Identification of Criticality in Neuronal Avalanches: I. A Theoretical Investigation of the Non-driven Case

A number of in vitro and in vivo studies [1–4] have shown neuronal avalanches—cascades of neuronal firing—that may exhibit power law statistics in the relationship between avalanche size and occurrence. This has been used as prima facie evidence that the brain may be operating near, or at, criticality [5, 6]. In turn, these results have generated considerable interest because a brain at or near criticality would have maximum dynamic range [7–10] enabling it to optimally react and adapt to the dynamics of the surrounding environment [5, 11] whilst maintaining balanced neuronal activity [12–14]. Neuropathological states (e.g., epileptic seizures) could then be conceptualised as a breakdown of, or deviation from, the critical state; see [15], for example. Furthermore, these findings have led to the notion that the brain may self-organise to a critical state [16], i.e., its dynamics would be driven toward the critical regime by some intrinsic mechanism and not be dependent on external tuning. In support of this view, Levina and colleagues [17] showed analytically and numerically that activity-dependent depressive synapses could lead to parameter-independent criticality.

The interpretation that neuronal activity is poised at a critical state appears to be mostly phenomenological whereby an analogy has been developed between the propagation of spikes in a neuronal network and models of percolation dynamics [18] or branching processes [19, 20]. Remarkable qualitative similarities between the statistical properties of neuronal activity and the above models have given credence to this analogy, however, the question remains as to whether it is justified. Indeed, various key assumptions underlying percolation dynamics and branching processes are typically violated in the neuroscience domain. For example, full sampling, which is required in order to assess self-organised criticality, is unattainable even in the most simple in vitro scenario and yet it has been shown that sub-sampling can have profound effects on the distribution of the resulting observations [21]. On a related note, and more generally, the formal definition of a critical system as one operating at, or near, a second-order (continuous) phase transition is problematic since the concept of phase transition applies to systems with infinite degrees of freedom [22]. Many neuroscience authors address this by appealing to the concept of finite size scaling and many published reports implicitly assume that distributions are power law with truncation to account for the so-called finite size effect. Typically, such reports adopt an approach whereby (a) scale invariance is assessed through finite size scaling analysis, confirming that upon rescaling the event size, the curves collapse to a power law with truncation at system size (but see below regarding the definition of system size); (b) the parameters of statistical models are estimated, typically over a range of event size values that are rarely justified; and (c) the best model is determined by model selection, in which power law and exponentially truncated power law are compared to alternatives such as exponential, lognormal and gamma distributions; see [23] for a typical example. Whilst greater rigour in the statistical treatment of the assessment of the presence of power laws has been attained following Clauset and colleagues’ influential paper [24], what seems to be lacking is a rigorous treatment as to why a power law should be assumed to begin with. Although this question is particularly pertinent to the neurosciences, it should be noted that similar questions remain open in the field of percolation theory (e.g., [25, 26]), namely: (i) how does the critical transition behaviour emerge from the behaviour of large finite systems and what are the features of the transition? (ii) what is the location of the scaling window in which to determine the critical parameters?

One way to address these questions more rigorously is to use simplified, but therefore more tractable conceptual models (e.g., [35]). In this paper, we use a model of a purely excitatory neuronal system with simple stochastic neuronal dynamics that can be tuned to operate at, or near, a second-order phase transition (specifically, a transcritical bifurcation). The simplicity of the model allows us to analytically calculate the exact distribution of avalanche sizes, which we confirm through simulations of the system’s dynamics. We study our model at the critical point and compare our exact distribution to the explicit but approximate solution proposed by Kessler [36] in an analogous problem of modelling disease dynamics. We confirm that Kessler’s approximate solution converges to our exact result. Importantly, we show that, in the proposed finite-size system, this distribution is not a power law, thus highlighting the necessity of considering other markers of criticality. We thus analyse two potential markers of criticality: (i) the divergence of susceptibility that arises in the model as we approach the critical point, (ii) the slowing down of the recovery time from small disturbances as the system approaches the critical point. Finally, we speculate on a sufficient but not necessary condition under which our exact distribution could converge to a true power law in the limit of the system size.

where $s_i(t)={\sum }_j\frac{w_{ij}}{N}{a}_j(t)+h_i$ is the input to the neurone. Here, f is an activation function, $h_i$ is an optional external input, $w_{ij}$ is the connection strength from neurone i to neurone j, and $a_j(t)=1$ if neurone j is active at time t and zero otherwise. α is the de-activation rate and, therefore, controls the refractory period of the neurone.

