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Edited by: Tjeerd W. Boonstra, University of New South Wales, Australia

Reviewed by: Alain Destexhe, Information and Complexité Centre National de la Recherche Scientifique, France; Woodrow Shew, University of Arkansas, USA

*Correspondence: John M. Beggs, Department of Physics, Indiana University, 727 East, 3rd Street, Bloomington, IN 47405-7105, USA. e-mail:

This article was submitted to Frontiers in Fractal Physiology, a specialty of Frontiers in Physiology.

This is an open-access article distributed under the terms of the

Relatively recent work has reported that networks of neurons can produce avalanches of activity whose sizes follow a power law distribution. This suggests that these networks may be operating near a critical point, poised between a phase where activity rapidly dies out and a phase where activity is amplified over time. The hypothesis that the electrical activity of neural networks in the brain is critical is potentially important, as many simulations suggest that information processing functions would be optimized at the critical point. This hypothesis, however, is still controversial. Here we will explain the concept of criticality and review the substantial objections to the criticality hypothesis raised by skeptics. Points and counter points are presented in dialog form.

Critio: Hello professor. I enjoyed your presentation this morning. Your group is doing some fascinating work on synaptic plasticity. I was particularly interested in your thoughts on how synaptic changes underlie memory.

Mnemo: Thank you! I can see from your badge that you are in a physics department. What brings you to a neuroscience conference?

Critio: Well, I have been using ideas from statistical mechanics to try to explain how groups of neurons collectively behave. One of my primary research interests is determining whether or not the brain is operating at a critical point.

Mnemo: I’ve seen several papers in that area and they seem to show some interesting results. There also appears to be a great deal of controversy about criticality in biology (Gisiger,

Critio: It is definitely true that there is significant disagreement in the research community about the role criticality plays in neural dynamics (Stumpf and Porter,

Mnemo: Well, that’s to be expected. Many topics in science are hotly debated and that’s part of the fun of being a scientist!

Critio: Oh, I agree! I just want to say that, even given my view that criticality does play an important role in neural dynamics, I recognize that it is completely possible that criticality, in fact, does not play an important role in neural dynamics. Other methodologies, such as non-linear systems might better explain neural dynamics (May,

Mnemo: Well, this certainly sounds like an interesting topic. But since we have a few minutes here, why don’t you give me a quick description of your research? I probably won’t read a review, but I could learn a few things from you over lunch. Do you mind if I pick your brain, so to speak?

Critio: Not at all! I guess I could give you an overview of criticality and how it might apply to the brain. I am somewhat biased, but I’ll do my best to present arguments from researchers who disagree with my view of criticality in neural systems. You can help me by being as skeptical of my arguments as possible.

Mnemo: That sounds great! But before we get started, I would like to clear one thing up that has been bugging me. Several of the other researchers at my institution study network topology. I always hear them talking about scale-free networks, power laws, and criticality. Are those all the same thing?

Critio: That is an excellent question and I think it gets at a point that isn’t widely made in the literature. If we’re interested in network topology, we’re interested in how the nodes of a network are connected to each other. A scale-free network has nodes that are connected in a certain way. If we’re interested in criticality, we’re interested in how the network behaves. The two topics are certainly related, but it is possible for non-scale-free networks to exhibit critical behavior and it is possible for scale-free networks to not exhibit critical behavior. The network connectivity affects the critical behavior of that network (Haldeman and Beggs,

Mnemo: That sounds complicated! But, since I hear most people discussing criticality, let’s discuss that first. So, what is criticality?

Critio: Criticality is a phenomenon that has been observed in physical systems like magnets, water, and piles of sand. Many systems that are composed of large numbers of interacting, similar units can reach the critical point. At that point, they behave in some very unusual ways. Some people, including myself, suspect that cortical networks within the brain may be operating near the critical point.

Mnemo: This all sounds intriguing, but I have no idea what you mean by the critical point. Can you give me a simple example?

