Self-Organized Criticality in the Brain
Dietmar Plenz, Tiago L. Ribeiro, Stephanie R. Miller, Patrick A. Kells, Ali Vakili, Elliott L. Capek
Self-Organized Criticality in the Brain
Dietmar Plenz * , Tiago L. Ribeiro, Stephanie R. Miller, Patrick A. Kells, Ali Vakili, Elliott L. Capek Section on Critical Brain Dynamics, National Institute of Mental Health, National Institutes of Health, Bethesda, MD, USA * Correspondence:
Dr. Dietmar Plenz, [email protected] A BSTRACT
Self-organized criticality (SOC) refers to the ability of complex systems to evolve towards a second-order phase transition at which interactions between system components lead to scale-invariant events beneficial for system performance. For the last two decades, considerable experimental evidence has accumulated that the mammalian cortex with its diversity in cell types, interconnectivity, and plasticity might exhibit SOC. Here we review experimental findings of isolated, layered cortex preparations to self-organize towards the four dynamical motifs of up-states, oscillations, neuronal avalanches, and coherence potentials. During up-states, the synchronization observed for nested theta/gamma-oscillations embeds scale-invariant neuronal avalanches identified by robust power law scaling in avalanche sizes with a slope of -3/2. This precise dynamical coordination can be tracked in negative transients of the local field potential (nLFP), emerges autonomously in superficial layers of organotypic cortex cultures, is homeostatically regulated, exhibits separation of time scales, and reveals unique size vs. quiet time dependencies. A subclass of avalanches, the coherence potentials, exhibits precise maintenance of the time course in propagated local synchrony. Avalanches emerge in superficial layers of cortex under conditions of strong external driving. The balance of excitation and inhibition (E/I) and neuromodulators such as dopamine establish powerful control parameters for avalanche dynamics. This rich dynamical repertoire is not observed in dissociated cortex cultures, which lack the differentiation into cortical layers. The precise interactions between up-states, nested oscillations and avalanches in superficial layers of cortex provide compelling evidence for SOC in the brain. I NTRODUCTION
Brains are inherently complex. Composed of a vast number of cell types, orders of magnitudes larger number of connections, and a myriad of structural and functional networks that make up second messenger pathways affecting every spatial and temporal scale of brain organization – brains are deeply challenging to study. Yet, elaborate attempts to assemble the rich and detailed evidence on brain circuits have resulted in the emergence of relatively simple brain dynamics. Highly detailed brain models comprised of thousands of neurons exhibit relatively simple neuronal activity patterns that range from irregular firing to synchronized or oscillatory activity similar to what is measured in real brains (Markram et al., 2015; Dura-Bernal et al., 2019). However, one major aspect of brain dynamics so far has been difficult to understand, which is how the many neurons in the cortex can communicate over long distances and times in a coherent and selective manner. The aspect of many interacting elements leading to relatively few dynamical motifs is also a major appeal of self-organized criticality (SOC; (Bak et al., 1987)). SOC will drive a system towards a second order phase transition at which dynamics are dominated by universal properties (for review see (Jensen, 1998; Chialvo, 2010; Mora and Bialek, 2011;
Pruessner, 2012; Hesse and Gross, 2014; Marković and Gros, 2014; Muñoz, 2018). The universal property that is of particular interest to brain functions is scale-invariance or, alternatively, the system-wide correlations that can build up in a system exhibiting SOC. Such scale-invariance could be a hallmark of coordinated, yet adaptive neuronal activity that incorporates large number of brain cells. For the brain, and specifically the cortex, it still needs to be demonstrated whether certain aspects of brain dynamics are true aspects of SOC. Fortunately, numerous key features of SOC can be addressed experimentally in brain networks (Plenz, 2012). For example, one would expect an autonomous development of cortex tissue in isolation, i.e. in the absence of any instructive sensory and motor inputs, to develop scale-invariant properties in the emergent dynamics. One would also expect that the emergence of scale-invariance is highly regulated, for example it should be robust to slow driving, i.e., exhibiting a separation of time scales, to exhibit homeostatic regulation, i.e., the return to scale-invariance after lenz et al. 2021 SOC in the brain profound perturbations, and for these regulations to fail when essential circuit components are absent or suppressed. This review focusses and summarizes experimental findings on the emergent dynamics of immature and mature cortical networks when taken in isolation and thus disconnected from any external, structuring input or required outputs. It will be argued that the four dynamical motifs of up-states, nested oscillations, neuronal avalanches, and coherence potentials emerge in superficial layers of cortex as major hallmarks of SOC in the brain. S TRUCTURAL MOTIFS OF SELF - ORGANIZATION : LAYERS , PYRAMIDAL NEURONS , INTERNEURONS , AND GLIA CELLS
The organotypic cortex culture to date represents the most complex in vitro model of the cortex (Bolz et al., 1990; Gähwiler et al., 1997; Humpel, 2015). This preparation, where a slice of cortical tissue typically taken from a newborn rodent is grown in isolation for up to several months (Fig. 1), captures several core features of cortical organization. First, it exhibits the major division of the mammalian cortex into superficial and deep cortical layers (Fig. 1A, B; (Götz and Bolz, 1992; Plenz and Aertsen, 1996b; a; Gorba et al., 1999), which exhibit distinct functional properties (Luhmann et al., 2016; Molnár et al., 2020). Superficial layers 2/3, also called the associative layers, are composed of pyramidal neurons with reduced apical branching who preferentially connect to other pyramidal neurons (Fig. 1C). In contrast, pyramidal neurons from deep layer 5/6 typically feature elaborate apical dendritic trees and, besides selectively connecting with superficial layers, communicate in vivo with brain regions outside cortex (Fig. 1D; (Plenz and Kitai, 1998)). The layer 4, which in vivo receives sensory input from thalamus, has also been identified in organotypic co-cultures using thalamus and primary cortices (Bolz et al., 1990; Gähwiler et al., 1997; Humpel, 2015). The second important hallmark in cortical organization is the presence of three major interneuron classes identified as parvalbumin- (PV), somatostatin- (SST) and vasoactive intestinal peptide- (VIP) expressing neurons, which exhibit highly selective connectivity and specific firing patterns (for review see (Tremblay et al., 2016; Lim et al., 2018)). Several of these classes with their layer-specific distribution and electrophysiology have been demonstrated in organotypic cortex cultures using various immunochemical markers (Fig. 1B; (Götz and Bolz, 1989; Plenz and Aertsen, 1996a; Klostermann and Wahle, 1999)). The third and often overlooked hallmark of the cortical microcircuit is the up to 10x higher presence of non-neuronal cells, or glia cells, compared to neurons. Of the three types of glia cells, cortical astrocytes exhibit brain region specific control over neuronal excitation and dynamics amongst many other functions (Halassa et al., 2007; Fellin et al., 2009; Perea et al., 2014). For organotypic cortex cultures, glia cells have been demonstrated to protect the neuronal tissue from mechanical damage (Schultz-Süchting and Wolburg, 1994; Schmidt-Kastner and Humpel, 2002). Organotypic cortex cultures also show clear differences compared to in vivo cortex such as an overall reduced connectivity due to a reduction in the 3 rd dimension when preparing a slice (Cäser and Schüz, 1992) or change in glia protein expression (Staal et al., 2011). Organotypic cultures are typically prepared from newborn animals. Thus, the cortical section of postnatal brain which is taken for culturing is still immature particular with respect to the development of superficial layers. However, this immature cortex has benefited from structuring input during embryonic development, which has been shown to be important for, e.g., somatotopic map formation (Antón-Bolaños et al., 2019). This organotypic cortex culture thus should be best thought of as an in vitro system that has experienced a robust structural organization during embryogenesis and contains the blueprint for the organization of layered, cortical columns in isolation. The next section will summarize how the structural self-organization which continues as the cortex further matures in isolation supports the self-organization of basic dynamical motifs in neuronal population activity. T HE ST AND ND DYNAMICAL MOTIF OF SELF - ORGANIZATION : UP - STATES AND NESTED θ / γ – OSCILLATIONS
