Random Graph Theory and Neuropercolation

for Modeling Brain Oscillations at Criticality

Robert Kozma, and Marko Puljic

aDepartment of Mathematical SciencesUniversity of Memphis, Memphis, TN 38152, USA

Abstract

Mathematical approaches are reviewed to interpret intermittent singular space-time dynamicsobserved in brain imaging experiments. The following aspects of brain dynamics are con-sidered: nonlinear dynamics (chaos), phase transitions, and criticality. Probabilistic cellularautomata and random graph models are described, which develop equations for the proba-bility distributions of macroscopic state variables as an alternative to differential equations.The introduced modular neuropercolation model is motivated by the multilayer structureand dynamical properties of the cortex, and it describes critical brain oscillations, includ-ing background activity, narrow-band oscillations in excitatory-inhibitory populations, andbroadband oscillations in the cortex. Input-induced and spontaneous transitions betweenstates with large-scale synchrony and without synchrony exhibit brief episodes with long-range spatial correlations as observed in experiments.

Corresponding Author Robert KozmaDepartment of Mathematical Sciences373 Dunn Hall, University of MemphisMemphis, TN 38152, USAPhone: +1-901-678-2497Fax: +1-901-678-2480Email: rkozma@memphis.edu

HighlightsTransient brain dynamics as manifestation of a system at the edge of criticality.Rapid phase transitions between metastable states with cognitive content.Neuropercolation approach to criticality controlled by inhibition, rewiring, noise.Collapse of broad-band oscillations to highly synchronous narrow-band dynamics.Heavy-tail distributions at criticality resembling ”Dragon King” extreme events.

Preprint submitted to Current Opinions in Neuroscience, 31C:181-188. February 3, 2015

Introduction

Experimental results from depth unit electrodes, from surface evoked potential record-ings, and from scalp EEGs and MEG/fMRI images indicate the presence of intermittentsynchronization-desynchronization transitions across cortical areas (1; 2; 3; 4; 5; 6). Transi-tions in temporal and spatial dynamics provide the window for the emergence of meaningfulcognitive activity (7; 8; 9; 10). Gap junctions between interneurons have been shown to pro-mote intermittent synchronization-desynchronization of firing (11; 12). Transient sequentialneural dynamics is not unique to mammals and it has been observed in zebrafish (13),andin the navigation system of mollusks (14). We review various theoretical concepts to inter-pret experimental findings on rapid transitions in brains and cognition, including dynamicalsystems and chaos, models of criticality, and network science and graph theory. Neuroper-colation combines these concepts and provides a powerful tool for efficient model building.Advantages and disadvantages are summarized, and perspectives for future developments areoutlined.

Modeling Transient Brain Dynamics

Brains as dynamical systems

Basic models apply Kuramoto’s classical phase oscillator equations to cortical networks(15). Population models governed by neural mass equations have been used to describe tran-sient synchronization effects in brains (16; 17; 18). Complex spatio-temporal behaviors havebeen modeled using nonlinear ordinary and partial differential equations (19; 20). Theseapproaches view brains as dynamical systems with evolving trajectories over attractor land-scapes (21; 22). With a focus on transient brain dynamics, principles of metastability havebeen exploited (23). Chaotic itineracy and Milnor attractors are mathematical models de-scribing cognitive transients (12; 4). Metastable transients reflect sequential memory, andthey have been modeled successfully using stable heteroclinic cycles in competitive networkswith excitatory and inhibitory interactions (24; 25).

Criticality in brain operation

Brains can be modeled as dissipative thermodynamic systems that hold themselves neara critical level of activity that is a non-equilibrium metastable state. The mechanism ofmaintaining the metastable state can been described as homestasis (26), or alternativelyas homeodynamics and homeochaos, emphasizing the dynamic nature of the resting state(27). Criticality is arguably a key aspect of brains in their rapid adaptation, reconfiguration,high storage capacity, and sensitive response to external stimuli (28; 29). During recentyears, self-organized criticality (SOC) and neural avalanches became important concepts todescribe neural systems (4; 7; 30; 31; 32). In spite of the successes of SOC for brain dynamics,important questions remain unresolved regarding the generation of experimentally observedrhythms and sequences of transient dynamic patterns (33). There is empirical evidenceof the cortex conforming to self-stabilized, near critical state during extended quasi-stableperiods, and existence of rapid transitions exhibiting long-range correlations (2; 3; 34). Fora comprehensive overview of the state-of-art of criticality in neural systems, based on SOCand beyond, see (35).