Critio: Sure, let’s use a well explored model in this field: the Ising model (Brush,

This model will illustrate the critical point pretty well. See these circles? They represent lattice sites in a piece of iron. At each site, there is an electron whose “spin” is either up or down. You can think of these arrows as little bar magnets, with the arrowhead being the North pole of the magnet. In a piece of iron, these bar magnets influence their nearest neighbors to align in the same direction. I will represent their influence on each other by drawing lines between the circles. So, when the temperature T is low, as in the left panel of Figure

This movie shows a simulated piece of iron as the temperature is cooled. Each black square represents a spin pointed up, and each white square is a spin pointed down. See how, over time, all of the spins begin to point in the same direction? Pretty soon the whole sample will be either all black or all white. That behavior is caused by the nearest neighbor interactions.

Mnemo: So all iron is magnetic?

Critio: No, certainly not. Being ordered like that is just one phase that the piece of iron can be in. And that happens only at low T. If you heat it up, you can make it change into another phase.

Mnemo: Oh, I have heard some things about a “phase transition.” Is that where this is going?

Critio: Well, yes. If you heat up the iron quite a lot, then this increased thermal energy will begin to “jostle around” the spins. Even though they still have a tendency to align with each other, this will be overwhelmed by the added heat. [Critio sketches the right panel of Figure

Mnemo: So is this why a magnet loses its ability to stick when it is heated up too much?

Critio: Exactly. All the spins are pointing in different directions and they cancel each other. There is no net magnetic field produced by the sample any more.

Mnemo: So now you have shown me the ordered and the disordered phases. What happens between them, at the so called “phase transition point?” Is this the same thing as the “critical point?”

Critio: Yes it is. If you add just the right amount of heat to get to the critical temperature, then the tendency for the spins to align is exactly counterbalanced by the jostling caused by the heat. Now you no longer have global order. Instead, there will be local domains where a group of spins are pointed up, and other domains where the spins are pointed down (Stanley,

Mnemo: Wow, that is really interesting – some of the domains almost look like amoeba crawling across the screen, with boundaries that are extending and contracting. I can see that there are many different sized domains too. OK, you have been telling me a lot about this piece of iron, but how does this relate to the brain?

Critio: Good question, but before we get to neural data, we need to understand a few more things about criticality. You certainly must agree that communication between neurons is very important for the brain. If we continue with the magnet analogy, we could ask how two spins at different lattice sites might communicate with each other.

Mnemo: Ok, go on…

Critio: A simple way to measure this would be to look at the dynamic correlation between two lattice sites. This is not the correlation that is usually used in statistics, but something that depends on coordinated fluctuations. Here is the equation for the dynamic correlation. Critio then writes down Eq.

The angled brackets here indicate a time average, so 〈_{ij}

Mnemo: Ok, that seems to make sense. Now I see why it is called the dynamic correlation – if both

Critio: Great, you get it! Now let’s take a look at what happens to the dynamic correlation for the three different cases we talked about: low T, high T, and critical T. In the low T case, the piece of iron is extremely ordered and all the spins are pointed in the same direction. The dynamic correlation is low because there are no fluctuations and the terms in parentheses are both nearly 0 all the time. In contrast, for the high T case, there are plenty of fluctuations, as the spins are constantly deviating from their averages, but there is no coordination between sites

Mnemo: So there is dynamic correlation between spins only at the critical temperature?

Critio: Well, there might be some dynamic correlation in all three cases, but it is certainly strongest at the critical temperature. Another key difference is that the distance over which these correlations extend is greatest at the critical temperature.

Mnemo: Can you show me what you mean by that?