Borrowing from epidemiology, we define the threshold $R_0=\frac{w}{\alpha }$. If $R_0>1$, we see that $g^{\prime }(A^{\ast })=\alpha -w<0$ and the non-zero steady state is stable. Figure 1 illustrates the differing behaviour of the solution to the above ODE for $R_0<1$ (sub-critical), $R_0=1$ (critical), and $R_0>1$ (super-critical).

2.2 Tree Approach to the Avalanche Distribution

We begin by defining $q_i$ as the probability the next transition is a recovery (from A to Q) given i active neurones ($i>0$). The probability the next transition is an activation is then $1-q_i$ and we have:

$\begin{array}{rl}{q}_i& =\frac{\alpha N}{w(N-i)+\alpha N}=\frac{N}{R_0(N-i)+N},\\ 1-q_i& =\frac{w(N-i)}{w(N-i)+\alpha N}=\frac{R_0(N-i)}{R_0(N-i)+N}.\end{array}$

In order to calculate the avalanche size distribution, we adopt a recursive approach. We begin by considering the process unfolding in a tree-like manner with 1 initially active neurone. The tree can be divided into levels based on the number of transitions that have occurred and how the process is unfolding. Let level j contain the possible number of active neurones after j transitions. The recursive tree approach relates the probability of transition between levels to the final avalanche size. Figure 2 illustrates the initial levels of this process.

To continue we define $p_j^i$ as the probability of having i active neurones on level j with $i=0,1,2,\dots ,N$ and $j\in {\mathbb{N}}_0$. Assuming initially only one active neurone, we immediately see that $p_0^1=1$, $p_1^2=1-q_1$ and $p_1^0=q_1$. To proceed, we will consider the probability of having a particular number of active neurones on an arbitrary level. First, we note the following relation between levels:

$p_j^i=\{\begin{array}{cc}{p}_{j-1}^2{q}_2,\hfill & \text{if }i=1,\hfill \\ p_{j-1}^{i-1}(1-q_{i-1})+p_{j-1}^{i+1}{q}_{i+1},\hfill & \text{for }1<i<N,\hfill \\ p_{j-1}^{N-1}(1-q_{N-1}),\hfill & \text{if }i=N.\hfill \end{array}$

We now define:

$\mathbf{p}(l)=\left(\begin{array}{c}{p}_l^1\\ ⋮\\ p_l^N\end{array}\right).$

We can now write $\mathbf{p}(l+1)=\mathbf{A}\cdot \mathbf{p}(l)$ where matrix A is given by the following tridiagonal matrix:

$\mathbf{A}=\left(\begin{array}{cc}0& c_1\\ \ddots & \ddots & \ddots \\ \ddots & \ddots & \ddots \\ b_i& 0& c_i\\ \ddots & \ddots & \ddots \\ \ddots & \ddots & \ddots \\ b_N& 0\end{array}\right)$

with $b_i=(1-q_{i-1})$ and $c_i=q_{i+1}$.

On the j th level of the tree, the probability of only 1 neurone being active is given by $p_j^1$. As on level 0, we began with only a single active neurone then for j odd, $p_j^1$ is always equal to zero. For j even, say $j=2k$, then as we began with only one active neurone on level 0, to have only one active neurone on level j means that k firings must have occurred. We can then calculate the probability of zero active neurones after k firings as $q_1{p}_{2k}^1$; this is thus the probability, $P(k+1)$, of having an avalanche of size $k+1$ (or size k if we were not to include the initial active neurone). Setting $e={(1,0,0,\dots ,0)}^T$ and noting that $p_{2k}^1=e^T{A}^{2k}e$ we have $P(k+1)=q_1{e}^T{A}^{2k}e$. To calculate the distribution, we implemented this recursive method of calculation in the MATLAB^® environment. Whilst this result is exact, and will be referred to as such henceforth, it can only be calculated numerically via recursion and cannot be given in the form of a closed expression.

2.4 Exact Solution Compared to Simulation

Values of the threshold, $R_0$, were chosen less than, equal to and finally above 1. We will refer to these regimes as subcritical, critical, and supercritical, respectively. Figure 3 illustrates the, as expected, good agreement between the simulations and the exact result for the three different regimes of $R_0$.

