The Abelian distribution

TL;DR

This paper introduces the Abelian distribution, explores its fundamental properties, and demonstrates its relevance in modeling neural avalanches within fully-connected integrate-and-fire neural systems exhibiting self-organized criticality.

Contribution

It defines the Abelian distribution and investigates its properties, linking it to neural modeling and avalanche size distributions.

Findings

Abelian distribution models neural avalanche sizes.

Properties of Abelian distribution are characterized.

Application to neural systems demonstrates relevance.

Abstract

We define the Abelian distribution and study its basic properties. Abelian distributions arise in the context of neural modeling and describe the size of neural avalanches in fully-connected integrate-and-fire models of self-organized criticality in neural systems.

Equations

$$P_{\alpha ,N}(L)=C_{\alpha ,N}\left(\genfrac{}{}{0ex}{}{N}{L}\right){\left(L\frac{\alpha }{N}\right)}^{L-1}{\left(1-L\frac{\alpha }{N}\right)}^{N-L-1},$$

$$C_{\alpha ,N}=\frac{1-\alpha }{N-(N-1)\alpha },$$

$$\sum _{L=1}^N{C}_{\alpha ,N}\left(\genfrac{}{}{0ex}{}{N}{L}\right){\left(L\frac{\alpha }{N}\right)}^{L-1}{\left(1-L\frac{\alpha }{N}\right)}^{N-L-1}=1.$$

$$\sum _{L=1}^N\left(\genfrac{}{}{0ex}{}{N}{L}\right){\left(Lx\right)}^{L-1}{\left(1-Lx\right)}^{N-L-1}=\frac{1}{C_{\alpha ,N}},$$

$$\sum _{L=1}^{N-1}\left(\genfrac{}{}{0ex}{}{N}{L}\right){\left(Lx\right)}^{L-1}{\left(1-Lx\right)}^{N-L-1}=\frac{1}{C_{\alpha ,N}}-\frac{(Nx{)}^{N-1}}{1-Nx}.$$

$$\sum _{L=1}^{N-1}\left(\genfrac{}{}{0ex}{}{N}{L}\right){\left(Lx\right)}^{L-1}\sum _{m=0}^{N-L-1}(-1{)}^m(\genfrac{}{}{0ex}{}{N-L-1}{m})(Lx{)}^m.$$

$$\sum _{i=0}^{N-2}{x}^i\sum _{k=0}^i(-1{)}^{i-k}(\genfrac{}{}{0ex}{}{N}{k}\left)\right(\genfrac{}{}{0ex}{}{N-k-1}{i-k})(k{)}^i=\sum _{i=0}^{N-2}{P}_i(N)x^i,$$

$$P_i(N)=(-1{)}^{i-k}(\genfrac{}{}{0ex}{}{N}{N}\left)\right(\genfrac{}{}{0ex}{}{-1}{i-k})N^i=N^i\mathrm{for}N=1,\dots ,i+1\mathrm{and}i>0.$$

$$1+\sum _{i=1}^{N-2}{x}^i{N}^{i-1}=1+x\frac{1-(Nx{)}^{N-2}}{1-Nx}.$$

$$1+x\frac{1-(Nx{)}^{N-2}}{1-Nx}=\frac{N-(N-1)\alpha }{N(1-\alpha )}-\frac{(Nx{)}^{N-1}}{N(1-Nx)},$$

$$E\xi =\frac{N}{N-(N-1)\alpha }.$$

$$E\xi =\sum _{L=1}^N{L}^{L-1}\left(\genfrac{}{}{0ex}{}{N-1}{L-1}\right){\left(\frac{\alpha }{N}\right)}^{L-1}{\left(1-L\frac{\alpha }{N}\right)}^{N-L-1}\frac{N(1-\alpha )}{N-(N-1)\alpha }.$$

$$\sum _{L=1}^N{L}^{L-1}\left(\genfrac{}{}{0ex}{}{N-1}{L-1}\right){\left(\frac{\alpha }{N}\right)}^{L-1}{\left(1-L\frac{\alpha }{N}\right)}^{N-L-1}=\frac{1}{1-\alpha }.$$

$$\sum _{L=1}^N\left(\genfrac{}{}{0ex}{}{N-1}{L-1}\right)(Lx{)}^{L-1}{\left(1-Lx\right)}^{N-L-1}=\frac{1}{1-Nx}.$$

$$\sum _{L=1}^{N-1}\left(\genfrac{}{}{0ex}{}{N-1}{L-1}\right)(Lx{)}^{L-1}{\left(1-Lx\right)}^{N-L-1}+(Nx{)}^{N-1}(1-Nx{)}^{-1}=\frac{1}{1-Nx}.$$

$$\sum _{L=1}^{N-1}\left(\genfrac{}{}{0ex}{}{N-1}{L-1}\right)(Lx{)}^{L-1}{\left(1-Lx\right)}^{N-L-1}=\sum _{i=0}^{N-2}(Nx{)}^i.$$

$$\sum _{k=0}^i(-1{)}^{i-k}(\genfrac{}{}{0ex}{}{N-1}{k}\left)\right(\genfrac{}{}{0ex}{}{N-k-2}{i-k})(k+1{)}^i=N^i.$$

$$(-1{)}^{i-k}(\genfrac{}{}{0ex}{}{N-1}{N-1}\left)\right(\genfrac{}{}{0ex}{}{-1}{i-k})N^i=N^i.$$