Critio: Sure. If we were to measure the dynamic correlation between two spin sites

Critio: In this example from a simulation, the dynamic correlation at the critical temperature extends about 15 lattice sites before it drops down to near 0. We call the distance at which the dynamic correlation first reaches 0 the “correlation length” and it is often given by the Greek capital letter gamma, Γ; in this case the correlation length is 15 lattice sites long. We didn’t build this length into the model – it merely emerged at the critical temperature. At this temperature, when one spin flips from down to up, for example, it might influence one of its nearest neighbors to also flip, which might in turn influence one of its nearest neighbors and so on. In this way, the movement at one lattice site can propagate beyond the nearest neighbor length. You could draw the correlation length as a function of temperature, and it would show a sharp peak at the critical point. [Critio asks another person sitting at the table for a fresh napkin and draws Figure

Critio: Again, this shows the separation of phases nicely. On the left you have the ordered phase, with low temperature. This is sometimes called the subcritical regime. On the right you have the disordered phase, with high temperature, and this is sometimes called the supercritical regime. Between them you have the phase transition region, which is very narrow and occurs at the critical temperature.

Mnemo: I think I see what is going on. Only at the critical temperature can you have communication that spans large distances. So if I were to make an analogy with a neural network, it would be that at the critical point, the neurons can communicate most strongly and over the largest number of synapses, right?

Critio: Exactly!

Mnemo: But wait, what do you mean by “communication?” When the model is at low temperatures, the state of one lattice site strongly influences the state of lattice sites throughout the whole network. So, it would seem to me that communication is maximized when the temperature is low, not when the system is at the critical point.

Critio: Ah, that is a subtle point. Clearly, we haven’t been very rigorous with our definition of “communication,” but let me see if I can clarify my point. When the model is at low temperatures, the coupling between the lattice sites is strong, so coordination is high. However, the state of each lattice site doesn’t change very much through time, so fluctuations are low. Communication requires both coupling and variability, or in other words, both coordination and fluctuation. If communication is to take place, lattice sites must be able to influence each other and that influence must actually affect changes. Does that make more sense?

Mnemo: Yes, I see your point about the distinction between communication and coupling.

Critio: Great! So, at the critical point these two qualities of the system – coupling and variability – are balanced to produce long distance communication. And it turns out that it is not just communication that would be optimized at the critical point (Beggs and Plenz,

Mnemo: So this sounds pretty reasonable to me so far. But it is only an analogy. You haven’t shown me any evidence to suggest that the brain might be doing this. What evidence, if any, do you have to make me think that this is connected to real neurons?

Critio: Again, a very fair question. Before we can get to the neural data, I first need to show you how I got interested in this topic. Let me return for a moment to the plot of the average dynamic correlation length. If I were to change the axes by making them both logarithmic, then I would get something like this for the dynamic correlation, plotted now only for the critical case. [Critio draws Figure

Critio: When plotted this way, the dynamic correlation approximates a straight line over part of its range. This suggests that it could be described by a so-called “power law,” where the dynamic correlation,

Mnemo: Ok, for the moment I will assume you are right that this power law would extend to indefinitely large distances if the system were large enough. What is so special about a power law, besides the fact that it might suggest your system is critical?

Critio: An interesting feature of power laws is that they show no characteristic scale. When plotted in log-log coordinates, they produce a straight line that has the same slope everywhere. This implies that the data will have a fractal structure. For example, imagine what the distribution of correlation strength would look like if you were only able to sample separation distances from 10^{1} to 10^{2} units. It would be a straight line with a slope of −α when plotted logarithmically. Interestingly, this would look just like the distribution that you would observe if you were only able to sample separation distances from 10^{2} to 10^{3} units. Again, the exponent would be −α. This has caused some people to use the phrase “scale-free” when describing power law distributions (Stam and de Bruin,

Mnemo: So is where the name “scale-free” network comes from?

Critio: Yes! In scale-free networks, the degree distribution – the distribution of the number of connections each node possesses – follows a power law. But notice, in the Ising model, the nodes are connected in a lattice and the Ising model exhibits critical behavior. So, here we can see the distinction between criticality and scale-free networks in action. The nodes are not connected as in a scale-free network, yet the activity is scale-free.