The structural self-organization in organotypic cultures should parallel a self-organization of dynamical motifs found for the fully mature brain. One of these motifs, which is dominant in the electrocorticogram (ECoG) in humans in the wake state, is composed of transient nested oscillations in the theta ( θ : 8 – 12 Hz) and gamma ( γ : >25 Hz) range capturing the emergence of population synchrony at many local sites (Fig. 2A; (Canolty et al., 2006)). The nesting of high-frequency γ –oscillations to each θ -cycle has been proposed to be essential for working memory (Lisman and Idiart, 1995; Lisman and Jensen, 2013) and information transfer from lower to higher lenz et al. 2021 SOC in the brain cortical areas (e.g., (André et al., 2015; Lundqvist et al., 2016)). In mature organotypic slice cultures, detailed intracellular recordings demonstrated the emergence of nested θ / γ –oscillations during periods of pronounced depolarization, which establishes the dynamical motif of an “up-state” (Fig. 2B, C; (Plenz and Kitai, 1996; Klostermann and Wahle, 1999; Johnson and Buonomano, 2007)). The propensity of isolated cortex to produce up-states and nested oscillations is also demonstrated in the acute cortex slice, in which tissue is studied within hours after being taken from the brain. In the slice, synchronized nesting during up-states can be induced by an external, pharmacological drive which includes direct neuronal depolarization through glutamate receptor stimulation in combination with neuromodulation typically using a cholinergic agonist (Fig. 2D, E; (Buhl et al., 1998; Compte et al., 2008; Yamawaki et al., 2008)). Current- source density (CSD) analysis (Nicholson and Freeman, 1975; Mitzdorf, 1985; Plenz and Aertsen, 1993) demonstrate that nested θ / γ –oscillations originate in superficial layers 2/3 in both the acute cortex slice (Fig. 2E; (Oke et al., 2010)) and organotypic cortex culture (Fig. 3). Developmentally, these dynamical motifs occur in culture with a similar time course compared to in vivo when co-culturing isolated cortex with midbrain containing dopamine neurons from the ventral tegmental area (Fig. 3; (Gireesh and Plenz, 2008)). The dominance of up-state triggering nested θ / γ –oscillations has several profound implications when studying SOC in isolated cortex preparations. First, it is well known that up-states, particularly prolonged ones (>0.2 s), require stimulation of the fast-acting (<30 ms) AMPA-glutamate receptor as well as the slow-acting (>50 ms) NMDA-glutamate Figure 1. Superficial and deep layers in organotypic cortex cultures. A , Coronal sections from the brain in adult rats showing somatosensory cortex ( left ) and of motor cortex ( right ). Note high density of calbindin ( CB ) positive interneuron stain typical for superficial layers and the layer dependent bands of parvalbumin ( PV ) positive interneurons in deep layers. B , Organotypic cortex culture after ~4 weeks grown on a planar multielectrode array (MEA). Note transparent healthy neural tissue covering 4 mm of the array at a thickness of ~100 – 200 µm and electrodes ( black dots ), conductors of MEA. Composite images (red rectangles) indicating superficial layers ( L2/3 ) that contain PV and CB positive interneurons and deep layers ( L5/6 ) with their intense band of PV-positive interneurons. WM : white matter region. C , Typical cell body and dendritic morphology of pyramidal neurons from L2/3 and L5, the latter with their characteristiclly long and branched apical dendrite. Inset : Spiny dendrite typical for pyramidal neurons. [ A reprinted with permission from (Van Brederode et al., 1991).] [ B modified with permission from (Plenz and Aertsen, 1996a).] [ C modified with permission from (Plenz and Kitai, 1998). Copyright 1998 Society for Neuroscience.] WM L2/3L5/6 Pv A B C PV CB
50 µm PV
50 µm 0.1 mm
L 5/6
L 2/3
Adult rat cortex Mature organotypic cortex culture
PVCB lenz et al. 2021 SOC in the brain receptor. The prolonged time course of the NMDA-glutamate receptor reduces the precision in action potential timing (Harsch and Robinson, 2000), suggesting that the scaffolding of precise spatiotemporal events requires alternative mechanisms, e.g., interneuron firing. Indeed, pyramidal neurons tend to fire sparsely during up-states, whereas interneurons fire reliably during almost every γ –cycle, a robust finding established in organotypic cortex cultures (Plenz and Kitai, 1996; Klostermann and Wahle, 1999; Czarnecki et al., 2012) and the acute cortex slice (Compte et al., 2008). Second, the profound intracellular depolarization found in neurons during up-states indicates an overall increase in network activity. However, the up-state depolarization should not be equated with a higher excitability of individual neurons, which is implicitly assumed in neuronal models that do not take intracellular membrane Figure 2.
Self-organization into the 1 st and 2 nd dynamical motif of up-state and nested θ / γ - oscillations respectively in organotypic cortex cultures and acute cortex slices. A , The human brain displays nested θ / γ –oscillations at rest and during behavior. Left : Electrode array on the cortex surface ( circles ) recording the electrocorticogram.
Right : Corresponding power spectrum of fast γ –oscillations (>25 Hz; top ) phase-locked to the θ –oscillation ( bottom ). [Reprinted with permission from (Canolty et al., 2006).] B , Mature organotypic cortex cultures display nested θ / γ –oscillations during the up-state. Left : Intracellular membrane potential of sparsely firing pyramidal neuron during prominent up-state and nested θ / γ –oscillations. Right : Fast γ –oscillations are found in pyramidal neurons ( triangles ) and interneurons ( circles ) mainly in superficial ( open ), but not deep layers ( filled ). [From (Plenz and Kitai, 1996).] C , In organotypic cortex cultures, blocking the excitatory NMDA glutamate-receptor with the antagonist APV abolishes θ –oscillations during prolonged up-states, but not the initial up-state transition itself ( red ; PB). Left : Population activity time course. ctl : control.
Right : Summary of activity parameters.
Act : control. [Reprinted with permission from (Czarnecki et al., 2012).] D , In the acute cortex slice from adult ferret, spontaneous periods of fast γ –oscillations are found in the LFP. [Reprinted with permission from (Compte et al., 2008). Copyright 2008 Society for Neuroscience.] E , Fast γ –oscillations emerge in superficial layers of the acute cortex slice (CSD analysis). [Reprinted with permission from (Oke et al., 2010).] AB D D C D DD E DD lenz et al. 2021 SOC in the brain conductance changes, e.g., due to synaptic inputs, into account (Bernander et al., 1991). On the contrary, individual neurons change significantly how they respond to additional input during up-states (Petersen et al., 2003; Czarnecki et al., 2012; Reig et al., 2015), which is controlled by a rather expansive combination of a decrease in neuronal input resistance (Monier et al., 2008), a shortened synaptic integration windows (Bernander et al., 1991), transient changes in the balance of excitatory to inhibitory (E/I) synaptic transmission (Haider et al., 2006), active dendritic conductances (Ness et al., 2016), a critical slowing down of the threshold to action potential generation (Meisel et al., 2015) and other mechanisms (for further reading see (Fregnac et al., 2004)). Few neuronal simulations take these changes during the build-up of network activity into account (Markram et al., 2015; Dura-Bernal et al., 2019), potentially limiting insights that can be gained into these dynamical motifs from less biophysically oriented modelling. Third, nested θ / γ –oscillations during the up-state are not blocked by the gap junction blocker carbenoxolone (Gireesh and Plenz, 2008) and the activity propagates relatively fast, with a velocity >50 mm/s (Beggs and Plenz, 2003). These findings support the view that nested θ / γ –oscillations originate in superficial layers from synaptic interactions between local interneurons and pyramidal neurons. These nested oscillations are therefore considered to differ from so-called slow oscillations, which in vivo can be induced by deep but not superficial layer stimulation (Beltramo et al., 2013) and in vitro were shown to originate in deep layers, propagate significantly slower than θ / γ –oscillations locally yet can contribute to up-state initiation in superficial layers (Sanchez-Vives and McCormick, 2000; Wester and Contreras, 2012; Capone et al., 2017). Figure 3. Self-organization into nested θ / γ – oscillations in cortex slice cultures during the 2 nd week postnatal. A , Coronal sections of cortex and ventral tegmental area ( VTA ) are combined and slightly expand on the MEA over 2 weeks in culture.
