LITERATURE THESIS
Master Brain and Cognitive Sciences Track: Cognitive Neuroscience
Thomas Pfeffer 6317421
THE CRITICAL BRAIN
How critical network dynamics might shape perception
Advisor: Dr. Tobias H. Donner Co-Assessor: Dr. Tomas H. J. Knapen
University of Amsterdam, The Netherlands 21-10-2012
1
Introduction
Scaling laws are widely observed in nature and suggest similar underlying processes, repeated across various levels of observation (Kello, 2010). By scaling law behavior, one typically means that some variable is proportional to another variable raised to a power, e.g.,
where k is a constant and α denotes the exponent of the scaling law. Of all possible scaling laws, many take the form of a power law. Ever since power-law behavior has been first observed, an extraordinary number of natural phenomena have been reported to follow a power-law distribution (see Newman, 2005). Despite their ubiquity, there is still no general agreement about the underlying mechanism leading to this behavior. In the late 1980s, Per Bak, Kurt Wiesenfeld and Chang put forward the theory aiming to explain the origin of one of the most pervasive type of power-law distribution: the so-called 1/f noise. They argued that 1/f noise is a signature of complex dynamical systems, i.e., systems with a great number of nonlinearly interacting elements, that have organized into a state poised between order and chaos, the critical state. Importantly, such a system is not tuned to the cusp of a phase transition by external driving forces or external control parameters like other physical systems (e.g. the Ising model of spin magnetization). Consequently, Bak and colleagues termed the state
self-organized criticality. In principle, in this critical state, even smallest perturbations can eventually lead to reconfigurations of the system on a very large scale (Bassett et al., 2006). In the critical state a system is characterized by scale-invariant spatial properties as well as scale-free distribution of dynamic events. Ever since the publication of their theory, self-organized critical states have been described for a wide variety of complex phenomena in nature such as the distribution of earthquakes, fluctuations in financial systems, landscape formation and many more (see Bak, 1999). Without doubt the most complex system that displays 1/f behavior is the (human) brain. It has long been argued that being tuned to a critical point may be of special benefit for an organ like the brain. Already in 1950, Alan Turing, the pioneer of computer science and artificial intelligence, postulated that the brain must operate at a critical point to ensure fast dynamical transitions between different states and the ability to react to
novel input (see Werner, 2010). Indeed, the idea of critical brain dynamics has intuitive appeal: imagine the brain operating in a subcritical state. In an ever changing environment, it would not be able to adapt to novel situations as quickly as needed for the organism to survive, brain activity would be too static. On the other hand, in a supercritical state, the brain would not be able to form long-term memories and would consequently switch between different states constantly. Therefore, a tight balance between order and chaos seems of special benefit as it masters ensuring that an organism can adapt to novel situations while maintaining the ability to form long-term memories at the same time. Indeed, mathematical models and computer simulations suggest that information processing (Haldeman & Beggs, 2005; Kinouchi, 2006;Beggs, 2008) and learning (de Arcangelis et al., 2010) is optimized if a system is tuned towards criticality.
Despite the long history and the appeal of the idea, it took until about the 1990s that more research was deployed to the study of criticality in neural circuits. At first, the "criticality hypothesis" (Beggs, 2008) was mostly investigated by means of computational models of neural networks (e.g. Usher et al., 1995) and empirical evidence in its support remained sparse. However, the past decade has seen an increasing number of empirical investigations at different levels of observation which seem to support the claim of a critical brain. Despite the increasing interest, it still remains largely unknown, how critical network dynamics affect cognition and behavior. The goal of this review is to integrate evidence from several levels of observation and, ultimately, to draw conclusions about how critical brain activity influences cognitive phenomena in general, and it relates to perception in particular.
In the present review, I will first briefly outline the background of the concept of self-organized criticality by describing the most simple and famous model that has been developed by Bak and colleagues: the sand-pile model of criticality. In the second part, I will discuss current experimental findings showing how neural networks exhibit critical dynamics. This part will be structured in a hierarchical way, from microscopic evidence in neuronal tissue, to mesoscopic evidence from human neuroimaging studies to the macroscopic level of cognition and behavior. At the end of this chapter, I will try to show how critical network dynamics shape behavior and cognition in general, and how they might shape visual perception in particular. The last part of this thesis will summarize
the findings but also give a critical account of the reported results and will highlight some alternative interpretations.