Mnemo: That is certainly interesting, but I am still searching for a strong argument, not nice pictures. So power laws are an indicator of criticality? And you are going to tell me that you see some power laws in your neural data? This is the argument? It must be more substantial than that! After all, this is science, not just loose associations!

Critio: A critical system will produce power laws, yes, but power laws do not prove criticality! There are many ways to get power laws, and I can tell you more about that in a minute. The key thing to remember here is that exhibiting power laws is strongly suggestive of criticality. However, power laws alone are not sufficient to establish criticality.

Mnemo: Ok, I want to ask about these other ways to get power laws in a minute. But to return to the issue I raised earlier, you are going to tell me about some neural data that display power laws?

Critio: Yes, I can tell you about that first and then we can get to all the potential objections.

Mnemo: That sounds fine. Proceed with the data.

Critio: Well, there were several early reports that the nervous system could produce power law distributions (Chen et al.,

Since these initial results, power law distributions of avalanche sizes have been reported in awake monkeys (Petermann et al.,

Mnemo: That is an impressive list of neural systems in which power laws have been observed. However, I seem to recall hearing that other researchers have found that these power laws were actually better fit by exponentials. Is that true?

Critio: That is true. Using sophisticated statistical tests, several researchers have shown that some data sets, none of which were from neuroscience, that were previously thought to be power law distributed are actually better fit by an exponential distribution (Clauset et al.,

Mnemo: How do you escape that objection? It seems like so much of your argument is based on power laws. If those really aren’t there or if they are artifacts, then your system certainly isn’t operating at the critical point, is it?

Critio: You are right; this is a very important part of my argument. Let’s talk about each paper separately since they present distinct arguments and evidence. First, let’s discuss the papers that argue the observed power laws are artifacts. Some researchers have produced strong theoretical models that indicate that the extra-cellular medium may behave as a 1/

Mnemo: I see your point, what about the other papers?

Critio: Those papers argue that the power laws associated with neural phenomena that have been observed are not actually present. Several of the investigators who claimed to show that neural event size distributions were better fit by exponentials did not use many electrodes in their recordings. In some of their papers, they only had about eight electrodes (Bedard et al.,

Mnemo: Ok, but first let me understand this a bit more. You just told me about a magnetic model – the Ising model – and how that would settle into different equilibrium states at different temperatures. Now you are jumping to a network of neurons, where things do not settle at all. In fact, the Ising model seems like it would be pretty poor at describing how one neuron excites another, leading to cascades of activity spreading through the network.

Critio: As a neuroscientist, you have a very keen intuition for the physics! You are absolutely right to point out the potential problem. The Ising model is an equilibrium model, appropriate for describing how the system will settle at different temperatures, but this model does not explicitly account for time. To try to extend the Ising model into the range of dynamics, some people have applied a perturbation to the model – a slowly changing magnetic field for example – and watched how the system responds. Typically, when the model is at the critical temperature, applying a local magnetic field will cause several nearby spins to flip, so as to align with the applied field. These spins will in turn cause a change in the orientation preference for other nearby spins, and so will cause them to flip, leading to avalanches of spin flips. This is called the Barkhausen effect. In both theoretical work (Sethna et al.,

Mnemo: So to follow your analogy, the neurons in the brain could be thought of as spins in a magnet at the critical point. When something comes along and delivers an input, this propagates through the system with maximum distance, because the correlation length is greatest at the critical point. The avalanches of activity have sizes distributed as a power law, and you mentioned that some experimenters have observed power law distributions of avalanche sizes in neural tissue as well.

Critio: That is a good summary of what I have said so far. Even though impressive progress has been made recently in applying the Ising model to neuronal activity patterns found in actual data (Schneidman et al., ^{1}

Mnemo: I understand that no model is perfect and that it is easier to start with a simplified system, but I’m still dissatisfied.

Critio: What’s bothering you?