Left : Sketch of coronal cortex slice and midline crossing midbrain region containing the VTA at postnatal day (
PND ) 1 – 2, when taken into culture after 1 day in vitro (
DIV ; middle ), and about 2 weeks later ( DIV right ). Note flattening of the culture visible by the increased transparency and expansion of the dorsal tissue on the MEA as superficial layers develop (light microscopic image). B , Nested oscillations increase in power from 1 st ( white ) to 2 nd ( black ) postnatal week in vivo ( left ), a developmental time course mirrored in organotypic cortex-VTA co-cultures ( right ). C , Spontaneous nested θ / γ –oscillations distribute along the dorsal part of the cortex within superficial layers. D , CSD demonstrates θ / γ –oscillations to originate from synaptic sources ( sinks ) within superficial layers. E , Summary of average CSD with distance from dorsal border for 7 cultures separated into θ – and γ –oscillations activity. F , Separating the broadband LFP into spike information carrying high-frequency band (HP) and the population activity containing low-frequency band (LP) demonstrates local spiking (vertical bars) is phase-locked to γ –oscillations. G , Spike probability is highest at the negative peak of the γ –cycle at t = 0. [Reprinted with permission from (Gireesh and Plenz, 2008). Copyright 2008 National Academy of Sciences.] C D E GA B F lenz et al. 2021 SOC in the brain T HE RD DYNAMICAL MOTIF OF SELF - ORGANIZATION : N EURONAL AVALANCHES
Until now, the two dynamical motifs of up-states and nested oscillations have been treated from the point of view of averages, disregarding their variable spontaneous or evoked instantiation in cortical networks. In fact, when taking this variability into account, a third dynamical motif arises: neuronal avalanches. In a comparative in vivo and in vitro study on the developmental emergence of neuronal avalanches in superficial layers of cortex, Gireesh and Plenz (2008) used multielectrode array (MEA) recordings to demonstrate the nesting of spatiotemporal negative LFP deflections (nLFPs) into ongoing oscillations (Fig. 4). The emergence of power law distributions in size from nLFP avalanches was completed during the 2 nd week postnatal concomitant with significant peaks in coherence within ongoing θ –, β –, and γ –oscillations (Fig. 4A, B). This finding was contrasted by the variability in nLFP cascade sizes during the 1 st postnatal week, which revealed a more bimodal distribution in line with a preponderance towards system-wide activity (Fig. 4C). This complex developmental signature of avalanches and nested oscillations in vivo develops autonomously in organotypic cortex cultures with a similar developmental time course, i.e. it is established towards the end of the 2 nd week postnatal in the absence of any structuring sensory input or motor output (Fig. 4D, E). The precise match of the power law in avalanche sizes with slope of -3/2 that emerges from the variability of nested θ / γ –oscillations is no statistical coincidence. Besides both dynamical motifs being highly sensitive to fast inhibition via the GABA A receptor and slow excitation via the NMDA–glutamate receptor, this co-existence required fine-tuning via the dopamine D –receptor. Specifically, when the dopamine D – but not D –receptor was blocked, nested oscillations continued to emerge, yet the resulting nLFP cascades now exhibited a much steeper size distribution (Gireesh and Plenz, 2008). This complex regulation of avalanche size distributions to a slope of -3/2 as a function of NMDA/D –receptor co-stimulation has been Figure 4. Self-organization into the 3 rd dynamical motif of neuronal avalanches embedded in nested θ / γ –oscillations. A , Transient nested θ / γ –oscillation in the superficial layers of a 2-week old rat in vivo with LFP at single cortical site ( top ), corresponding simultaneous nLFP pattern on the MEA ( middle ) and expanded view of identified avalanches ( blue rectangles ) during each γ –cycle ( bottom ). B , Nested θ / γ –oscillations show high coherence in vivo ( black ), which is destroyed by random shifts in time ( red ). C , Development of power law in avalanche sizes identifies slightly bimodal form ( red arrow ) at the beginning of the 2 nd week postnatal ( blue ), which disappears at the end of the 2 nd week ( black ). D , Corresponding example of transient nested θ / γ –oscillation in organotypic cortex cultures grown for 2 weeks with LFP at single cortical site ( top ), corresponding simultaneous nLFP pattern on the MEA ( middle ) and expanded view of identified avalanches ( blue rectangles ) during each γ –cycle ( bottom ). E, Power law in avalanche size distribution measured in number of active cortical sites ( top ) and summed nLFP amplitude ( middle ) for a single culture and averaged over all cultures. [Reprinted with permission from (Gireesh and Plenz, 2008). Copyright 2008 National Academy of Sciences.]
A B C D E EE organotypic cortex culturein vivo cortex lenz et al. 2021 SOC in the brain confirmed for superficial layers in acute slices of prefrontal cortex taken from 2 months old, adult rats (Stewart and Plenz, 2006; Bellay et al., 2021) (cf. Fig. 11). Recent analysis in vivo in the prefrontal cortex of awake nonhuman primates further confirmed this precise relationship between avalanche dynamics and Figure 5.
Developmental time course for the self-organization of neuronal avalanches in isolated cortex preparations. A , Overview picture of a custom-made incubator for long-term recordings of individual organotypic cortex cultures on a MEA in a chronic, sterile chamber with head stage ( left ) and off-recording storage racks ( right ). For details see (Plenz et al., 2011).[Reproduced with permission from (Pfeffer et al., 2004).] B , In organotypic cortex cultures, avalanches are absent during the 1 st postnatal week in vitro , but increase in rate during the 2 nd and 3 rd postnatal week in line with in vivo maturation of superficial layers. Highlighted periods : equivalent postnatal week in vivo given cultures are taken from pups at PND 2. C , Raster plots of spontaneous nLFPs increase in complexity from 1 st to 5 th week ( rows ) postnatally in vitro . Top of each row : 5 hr raster.
