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The brain as a critical system – Simon-Shlomo Poil

Many studies have suggested that the brain is in a so-called self-organized critical state (Bak, 1996; Chialvo and Bak, 1999; Linkenkaer-Hansen et al., 2001; Beggs and Plenz, 2003; Chialvo, 2004, 2007, 2010; Poil et al., 2008; He et al., 2010; Shew and Plenz, 2012; Beggs and Timme, 2012); a state where activity is qualitatively characterized by a compromise between uncorrelated and strongly correlated dynamics—between—disorder and order—and quantitatively by its scale-free dynamics. The concept of criticality comes from statistical physics, where certain systems, such as, the Ising model of ferromagnetism, undergoing a phase transition and exhibit scale-free behaviour (Christensen and Moloney, 2005). Bak et al. (1987) discovered a wide class of systems that reach a critical state through a process of self-organization (Bak et al., 1987; Bak, 1996; Jensen, 1998), which makes the dynamical state of self-organized critical systems robust.

The prototype self-organized critical system is the sand pile model (Bak et al., 1987). Imagine grains of sand dropping slowly on a sand pile. Each time we drop a grain of sand, the energy landscape of the pile is changed; this leads to avalanches of sand moving down the slope of the pile. The dynamics of these avalanches is determined by the angle of the slope. Above a certain slope angle, the sand pile is in a super-critical state in which the activity is dominated by massive avalanches. Below a certain slope angle, it is in a sub-critical state in which the activity is dominated by very small avalanches. In between these slope angles, we have the critical state, where avalanches of all sizes occur — the so-called “scale-free” activity. This means that if we plot the probability distributions of occurring sand avalanche sizes and durations we will observe that these vary as a power of the sizes of the sand avalanches. This means sand avalanches follow power-laws at the critical sand pile slope-angle; no single scale of sand avalanches size is dominating, small, or large.

What does a sand pile have in common with our brain? In the sand pile, the scale-free dynamics at the critical angle occur because of thresholds effects, small pockets of sub-critical and super-critical slopes can be found on the pile. However, as long as the pile is driven slowly neither the sub-critical nor the super-critical pockets manage to grow and dominate the system. This is the self-organized criticality of the sand pile, it is attracted to the critical state, and it would come back to this state if we perturbed it by adding a lot of sand. In the brain, we also have networks that can be described to be threshold systems, supporting a build up of pockets of sub-or super-criticality of activity. Indeed, it is meaningless to imagine the brain with an energy landscape without thresholds, because that would mean no memory, no learning could be obtained (Dayan and Abbott, 2001; Chialvo, 2006). Memory formation shapes the neuronal network activity by regulating, using synaptic mechanisms, the threshold for neuronal spiking, or the balance between excitatory versus inhibitory populations. In the brain, we can have either low excitability — sub-critical state — or high excitability — super-critical state, or the mixture of pockets of each — the critical state.

It seems the brain could be in any of these states, sub-critical, super-critical, or critical. However, I suggest that the brain is in the critical state, because in this state information is neither damped nor diluted by hyperactivity but a balanced activity which is the most optimal for handling information (Beggs and Plenz, 2004; Haldeman and Beggs, 2005; Kinouchi and Copelli, 2006; Beggs, 2008a; Shew et al., 2009, 2011; Yang et al., 2012). This balanced activity maximizes the dynamical range, the range of inputs to a network that can uniquely be differentiated (Kinouchi and Copelli, 2006). I further claim that mechanisms at the single-neuron level, such as homeostatic plasticity, will keep the brain in the critical state, and thus, insure the emergence of a stable self-organized critical state (Bornholdt and Röhl, 2003; Abbott and Rohrkemper, 2007; Turrigiano, 2012).

The empirical evidence of criticality in the brain has developed into two apparently distinct directions. One is based on the robust observation of a power-law decay of temporal autocorrelations in the amplitude modulation of ongoing oscillations, also referred to as long-range temporal correlations (Linkenkaer-Hansen et al., 2001). The other is based on the observation of so-called neuronal avalanches in the spreading of local field potentials in vitro (Beggs and Plenz, 2003) and in vivo (Gireesh and Plenz, 2008; Petermann et al., 2009; Hahn et al., 2010). A recent modelling study has showed that neuronal avalanches and long-range temporal correlations can emerge in the same oscillatory network (Poil et al, 2012).

Scale-free dynamics has also been observed in other parameters (Kello et al., 2010), such as in the decay of temporal correlations in behavioral performance (Gilden, 2001), in the phase-locking intervals between brain regions (Gong et al., 2003; Kitzbichler et al., 2009), and in fMRI (He et al., 2010).

Solid empirical evidence for the existence of long-range temporal correlations has been gained over the past decade from various laboratories. In 2001, Linkenkaer-Hansen et al. (2001) showed for the first time long-range temporal correlations in the amplitude fluctuations of ongoing brain oscillations (Linkenkaer-Hansen et al., 2001). The long-range temporal correlations were found using detrended fluctuation analysis, and the DFA scaling exponent was introduced as a measurement of the state in the underlying neuronal network (Hardstone et al, 2012). The test-retest reliability of the DFA scaling exponent was confirmed in a study by Nikulin et al. (2004) (Nikulin and Brismar, 2004). Later, it was also shown that LRTC is highly heritable, and uncorrelated with power (Linkenkaer-Hansen et al., 2007), thus showing that the DFA quantify a unique feature of ongoing oscillations. Long-range temporal correlations has also been found in fast network oscillations in in vitro hippocampal slices (Poil et al, 2011).

Long-range temporal correlations have been found to be impaired in major depressive disorder (Linkenkaer-Hansen et al., 2005), epilepsy (Monto et al., 2007), Alzheimer’s disease (Montez et al., 2009), and schizophrenia (Nikulin et al., 2012). Recently, Smit et al. (2011) also showed that the strength of LRTC in alpha and beta oscillation increases in the age range from 5 to 25 years, and thereafter remains stable (Smit et al., 2011). This suggests that LRTC can be used to study abnormal brain development in, for instance, ADHD or schizophrenia.

In 2003, Beggs and Plenz reported measurements of the spontaneous activity of local LFP in organotypic cultures from coronal slices of rat somatosensory cortex and from acute coronal slices of rat brains. (Beggs and Plenz, 2003). They showed that the LFPs propagated in what was coined “neuronal avalanches”, and that the number of avalanches relative to the avalanche sizes was power-law scaled with the scaling exponent 1.5. Similar results have later been found in other studies (Mazzoni et al., 2007; Pasquale et al., 2008; Shew et al., 2011; Yang et al., 2012). Recent in vivo studies on monkeys also show power-law distributions of avalanche size, suggesting a universal appearance of avalanches in neuronal networks (Plenz and Thiagarajan, 2007; Petermann et al., 2009).

See my overview of self-organized criticality


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The brain as a critical system