Mnemo: You’ve given me a nice story, but this is hardly proof. As you said, the existence of power laws is a necessary, but not sufficient condition for criticality. So, just because we’ve found some power laws in neural data, the existence of those power laws not prove that the neural systems are operating at the critical point. I don’t know about you, but I don’t like to affirm the consequent.

Critio: You are right to be skeptical. As I said, the power laws are

Mnemo: Sure, it seems like now would be a good time for you to tell me about the many other ways in which power laws can be generated.

Critio: There are so many ways to generate power laws that it is hard to know where to begin. People have written entire articles devoted largely to this topic (Mitzenmacher,

Mnemo: It now seems that you have dug yourself into a hole from which you cannot escape. If power laws are so unexceptional, then why should I be so excited about seeing them in neural data?

Critio: The fact that other non-critical systems also produce power laws is very important. Fortunately, recent experiments by several groups have addressed this issue directly. There are three main ways to demonstrate that the power laws observed in neural tissue are the result of a critical mechanism: the ability to tune the network from a subcritical regime through criticality to a supercritical regime, the existence of mathematical relationships between the exponents of the power laws for a system, and the existence of a data collapse within neural data.

Critio: First, recall that in a system that displays criticality, the power law will only occur when the system is between phases, in other words, at the phase transition point. So, for systems that really are critical, we should be able to observe different phases on either side of the critical point and get distributions there that do not follow power laws.

Mnemo: And you have evidence of this?

Critio: Actually, yes. By blocking excitatory synaptic transmission, you can dampen network excitability, leading to smaller avalanches (Mazzoni et al.,

In Figure

Critio: Related to this, there have been some very elegant experiments that have shown how information processing functions in the tissue approach optimal behavior near the critical point (Shew et al.,

Mnemo: What do you mean by that? And how is it related to the idea of phases?

Critio: Shew and colleagues looked at information transmission through cortical slice networks under three different conditions: where excitatory transmission is reduced; where there is no manipulation; and where inhibitory transmission is reduced. They showed that there was a peak in information transmission in the unperturbed condition, and that information transmission fell to either side of this point as perturbations increased. In many ways, they observed behavior just like that seen in the correlation function in the Ising model that we talked about earlier from Figure

Mnemo: So it seems that you need to be able to move the system from one phase to another if it is going to show a critical point. What you have been telling me is that these neural systems can be moved in this way.

Critio: That’s right. If a system displays criticality, then it must be tunable in some sense. Typically, a “control parameter” can be adjusted to determine the phase of the system. In the Ising model that we discussed earlier, the temperature is the control parameter. Sweeping the temperature from 0 to some high value would bring the system from the subcritical, ordered, phase, up to the critical point, and then into the supercritical, disordered, phase. The “order parameter” is what tells you the phase. In the case of the Ising model, the order parameter would be the net magnetic field produced by all the spins, called the magnetization. In the subcritical phase, all the spins are aligned and the magnetization has a large magnitude. In the supercritical phase, all the spins are pointing in random directions and the magnetization is 0. Near the critical point, we see the transition of the magnetization from some large magnitude toward 0. If a system is indeed critical, then all of the variables that could indicate its phase will depend on the control parameter.

Mnemo: To continue with the analogy, what would be the control parameter in neural systems?

Critio: That is a very good question. At the moment, it seems that the balance between excitation and inhibition can serve as a control parameter (Mazzoni et al.,

Mnemo: So you cannot tune the other, non-critical stochastic systems, like successive fractionation, or a combination of exponentials?

Critio: Well, you could tune them in some sense, but such tuning would only change the exponent of the resulting power law distribution. For example, let’s return to the combination of exponentials model proposed by Reed and Hughes (

Mnemo: I think I get it: if they don’t have different phases, then they are not operating at a phase transition point, even though they may produce power laws. That all sounds reasonable. But you told me that there were additional arguments to support your point, right?