2.5 Comparing the Exact Solution to a Closed Form Approximation

In [36], Kessler proposed a closed solution to the analogous susceptible-infected-susceptible (SIS) problem where he was interested in the number of infections (including reinfections) occurring over the course of an epidemic. For small avalanche sizes where the number of infectives is negligible compared to the network size, the transition probabilities can be approximated as

$\begin{array}{rl}{q}_i& =\frac{N}{R_0(N-i)+N}\approx \frac{1}{R_0+1},\\ 1-q_i& =\frac{R_0(N-i)}{R_0(N-i)+N}\approx \frac{R_0}{R_0+1}.\end{array}$

In the critical regime $R_0=1$, the problem reduces to calculating the distribution of first passage times of a random walk with equal transition probabilities. Thus, for avalanche sizes in the range, $1\ll n\ll \sqrt{N}$, Kessler [36] gave the following distribution based on Stirling’s approximation:

$P(n)=\frac{1}{2^{2n-1}}[\left(\genfrac{}{}{0ex}{}{2n-2}{n-1}\right)-\left(\genfrac{}{}{0ex}{}{2n-2}{n}\right)]\approx \frac{1}{\sqrt{4\pi n^3}}.$

(1)

We note however that the range over which the distribution can be shown to be a power law is rather limited and for small networks will not hold. Using the theory of random walks and a Fokker–Planck approximation, Kessler also derived the following closed solution to the probability distribution of infections in the critical regime ($R_0=1$) for larger sizes:

$P(n)=\frac{1}{\sqrt{4\pi N^3}}exp(n/2N){sinh}^{-3/2}(n/N)(n\gg 1).$

(2)

Figure 4 plots this approximation against our exact solution for a network of size $N=800$. To more formally assess the convergence of the approximate solution to that of our exact solution, we considered the probabilities of avalanches from size $N/10$ to 20N and measured the difference between the distributions using two different metrics. Letting $P_e(n)$ be the exact probability of an avalanche of size n and $P_k(n)$ be the Kessler approximation to this, we first considered the standard mean square error given by

$\mathit{Error}(N)=\frac{1}{R}\sum _{n=N/10}^{20N}{(P_e(n)-P_k(n))}^2\text{where }R=20N-N/10+1.$

Secondly, we considered a more stringent measure of the error by looking at the supremum of difference between the same range of avalanches

$\mathit{Error}(N)=\underset{n}{sup}|P_e(n)-P_k(n)|.$

Figure 5 illustrates the two errors for increasing network size and both show how the proposed closed solution is indeed converging to that of the exact.

The fact that we have an exact form for the distribution allows us to make further important observations about some of its characteristics. Here, we explore the origin of the distribution’s truncation. Let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_N$ be the eigenvalues of A with the corresponding eigenvectors $u_1,u_2,\dots ,u_N$. The initial condition can then be given as $\mathbf{p}(0)=c_1{u}_1+c_2{u}_2+\cdots +c_N{u}_N$. As the matrix A is similar to a symmetric tridiagonal matrix with real entries (consider the diagonal similarity transformation matrix D, with $D_1=1$ and $D_j=\sqrt{(b_j{b}_{j-1}\cdots b_2)/(c_{j-1}{c}_{j-2}\cdots c_1)}$), we know that its eigenvalues are real.

where $q_1$ is the probability that the next transition is a recovery (from A to Q) given 1 active neurone (as defined earlier), ${\lambda }_i$ are the eigenvalues of the transition matrix A and $d_i$ are specified by the eigenvectors of the transition matrix and the initial conditions. We note that the earlier equation, $\mathbf{p}(0)=c_1{u}_1+c_1{u}_1+\cdots +c_N{u}_N$, can be solved to obtain $c_i$. Using this, we can then calculate $d_i$ as the first entry of the vector $c_i{u}_i$. Equation (3), which is exact, thus demonstrates that the distribution of avalanche sizes is a linear combination of exponentials.