Bottom of each row : Higher temporal resolution for periods indicated by the blue rectangle. D , E, Bimodal avalanche size distribution during the 1 st week postnatally changes to a power law during the 2 nd week postnatal with a slope at -3/2 in organotypic cortex cultures, which correlates with an~10x increase in LFP activity ( F ). [ B – F modified and reproduced with permission from (Stewart and Plenz, 2007).] G , In dissociated cultures taken at PND 0 – 1, power laws tend to be reported after ~4 weeks in culture. [Reproduced under CC-NY license from (Tetzlaff et al., 2010).] H , Neuronal activity reaches steady state in dissociated cortex cultures after ~4 weeks in vitro . [Reproduced with permission from (Pasquale et al., 2008).] I , Transition of avalanche size distributions from exponential to bimodal in dissociated cortex cultures. [Reproduced under CC-NY license from (Levina and Priesemann, 2017).] A D EF s t w ee k nd w ee k B G e l e . C G
I H lenz et al. 2021 SOC in the brain γ – oscillations (Miller et al., 2019). To summarize, in vivo experiments in rodents and nonhuman primates, as well as developmentally well controlled in vitro experiments using organotypic cortex cultures and acute cortex slices demonstrate a precise regulation between up-states, nested oscillations and neuronal avalanches that involves fast GABA-mediated inhibition, slow-glutamate mediated excitation and the neuromodulator dopamine. D EVELOPMENTAL SELF - ORGANIZATION OF ROBUST AVALANCHE DYNAMICS IN ORGANOTYPIC CORTEX CULTURES
The previous sections demonstrated the emergence of neuronal avalanches around the 2 nd week postnatally in culture and in adult slices when tested in isolation. It is well understood that cortical development in vivo involves intrinsically maturing cellular properties and microcircuits in a complex interplay with structuring sensory inputs and motor outputs (Molnár et al., 2020). Many of these embryonic and neonatal intrinsic dynamics are found to arise autonomously in isolated cortex preparations (Allene and Cossart, 2010; Luhmann et al., 2016). Yet, so far only a few studies have reliably covered the time course of avalanche emergence during development over prolonged periods. In a first study of postnatal in vitro maturation of avalanches, Stewart and Plenz (2007) grew individual organotypic cortex cultures on a planar MEAs in sterile chambers over many weeks (Fig. 5A). Spontaneous LFP activity emerged towards the beginning of the 2 nd week postnatally with typical bimodal distribution in cascade sizes (Fig. 5B, C) indicating a bias towards system-wide population bursts before the time of superficial layer maturation. During the end of the 2 nd week, stable power laws in avalanche activity emerged particularly in those cultures that reached a high level of spontaneous activity (Fig. 5B – F). Given the late development of superficial layers, and the well-known preponderance of deep layer gap-junctions during the first week postnatal (Dupont et al., 2006), the initial bimodal distribution in cascade sizes might reflect system-wide deep-layer synchronization supported by extensive gap-junction coupling (Kandler and Katz, 1998) potentially facilitated by transient hyperconnectivity which reduces towards the end of the second week postnatal in vivo (Meng et al., 2019). This ability of young cortex to express neuronal avalanches towards the end of the 2 nd week postnatally was recently confirmed for superficial layers in young acute cortex slices (Bellay et al., 2021). A second developmental study followed avalanche emergence in dissociated cortex cultures grown on planar microelectrode arrays starting with neonatal cortex tissue around postnatal day (PND) 0 – 1 (Tetzlaff et al., 2010). This study also described an initial bimodal size distribution, characterized as ‘supercritical’, followed by a pronounced ‘subcriticality’ and eventually, after more than 5 weeks in culture, a ‘critical’ condition characterized by stable power laws in size distribution (Fig. 5G). While both culture systems capture an initial bimodal activity state, the developmental time course in dissociated cultures appears to be delayed by more than 3 weeks with respect to the build-up of neuronal activity (Fig. 5H) as well as power law formation when compared to in vivo (Gireesh and Plenz, 2008). Recently, Levina and Priesemann (2017) demonstrated that the bimodal distribution in avalanche sizes is maintained in dissociated cultures over long periods, questioning the robustness of power laws identified by previous studies for that system (Fig. 5I; see also below). P OWER LAW OPERATIONS AND BRANCHING PROCESS IN THE ORIGINAL IDENTIFICATION OF NEURONAL AVALANCHES
When analyzing dynamical motifs, the identification of avalanches and their implication for SOC has been a particular difficult challenge. Besides the structural constraints of superficial layers and developmental period that need to be considered, there are additional specific aspects in the emergence of neuronal avalanches themselves that are of importance and which will be addressed in the next sections. The original identification of neuronal avalanches (Beggs and Plenz, 2003) involved numerous controls to demonstrate that power laws identified in propagated neuronal activity were robust to obvious choices in the experimental setup. In short, tracking the spatiotemporal spreading of an avalanche when using discrete, spatial sensors such as MEAs, requires the appropriate choice of a discrete time interval Δ t (Fig. 6A). This choice of Δ t is imposed by the finite propagation velocity < v > for neuronal activity in the system and the introduction of a discrete sampling distance of Δ d by the MEA. Of note, increasing Δ t while keeping Δ d constant biases towards avalanche concatenation leading to a more shallow slope without change in power law form allowing for scaling collapse in vitro (Fig. 6B) as well as in vivo (Petermann et al., 2009). This robust scaling operation is not observed in dissociated culture experiments, where an increase in Δ t typically steepens the initial lenz et al. 2021 SOC in the brain slope and uncovers a bimodal cascade size distribution (see also Fig.13). Of note, this change from a power law form towards a bimodal distribution was found in the original analysis by Beggs and Plenz (2003) only under disinhibited conditions (Fig. 6C). Importantly, when changing Δ d and adjusting Δ t = < v >* Δ d accordingly, the size exponent of -3/2 is preserved (Fig. 6D). Equally important, for a given Δ d , a critical branching parameter close to 1 is identified at the corresponding Δ t (Fig. 6E). Finally, it was shown that the compact size of the MEA used to identify avalanches only affects the cut-off of the power law in size, but not the power law form itself (Fig. 6F). These scaling operations for avalanches involved >10 hr of continuous recordings in vitro , which is difficult to achieve under standard experimental conditions. Recently, Miller et al. (Miller et al., 2019; 2021) extended this scaling analysis of LFP based avalanches to identify a critical scaling exponent of 2 for avalanche waveform and mean size vs. duration relationships in line with expectations for a critical branching process. T HE TH DYNAMICAL MOTIF OF SELF - ORGANIZATION : COHERENCE POTENTIALS
As the previous sections demonstrated, considerable effort has been placed on identifying a power law distribution in avalanche sizes from isolated cortex preparations. Importantly, the presence of such a power law is linked to numerous additional aspects in the emergence and propagation of neuronal activity. It was found that avalanche dynamics implicitly contains a local synchrony threshold that identifies a subclass of avalanches, the coherence potential (Thiagarajan et al., 2010; Plenz, 2012). Coherence potentials are avalanches with nLFPs above a minimal amplitude threshold typical about 3 – 4 SD of the ongoing LFP fluctuations (Yu et al., 2011). Importantly, coherence potentials are composed of identical nLFP waveforms
Figure 6.
Overview of structural proof in the original identification of neuronal avalanches by (Beggs and Plenz, 2003). A , The interelectrode distance of the MEA used to identify propagated activity introduces the discrete spatial scale Δ d . For different spatial scales, ( black and red grid) and a finite propagation velocity for neuronal activity of < v >, the time Δ t to wait in order to identify propagation towards a nearby electrode is approximately Δ t ≅ < v >* Δ d . B , At fixed Δ d = 200 µm, an increase in Δ t changes the power law slope towards more shallow values. Avalanche size based on number of active electrodes ( left ) or summed absolute nLFP amplitudes ( right ). Note the absence of any change in the power law form itself. C , Pharmacologically reducing fast inhibition changes the power law to a bimodal distribution ( red ) with an initial, steeper slope close to ~-2 and a preference for large, i.e. system-wide propagated population events. D , A change in Δ d compensated by corresponding estimates in Δ t ≅ < v >* Δ d maintains the size distribution slope of -3/2 ( broken line ). Sizes based on electrodes ( left ) or nLFP amplitudes ( right ). E , Avalanche dynamics crosses the critical point (1, -3/2) predicted for a critical branching process with change in Δ t at fixed Δ d . F , Finite-size scaling using compact sub- arrays only affects the power law cut-off. [Subpanels B – F reprinted with permission from (Beggs and Plenz, 2003). Copyright 2003 Society for Neuroscience.] AD F B E ∆ d = 800 µ m ∆ d = 200 µ m σ C lenz et al. 2021 SOC in the brain extending the simple notion of a critical branching process to that of high-fidelity propagation, summarized in figure 7. In a first step, it was demonstrated that the are of individual nLFPs that participate in a single coherence potential on average does not grow nor decay as the avalanche unfolds and this aspect is independent of the size of the initiating nLFP (Fig. 7A – D). This property of preserving the local group size initiating an avalanche was lost when the cortex was even mildly disinhibited, upon which propagated activity displayed a within-cascade explosive growth (Fig. 7D). Importantly, the preservation of initiating event properties is extended to the nLFP waveform itself. That is, nLFP waveforms were highly similar within each avalanche but they varied considerably between successive avalanches (Fig. 7E, F). Again, this feature of maintained within-nLFP waveforms was lost under disinhibited conditions (Fig. 7G). This particular analysis approach also extends the identification of the critical branching parameter which in the original work (Beggs and Plenz, 2003) was based on the ratio in the number of nLFPs between the second (‘descendants’) and first (‘ancestors’) time bin in the unfolding of an avalanche. The analysis in Fig. 7B, D is more precise by including Figure 7. The 4 th dynamical motif of self-organization: The coherence potential. A , Area a and waveform of a local nLFP identified by threshold ( thr ) crossing. B , nLFP areas normalized by initial nLFP area (red dots) distribute around a median value (blue distribution) after n time bins of duration Δ t . C , In vitro avalanches reveal distribution around log(1) = 0 mode demonstrating that, as an avalanche unfolds, nLFPs on average do not grow nor decay in area in line with expectations for a critical branching process. D , In slightly disinhibited cultures, expansion of nLFP area with time from cascade ignition is found indicative of a supercritical branching process. E , Example of within avalanche waveform ( green ) and between avalanche waveform comparison ( red ). F , nLFP waveforms within a coherence potential are highly similar. Overplot matrix of waveform examples for coherence potentials of size 4 – 6 for nLFP areas in three size categories. Right : Distribution of deviation in waveforms for within and between coherence potentials. G , Under disinhibited conditions ( PTX ), coherence potentials are absent. The deviation from the initiating nLFP waveform increases with successive propagation steps within a cascade, significantly outpacing deviations found for coherence potentials. [Reproduced with permission from (Plenz, 2012).]