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Graph theory for brain networks

Random graph theory and percolation dynamics are fundamental mathematical approachesto model critical behavior in spatially extended, large-scale networks (36). There has beenintensive research in the past decade to develop efficient algorithms evaluating key statisticalproperties of structural and functional brain networks (37; 38), including hub structures andrich club networks (39; 40), and networks with causal links (41). The relation of fMRI-basedslow network dynamics to cognitive processes, their relation to much faster non-stationaritiesin synchronization patterns measured with EEG and MEG, and their potential significancefor clinical studies remain to be explored (42; 43). The presence or absence of scale-freeproperties is a contentious issue (16; 44; 45). Deviation from scale-free behavior has beendemonstrated, for example, in rich club networks (37; 42). There is a dominant view thatbrains are not random and one should not use the term random graphs and networks forbrains. Without going into a metaphysical debate at this point, it can be safely assumedthat brains, viewed either as complex deterministic machines or as random objects, can ben-efit from the use of statistical methods in their characterization (46; 9). The identification ofpercolation transitions in living neural networks (47) pints at the potential relevance of thecorresponding mathematical concepts of percolation theory to brains (48). Neuropercolationbuilds on these advances and establishes a link between structure and function of cortical andcognitive networks by filling in the models with pulsing, dynamic, living content (29; 49).

Summary of neuropercolation approach to brains

Neuropercolation combines three fundamental mathematical concepts: (i) complex dy-namics and intermittent chaos; (ii) geometric graphs and percolation theory; and (iii) phasetransitions at critical states. Specifically, the neuropercolation model uses geometric randomgraphs tuned to criticality to produce transient dynamical regimes with intermittent chaosand synchronization-desynchronization transitions. It is based on the premise that the repet-itive sudden transitions observed in the cortex are maintained by neural percolation processesin the brain as a large-scale random graph near criticality, which is self-organized in collectiveneural populations formed by synaptic activity. Neuropercolation addresses complementaryaspects of neocortex, manifesting complex information processing in microscopic networks ofspecialized spatial modules, and developing macroscopic patterns evidencing that brains areholistic organs.

Neuropercolation: A Hierarchy of Probabilistic Cellular Automata

Conceptual outline

The main components of the approach are summarized in Fig. 1, including (a) exper-iments, (b) model development, (c) model validation, (d) adaptation. Cognitively rele-vant transient brain dynamics is monitored using high-resolution multichannel experiments.Graph theory reproduces experimentally observed synchronization-desychronization episodesat alpha-theta rates, leading to the interpretation of the measured transients as critical phe-nomena and phase transitions in the cortical sheet. Model predictions are tested and vali-dated via various quantitative metrics involving transient desynchronization, input inducednarrow-band oscillations, and scale-free PSD functions (49).

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COMPARE    

HIGH-­‐DENSITY  EXPERIMENT  (ECOG/fMRI/MEG/EEG)    

RANDOM  GRAPH  MODEL  -­‐  NEUROPERCOLATION  

       Mul%channel    Time/frequency        Transient              Brain  Monitoring          Domain  Data        Desynchroniza%on  

       Mul%layer              Simulated  Transient        Phase  Transi%ons    Neuropercola%on            Time  series              near  Cri%cality                            

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 Metrics  (PSD  slope,  freq.,  …)  

MODEL  ADAPTATION,  TUNE  TO  CRITICALITY  Control  parameters  (Inhibi%on,  Noise,  Rewiring),  Learning    

(a)  

(b)   (c)  

(d)  

Figure 1: Schematics of the critical brain approach with components: (a) experiments, (b) modeling, (c)model validation, (d) adaptation. The 3D plots in (a) and (b) use x-axis for time, y-axis for linear space acrossthe cortical surface, and z-axis as phase synchronization index. Synchrony is marked by blue; brief desyn-chronization episodes are shown in green, yellow, and red; adopted from (3) and (49). The experimentallyobserved large-scale synchronization-desynchronization transitions are reproduced by the neuropercolationmodel. Validation metrics include the slope of the scale-free power spectral density (PSD), input-inducedcollapse of the broad-band oscillations to a narrow-band carrier wave. By tuning control parameters, variousoperating modes are simulated.