Hence, $P(n)\sim q_1{e}_1{\lambda }_1^{2n}$ and for larger avalanche sizes we have the leading eigenvalue dominating thus giving the exponential cutoff observed. We illustrate this convergence in Fig. 8 where we plot the exact avalanche distribution, $P(n)$, against $q_1{e}_1{\lambda }_1^{2n}$. This figure also illustrates that the leading eigenvalue begins to dominate for avalanches just over the system size. It is for this reason that we chose an upper bound of $\frac{9N}{10}$ when fitting a power law to the distribution of avalanche sizes in the previous section.

where D is an operator such that $D(f(z))=z\frac{d(f(z))}{dz}$. For a fixed integer value of q, an approximation for $P(n)$ can be obtained by simply applying the operator as many times as necessary and then substituting $z=e^{\mu n}$. For $q=1$, for example, $P(n)\propto \frac{e^{\mu n}}{{(e^{\mu n}-1)}^2}$ which for small values of μ is well approximated by $\frac{1}{{\mu }^2}\frac{1}{n^2}$.

It is worth noting that this result is consistent with that obtained for integer values of q.

For a simple empirical verification of this conjecture, we determined the values of K, μ, L, and q in the above conditions through numerical fitting of the first 23 eigenvalues and e constants of the exact distribution for a network of size $N=800$ (see Fig. 9(a), (b)) and compared the resulting probability distribution with the exact distribution. Whilst the lesser valued eigenvalues and larger e values were not fitted well, Fig. 9(c) shows there is still remarkable agreement between both curves over a broad range of values, including the range [$\frac{1}{10}N,\frac{9}{10}N$] over which a power law like behaviour was established earlier (see Fig. 7). This result clearly illustrates the dominance of the larger eigenvalues and, given that the fitted distribution converges to a power law, gives support to the conjecture that the exact distribution would do so in the limit of an infinite network.

Since the distribution of avalanche sizes in the finite-size critical system does not necessarily follow a true power law, the application of robust statistical testing in experimental conditions could well lead to rejecting the hypothesis that the data may come from a system operating in the critical regime. Therefore, in this section, we consider two experimentally testable markers of criticality: critical slowing down and divergence of susceptibility. We will define those concepts below but first we briefly summarise Van Kampen’s system size expansion [40], which we use to illustrate those markers on our system.

5.1 System Size Expansion

For generality, we now assume that each neurone receives a constant external input and that the activation function can take forms other than the simple identity function. We define the probability that the number of neurones active at time t is n as $P_n(t)$. Then the master equation can be given as

$\begin{array}{rl}\frac{d{P}_n(t)}{dt}=& \alpha (n+1)P_{n+1}(t)\\ -\alpha n{P}_n(t)\\ +f(\frac{w(n-1)}{N}+h)(N-(n-1))P_{n-1}(t)\\ -f(\frac{wn}{N}+h)(N-n)P_n(t).\end{array}$

The idea of the system size expansion is to now model the number of active neurones as the sum of a deterministic component scaled by N and a stochastic perturbation scaled by $\sqrt{N}$, i.e.,

$n(t)=N\mu (t)+N^{1/2}\xi (t).$

A more detailed explanation of this can be found in [12] and [40], but importantly what is obtained is the following set of equations for μ (which is the solution to the mean field equation of the proportion of active neurones), $〈\xi 〉$ (the expected value of the fluctuations) and ${\sigma }^2=〈{\xi }^2〉-{〈\xi 〉}^2$ (the variance of the fluctuations)

$\frac{\partial \mu }{\partial t}=-\alpha \mu +(1-\mu )\hat{f},$

(10)

$\frac{\partial 〈\xi 〉}{\partial t}=-(\alpha +\hat{f}-w{\hat{f}}^{\prime }(1-\mu ))〈\xi 〉,$

(11)

$\frac{\partial 〈{\sigma }^2〉}{\partial t}=-2(\alpha +\hat{f}-w{\hat{f}}^{\prime }(1-\mu ))〈{\sigma }^2〉+(\alpha \mu +(1-\mu )\hat{f}).$

(12)

Here, $\hat{f}=f(w\mu +h)$ and ${\hat{f}}^{\prime }=f^{\prime }(w\mu +h)$. These equations, in turn, give the following equations for the mean, A, and variance, $A_{\sigma }$, of the number of active neurones:

$\begin{array}{rl}A& =N\mu +N^{-1/2}〈\xi 〉\\ =N\mu \text{(assuming we know the initial number of active neurones)},\end{array}$

(13)

$A_{\sigma }=N〈{\sigma }^2〉.$

(14)

We make use of these equations in the following two sections.