2 Background
1/f noise (sometimes also called "pink noise") describes a special form of a power-law which follows the form
where α denotes the so-called scaling-parameter. Typically, this parameter is in the range from 0 to 2. Such a distribution has a so-called "heavy tail", which means that extreme values are far more likely than would be assumed for a (e.g.) normal distribution. Since its first discovery, 1/f distributions have been widely observed in nature (Bak, 1999; Ward & Greenwood, 2007). Despite this pervasiveness, scientists have not been able to agree on an underlying mechanism generating 1/f behavior (Ward & Greenwood, 2007). Another frequent empirical observation are those of self-similar or
fractal structures (Mandelbrot, 1982). Self-similarity refers to the phenomenon that a
spatial structure looks roughly the same, independent of the scale of observation, i.e., they are scale-free or scale-invariant. In their influential article from 1987, Per Bak, Chao Tang and Kurt Wiesenfeld (BTW) provide a theory that aims to explain the generation of the ubiquitous 1/f noise as well as the preponderance of fractal structures in nature. They argue that large systems of interacting elements with a high number of degrees of freedom naturally evolve into a critical state, independent of the initial conditions and without external tuning factors. This means they self-organize to a state where they are poised between order and chaos. They claim that the intrinsic dynamics of critical systems are the driving force behind all phenomena displaying 1/f noise and spatial self-similarity (Bak et al., 1988). Using a simple mathematical model, the so-called BTW sand pile model, they could illustrate how a dynamical system with slow energy accumulation slowly drifts into a critical state where it displays fast energy redistributions, a cascade of energy dissipation, called "avalanches":
"Suppose we start from scratch and build the pile [of sand] by randomly adding sand, a grain at a time. The pile will grow, and the slope will increase. Eventually, the slope will reach a critical value; if more sand is added it will slide off. Alternatively, if we start from a situation where the sand is too steep, the pile will collapse until it reaches the critical state, such that it is barely stable with respect to further perturbations. The critical state is an attractor for the dynamics." (Bak et al., 1988)
Importantly, the size of these avalanches described above has no typical scale, meaning that the system displays avalanches of all sizes, up to the size of the system itself. This means, the size of those events is scale-free or fractal and follows a power-law distribution with a scaling parameter close to -3/2. Critical systems showing this power-law behavior have another interesting feature: they are by definition long-range dependent. This means, they show persistent serial correlations (Wagenmakers et al., 2004), which means that even small perturbations of the system in the distant past can lead to events on a macroscopic level in the future.
A common approach to identify a critical system in models as well as empirically is by determining the so-called "branching ratio" of a network (e.g. Kello & Mayberry, 2010). The branching ratio describes how activity is propagated in a network, i.e., how many descendent units are activated by a single ancestor unit. Imagine a very simplified and small population of about 100 completely interconnected neurons that produce action potentials. Each action potential at neuron A causes a connected neuron to fire with a certain probability p. If we define p = 0.5, this means just a single action potential at neuron A is enough to activate on average half of the whole population which in turn activate all remaining neurons, i.e., within only two time step, the whole neural network is firing. Even with a slightly lower value for p, a chain of activity will proceed until all neurons of this small example population will fire and this activity will never die out. In the other extreme, p is very small, for example p = 0.001. Now, one spike in neuron A will activate on average not even one other neuron. This means, ten spikes in A would be required in order to activate on average one other neuron, which in turn - on average - does not elicit any other units to become active. In this setting, activity dies out very quickly. Generally speaking, if a neuron A connects to N postsynaptic neurons, to achieve balance, P needs to be tuned to 1/N (Shew & Plenz, 2012). In the example
above, this means that if A on average activates one connected neuron and this in turn activates one neuron, activity spreads for very long time scales.
In general, a network with p smaller than 1/N is in a "subcritical state" where one active neuron activates on average less than one descendent neuron. A supercritical system is characterized by values for p larger than 1/N, which means that one active neuron activates on average more than one connected element. A system with p = 1/N is poised at a critical state. The advantages of such a system are obvious: in a subcritical system, activity would die out very soon and therefore, there would be no long-term dependencies. However, these long-range temporal dependencies (or correlations) are crucial for a system to "remember" past events. In a supercritical system, on the other hand, activity spreads immediately and leads to the joint activation of large subsets of neuronal population. Such a spread of neuronal activation resembles the spontaneous synchronization of neuronal populations during an epileptic seizure and is far from optimal for an organism. Therefore, a system tuned to the critical state between these two extremes seems to be best suited to ensure quick information transmission and adaptation to novel situations, but at the same time be stable enough to form long-term memories.
3 Experimental Evidence for Criticality
3.1. The microscopic level: neuronal avalanches
In a series of studies, Dietmar Plenz and his colleagues identified and characterized what they claim to be a "new mode" of neural activity in cortical networks. Using microelectrode arrays, they recorded spontaneous negative local field potentials (nLFPs) from acute coronal slices of the rat cortex (Beggs & Plenz, 2003). They report that the propagation of these nLFPs is best described by equations that govern avalanche dynamics as they have been reported in models of self-organized criticality (e.g. Bak, 1999). This means, size and duration of those "neuronal avalanches" follow a power-law distribution with a scaling parameter close to -3/2, as expected from a critical branching process, which suggests that the slice preparations of rat cortex
indeed operate at criticality. Theoretical work and computer simulations support this claim (e.g. Haldemann & Beggs, 2005), showing that power-law distributions of avalanches are a characteristic feature of critical, but not of sub- or supercritical networks. Until today, these results have been successfully replicated at various organizational levels, from the isolated leech ganglion (Mazzoni et al., 2007) to sensory areas of anesthesized cats (Hahn et al., 2010) and (pre-) motor areas of the awake rhesus monkey (Petermann et al., 2009), whereas the latter was the first study to demonstrate the existence and functional relevance of neuronal avalanches also in a fully functional cortical system.