A B ECD
In vitro FG Within coherence potential between coherence potentials lenz et al. 2021 SOC in the brain nLFP area and waveform and considers all avalanches in their full duration. The critical branching parameter is reflected in the finding that normalized distributions have a stable mode of 1, i.e., log(1) = 0, for up to 20 ms of propagation, which typically covers the full area of recording. These spatiotemporally extended coherence potentials were also demonstrated in vivo in the awake nonhuman primate cortex (Thiagarajan et al., 2010), which allowed for the demonstration that nLFP waveform identity correlates with local spike sequence identity. In the human ECoG, coherence potentials were found to initiate finger tapping (Parameshwaran et al., 2012). The emergence of coherence potentials in cortical networks with avalanche dynamics has been compared to the emergence of ‘gliders’ in cellular automata and hypothesized to be a vehicle of information transfer within cortex at the network level (Plenz, 2012). S EPARATION OF TIME SCALE AND HOMEOSTATIC REGULATION OF NEURONAL AVALANCHES
A separation of time scales, in which the time course of driving the system is slow enough as to not interfere with the fast avalanching process itself, is of essence in some models of SOC (e.g., (Hesse and Gross, 2014)). This concept was tested in organotypic cortex cultures by periodically alternating between submerged and exposed environmental conditions which induces concommittantly large changes in neuronal activity (Plenz, 2012). The resulting avalanche size distributions were power law distributed despite strong common, correlated external driving (Fig. 8A). In a second series of experiments, the well established effect of rebound activity and rebound bursts after prolonged periods of suppression in excitatory synaptic transmission (Fig. 8B, C; (Corner et al., 2002; Turrigiano and Nelson, 2004)) was used to study the robustness of avalanche dynamics. Excitatory synaptic transmission was mildly reduced in organotypic cultures by adding a low amount of the fast glutamate receptor DNQX to the culture medium for 24 hr. This reduction in excitatory transmission steepened the distribution in cascade sizes. Importantly, after removing the brake on excitatory transmission, cascade size distributions rapidly became bimodal, but recovered within 24 h towards the power law distribution observed prior to the perturbation (Fig. 8D). These experiments demonstrate homeostatic regulation of avalanche dynamics from a supercritical state in the absence of any structuring external inputs. A recent study by Ma et al. (2019) demonstrated recovery to power law distributed avalanches during monocular deprivation in vivo over the course of several days, suggesting that
Figure 8. Separation of time scales and homeostasis of neuronal avalanches in vitro . A , Power law in avalanche sizes is preserved during externally induced slow changes in neuronal activity.
Top : Periodic change from submerged to exposed culture medium condition ( angle ) induces large changes in nLFP activity in an organotypic slice culture.
Bottom : Power law in avalanche sizes for n = 7 cultures under period driving ( red distribution from single culture shown on top).
Inset : average autocorrelation of population activity. B , Intracellular recording demonstrating rebound hyperactivity when neuronal activity is suppressed in dissociated cortex cultures. [Reproduced with permission from (Turrigiano and Nelson, 2004).] C , Increase in rebound population bursts after 24 hr of excitatory glutamate receptor suppression in organotypic cortex cultures. [Reproduced with permission from (Corner et al., 2002).] D , Homeostatic regulation of avalanches in organotypic cortex cultures after 24 hr reduction in excitatory transmission. Successive 1 hr recordings from single culture. [ A , D reproduced with permission from (Plenz, 2012).] A BD C lenz et al. 2021 SOC in the brain recovery can be initiated from the subcritical phase too. T HE TEMPORAL ORGANIZATION OF NEURONAL AVALANCHES AND NESTED θ / γ - OSCILLATIONS IN ISOLATED BRAIN CIRCUITS
Population activity that spontaneously forms in isolated cortex preparations has been typically described as intermittant bursts of variable length, intensity and pauses in between (cf. Fig. 5C). When analyzing the summed population activity of avalanche activity more closely, the picture of ‘avalanches within avalanches’ readily emerges (Fig. 9A, B) which also dominates rhythmically driven cultures (Plenz, 2012). Based on the observation of an avalanche, the average time to wait before observing a future avalanche is known as the waiting time distribution and was found to reflect the characteristic time scales of Θ / γ –oscillations and up-states (Fig. 9C; (Lombardi et al., 2014)). This was true for avalanches independent of minimal size and to depend strongly on the E/I balance (Lombardi et al., 2012). By calculating the conditional probabilities of future avalanches given the minimal size of the current avalanche, Lombardi et al. (2016) obtained precise functions capturing the nesting of avalanches and demonstrated their reversal under disinhibited conditions (Fig. 9D). C ONTROL PARAMETERS IDENTIFIED IN THE REGULATION OF NEURONAL AVALANCHES
The core requirement for SOC is the ability of the system to adjust a control parameter which allows the system to reside near the critical point (Hesse and Gross, 2014; Chialvo et al., 2020). Given the complexity of cortical microcircuits regarding neurotransmitter categories (excitatory, inhibitory), neuromodulators (e.g., dopamine, acetylcholine, serotonin) and brain states (e.g., wakefulness, sleep, attention), there could be many control parameters that cortical networks might use to tune themselves towards or away from criticality. Of those potential regulatory mechanisms, few have been experimentally examined so far. Of common focus, the E/I balance establishes an important control parameter, first demonstrated for avalanches in organotypic cortex cultures (Beggs and Plenz, 2003). Specifically, reducing fast inhibitory synaptic transmission non-selectively by pharmacological means rapidly destroys the power law in LFP-based avalanches and causes bimodal distributed cascade sizes ( cf . Fig. 6C). Figure 9. Temporal self-organization of neuronal avalanches in organotypic cortex cultures. A , Time course of integrated avalanche activity at three different resolutions for a single culture. B , Sketch of the current avalanche begetting future avalanches. [ A, B
Modified and reproduced with permission from (Plenz, 2012).] C , Quiet time distribution revealing power law decay with θ –oscillation peak and indicated γ –oscillation and up-state regime. [Modified and reproduced under CC-BY license from Lombardi et al. 2014).] D , Quantification of the dependency in pre-avalanche size and successive quiet time. Avalanches beget future avalanches ( left ) absent driving, and during periodic, slowly driven condition ( middle ). This relationship is reversed in disinhibited cultures ( right ). [Reproduced under CC-BY license from (Lombardi et al., 2016).] Up- state
Θ γ