5.1.1 Critical Slowing Down

In dynamical systems theory, a number of bifurcations, including the transcritical bifurcation in our system, involve the dominant eigenvalue characterising the rates of changes around the equilibrium crossing zero. As a consequence, the characteristic return time to the equilibrium following a perturbation increases when the threshold is approached [41]. This increases has led to the notion of critical slowing down as a marker of critical transitions [42]. Here, we illustrate the critical slowing down of our model with the analytic derivation of the rate of convergence to the steady state (this derivation has been previously shown by [43]). We first begin by calculating the analytic solution to Eq. (10) for the percentage of active neurones. We again consider the case where f is the identity function and can thus write

$\frac{\partial \mu }{\partial t}=-\alpha \mu +(1-\mu )f(w\mu +h)=-\alpha \mu +(1-\mu )(w\mu +h).$

(15)

Assuming zero external input ($h=0$), we have

$\frac{\partial \mu }{\partial t}=-\alpha \mu +(1-\mu )(w\mu +h)=-\alpha \mu +(1-\mu )w\mu .$

(16)

We are interested in the solution of this equation and consider the result for different values of α. Firstly, we consider $\alpha \ne w$. In this case, we have

$\frac{\partial \mu }{\partial t}=-\alpha \mu +(1-\mu )w\mu =\mu (w-w\mu -\alpha ).$

(17)

Integrating this using separation of variables and the initial condition $\mu (0)={\mu }_0$, we find

$\mu (t)=\frac{w-\alpha }{A{e}^{(\alpha -w)t}+w}\text{where }A=\frac{{\mu }_0}{w-w{\mu }_0-\alpha }.$

(18)

The solution to this depends on whether $\alpha <w$ or $\alpha >w$ ($R_0>1$ and $R_0<1$, respectively). If $\alpha <w$, then as $t\to \mathrm{\infty }$, $\mu \to \frac{w-\alpha }{w}$. If $\alpha >w$ then as $t\to \mathrm{\infty }$, $\mu \to 0$. Note that in both cases, convergence of the number of active neurones to the steady state solution is exponential.

Now we consider the solution when $\alpha =w$, i.e., the critical regime

$\frac{\partial \mu }{\partial t}=-\alpha \mu +(1-\mu )\alpha \mu =-\alpha {\mu }^2\Rightarrow \mu (t)=\frac{1}{\alpha t+{\mu }_0^{-1}}.$

Thus, as $t\to \mathrm{\infty }$ we find $\mu (t)\to 0$. However, unlike for $R_0\ne 1$, convergence to the steady state exhibits a power law dependence on time [43].

5.1.2 Divergence of Susceptibility

A correlate of the phenomenon of critical slowing down is that of the divergence of susceptibility of the system as the system approaches the bifurcation [42]. In this section, we investigate the behaviour of the equation for the variance. For simplicity, we consider again the case of the identity activation function and a non-driven system $h=0$. First, we use Eq. (12) to calculate the variance in the percentage of active neurones:

$\begin{array}{rl}\frac{\partial {\sigma }^2}{\partial t}& =-2(\alpha +\hat{f}-w{\hat{f}}^{\prime }(1-\mu )){\sigma }^2+(\alpha \mu +(1-\mu )\hat{f})\\ =-2(\alpha +w\mu +h-w^2(1-\mu )){\sigma }^2+(\alpha \mu +(1-\mu )(w\mu +h))\\ =-2(\alpha +w\mu -w^2(1-\mu )){\sigma }^2+(\alpha \mu +(1-\mu )w\mu ).\end{array}$

Setting this equal to zero and rearranging, we obtain

${\sigma }^2=\frac{(\alpha \mu +(1-\mu )w\mu )}{2(\alpha +w\mu -w^2(1-\mu ))}=\frac{(\mu +(1-\mu )R_0\mu )}{2(1+R_0\mu -R_{0}w(1-\mu ))}.$

Here, we note that unlike the equation for μ where there was only the single bifurcation parameter $R_0$, we now have the additional dependence on w. To maintain consistency with earlier results, we now set $w=1$ to obtain

$\underset{t\to \mathrm{\infty }}{lim}{\sigma }^2(t)=\{\begin{array}{cc}\alpha \hfill & \text{if }\alpha <1(R_0>1),\hfill \\ \frac{1}{2}\hfill & \text{if }\alpha =1(R_0=1),\hfill \\ 0\hfill & \text{otherwise }(R_0<1).\hfill \end{array}$

Using Eq. (14), we obtain

$\underset{t\to \mathrm{\infty }}{lim}{〈A〉}_{\sigma }=\underset{t\to \mathrm{\infty }}{lim}N〈{\sigma }^2〉=\{\begin{array}{cc}\frac{N}{R_0}\hfill & \text{if }{R}_0>1,\hfill \\ \frac{N}{2}\hfill & \text{if }{R}_0=1,\hfill \\ 0\hfill & \text{otherwise }(R_0<1).\hfill \end{array}$

Figure 10 illustrates the jump to a non-zero steady state when the critical value $R_0=1$ is approached from below, and the divergence in variance when it is approached from above.