Plenz and colleagues argue that neuronal avalanches represent a fundamental organizational principle in the brain, capable of producing a great repertoire of diverse activity patterns that can be reliably reconstructed over time (Plenz & Thiagarajan, 2007). More specifically, they hypothesize that with help of neuronal avalanches, functional coalitions of neurons, or cell assemblies, are formed. This means, neuronal avalanches might allow for the transient synchronization of populations of neurons. Due to their scale-invariant nature, avalanches can potentially span the whole brain and could therefore play an essential role in the binding or the integration of information represented in distant areas of the brain.
Another noteworthy finding in this context is that many avalanches form temporal activation patterns to which the neural network returned repeatedly, i.e., the network seemed to have preferences for certain activation states (Beggs & Plenz, 2004; Plenz & Thiagarajan, 2007). A preference for, or "replay" of certain activation patterns has previously been reported in studies of memory reconsolidation during sleep (e.g. Louie & Wilson, 2001) and is indicative of a potential role of neuronal avalanches in the storage or the consolidation of information. Consistent with this claim, Beggs and Plenz (2004) found that certain "families" of neuronal avalanches, i.e., distinct variants of activity patterns, exhibited a remarkable stability over time: 78% of the avalanches produced by the neural network after 10 hours were highly similar to those recorded during the first hour. These results are in line with early computer simulations of neural networks showing that a critical system that exhibits power-law temporal correlations and 1/f behavior has a preserved ability to "remember" past events (Usher et al., 1995). Given the very short temporal scale of neuronal avalanches (Beggs & Plenz, 2003), it seems as if critical networks balance the need for flexible adaptation, while maintaining
sufficient network stability at the same time. More recent work has further demonstrated theoretically as well as empircally the benefits of critical networks for learning (de Arcangelis et al., 2010) and various other aspects of information processing (Haldemann & Beggs, 2005; Kinouchi & Copelli, 2006; Shew et al., 2009; Shew et al., 2011).
However, it is important to note that self-organized criticality, though appealing, is not the only process capable of explaining 1/f distribution in cortical tissue. It has been shown that poisson (Bédard et al., 2006) but also a superposition of several short-range processes can lead to 1/f scaling (Wagenmakers et al., 2004). Despite the intuitive appeal of the "criticality hypothesis" (Beggs, 2008), it is therefore important to remember that 1/f scaling may be highly suggestive of (Sporns, 2010), but does not imply criticality.
Another open question remains what requirements have to be met for a network to reach a critical state. It has been shown in models of spiking neural networks that neural plasticity plays an important role in the development of critical states (Levina et al., 2007). Moreover, neuronal avalanches seem to crucially depend on the right balance between inhibition and excitation (Poil et al., 2012). Another idea is that neuromodulators play a role in tuning a system towards a point of a phase transition (Chialvo, 2010), an idea which has also received some experimental support (Steward & Plenz, 2006).
3.2. The mesoscopic level: critical small worlds
Statistical physics states that microscopic changes can propagate over several scales and eventually lead to macroscopic events, evolving on many time scales when a complex system is operating at criticality (Kello, 2010). Therefore, if neuronal populations in vitro as well as in vivo display critical dynamics, this should also be visible at a bigger scale such as large populations of neurons. And ultimately, if criticality and neuronal avalanches are indeed a fundamental mode facilitating the formation of transient functional cell assemblies, evidence should also be found in the human brain. Converging lines of research support this claim.
In recent years, the application of graph theory to the investigation of different types of networks received an increasing amount of attention (Newman, 2010; Sporns, 2010). At
many different levels of organization, it has been shown that small-world network characteristics are prevalent. Small-world networks have some interesting features: theory and computer simulations predict that this type of network architecture is more energy efficient than other types of networks (Sporns, 2010), making them also attractive for a highly optimized network like the brain. Therefore, in recent years, graph theoretical analyzes have also been applied to the field of (cognitive) neuroscience. Crucially, these studies bear some important implications for the study of self-organized criticality as it has been argued that fractal (or small-world) organization of a network is a prerequisite for a system to be tuned towards criticality, as only those types of networks provide the necessary stability and allow for a rich diversity of different activity patterns at the same time (Kaiser & Hilgetag, 2010). Consistent with this notion, a number of recent studies have indeed demonstrated the fractal organization of structural as well as functional brain networks in general (Bullmore & Sporns, 2009; Sporns, 2010) and in the context of avalanche dynamics in particular (Pajevic & Plenz, 2009).
Although a necessity, fractal organization of the brain does not imply critical network dynamics per se. However, also in another line of research, evidence for a critical operation mode of brain activity on a "macroscopic" level is now accumulating.
A very early study to demonstrate a possible signature of criticality in the human brain was carried out by Kelso et al. (1992). In this experiment, the authors used a 32-sensor squid-array to measure neuronal activity in the human brain, while manipulating the speed at which a finger flexion movement had to be performed. They observed a rapid transition of sensorimotor coordination behavior in all subjects which was accompanied by large and sudden changes in neuromagnetic field patterns. They conclude with the hypothesis that the brain might be a pattern forming system operating at point of critical instability to flexibly and spontaneously switch between different possible activation states. Importantly, they were the first to also link these rapid transitions in neuronal activation patterns to human behavior, thereby demonstrating the potential relevance of critical network dynamics for behavior and cognition.