A BD normal disinhibited Θγ Up-state slowly driven C lenz et al. 2021 SOC in the brain Similar results have been obtained in dissociated cultures in which a power law distribution in avalanches changed to a bimodal distribution when inhibition was blocked (e.g., (Pasquale et al., 2008)). In more detailed follow up studies, a reduction in fast synaptic inhibition or in fast and slow synaptic excitation changes the dynamics from avalanches to a ‘supercritical’ or ‘subcritical’-like conditions (Fig. 10; (Shew et al., 2009; Shew et al., 2011; Yang et al., 2012; Shew and Plenz, 2013)). These studies demonstrated that numerous network parameters are maximized at the E/I balance at relatively low level of synchronization and where avalanche dynamics reigns (Fig. 10A – D). A power law in avalanche sizes does not simply emerge from firing rate statistics of individual neurons itself. Blocking of excitatory and inhibitory synaptic transmission in dissociated cultures uncovers an exponential size distribution expected for uncorrelated activity (Fig. 10E; (Yada et al., 2017)). Dopaminergic modulation has been identified as a second control parameter for the regulation of neuronal avalanches in prefrontal cortex. Dopamine is crucial for working memory performance which in turn requires prefrontal cortex functioning (Durstewitz and Seamans, 2008; Arnsten, 2011). Acute prefrontal cortex slices taken from adult rats and exposed to a moderate external excitatory drive rapidly respond to the presence of dopamine with the emergence of nLFP activity (Fig. 11; (Stewart and Plenz, 2006)). At intermediate levels, but not low or high levels of dopamine, nLFPs formed a power law in avalanche sizes with slope of -3/2. The activity was selective for the dopamine D -receptor and required NMDA-glutamate receptor stimulation thus matching the pharmacological inverted-U profile reported for working memory performance in prefrontal cortex (Cai and Arnsten, 1997). Intracellular analysis during LFP avalanches at optimal NMDA/dopamine concentrations demonstrated that even large avalanches engage individual pyramidal neurons selectively and this selectivity breaks down when inhibition is reduced (Bellay et al., 2021). These results taken together suggest that the control parameter dopamine maximizes the spatial extent and Figure 10. The E/I balance is a control parameter for the emergence of avalanches. A , The dynamic range is maximized when avalanche size distributions are closest to a power law ( κ =1). B , The information capacity is maximized close to κ = 1. C , Synchronization exhibits a phase transition at κ = 1 ( top) , at which the entropy of synchronization is maximal ( bottom ). D , In dissociated cultures, blocking fast inhibition and fast/slow excitation uncovers irregular spiking activity ( top ) leading to an exponential distribution in cascade sizes ( bottom ). [Subpanels A , B , C reproduced from (Shew et al., 2009; Shew et al., 2011) and (Yang et al., 2012) respectively. Copyright 2009, 2011, 2012 Society for Neuroscience.] [Subpanels in D reproduced with permission from (Yada et al., 2017).] A v a l an c he s Synchrony
EntropySRH A v a l an c he s Excitation / Inhibition ratiog
AB C D
Time (s) c h a nn e l Bic/AP5/CNQXSynchronyEntropy lenz et al. 2021 SOC in the brain occurrence frequency of system-wide avalanches formed by selective activation of distributed pyramidal neurons in the network. The LFP is a continuous time-varying signal, which for avalanche processing requires a threshold operation to convert this signal into point process-like data. Such thresholding preserves essential avalanche information in a discretized spatiotemporal raster, as, e.g., shown for human avalanches in the fMRI (Tagliazucchi et al., 2012). Yet, thresholding is a non-linear operation and for single timeseries can affect scaling regimes (Francesc et al., 2015; Villegas et al., 2019). On the other hand, if thresholding were the underlying cause to observe scale-invariant avalanches, one would expect power law characteristics of avalanche dynamics to be robust to pharmacological manipulation. In contrast, the high sensitivity of the power law to the reported control parameters demonstrates that thresholding is unlikely to play a major role in the origin of this scale-invariance. Changes in network connectivity based on local plasticity rules have been demonstrated to establish SOC in models (Bornholdt and Rohlf, 2000), suggesting that plasticity could function as a control parameter. Network connectivity itself could be a control parameter as it was found to support avalanche dynamics in dissociated cultures (Massobrio et al., 2015). On the other hand, measurements in organotypic cortex cultures and in nonhuman primates in vivo demonstrate avalanches to establish integrated network architectures that are robust to certain plastic changes (Pajevic and Plenz, 2009; 2012; Miller et al., 2021). Of note, in vivo studies have shown avalanches to be exquisitely sensitive to the sleep/wakefulness transition (Ribeiro et al., 2010; Scott et al., 2014; Bellay et al., 2015; Fagerholm et al., 2016; Ribeiro et al., 2016) suggesting sleep (Meisel et al., 2013; Meisel et al., 2017) and sleep-arousal transitions (Lombardi et al., 2020) as a behavioral state control parameter. L ACK OF SCALE - INVARIANT
LFP
AVALANCHES IN DEEP LAYERS
Results summarized here were based on LFP recordings taken from high-density arrays orientated in a specific manner with respect to the underlying cortical column. The planar projection of the array was aligned such as to be able to monitor propagation of activity in all layers of cortex. Even under those carefully chosen projection conditions, deep layer LFP activity was either strikingly absent, e.g., during spontaneous avalanche emergence in organotypic cortex cultures (Fig. 3) or during external glutamate mediated depolarizations which induces avalanche activity in superficial layers in the acute cortex slice (Fig. 11). The absence of LFP-based avalanches in
Figure 11. Dopamine is a control parameter for the emergence of neuronal avalanches in superficial layers of cortex. A , Externally driven acute cortex slice using weak excitatory drive (NMDA) maximizes avalanche activity at intermediate dopamine concentrations. B , Externally driven neuronal avalanches emerge in superficial layers 2/3. C , Activity consists of neuronal avalanches with a power law slope of -3/2 at moderate dopamine concentration. Size distributions ( top ) and corresponding slopes (bottom) as a function of dopamine concentration. [Reprinted with permission from (Stewart and Plenz, 2006). Copyright 2006 Society for Neuroscience.] A CCCB C lenz et al. 2021 SOC in the brain deep layers in vitro could have various causes. First, deep layers could mature incompletely in organotypic cultures preparations, e.g., due to lack of subcortical inputs from thalamus or lack of subcortical targets, However, this argument does not apply to the acute cortex slice. Second, deep layers might require additional neuromodulatory signals such as acetylcholine and neurotensin, which regulate the amount of bursting in deep layer pyramidal neurons (McCormick and Prince, 1987; Case et al., 2016). However, even in the awake nonhuman primate, the LFP activity in deep layers does not establish power laws even when avalanche activity propagates simultaneously in superficial layers (Fig. S4 of (Petermann et al., 2009)). Fourth, avalanches in deep layers could be composed of spatially distributed neurons that are is difficult to track in the LFP. However, local cortical connectivity favors connections between nearby pyramidal neurons (Braitenberg and Schüz, 1991) expecting avalanche activity to sum in the LFP. Taken together, these arguments suggest that deep layers might not be able to support scale-invariant avalanche dynamics in general. Even advanced recording techniques in vivo Figure 12. Experimental demonstration of the transition from single nLFP avalanches to spike avalanches in organotypic cortex cultures. A , Sketch of spatial transformation of propagated nLFPs on the electrode array to propagated spike activity in local neuronal groups ( circles ; red : spiking; blue : quiet) within the neighborhood of a single electrode ( zoom ). B , Distribution of nLFP amplitudes at single electrodes ( red : average; black : example single electrodes). Power law-like distribution changes into bimodal distributions when reducing inhibition ( disinhibited ) or excitation ( disfacilitated ). [Reproduced with permission from (Plenz, 2012).] C – F , Reconstruction of spike avalanches in organotypic cortical culture. C , Single organotypic co-culture ( a ) expressing the GECI YC2.6 in superficial cortex layers using E16.5 electroporation. ( b ) and ( c ) are successive zooms. D , Raster of spontaneous spike density monitored with 2-photon imaging and obtained through deconvolution (n = 40 pyramidal neurons). Top : Binarized raster.