Here, it should be noted that any finite-size network has a zero absorbing state so that eventually all activity will die out irrespective of the value of $R_0$. However, it has been shown that the ODE limit is a valid approximation to the solution of the master equation for reasonably sized systems with values of $R_0$ greater than 1 and only over a finite time horizon (see [44] for further discussion). Defining the true (i.e., calculated directly from the master equation for $P(n)$) expected value of active neurones at time t as $\tilde{A}(t)$, the convergence of the ODE approximation for $A(t)$ given by Eq. (13) is such that for any $t\ge 0$${lim}_{N\to \mathrm{\infty }}|A(t)-\tilde{A}(t)|=0$[45].

Over the last decade or so, the search for evidence that the brain may be a critical system has been the focus of much research. This is because it is thought that a critical brain would benefit from maximised dynamic range of processing, fidelity of information transmission and information capacity [46]. Whilst support for the critical brain hypothesis has emerged from comparing brain dynamics at various scales with the dynamics of physical systems at criticality (e.g., [31, 34, 47–50]), in this paper, we focus on the important body of work that has relied on characterising power laws in the distributions of size of neuronal avalanches [8, 30]. Our focus on this scale is motivated by empirical considerations regarding how one can go about demonstrating the above functional properties. Shew and Plenz [46] remark that any research strategy to test whether these properties are optimal near criticality will have to achieve two criteria: a means of altering the overall balance of interactions between neurones and a means of assessing how close to criticality the cortex is operating. As argued by these authors, the study of neuronal avalanches offers the greatest likelihood of achieving those two criteria.

The importance of a robust assessment of the statistical properties of the avalanche size is therefore two-fold: on the one hand, it is about ascertaining the extent to which the system being studied has the statistical properties expected of a system operating at, or near, criticality; on the other hand, it is about being able to confirm that a manipulation/perturbation of the system aimed to push the system away from this critical regime has been effective. This consideration therefore puts a lot of importance on the description of the statistics one should expect in such a system. In the current literature, the assumption of the distribution of avalanche sizes taking a power law functional form relies on an analogy between the propagation of spikes in a neuronal network and models of percolation dynamics or branching processes for which exact power laws have been demonstrated in the limit of system size. As a result of the importance of having a robust assessment of the expected presence of a power law, greater emphasis has recently been put on using a sound statistical testing framework, e.g., [24]. Whilst we are unaware of any study in which the criticality hypothesis was rejected due to failure of rigorous statistical testing (which we suspect is due to the necessarily small number of observations, as we will argue below), there is clear evidence that many authors are now using the methods of Clauset et al. [24] to confirm the criticality of their experimental findings, e.g., [12, 23, 29]. As a result, we feel that it is all the more important to confirm that the assumed power law functional form is indeed a sensible representation of what one should expect in in vivo and in vitro recordings, which, unlike the physical systems considered when deriving the power law statistics, are finite-size systems. The aim of the paper was therefore to consider a model of neuronal dynamics that would be simple enough to allow the derivation of analytical or semi-analytical results whilst (i) giving us a handle on the parameter controlling the fundamental principle thought to underlie criticality in the brain, namely, the balancing between processes that enhance and suppress activity (note that we are intentionally not referring to excitation and/or inhibition—we will return to this below) and (ii) allowing us to determine its distribution of avalanche sizes when operating in the critical regime. Note that because we are using a finite-size system, we are appealing to a normal form of standard bifurcation, here, a transcritical bifurcation, because it embodies all that needs to be known about the ‘critical’ transition (Sornette, private communication).