Significant advances in technology during recent decades, allowed for a more detailed investigation of the proposal in a recent study using magnetoencephalography (MEG). MEG also makes use of SQUID arrays, however, with higher sensor density and
whole-head coverage. Inspired by the avalanche analysis of Beggs and Plenz (2003), Poli et al. (2008) looked for similar critical dynamics in neuronal oscillations, which are thought to reflect fluctuations in the excitability of local neuronal populations (Buszaki et al., 2012) and have been linked with a variety of cognitive functions (Engel et al., 2001; Siegel et al., 2012). In the context of this review, neuronal oscillations are of special interest as it has been shown that their power spectrum is inversely proportional to the frequency, i.e., it follows a 1/f distribution (Buszaki & Draguhn, 2004) and thought to depend, just like critical networks, on balanced excitation and inhibition (Beggs & Plenz, 2003; Poil et al., 2012). Therefore, they might emerge from a similar underlying mechanism (Poli et al., 2012), although alternative explanations are also conceivable (Buzsaki et al., 2012). Using a combination of computational modeling and magnetoencephalography (MEG), Poli et al. (2008) identified avalanche-like oscillation bursts specifically for activity in the alpha-range (8-12 Hz), showing long-range temporal correlations. The life-time probability of these events followed a power-law distribution with a scaling parameter indicative of an underlying critical process. These results are generally consistent with the finding that amplitude fluctuations of alpha and beta oscillations are correlated over thousands of cycles and follow a power-law with a scaling parameter that is highly invariant across subjects (Linkenkaer-Hansen et al., 2001). Therefore, this power-law scaling might indeed reflect macroscopic critical network dynamics. However, it has been criticized that both long autocorrelation times as well as power-law scaling are only suggestive of, but do not imply criticality, as even simple stochastic models (Aburn et al., 2012) or a combination of several short-range processes (Wagenmakers et al., 2004) can potentially exhibit these features under certain circumstances (but see the last section for details).
Another complementary piece of evidence stems from a study (Kitzbichler et al., 2009) which investigated the phase lock interval between different areas of the brain by means of MEG as well as fMRI. They hypothesized that if neuronal avalanches indeed organize populations of neurons into transient assemblies, reflected by rapid transitions between activation patterns, this should also be evident in occasional large-scale changes in the functional coupling between various areas., i.e. changes in synchronization between different regions of the brain. These changes in the coupling should follow similar distributions as observed in the avalanche studies reviewed above (e.g. Beggs & Plenz, 2003). Indeed, they could demonstrate that network
synchronization follows a power-law and with the help of computational models link this to an underlying critical mechanism. Importantly, Kitzbichler and colleagues demonstrate power-law scaling for phase lock intervals, which means that rapid reconfigurations of synchronization patterns occurred throughout the recording session. These fundamental changes in synchronization patterns might indeed reflect the formation of transient functional coalitions of neuronal populations, as proposed by Plenz and Thiagarajan (2007). The short temporal scale in which these patterns can emerge might ensure that network reconfigurations occur as rapidly as needed for the brain to remain responsive to novel and unexpected events. Alternatively, one could think of these transitions as being reflective of sudden changes in global mental state of a subject. In any case, large-scale network reconfigurations as the ones reported should have behavioral consequences and it will be an obvious future challenge to demonstrate the behavioral and cognitive significance of these.
3.3. The macroscopic level: emergent properties of behavior
While for a long time it has been thought that the brain only translates external input into some sort of output, it is now clear that every cognitive act is to a certain extent also a product of ongoing, endogenous activity in the brain (Raichle, 2010) and therefore partly independent of external physical stimulation. In a recent study (Fox et al., 2007), it has been shown that a great part of the variability in a simple motor behavior can be attributed to intrinsic fluctuations of the BOLD signal, demonstrating that these fluctuations are more than just a mere physiological artifact. Further, as depicted in earlier sections, it has been shown that ongoing activity in the brain exhibits critical dynamics on every level of observation. One of the appeals of a scaling law like the one discussed in this article is that it suggests a similar underlying process at all of these levels (Kello, 2010). Hence, given that behavior does depend on the collective critical dynamics of large populations of interacting neurons, signatures of this process should as well be apparent on a behavioral level.