Critio: Yes, the second main argument comes from a slightly different aspect of critical phenomena. It will take me a minute or two to explain, but I think it will be helpful. As I said previously, if a system is truly critical, it will display power law distributions in more than one variable of interest (Stanley,

Mnemo: Why are there multiple power laws?

Crito: Remember how I said that the phase of a critical system can be determined by a control parameter? Let me describe how important that parameter is. If we go back to that curve of the correlation length, recall that it had a sharp peak near T_{c}, the critical temperature. This type of curve is observed experimentally in diverse critical systems (Stanley, _{c} if you had an infinitely large system. A simple way to describe such a curve would be with an equation like this:

As T approaches T_{c}, the denominator goes to 0, and the correlation length, Γ, shoots up to infinity. The exponent ξ is another value that would be obtained from experimental data, and in general it would not always be 1. For convenience, physicists often use something called the “reduced temperature” given here by t, in describing critical phenomena:

In general, we don’t know precisely how the correlation length will depend on the reduced temperature, but I am able to write the correlation length as a power series in t, like this:

Near the critical point, the reduced temperature t approaches 0, so all the higher-order terms of this series become very small. We can then approximate the whole power series by something like this:

And you should recognize that this as a power law relationship. Using similar methods, other power laws can be found that relate other variables associated with the system, such as the relationship between the dynamic correlation value and distance between lattice sites in the Ising model (Figure

Mnemo: Why wouldn’t successive fractionation produce a relationship between exponents?

Critio: In that simple, one-dimensional system, there is only one power law, and that is related to the lengths of the sticks. There is only one exponent, so it can’t be related to other exponents.

Mnemo: But what about something like a combination of exponentials?

Critio: Recall that in that model the exponents α and β are the rates at which exponential processes increase and decrease, not exponents of power laws observed in variables associated with the system. The event size distribution is a power law whose exponent is related to the ratio of α/β. So, again there is only one exponent, so it can’t be related to other exponents. In addition, α and β are independent input parameters in the model, so there can be no relationship between them.

Mnemo: Let us assume for the moment that I agree that you should have exponent relationships if your system is truly critical. Is there any evidence for this type of relationship in neural data collected so far?

Critio: In fact there is. There is a recent article (Friedman et al.,

They found that the exponent γ, predicted in this way, fit reasonably well to the actual data. So, this is another piece of evidence suggesting that the system can display critical behavior (Friedman et al.,

Mnemo: Alright, this makes sense. It seems to be another way to assess whether or not the system is critical. But I would still like to hear more. What is your third argument that the neural data are collected from a critical process?

Critio: Remember when I said that power law distributions were scale-free? Recall that this was related to fractals that showed self-similarity?

Mnemo: Yes, I do. I have read some popular articles about fractals, so I am not completely new to this (Mandelbrot,

Critio: Good, then I can build on your existing knowledge to explain my last argument about why the neural data suggest criticality. It goes like this: The critical point is characterized by power laws in many variables, all of which express fractal relationships. We know that neural activity propagates dynamically through networks of neurons in cascades of activity. If these cascades, or avalanches, are truly critical then there should be some way to capture a relationship between the avalanches in a fractal way. What if we could take something like avalanche shapes and show that they were fractal? If we could do this, it would allow us to go beyond power laws, and show a scaling relationship that captured the dynamics of these non-equilibrium systems.

Mnemo: This sounds pretty abstract! Could you give me a more concrete example of what you are talking about?

Critio: Yes, of course. Let me describe what I mean by the avalanche shape. Consider how an avalanche of neural activity might evolve. It could start with one or a few spiking neurons. These could activate others, so the number of active neurons would increase over time. Eventually this would decline to 0, marking the end of the avalanche. If we plotted the average number of active neurons over time, we might get something that looked like an inverted parabola. This is what I mean by the average avalanche shape. Now if the network is at the critical point, then I should be able to take average avalanche shapes from different durations and show that they are all fractal copies of each other. In other words, I should be able to rescale them with the appropriate critical exponents and get them all to lie on top of each other, in what is called a data collapse. [Critio sketches Figure

Critio: These are average avalanche shapes taken from avalanches of different durations. See how they look like they might have roughly the same shape?