Our semi-analytic derivation of the true distribution of avalanche sizes in a finite-size system suggests that, even though it is approximately scale free over a limited range, the distribution is not a true power law. First, this has important implications for the interpretation of results from a robust statistical assessment of the distribution. Indeed, as has been discussed by Klaus and Plenz [23], with a large number of samples, any distribution that deviates from the expected distribution by more than noise due to sampling, will eventually yield a p-value such that the power law hypothesis will be rejected, thus leading to the potentially incorrect conclusion that the system is not critical. This is the case in our scenario where using 10⁶ avalanches lead to a rejection of the criticality hypothesis even though the system is tuned to the critical regime. In contrast, with 10⁵ avalanches (which is consistent with empirical observations), a p-value above threshold leads to not rejecting the hypothesis that the distribution is a power law even though we established it is not one.¹ This finding therefore provides an important counterpart to the analytical results of Touboul and colleagues [29] who showed that thresholded stochastic processes could generically yield apparent power laws that only stringent statistical testing will reject. Whilst the stringent testing will reject the hypothesis of criticality for a system that is not necessarily critical, it may also reject the hypothesis of criticality for a system that is critical only because the actual distribution is not actually a power law. This ambiguity of the avalanche distribution in the finite-size system therefore requires that one should carefully consider to what fundamental property the idea of a critical brain actually appeals to. We suggest that the key appeal is that the brain can exhibit long-range correlations between neurones without it ever experiencing an over saturation of activity or long periods of inactivity. It then follows that the importance is not in the exact distribution obtained but in the approximately scale-free behaviour it exhibits. In turn, this highlights the importance of looking at other markers of criticality (which we will discuss below).

Another important result of this work is to provide the beginning of a mechanistic explanation for an often alluded to (e.g., [51]) but never properly treated (as far as we are aware) observation that whereas avalanches in a critical system with re-entrant connections could in principle be arbitrarily long, and certainly, exceeding the number of recording sites, neuronal avalanches in in vitro or in vivo systems (and many computational models of self-organised criticality) often show a cut-off at the number of sites. Our work suggests that the lead eigenvalue of the transition matrix between states fully determine the location of this cut-off, which turns out indeed to be at about the system size, even if avalanches of up to 20 times the system size can be observed. This finding therefore provides some justification for setting, or accepting, a bound within which to apply a Clauset-type methodology (we note that various reports use different ranges, e.g., 80 % of system size in [17], roughly system size in [51]). It is worth remembering that the number of recording sites can have profound implications on the nature of the distribution observed [21].

In addition to providing results on the distribution of avalanche sizes, we also sought to explore other potential markers of criticality. We provided results on two other markers of criticality—critical slowing down and divergence of susceptibility—both of which again follow from a dynamical systems appreciation of a critical bifurcation, i.e., the behaviour of a system whose lead eigenvalue crosses zero. The appeal of those markers, which have been documented in many other natural processes, e.g., [42, 52], but seldom at the mesoscopic brain level² (see [53] for a rare example) is that (a) they strengthen the assessment of the system being critical and (b) may contribute to achieving the second criterion of Shew and Plenz [46]. Although the authors are not in a position to provide explicit recommendations for an experimental design, we believe that these markers are amenable to robust experimentation, e.g., through pharmacological manipulation.

Whilst we hope we have convinced the reader of the potential importance of these findings, we also need to recognise that the very simplicity that makes analytical work possible does also raise questions regarding how physiologically plausible such a model is and, therefore, whether its conclusions should be expected to hold. Below, we address a few of the points worthy of further consideration.

6.1 Validity of Inferring Criticality in a Finite Network

In using the meanfield equations, it is important to understand how well they capture the behaviour and bifurcation structure of the stochastic process they are approximating. Whilst it is known that on the complete graph (see [54] for instance) and in the limit $N\to \mathrm{\infty }$ the steady state solution of the ODE will converge to the expected value of the comparable stochastic process, it is unclear whether the critical point of the infinite system corresponds to that in the finite system. Furthermore, it is unclear whether a finite system can truly have a critical point and we must be cautious in claiming one exists. Importantly, however, it has been shown in [55] that for a complete graph, $R_0\approx 1$ (the paper proves the result for α fixed as 1 but the result is generalisable for any α) is the threshold below which the disease will die out quickly (expected time to extinction $O(log(n))$), and above which it dies out slowly (expected time to extinction $O(n^a)$ for some a). Simulating the steady state of the network for increasing $R_0$ also shows (see Fig. 11) the characteristic feature of a second-order phase transition found at a critical point. For these reasons, whilst acknowledging the problem of inferring criticality in a finite regime, we feel justified in claiming $R_0=1$ as the critical point for the process unfolding on our finite network.