Indeed, the first evidence of a behavioral signature of critical network dynamics reach back to the beginning of psychophysics about 150 years ago. Ernst Weber was among the first to characterize the relation between a physical stimulus and a behavioral response in a quantitative fashion (Chialvo, 2006). He found that the threshold to
perceive the difference between two stimuli is proportional to the magnitude of those stimuli. This means, the larger the magnitude of the stimuli, the larger the difference of the to-be-compared stimuli needs to be in order to be correctly differentiated. This relationship was later on formulated more generally as Steven's power-law. It states that the magnitude of a stimulus S is proportional to its perceived intensity raised to a power of α (see Kello, 2010). Ever since, fluctuations in a vast variety of psychophysical tasks have been reported to follow a 1/f distribution (Gilden, 1997; Gilden, 2001), which may be the consequence of a underlying critical neuronal activity (Werner, 2010). Gilden (2001) also emphasizes that the prevalence of 1/f noise in behavior is not attributable to regular kinds of correlative processes usually invoked in psychological theory, but rather reflects "a kind of memory that arises in dynamical systems" (Gilden, 2001). Unfortunately, psychology and the neurosciences have mostly ignored potential underlying mechanistic theories, which could explain the origin of these pervasive relations (Kello, 2010). Self-organization might serve as such an overarching explanation: another intriguing aspect of psychophysics is that the intensity of physical stimuli varies greatly, over several orders of magnitude (Chialvo, 2010). Interestingly, it has not been shown yet that the dynamic range of single cells is large enough in order to explain such sensitivity (Chialvo, 2006). However, in a recent study, Kinouchi and Copelli (2006) presented a population model consisting of elements, which on their own have a limited dynamic range, but collectively respond with a much broader dynamic range and very high detection sensitivity. The authors claim that Steven's law can be attributed to critical dynamics, at which the network shows a maximized dynamic range, suggesting that scaling laws in psychophysics could be another signature of critical network activity.
4 How critical network dynamics shape perception
The perception of a sensory stimulus presented at the so-called detection threshold fluctuates. While on some trials, a subject will successfully report the presence of the stimulus, on others the subject will fail to do so. In the past, it has been thought that these fluctuations in perception arise due to imperfections of the system, such as noise
at different levels of the processing stream. In the past years, this claim has been challenged by various theories claiming that perceptual fluctuations and neronal "noise" have a functional role and might render the system more susceptible to sensory stimulation (Deco & Romo, 2008; McDonnell & Ward, 2011). A concept known as "stochastic resonance" describes the seemingly paradoxical phenomenon that in certain settings, a higher baseline level of noise can actually lead to an increase in sensitivity of a system towards weak levels of stimulation (Rouvas-Nicolis & Nicolis, 2007). Recently, evidence for stochastic resonance in the brain has been found in humans (Linkenkaer-Hansen et al., 2001). Hence, it is becoming increasingly clear that neuronal fluctuations and neuronal "noise" are not reflecting imperfection, but may indeed have an important role in the brain (Ermentrout et al., 2008).
It has been suggested that perceptual fluctuations are due to transitions in underlying "attractor states" (see Braun & Mattia, 2010). The concept of (biophysically plausible) attractor networks has been derived from statistical physics and has been successfully applied to multistable perception and decision-making in previous investigations (see Deco et al., 2009). Attractor states can be thought of as points to which a system automatically converges over time. A system is thereby not limited to only one attractor state - several such states are possible at the same time. The idea of attractor states are reminiscent of the report of neuronal avalanche patterns that showed a very consistent recurrence over time in the previously reported study of neuronal tissue (Beggs & Plenz, 2003). These repeating activity patterns resemble preferred metastable states, generated by neural attractor network models (Haldeman & Beggs, 2005). This suggests that the concept of neuronal avalanches and, consequently, criticality, might be inherently related to attractor dynamics (Brain & Mattia, 2010), which opens up the exciting idea that fluctuations in decision-making or perception, such as the transition from one percept of a bi- or multistable stimulus to another, is actually a signature of critical network dynamics in the brain. In fact, there are several reasons to believe that self-organized criticality is indeed the mechanism underlying these irregular perceptual fluctuations. Attractor models have been successfully applied in order explain transitions in the perception of a bistable stimulus (Deco et al., 2009). Neuronal avalanches were interpreted in terms of reflecting a switch in global brain state or the formation of functional assemblies. However, one could also understand these rapid reconfigurations as changes in the "attractor landscape", preventing "deadlock" of the
system in a certain attractor state and thereby, enhancing the flexibility and its sensitivity of the perceptual system (Deco & Romo, 2008) towards novel input.
Another interesting feature of neural attractor models is that they can be reduced to a one-dimensional diffusion equation (see Deco & Romo, 2008). These diffusion models have been extremely successful in explaining a wide variety of behavior (Smith & Ratcliff, 2004).
Ongoing activity and neuronal avalanches might serve the purpose of the continuous exploration of activity patterns in some of which the brain gets locked for a while as a task needs to be executed or as long as the sensory system is stimulated (Taggliazucchi & Chialvo, 2011). One obvious way to test this idea is to investigate how the tuning of the network dynamics from a subcritical to a supercritical regime influences the distribution of perceptual fluctuations. Only at the critical state, the system should exhibit signatures such as power-law scaling. In general, the investigation multistable perception may possibly offer a window into how neuronal avalanches might optimize behavior on a macroscopic scale.