Bottom : Temporally expanded raster segmented with color coded spike intensity. E , Normal ongoing activity reveals power laws in spike avalanches independent of threshold at the single neuron level, which are destroyed by shuffling. F , The power law in avalanche size is also destroyed when reducing excitation ( left ) or inhibition ( right ). [Modified for C and C – F reproduced under CC0 license from (Bellay et al., 2015).] Disfacilitated
A B FC
Electrode array Single electrode
Disinhibited DisfacilitatedDisinhibited E
10 s D lenz et al. 2021 SOC in the brain in the awake rodent demonstrate absence of avalanches in deep layers. Using 2-photon imaging in vivo , power laws in spike-based avalanches were identified in cortical layer 2/3 and layer 4 (Bowen et al., 2019), but seem to be absent in deep layers (Ma et al., 2020). T HE N
LFP
IS THE AVALANCHE – A LOCAL RECONSTRUCTION FROM SPIKE AVALANCHES USING PHOTON IMAGING
In the LFP, the structural and dynamical heterogeneity of the network is summed to form a local point source, which does not allow for the identification of the network elements contributing to the LFP (Buzsáki et al., 2012). While many experimental findings on avalanches have utilized spatially expansive MEAs, scale-invariance predicts that avalanche dynamics should be observable even within the local neighborhood of a single electrode as the spatial resolution increases. This in turn should allow for a more detailed analysis of the underlying network components contributing to scaleinvariance. In this scenario, the nLFP amplitude should reflect the local neuronal group activity governed by avalanche dynamics (Fig.12A). Accordingly, it was found that already the nLFP amplitude distribution at a single electrode approximates a power law with slope of -3/2, which is destroyed when pharmacologically changing the E/I balance (Fig. 12B; (Plenz, 2012)). That indeed the local summed activity of neuronal group firing constitutes avalanche dynamics was first demonstrated directly with 2-photon imaging using the genetically encoded calcium indicator (GECI) YC2.60 which exhibits single spike sensitivity (Yamada et al., 2011; Bellay et al., 2015). The indicator was selectively expressed in pyramidal neurons from superficial layer in organotypic cortex cultures using electroporation (Fig. 12C; (Bellay et al., 2015)). Avalanche analysis of ongoing spike activity from pyramidal neurons revealed a power law in avalanche sizes, which was transformed to a bimodal distribution after pharmacologically reducing either excitation or inhibition (Fig. 12, D – F). These results demonstrate that the nLFP reflects local avalanche activity and should not be equated to single spikes. These findings also identify pyramidal neuron activity in superficial cortex layers to carry avalanche signature, which has recently been confirmed in vivo (Bellay et al., 2015; Karimipanah et al., 2017; Bowen et al., 2019; Ma et al., 2020). N OT ALL AVALANCHES ARE AVALANCHES - S YSTEM - WIDE POPULATION EVENTS GOVERNED BY A ST ORDER PHASE TRANSITIONS IN D ISSOCIATED NEURONAL CULTURES
Dissociated cultures (Dichter, 1978) have been used for decades to study the autonomous development in structure and dynamics of cortical microcircuits. As a complementary approach to organotypic cultures (Gähwiler et al., 1997), dissociated neuronal cultures are prepared from cortical tissue typically taken from an embryo at embryonic day 15 – 18, that is 3 – 6 days before birth, which is then mechanically and enzymatically disintegrated, and finally the remaining neuronal cell bodies and pre-cursor cells are re-seeded on a glia-feeder layer and grown for up to several months in vitro (Dichter, 1978). Dissociated cultures appeal by focusing on the de novo formation of neuronal connections yet they require careful attention to the design of glia-feeder layers and the culture medium composition. They lack cortical layers and a classification of pyramidal neurons and interneurons into subtypes, which, in contrast, are well established in vivo at various developmental stages and largely preserved in organotypic cortex cultures (see introductory sections). Since its first observation using MEAs, synchronized bursting has been the hallmark of developing population activity in dissociated cortex cultures (Maeda et al., 1995). Despite its apparent simplicity, when systematically studied using a large number of cultures grown on MEAs over many weeks, highly variable outcomes in neuronal synchronization have been documented that depend on plating density, which affects the number of neurons per area, and developmental trajectory (Wagenaar et al., 2006). Accordingly, the application of avalanche analysis to these synchronized bursts has yielded heterogenous outcomes across and within studies (Fig. 13). Nevertheless, a consistent finding emerges across these studies, which deviate from results reported for LFP- and spike based avalanches in organotypic cortex cultures, pointed out in detail in the following paragraphs. Pasquale et al. (2008) were the first to report spike-avalanches in 6 dissociated cultures with size distribution of either exponential, bimodal or power law form. Two out of 6 cultures displayed the power law in avalanche size, yet only at sub-millisecond bin width. In fact, increasing the temporal bin width to 1 ms rapidly uncovered a bimodal distribution, explained in their model as explosive growth introduced by neuronal hubs (Fig. 13A). Tetzlaff et al. (2010) using neonatal tissue from P0 tracked spike avalanche distributions during development, finding lenz et al. 2021 SOC in the brain for mature cultures a relatively mild bimodality with increasing bin size (Fig. 13B). Levina and Priesemann (2017) using dissociated cultures prepared from E18 tissue reported spike avalanches that distributed exponentially early but changed to bimodal late during culturing. Those bimodal distributions were robust to further increases in bin width (Fig. 13C). Yada et al. (2017) also tracked the development of spike avalanches in 6 cultures revealing rapid changes to a bimodal form with slight increases in bin width (Fig. Figure 13. Spike ‘avalanches’ are mostly bimodal in dissociated cultures. A , Spike avalanches in dissociated cultures become more bimodal with increase in Δ t . Left : cultures;
Right : model. [Reproduced with permission from (Pasquale et al., 2008).] B , Change from exponential to bimodal size distribution with increase in Δ t . [Reproduced under CC-NY license from (Tetzlaff et al., 2010).] C , Initial steep slope of -2 and bimodality with increase in Δ t . [Reproduced under CC-NY license from (Levina and Priesemann, 2017).] D , Bimodal size distributions around the mean interspike interval Δ t . [Reproduced with permission from (Yada et al., 2017).] E , Bimodality and linear mean size to duration at Δ t close to the mean interspike interval. [Reproduced under CC-NY license from (Shaukat and Thivierge, 2016).] F , Spike avalanches of unknown layer origin in organotypic cortex cultures show bimodal size distribution with increase in Δ t and near linear mean size vs. duration scaling similar to spike avalanches from dissociated cultures in E . [Reproduced under CC-BY license from (Friedman et al., 2012).] G , Mean size to duration slope close to 2 is found in dissociated cultures prepared from postnatal tissue. [Reproduced under CC-BY license from (Yaghoubi et al., 2018).] Subpanels A -F have been modified by adding a red arrow emphasizing bimodal feature in each size distribution.