Mnemo: Yes, sort of. They could be copies of one another at different scales, but how are you going to show this?

Critio: Well, if we divide each curve by its duration, then they will be rescaled to all have the same length. Then if we rescale their heights by their duration raised to an exponent, γ from Eq.

Mnemo: The curves do seem to lie on top of one another pretty closely. Each curve is an average of how many avalanches?

Critio: Yes, each average avalanche shape is produced by hundreds of avalanches. So this data collapse is highly unlikely to have occurred by chance. In fact, when the spike train times from the original data are randomly jittered by 50 ms, the curves no longer look like copies of each other, suggesting that this scaling relationship has relatively tight temporal precision (Friedman et al.,

Mnemo: Although I can’t claim to understand all the math behind this, it certainly seems like your argument does not now rest on power laws alone. You have shown me a fractal relationship that ties together both space and time in the dynamic evolution of the avalanches. From all that you have told me, this should only occur near the critical point.

Critio: Yes, but it again sounds like you are not fully convinced!

Mnemo: You are correct – I still have another question about all this. In particular, I seem to recall reading somewhere that fractals are everywhere in neuroscience.

Critio: That’s right. Some have shown that a plot of the number of spikes produced by a neuron looks roughly the same at all intervals (Teich et al.,

Mnemo: If all this is true, then I guess I shouldn’t be so surprised when you tell me that some networks of neurons also display fractal behavior. The activity in the network could just be reflecting power law statistics that appear at other scales below it.

Critio: You are right to bring this up – with so many fractals out there, why should I get excited about a power law distribution of activity in small, local networks of neurons? Well, I have two answers to this. First, I could say that all the evidence I just mentioned about fractals in phenomena related to individual neurons is actually in favor of my general argument. We might expect the brain and its underlying systems to operate near a critical point to optimize information processing. However, the existence of the expectation is certainly not an argument against that which is expected. It seems that many biological systems would approach optimality by operating in a regime where they produce power laws (Mora and Bialek,

Mnemo: Yes, I think that is a fair description of my objection.

Critio: Ok, let us assume for the sake of argument that power laws in spike counts, transmitter secretion and channel dynamics are all produced by processes that are not critical. Is it really clear that if we combined such processes that the resulting cascades of activity on a network also would have to follow a power law? Would the resulting network therefore not be critical? We know from computer simulations that the pattern of network connectivity can have a profound effect on whether the network produces power laws or not (Teramae and Fukai,

Mnemo: Oh, is this where all that “self-organized criticality” literature comes in (Bak et al.,

Critio: It is still an open question as to how the network operates at the critical point, if it is indeed operating a critical point, and there have been several interesting proposals and experiments related to this topic (Bienenstock,

Mnemo: I suppose we will have to settle this over another lunch, as I have to go to another talk!

Critio: Wow, it is late! Hey, do you mind if I write this up and submit it to a journal? I think you have raised some very interesting objections, and you have forced me to think through my positions more thoroughly.

Mnemo: Sure, go ahead. But I am still skeptical, so don’t plan to include me as a co-author.

Critio: Not a problem. Thanks for sharing lunch.

Mnemo: My pleasure. Good bye.

The Movies S1–S3 for this article can be found online at

Simulation of an Ising model at low temperature.

Simulation of an Ising model at high temperature.

Simulation of an Ising model at the critical temperature.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

^{1}Critio: As a brief aside, I’m very interested in models of Epilepsy. In epileptic tissue, seizures exist that take the form of widespread coordinated activity. So, when modeling epileptic neural activity, we must be careful to incorporate seizures into our understanding of when the model is critical. For instance, during seizures, the activities of many neurons are highly correlated, so the dynamic correlation between model neurons is very high, but, by examining other parameters of the network, the network is not at a critical point.