It has been proposed that neuromodulators act as control parameters, tuning the cortical networks towards criticality (Chailvo, 2010). Interestingly, it has been shown that activity of the locus coeruleus, the major source of norepinephrine in the brain, modulates decision behavior. While the phasic activity mode is associated with maintaining a momentary behavioral state in order to optimize gain, a tonic level is associated with the disengagement from the current task and the exploration of alternative behaviors, when no gain is expected from the momentary behavior any longer (Aston-Jones & Cohen, 2005). Although very speculative, it may be possible to frame this theory also from the perspective of self-organized criticality, where neuronal avalanches serve to explore new behavioral patterns ("exploration state", corresponding to tonic activity) only to get locked into an attractor state for a while ("exploitation state", corresponding to phasic activity), until a macroscopic event, large enough to push it out of this state, occurrs. Hence, it would be exciting to see, if certain characteristics of critical network dynamics such as the scaling parameter of the event size and life-time of neuronal avalanches or the phase-locking index (Kitzbichler et al., 2009) changes, while a system undergoes a transition from the tonic to phasic activity mode.
5 Conclusion and critical notes
In this review, I have pointed out the benefits of critical network dynamics for the brain and have discussed several lines of research that support the idea that a critical operation mode of the brain, ranging from the demonstration of neuronal avalanches in neuronal tissue to observations of scale-free fluctuations in behavior. In general, the "criticality hypothesis" (Beggs, 2008) provides an attractive framework to explain findings at various scales and integrate several, seemingly disparate, behavioral and neuronal phenomena. Neuronal avalanches open the possibility for transient patterns of synchrony to form neuronal assemblies that occur over multiple time and spatial scales. Their scaling behavior and dynamics are highly suggestive of an underlying self-organized critical network state. This is supported by macroscopic observations of scale-invariant fluctuations in behavior and perception.
However, despite the appeal of this framework, alternative explanations have to be considered as well and one generally needs to be careful when interpreting the results reviewed above. Therefore, in this last section of this review, I would like to discuss some alternative interpretations and potential challenges for the criticality framework. Most of the reviewed studies are reliant upon the finding of a power-law scaling of a dependent measure such as neuronal spiking or population activity. Although physical systems poised at a phase transition display power-law behavior in various aspects (Beggs & Timme, 2012), power-law scaling is by no means sufficient but only suggestive of criticality (Wagenmakers et al., 2004; Wagenmakers et al., 2005) as already pointed out earlier. For example, it has been shown that also by a combination of several exponential processes that terminate at random times, one can produce power-law scaling that mimics the behavior of critical dynamics (Newman, 2005).
Moreover, in many cases, researchers already have the prior assumption of a 1/f scaling but do not test alternative models (Wagenmakers et al., 2004). For example, it has been argued that exponentials, lognormal functions or even just a straight line might give an even better fit to some of the reported data compared to power-law functions (Wagenmakers et al., 2004). In some cases, the presence of long-range dependency in the form of a power-law scaling is only determined by fitting a straight line to the log-log
plots by eye (Wagenmakers et al., 2004), which makes it obvious, why alternative models have never been assessed.
Furthermore, Wagenmakers et al. (2004) point out that not only long-range dependent processes can lead to a 1/f scaling but also a superposition of several short-range processes lead to 1/f noise. Even the most pervasive feature of the studies reviewed above, neuronal avalanches, can, in principle, arise from stochastic processes (Touboul & Destexhe, 2010), for example independent poisson processes (Bédard et al., 2006). Finally, true criticality can only be measured in systems of infinite size (see Stumpf & Portner, 2012). All discussed studies are usually limited by the amount of sensors from which neural activity is recorded. This makes it not impossible to identify critical processes, but makes it harder to conjecture from scaling parameters to underlying critical processes (Beggs, 2008).
It is obvious that these are serious issues that need to be addressed in future research. Some might be clarified just as methods (Klaus et al., 2012) and technologies are advancing (Kello, 2010), some perhaps require more sophisticated models to explain the observed behavior on all scales (Plenz, & Thiagarajan, 2007; Wagenmakers et al., 2004). Further, critical network dynamics challenge the functional approach of cognitive neuroscience, as in this context, behavior is no longer attributable to individual components, but rather an emergent property of collective activity (Kello et al., 2007). Therefore, mere isolated physiological would not be sufficient to explain isolated cognitive functions. However, if these obstacles can be overcome in the future, self-organized criticality provides an exciting framework for human brain function and might be a theory capable of integrating findings from investigations of spiking neural networks in animals on the one hand, with behavioral and cognitive observations of humans on the other.
6 References
Aburn, M. J., Holmes, C. A., Roberts, J. A., Boonstra, T. W., Breakspear, M. (2012). Critical Fluctuations in Cortical Models Near Instability. Frontiers in Physiology, 3(331), 10.3389/fphys.2012.00331.
Aston-Jones, G., & Cohen, J. D. (2005). An Integrative Theory of Locus Coeruleus-Norepinephrine Function: Adaptive Gain and Optimal Performance. Annual Reviews in Neuroscience, 28, 403-450.
Bak, P. (1999). How Nature Works. The Science of Self-Organized Criticality.
Bak, P., Tang, C., & Wiesenfeld, K. (1987). Self-organized criticality: an explanation of the 1/f noise.
Physical Review Letters, 59, 381-384.