Full text

power-law distribution; stable distribution; Abelian sum

I Introduction

In the present manuscript we introduce Abelian distributions. We have called the distribution Abelian because of a number of identities that arise in analysis and that resemble the Abel identity $(x+y)^{n}=\sum_{i=0}^{n}{n\choose i}x(x-iz)^{i-1}(y+iz)^{n-i}$ Saslaw1989 . This distribution appeared in 2002 in the study of a fully connected neural network Eurich2002 as a distribution of sizes of “avalanches” of neural activity. Apart from Ref. LevinaDiss , so far there no systematic and accessible study of the distribution has been published. The related results that were reported in the context of Cayley’s theorem Denker2011 are also based on Ref. LevinaDiss . Here we will discuss the basic properties of this probability mass distribution and describe its importance for the applications in theoretical physics and biology.

II Definition

Definition II.1**.**

Let $N\in\mathbb{N}$, $\alpha\in(0,1)$. The Abelian distribution is defined for $0\leq L\leq N$ by

[TABLE]

where

[TABLE]

is a normalizing constant.

Because $P_{\alpha,N}(0)=0$ we will in the following often assume that $L>0$.

Lemma II.1**.**

The Abelian distribution defined by (1),(2) is a probability distribution.

Proof.

We have to show that

[TABLE]

Introducing a new continuous variable $x$ instead of $\alpha/N$, we get

[TABLE]

which is equivalent to

[TABLE]

We can expand the sum on the left side of (3) and obtain

[TABLE]

Introducing $k=L$ we can rewrite the sum in the previous expression as a polynomial in $x$

[TABLE]

where $P_{i}(N)$ is a polynomial in $N$ of degree $i$. For every $N$ we have $P_{0}(N)=1$. Consider now $i>0$. To identify uniquely the polynomial $P_{i}(N)$ it is sufficient to find its values in $i+1$ different points that we select to be $N=1,\ldots,i+1$. Because ${{N-1\choose k}}=0$ for $k>N-1$, we have also ${N-k-2\choose i-k}=0$ for $N<i+2$ for any $k<N-1$. Hence,

[TABLE]

This means that $P_{i}(N)=N^{i-1}$ for any $N$ and $i>0$. Therefore the left side of (4) is

[TABLE]

Inserting (5) and (2) into (3) we arrive at

[TABLE]

which holds for any $N$ and $\alpha<1$. ∎

The authors of Ref. Denker2011 mention that the theorem can also be proved by using a generalized binomial theorem.

An Abelian-distributed probability mass function is shown in Fig. 1 for several values of the parameter $\alpha$. For small values of parameter $\alpha<0.9$ distribution is monotone and is dominated by approximately exponential decay, for $\alpha\lessapprox 1$ distribution is non-monotonous. For some small interval of parameter values $\alpha\approx 0.9$ the distribution closely resembles a power-law (with exponential cutoff at large $L$), see the double logarithmic plot in the inset. If a sample of data-points of size $10^{5}$ is drawn from this distribution, the hypothesis of an underlying power-law distribution cannot be rejected LevinaDiss .

The shape of the distribution varies in a similar way for all $N$, although for large $N$ the non-monotonous regime is present only for $\alpha\in\left(\alpha_{\mathrm{crit}}(N),1\right)$ where the value of $\alpha_{\mathrm{crit}}(N)$ has been numerically found to behave roughly as $1-\frac{1}{\sqrt{N}}$.

III Expected value

We will now consider the moments of the Abelian distribution.

Theorem III.1**.**

Suppose $\xi$ has an Abelian distribution with parameters $\alpha$ and $N$, then

[TABLE]

Proof.

From (1) and Lemma II.1 we have

[TABLE]

We have to prove that

[TABLE]

Using again $x=\alpha/N$ we can rewrite this equation as

[TABLE]

Transforming the sum in (6) we obtain

[TABLE]

which is equivalent to

[TABLE]

Both the left and the right side of the equation (7) are polynomials in $x$ of degree $N-2$. Hence in order to prove that equation (7) is an identity it is sufficient to show that the coefficients of $x^{i}$ on the both sides are equal for every $i$. In other words, we have to show that

[TABLE]

Again, both sides of (8) are polynomials of $N$ of the degree $i$. It is sufficient to prove that both sides of (8) are equal for $i+1$ different points. We can select these points to be $N=1,\ldots,i+1$.