AB BC D E
F G lenz et al. 2021 SOC in the brain Δ t. Spatial subsampling decorrelates activity leading to exponential distributions in cascade sizes (Ribeiro et al., 2014). However, in the present cases, a power law-like or exponential distribution is observed at the outset for spike avalanches at small Δ t, which changes to a bimodal distribution with increasing Δ t (Fig. 13A – D). Importantly, such sensitivity to Δ t is not observed for LFP-avalanches in superficial layers from organotypic cortex cultures, where the power law is robust to a large range of temporal integration windows allowing for scaling collapse (Beggs and Plenz, 2003). Instead, the sensitivity of the distribution form to Δ t suggests population activity in dissociated cultures to differ from critical avalanche dynamics and instead to follow transitions between local activity and system-wide synchronized events. The initial slopes of those bimodal size distributions were reported to be close to -2 (e.g., Fig. 13 C, D) and system-wide synchronized events constitute the ‘bump’ towards the end of those size distributions (Fig. 13; red arrows ). Steep initial slopes around -2 in addition to a ‘bump’ of system-wide events are hallmarks for disinhibited synchronized population activity (Beggs and Plenz, 2003; Shew et al., 2009). This is in line with a recent report that increasing inhibition by adding a GABA-agonist reduces bimodal size distributions in dissociated cultures, bringing them closer to a power law (Heiney et al., 2019). Further support that spike avalanches in dissociated cultures differ from LFP based avalanches in vivo and in organotypic cortex cultures comes from the mean size vs. duration scaling exponent. This exponent was found to be 2 in line with a critical branching process, but is between 1 – 1.5 in spike-avalanches with bimodal distributions even at large Δ t (Shaukat and Thivierge, 2016; Yada et al., 2017) in line with expectations of a noise process wherein size simply grows more linearly with duration (Touboul and Destexhe, 2017; Villegas et al., 2019). These statistics have been well confirmed in embryonic seeded dissociated cultures grown for up to 3 weeks in which population intracellular calcium imaging tracked neuronal activity and an avalanche algorithm was applied (Orlandi et al., 2013). Results established local, small population events with an initial slope in cascade size distribution around -2 and average size vs. duration scaling slightly larger than 1, as well as a separate group of system-wide population events. Could the latter indicate co-existence of two subnetworks with 2 nd order (avalanches) and 1 st order system-wide events? The fact that the slope of the ‘background’ avalanche distribution <<3/2 and pharmacological inhibition does not further change the bimodal distribution obtained suggest that the system was supercritical to begin with. It is striking that in some dissociated cultures power laws were found mainly at sub-millisecond Δ t (Pasquale et al., 2008). Given that synaptic integration times between neurons are of the order of 2 – 3 ms, a sub-millisecond temporal integration window will artificially promote premature cascade termination, thereby partitioning synchronous activity potentially supporting a regime for heavy-tail cascade sizes as found for, e.g., 1 st order phase transitions in the presence of noise (Scarpetta and de Candia, 2013; Scarpetta et al., 2018). Accordingly, when taking synaptic integration times into account, at temporal resolutions >3 – 4 ms, the true bimodal population dynamics of local, non-propagated activity and global, propagated activity is revealed, just as was captured in early models that feature a 1 st order phase transition of all-or-none propagation for dissociated cortex cultures (Giugliano et al., 2004) and up-state generation in deep layers of cortical slices (e.g., (Capone et al., 2017)), and which have been recently revived within the framework of self-organized bi-stability (Scarpetta and de Candia, 2013; di Santo et al., 2016; Scarpetta et al., 2018; Buendía et al., 2020) or quasi-criticality (Kinouchi et al., 2020). The study by Friedman et al. (2012) calculated spike avalanche distributions from 10 cortex slice cultures of which 3 were bimodal, 4 exponential and 2 were reported ‘critical’, i.e. power law-like. Similar to spike avalanches in dissociated cultures, power law distributions became bimodal with increasing bin width (Fig. 13F; (Friedman et al., 2012)) and a mean size vs. duration exponent close to 1. The loss of the power law at low temporal resolutions supports the interpretation of this activity to be of a 1 st order phase transition either from preferentially recording spikes from deep layers or from networks generally lacking inhibition. lenz et al. 2021 SOC in the brain D EVELOPMENTAL DIFFERENCES BETWEEN ORGANOTYPIC CULTURES AND DISSOCIATED CULTURES OF CORTEX
Organotypic cortex cultures that are grown from postnatal brains demonstrate up-states and nested θ / γ –oscillations in their superficial layers, which give rise to avalanche scaling (see Figs. 2 & 3). The conspicuous absence of these dynamics in dissociated cultures suggest an incomplete maturation of superficial layer circuitry, which is supported by several arguments, the most obvious one being that the standard protocol for dissociated cortex cultures biases towards the formation of deep layer circuits. Dissociated cultures are typically prepared from embryonic cortex at E18 (Pasquale et al., 2008), which is dominated by deep layer neurons known to autonomously generate population burst activity also called ‘delta’ brushes (Khazipov and Luhmann, 2006). In contrast, superficial precursor neurons develop relatively late (Luhmann et al., 2016; Molnár et al., 2020), and at E15 - 16 are still on their migration towards the cortex along the periventricular wall, a developmental feature that can be used to selectively transfect superficial cells at that developmental stage (Saito, 2006; Bellay et al., 2015) ( cf . Fig. 12). In addition, late migrating interneurons will be absent in dissociated cultures prepared from the embryonic cortical mantle only. Without endogenous neurotransmitter such as acetylcholine, which is lacking in vitro , deep layer pyramidal neurons exhibit intrinsic bursting (McCormick and Prince, 1987; Wang and McCormick, 1993; Compte et al., 2003) that can result in network-wide events (Sanchez-Vives and McCormick, 2000; Wester and Contreras, 2012; Capone et al., 2017). Importantly, organotypic cortex cultures are typically prepared from postnatal cortex between PND 1 – 2 at which most precursor neurons required for establishing superficial pyramidal and interneurons have already arrived in cortex, allowing autonomous assembly of superficial layers in the isolated local cortical culture (see above). This sensitivity to the developmental timepoint of neuronal harvest is further exemplified by dissociated hippocampus cultures which, when taken at E18, reveal avalanche size distribution slope close to -2 and a supercritical branching parameter at 3 ms bin width (Pu et al., 2013). In contrast, hippocampal cultures made from newborn pups reveal non-trivial mean size vs. duration scaling exponents of 2, not found for supercritical dynamics (Fig. 13G; (Yaghoubi et al., 2018)). Similarly, Tetzlaff et al. (2010) prepared dissociated cultures from PND1 – 2 resulting in relatively mild bimodality with increasing bin width (Fig. 13B). Preparing dissociated cultures from postnatal tissue, expansion towards 3-dimensional scaffolding using microbeads, and co-culturing with other brain regions, e.g., the hippocampus, might introduce structural heterogeneity that stabilizes avalanche dynamics in future analysis (Massobrio et al., 2015; Brofiga et al., 2020). To summarize, most avalanche analyses in dissociated cortex cultures reveal power laws that change to a bimodal distribution at longer integration windows, which seems to reflect a 1 st order phase transition commonly found for predominantly deep layer pyramidal networks. These findings suggest that activity in dissociated cultures do not compare well with neuronal avalanche dynamics originally described in organotypic cortex cultures, acute cortex slices and further established in awake in vivo preparation that feature superficial layer activity. S UMMARY AND C ONCLUSION
Experimental evidence for SOC in the brain points to the presence of at least 4 dynamical motifs – up-states, nested oscillations, neuronal avalanches, and coherence potentials. These motifs have been robustly reported for the intact brain and in isolated mammalian cortex with its layered structure and cell type diversity largely preserved, specifically the organotypic cortex culture and acute cortex slice. The co-emergence of scale-invariant neuronal avalanches with oscillations during up-states should encourage future work on SOC in the brain at a disorder-synchronization phase-transition. Neuronal population activity measured in dissociated cortex cultures typically differs from that reported for layered cortex preparations. Identifying the precise structural and dynamical constrains responsible for these differences might provide important insights into the mechanisms supporting SOC in the brain. C ONFLICT OF I NTEREST
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. A UTHOR C ONTRIBUTIONS
DP took the lead in writing the manuscript and in discussions with all other authors. lenz et al. 2021 SOC in the brain F UNDING
This research was supported by the Division of the Intramural Research Program of the National Institute of Mental Health (NIMH), USA, ZIAMH002797. This research utilized the computational resources of Biowulf (http://hpc.nih.gov) at the National Institutes of Health (NIH), USA. A CKNOWLEDGMENTS
We thank Drs. Dante R. Chialvo and Patrick Kanold for comments and support during writing of this manuscript. R EFERENCES
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