Bak, P., Tang, C., & Wiesenfeld, K. (1988). Self-organized criticality. Physical Review A, 38, 364-375.
Bassett, D. S., Meyer-Lindenberg, A., Achard, S., Duke, T., & Bullmore, E. (2006). Adaprtive reconfiguration of fractal small-world human brain functional networks. Proceedings of the National
Academy of Sciences, 103(51), 19518-19523.
Bédard, C., Kröger, H., & Destexhe, A. (2006). Does the 1/f Frequency Scaling of Brain Signals Reflect Self-Organized Critical States? Physical Review Letters, 97(11), 118202(4).
Beggs, J. M., & Plenz, D. (2003). Neuronal avalanches in neocortical circuits. Journal of Neuroscience,
23(25), 1167-77.
Beggs, J. M., & Plenz, D. (2004). Neuronal avalanches are diverse and precise activity patterns that are stable for many hours in cortical slice cultures. Journal of Neuroscience, 24, 5216-5229.
Beggs, J. M. (2008). The criticality hypothesis: how local cortical networks might optimize information processing. Philosophical Transactions of the Royal Society A, 366, 329-343.
Braun, J., & Mattia, M. (2010). Attractors and noise: Twin drivers of decisions and multistability.
NeuroImage, 57, 740-751.
Bullmore, E. T., & Sporns, O. (2009). Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience, 10, 186-198.
Buzsáki, G., & Draghun, A. (2004). Neuronal Oscillations in Cortical Networks. Science, 304, 1926-1929. Buszaki, G., Anastassiou, C. A., & Koch, Christoph (2012). The origin of extracellular fields and currents - EEG, ECoG, LFP and spikes. Nature Reviews Neuroscience, 13, 407-420.
Chialvo, D. R. (2006). Are our senses critical. Nature Physics, 2, 301-302.
Chialvo, D. R. (2010). Emergent complex neural dynamics. Nature Physics, 6, 744-750.
de Arcangelis, L., & Herrmann, H. J. (2010). Learning as a phenomenon occuring in a critical state.
Proceedings of the National Academy of Sciences, 107(9), 3977–3981.
Deco, G., & Romo, R. (2008). The role of fluctuations in perception. Trends in Neurosciences, 31(11), 591-598.
Deco, G., Rolls, E. T., Romo, R. (2009). Stochastic dynamics as a principle of brain function. Progress in
Engel, A. K., Fries, P., & Singer, W. (2001). Dynamic predictions: oscillations and synchrony in top-down processing. Nature Reviews Neuroscience, 2, 704-716.
Ermentrout, G. B., Galán, R. F., & Urban, N. N. (2008). Reliability, synchrony and noise. Trends in
Neuroscience, 31(8), 428-434.
Fox, M. D., Snyder, A. Z., Vincent, J. L., & Raichle, M. E. (2007). Intrinsic Fluctuations within Cortical Systems Account for Intertrial Variability in Human Behavior. Neuron, 56, 171-184.
Gilden, D. L. (1997). Fluctuations in the time required for elementary decisions. Psychological Science,
8(4), 296-301.
Gilden, D. L. (2001). Cognitive Emissions of 1/f Noise. Psychological Review, 108(1), 33-56.
Hahn, G., Peterman, T., Havenith, M. N., Yu, S., Singer, W., Plenz, D., & Nikolic, D. (2010). Neuronal avalanches in spontaneous activity in vivo. Journal of Neurophysiology,
Haldeman C, Beggs JM (2005) Critical branching captures activity in living neural networks and maximises the number of metastable states. Physical Review Letters, 94, 058101.
Kaiser, M., & Hilgetag, C. C. (2010). Optimal hierarchical modular topologies for producing limited sustained activation of neural networks. Frontiers in Neuroinformatics, 4(8).
Kello, C. T., Beltz, B. C., Holden, J. G., Van Orden, G. C. (2007). The Emergent Coordination of Cognitive Function. Journal of Experimental Psychology: General, 136(4), 551-568.
Kello, C. T., Brown, G. D. A., Ferrer-i-Cancho, R., Holden, J. G., Linkenkaer-Hansen, K., Rhodes, T., & van Orden, G. C. (2010). Scaling laws in cognitive sciences. Trends in Cognitive Sciences, 14(5), 223-232. Kello, C. T., & Mayberry, M. R. (2010) Critical branchin neural computation. IEEE World Congress on
Computational Intelligence, Barcelona, Spain, 1475-1481.
Kelso, J. A. S. (1992). A phase transition in human brain and behavior. Physics Letters A, 169(3), 134-144. Kinouchi, O., & Copelli, M. (2006). Optimal dynamical range of excitable networks at criticality. Nature
Physics,
Kitzbichler, M. G., Smith, M. L., Christensen, S. R., & Bullmore, E. (2009). Broadband Criticality of Human Brain Network Synchronization. Plos Computational Biology, 5(3), e1000314.
Klaus, A., Yu, S., Plenz, D. (2012). Statistical analyses support power law distributions found in neuronal avalanches. PLoS One, 6(5), e19779.
