanding physical systems, and they are proving to be useful for describing biology as well, but a deeper understanding of the full solution space is needed as we explore systems less amenable to arguments involving ergodicity or equally accessible states. Here we have shown that second order statistics do not significantly constrain the range of allowed entropies, though other constraints, such as exchangeability, do guarantee extensive entropy (i.e., entropy proportional to system size N).

We have shown that in order for the constraints themselves to impose a linear scaling on the entropy, the number of experimentally measured quantities that provide those constraints must scale exponentially with the size of the system. In neuroscience, this is an unlikely scenario, suggesting that whatever means we use to infer probability distributions from the data (whether maximum entropy or otherwise) will most likely have to agree with other, more direct, estimates of the entropy [67,68,69,70,71,72,73]. The fact that maximum entropy models chosen to fit a somewhat arbitrary selection of measured statistics are able to match the entropy of the system they model lends credence to the merits of this approach.

Neural systems typically exhibit a range of values for the correlations between pairs of neurons, with some firing coincidently more often than chance and others firing together less often than chance. Such systems can exhibit a form of frustration, such that they apparently cannot be scaled up to arbitrarily large sizes in such a way that the distribution of correlations and mean firing rates for the added neurons resembles that of the original system. We have presented a particularly simple example of a small system with uniform pairwise correlations and mean firing rates that cannot be grown beyond a specific finite size while maintaining these statistics throughout the network.

We have also indicated how, in some settings, minimum entropy models can provide a floor on information transmission, complementary to channel capacity, which provides a ceiling on system performance. Moreover, we show how highly entropic processes can be mimicked by low entropy processes with matching low-order statistics, which has applications in computer science.