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Probabilistic cellular automata in 2D

When modeling the cortical sheet, the employed graph lives in the geometric space, e.g.,over a two dimensional lattice, see Fig. 2. The corresponding mathematical objects arecellular automata, related to Ising spin lattices, Hopfield nets, and cellular neural networks(50; 51; 52). In the original bootstrap percolation, lattice sites are initialized as activeor inactive, and their activation evolves according to some deterministic rule. Majorityvoting rule declares that inactive sites become active if the majority of their neighbors areactive, while the bootstrap property requires that an active site always remains active. Ifthe iterations ultimately lead to a configuration when almost all sites become active, it issaid that there is percolation in the lattice. A crucial result of percolation theory states thaton infinite lattices, there exists a critical initialization probability separating percolating andnon-percolating conditions (36).

0 1 2 3

(a)

4 5 6 78 9 10 1112 13 14 15A

A

A

A

A

A

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Figure 2: Illustration of a 4 × 4 2D lattice with periodic (toroidal) boundary conditions (a), with verticeslabeled from 0 to 15; (b) in simulations, the 2D torus is approximated by a circular arrangement of thevertices.

Neuropercolation as generalized percolation

In neuropercolation, the bootstrap property is relaxed, i.e., a site is allowed to turn fromactive to inactive. Neuropercolation incorporates the following major generalizations basedon the features of the neuropil, the filamentous neural tissue in the cerebral cortex (52):

• Noisy interactions: Neural populations exhibit dendritic noise and other random ef-fects. Neuropercolation includes a random component (ε > 0) in the majority votingrule, demonstrating that microscopic fluctuations are amplified to macroscopic phasetransitions near criticality.

• Long axonal effects: In neural populations, most of the connections are short, butthere are a relatively few long-range connections mediated by long axons, related tosmall-world phenomena (53).

• Inhibition: Interaction between excitatory and inhibitory neural populations contributeto the emergence of sustained narrow-band oscillations.

These parameters can control the system and lead to complex spatio-temporal dynamics.For example, as the noise component approaches a critical value ε0, statistical propertiessuch as correlation length diverge and scale as (ε − ε0)

β , where β is the critical exponent(52).

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a  

           1      <a>                      0                1      <a>                      0                1      <a>                      0    

Time  step              Time  step                Time  step  

0                                                                                              106  

0                                                                                              106  

0                                    5x105                                    106  

(a)  (b)  (c)  

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Cluster  size  

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Figure 3: Illustration of critical behavior in the neuropercolation model; plots (a), (b), and (c) show examplesof time series of average activation < a > for noise levels near criticality [64]. Case (c) depicts a supercritical(unimodal) regime without phase transitions, while (a) and (b) critical (bimodal) oscillations. Diagram (d)shows the distribution of cluster sizes in the various models; (a) scale-free distribution over a broad range ofpositive cluster sizes, characteristic of SOC; (b) and (c) deviate from scale-free statistics at the tail of thedistribution with high cluster sizes.

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Neuropercolation describes the evolution of the cortex at criticality through a sequence ofmetastable states. The system stays at a metastable state for exponentially long time, and itflips rapidly to another state, in polynomial time (36; 53). During the transition, the activityeffectively percolates through the system starting from certain well-defined configurations(percolating sets) (54). Metastable states may be approximated as self-organized criticalitywith scale-free behaviors. However, rapid transitions from one metastable pattern to the otherare percolation processes, extending beyond SOC dynamics. This important fundamentaltheoretical result is exploited in neuropercolation models. Fig. 3 illustrates this view usingneuropercolation simulations near criticality. The rapid switch from one state to the next isclearly seen in Fig. 3(a-b). Phase transition generate deviations of cluster size distributionfrom scale-free law at high-size limit, see Fig. 3(d). Such behavior resembles Dragon Kings(55), which describe extreme events deviating from SOC scale-free property.