Levina, A., Herrmann, J. M., & Geisel, T. (2007). Dynamical synapses causing self-organized criticality in neural networks. Nature Physics, 3, 857-860.
Linkenkaer-Hansen, K., Nikouline, V. V., Palva, J. M., & Ilmoniemi, R. J. (2001). Long-range temporal correlations and scaling behavior in human brain oscillations. Journal of Neuroscience, 21, 1370-1377. Louie K., & Wilson, M. A. (2001). Temporally Structured Replay of Awake Hippocampal Ensemble Activity during Rapid Eye Movement Sleep. Neuron, 29(1), 145-156.
Mazzoni, A., Broccard, F. D., Garcia-Perez, E., Bonifaz, P., Ruaro, M. E., & Torre, V. (2007). On the Dynamics of the Spontaneous Activity in Neural Networks. PLoS One, 2(5): e439. doi:10.1371/journal.pone. 0000439.
McDonell, M. D., & Ward, L. M. (2011). The benefits of noise in neural systems: bridging theory and experiment. Nature Reviews Neuroscience, 12, 415-426.
Newman, M. E. J. (2005). Power laws, Pareto distributions and Zipf's law. Contemporary Physics, 46(5), 323-351.
Newman, M. E. J. (2010). Networks: An Introduction.Oxford: Oxford University Press.
Pajevic, S., & Plenz, D. (2009). Efficient Network Reconstruction from Dynamical Cascades Identifies Small-World Topology of Neuronal Avalanches. PLoS Computational Biology, 5(1), e1000271. doi: 10.1371/ journal.pcbi.1000271.
Petermann, T., Thiagarajan, T. C., Lebedev, M. A., Nicolelis, M. A. L., Chialvo, D. R., & Plenz, D. (2009). Spontaneous cortical activity in awake monkeys composed of neuronal avalanches. Proceedings of the
National Academy of Sciences, 106(37), 15921-15926.
Plenz, D., & Thiagarajan, T. C. (2007). The organizing principles of neuronal avalanches: cell assemblies in the cortex? Trends in Neuroscience, 30, 3, 101-110.
Poil, S., van Ooyen, A., & Linkenkaer-Hansen, K. (2008). Avalanche Dynamics of Human Brain Oscillations: Relation to Critical Branching Processes and Temporal Correlations. Human Brain Mapping,
29, 770-777.
Poil, S., Hardstone, R., Mansvelder, H. D., & Linkenkaer-Hansen, K. (2012). Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks.
Journal of Neuroscience, 32(29), 9817-9823.
Raichle, M. E. (2010). Two views of brain function. Trends in Cognitive Sciences, 14(4), 180-190. Rouvas-Nicolis, C., & Nicolis, G. (2007). Stochastic resonance. Scholarpedia, 2(11):1474.
Siegel, M., Donner, T. H., & Engel, A. K. (2012). Spectral fingerprints of large-scale neuronal interactions.
Nature Reviews Neuroscience, 13, 121-134.
Shew, W. L., Yang, H., Petermann, T., Roy, R., & Plenz, D. (2009). Neuronal avalanches imply maximum dynamic range in cortical networks at criticality. Journal of Neuroscience, 29, 15595-15600.
Shew, W. L., & Plenz, D. (2012). The Functional Benefits of Criticality in the Cortex. The Neuroscientist,
XX(X), 1-13.
Shew, W. L., Yang, H., Yu, S., Roy, R., & Plenz, D. (2011). information capacity and transmission are maximized in balacned networks with neuronal avalanches. Journal of Neuroscience, 31(1), 55-63.
Sporns, O. (2010). Networks of the Brain. Boston: The MIT Press.
Stumpf, M. P. H., & Porter, M. A. (2012). Critical Truths About Power Laws. Science, 335, 665-666.
Taggliazucchi, E., & Chialvo, D. R. (2011). Criticality in large-scale brain fMRI dynamics unveiled by a novel point process analysis. Frontiers in Fractal Physiology. doi: 10.3389/fphys.2012.00015.
Touboul, J., & Destexhe, A. (2010). Can Power-Law Scaling and Neuronal Avalanches Arise from Stochastic Dynamics? PLoS One, 5(2): e8982.doi:10.1371/journal.pone.0008982.
Usher, M., Stemmler, M., Olami, Z. (1995). Dynamic Pattern Formation Leads to 1/f Noise in Neural Populations. Physical Reveiw Letters, 75(2), 326-330.
Wagenmakers, E. J., Farrell, S., & Ratcliff, R. (2004) Estimation and interpretation of 1/f^a noise in human cognition. Psychonomic Bulletin & Review, 11(4), 579-615.
Wagenmakers, E. J., Farrell, S., Ratcluff, R. (2005). Human Cognition and a Pile of Sand: A Discussion on Serial Correlations and Self-Organized Criticality. Journal of Experimental Psychology: General, 134(1), 108-116.
Ward, L. M., & Greenwood, P. E. (2007). 1/f noise. Scholarpedia, 2(12):1537.
Werner, G. (2010). Fractals in the nervous system: conceptual implications for theoretical neuroscience.