Obviously, if $k>N-1$, then ${{N-1\choose k}}=0$, but also ${N-k-2\choose i-k}=0$ for $k<N-1$ because $N<i+2$. Hence the only non-zero item of the sum is the one corresponding to $k=N-1$, in this case we have

[TABLE]

∎

IV Motivation

Power-law distributions have been studied in the sciences for a long time, the most prominent example being the Gutenberg-Richter law which describes the energy distribution in earthquakes Gutenberg1954 . Other examples Bakb include forest fires, migratory patterns, infectious diseases, solar flares, sandpiles Bak1987 and neural activity dynamics Eurich2002 ; Beggs2003 ; Levina2007 ; Levina2009 . Some of these examples can be related to critical branching processes kolmogorov1947branching which are known to produce power-law event distributions Otter1949 . The relation between power-laws and branching processes usually requires a limit of large systems size Levina2008 which is, however, not relevant when a comparison to numerical computations or mesoscopic experiments is desired. Nevertheless, the Abelian distribution converges to a power-law (asymptotically for large event sizes $L\to\infty$ or as an event density) in the exchangeable limits $N\to\infty$ and $\alpha\to 1$. The exponent of the power-law $\gamma=-\frac{3}{2}$ is closely obeyed even for small $L$. Criticality being defined as the divergence of certain physical quantities (such as the mean event size) cannot occur in finite systems. Therefore it is tempting to use the Abelian distribution to define an analogon of criticality also for finite systems. Depending on the parameters the Abelian distribution has monotonic or non-monotonic behavior, the latter being characterized by a relative dominance of events with a size near the size of the system. The two behaviors, the sub- and the supercritical regime are separated by a “critical” distribution, which is, however, unambiguously defined in terms of a power-law only for large systems. Avoiding the dependence of the critical parameters on the sample size that may arise when using a test (e.g. Kolmogorov-Smirnov) in order to determine the likelihood of criticality, we propose instead to define criticality by qualitative criteria implied by the local similarity to a power-law. Consider the set $A(N)$ of parameters $\alpha$ for which the equation $\frac{d^{2}\log P_{\alpha,N}(L)}{d(\log L)^{2}}=0$ has a solution $L\in\{1,\dots,N\}$ for fixed $N<\infty$. Expecting $A(N)$ to contract into $\{1\}$ for $N\to\infty$, we can define $A(N)$ as the critical region for a finite system. Another possibility is to define a single critical value $\alpha_{\mathrm{crit}}$ as an $\sup_{\alpha<1}\left\{\alpha:\frac{d\log P_{\alpha,N}(L)}{dL}<0,\forall L<N\right\}$. This definition uses the property of a critical state to stay between strictly monotonous and non-monotonous regimes. For all our numerical evaluations we found $\alpha_{\mathrm{crit}}\in A(N)$. Thus, the Abelian distribution is one of the few cases where the emergence of criticality in an infinite system can be studied explicitly as a limit of finite systems which enables a direct comparison with numerical computations or mesoscopic experiments.

The Abelian distribution has been studied mainly in the context of neural avalanche dynamics Eurich2002 ; LevinaDiss , where it not only turned out be successful in predicting an experimental result from neuroscience Beggs2003 , but also allowed for an explicit and exact study of finite size effects. It is interpreted in this context as the conditional probability of $L-1$ other neurons being activated given that one neuron just became spontaneously active, thus forming an avalanche of $L$ neural action potentials. From Theorem III.1 follows that the expectation exists also in the limit of large $N$ if $\alpha<1$ as required by Definition 1. Correspondingly, in the neural system, a single nonterminating avalanche is observed at $\alpha=1$.

The application of the Abelian distribution as an event size distribution may require an appropriate definition of events. Although neurons produce quasi-discrete action potentials, in the experiments Beggs2003 events have been defined by threshold crossings, where an invariance of the distribution of the choice of the threshold is required for justification. In other time series, events can be defined either in a similar way. While the parameter $N$ has usually a natural interpretation as the size of the system, for e.g. financial time series its meaning is less obvious. If $N$ can be found by maximum likelihood, it can be interpreted as an effective system size. The parameter $\alpha$ describes in all cases the strength of the interaction between the elements in the system. If the elements are not all connected or if the system is heterogeneous, it seems reasonable to use, respectively, connectivity-rescaled parameters or an average interaction strength to determine estimates of this parameter.

V Open questions

A large number of questions related to the Abelian distribution are left for future investigation. Most important among them are the higher moments, characteristic function, stability and properties related to parameter estimation. Especially interesting for the application to critical system would be a scaling law for the critical value $\alpha_{\mathrm{crit}}$ and relation between different possibilities to define ctiticality for finite system.

Acknowledgments

The authors wish to thank Zakhar Kablutschko and Theo Geisel for helpful discussions and Manfred Denker for valuable comments, help and support. Supported by the Federal Ministry of Education and Research (BMBF) Germany under grant number 01GQ1005B.