Coupled oscillators with alternating narrow-band and broad-band dynamics

Neuropercolation implements a hierarchical approach to neural populations, illustratedin Fig. 4, employing Freeman K sets (22). Two coupled layers of excitatory-inhibitorypopulations (KII), see Fig. 4(a), exhibit narrow-band, bimodal oscillations for a specificcritical range of control parameters, with clear boundaries marking the region of critical-ity with prominent bimodal oscillations (52). Recently the term extended criticality hasbeen used for conditions when criticality exists over an extended range of parameters (56).Stretching criticality is yet another related concept introduced for neural systems (57).Stretching criticality may contribute to the emergence of hierarchical modular brain net-works identified by fMRI brain imaging techniques (58). Fig. 4(c) illustrates KIII sets withthree coupled oscillators. Fig. 4(d) depicts the ensemble average time series produced by eachof the oscillators. As a result of the winnerless competition between the oscillators, complextransient dynamics emerges, see bottom plot in Fig. 4(d). Neuropercolation produces inter-mittent synchronization effects (49) in line with experimental findings.

Pros and Cons

Criticality in brains is extensively studied in the literature, with SOC being a highly pop-ular model of criticality reproducing various experimentally observed properties. Its cons arethat it cannot produce the sequence of transient patterns observed in cognitive experiments.Neuropercolation reproduces important experimental observations, including deviation fromstrict scale-free SOC behavior, resembling Dragon King effects (55). Neuropercolation hasdemonstrated biologically feasible adaptation using Hebbian learning with reinforcement,when Hebbian cell assemblies respond to stimuli by destabilizing broad-band chaotic dynam-ics via narrow-band oscillation at gamma frequencies (49).

Differential equations require some degree of smoothness in the described process, whichposes difficulties when describing sudden changes and phase transitions in neurodynamics.The advantage of neuropercolation is that it can produce rapid spatio-temporal transitionswith possible singular space-time dynamics. Models based on stable heteroclinic cycles canproduce the required switching effects as well [24-25], and they may be viewed as a high-levelformulation of the symbolic dynamics, which may emerge from the population dynamics ofneuropercolation approach.

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Figure 4: Hierarchy of the neuropercolation models; (a) coupled excitatory-inhbitory layers (KII); (b)impulse response of KII: average density < a > of inhibitory population follows excitatory population witha quarter-cycle delay; (c) three coupled oscillators (KIII), O0, O1, and O2, each with a pair of excitatory-inhibitory layers; (d) top 3 diagrams: examples of time series of three isolated oscillators; (d) bottom panel:broad-band chaotic time series produced by interconnected six-layer oscillators; adopted from (49).

A potential shortcoming of neuropercolation is that it requires massive computationalresources to achieve the needed spatial and temporal resolution with proper accuracy. Thisshortcoming can be mitigated by dedicated computational resources, as cellular automataallow massively parallel implementations. In addition, analog platforms can be explored,which benefit from the intrinsic noise and memristve dynamics in such systems (59). Neu-ropercolation uses the constructive role of noise to tune the system to criticality, much like

8

the temperature can serve as a macroscopic control parameter in non-equilibrium thermody-namic systems (60).

Conclusions

There are several fundamental theoretical paradigms used in modeling experimentally ob-served transient brain dynamics and rhythms, including nonlinear dynamics with metastablestates, phase transitions, and criticality. The field is rapidly developing with frequent new dis-coveries in all of these areas. Neuropercolation is a natural mathematical domain for modelingcollective properties of networks, when the behavior of the system changes abruptly with thevariation of some control parameters. It provides a convenient framework to describe phasetransitions and critical phenomena in spatially distributed large-scale networks, in particularin brain networks with transient dynamics.

Acknowledgments

This material is based upon work supported by NSF CRCNS Program under Grant Num-ber DMS-13-11165.

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