Criticality between Cortical States

Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs

The hypothesis, that cortical dynamics operates near criticality also suggests, that it exhibits universal critical exponents which marks the Kuramoto equation, a fundamental model for synchronization, as a prime candidate for an underlying universal model. Here, we determined the synchronization behavior of this model by solving it numerically on a large, weighted human connectome network, containing 836733 nodes, in an assumed homeostatic state. Since this graph has a topological dimension d < 4, a real synchronization phase transition is not possible in the thermodynamic limit, still we could locate a transition between partially synchronized and desynchronized states. At this crossover point we observe power-law–tailed synchronization durations, with τ_t ≃ 1.2(1), away from experimental values for the brain. For comparison, on a large two-dimensional lattice, having additional random, long-range links, we obtain a mean-field value: τ_t ≃ 1.6(1). However, below the transition of the connectome we found global coupling control-parameter dependent exponents 1 < τ_t ≤ 2, overlapping with the range of human brain experiments. We also studied the effects of random flipping of a small portion of link weights, mimicking a network with inhibitory interactions, and found similar results. The control-parameter dependent exponent suggests extended dynamical criticality below the transition point.

Spike-Timing-Dependent Plasticity With Axonal Delay Tunes Networks of Izhikevich Neurons to the Edge of Synchronization Transition With Scale-Free Avalanches

Critical brain hypothesis has been intensively studied both in experimental and theoretical neuroscience over the past two decades. However, some important questions still remain: (i) What is the critical point the brain operates at? (ii) What is the regulatory mechanism that brings about and maintains such a critical state? (iii) The critical state is characterized by scale-invariant behavior which is seemingly at odds with definitive brain oscillations? In this work we consider a biologically motivated model of Izhikevich neuronal network with chemical synapses interacting via spike-timing-dependent plasticity (STDP) as well as axonal time delay. Under generic and physiologically relevant conditions we show that the system is organized and maintained around a synchronization transition point as opposed to an activity transition point associated with an absorbing state phase transition. However, such a state exhibits experimentally relevant signs of critical dynamics including scale-free avalanches with finite-size scaling as well as critical branching ratios. While the system displays stochastic oscillations with highly correlated fluctuations, it also displays dominant frequency modes seen as sharp peaks in the power spectrum. The role of STDP as well as time delay is crucial in achieving and maintaining such critical dynamics, while the role of inhibition is not as crucial. In this way we provide possible answers to all three questions posed above. We also show that one can achieve supercritical or subcritical dynamics if one changes the average time delay associated with axonal conduction.

Oscillations and collective excitability in a model of stochastic neurons under excitatory and inhibitory coupling

We study a model with excitable neurons modeled as stochastic units with three states, representing quiescence, firing, and refractoriness. The transition rates between quiescence and firing depend exponentially on the number of firing neighbors, whereas all other rates are kept constant. This model class was shown to exhibit collective oscillations (synchronization) if neurons are spiking autonomously, but not if neurons are in the excitable regime. In both cases, neurons were restricted to interact through excitatory coupling. Here we show that a plethora of collective phenomena appear if inhibitory coupling is added. Besides the usual transition between an absorbing and an active phase, the model with excitatory and inhibitory neurons can also undergo reentrant transitions to an oscillatory phase. In the mean-field description, oscillations can emerge through supercritical or subcritical Hopf bifurcations, as well as through infinite period bifurcations. The model has bistability between active and oscillating behavior, as well as collective excitability, a regime where the system can display a peak of global activity when subject to a sufficiently strong perturbation. We employ a variant of the Shinomoto-Kuramoto order parameter to characterize the phase transitions and their system-size dependence.

The scale-invariant, temporal profile of neuronal avalanches in relation to cortical γ-oscillations

Activity cascades are found in many complex systems. In the cortex, they arise in the form of neuronal avalanches that capture ongoing and evoked neuronal activities at many spatial and temporal scales. The scale-invariant nature of avalanches suggests that the brain is in a critical state, yet predictions from critical theory on the temporal unfolding of avalanches have yet to be confirmed in vivo. Here we show in awake nonhuman primates that the temporal profile of avalanches follows a symmetrical, inverted parabola spanning up to hundreds of milliseconds. This parabola constrains how avalanches initiate locally, extend spatially and shrink as they evolve in time. Importantly, parabolas of different durations can be collapsed with a scaling exponent close to 2 supporting critical generational models of neuronal avalanches. Spontaneously emerging, transient γ-oscillations coexist with and modulate these avalanche parabolas thereby providing a temporal segmentation to inherently scale-invariant, critical dynamics. Our results identify avalanches and oscillations as dual principles in the temporal organization of brain activity.

Avalanches and criticality in self-organized nanoscale networks

Current efforts to achieve neuromorphic computation are focused on highly organized architectures, such as integrated circuits and regular arrays of memristors, which lack the complex interconnectivity of the brain and so are unable to exhibit brain-like dynamics. New architectures are required, both to emulate the complexity of the brain and to achieve critical dynamics and consequent maximal computational performance. We show here that electrical signals from self-organized networks of nanoparticles exhibit brain-like spatiotemporal correlations and criticality when fabricated at a percolating phase transition. Specifically, the sizes and durations of avalanches of switching events are power law distributed, and the power law exponents satisfy rigorous criteria for criticality. These signals are therefore qualitatively and quantitatively similar to those measured in the cortex. Our self-organized networks provide a low-cost platform for computational approaches that rely on spatiotemporal correlations, such as reservoir computing, and are an important step toward creating neuromorphic device architectures.

Critical Dynamics Mediate Learning of New Distributed Memory Representations in Neuronal Networks

We explore the possible role of network dynamics near a critical point in the storage of new information in silico and in vivo, and show that learning and memory may rely on neuronal network features mediated by the vicinity of criticality. Using a mean-field, attractor-based model, we show that new information can be consolidated into attractors through state-based learning in a dynamical regime associated with maximal susceptibility at the critical point. Then, we predict that the subsequent consolidation process results in a shift from critical to sub-critical dynamics to fully encapsulate the new information. We go on to corroborate these findings using analysis of rodent hippocampal CA1 activity during contextual fear memory (CFM) consolidation. We show that the dynamical state of the CA1 network is inherently poised near criticality, but the network also undergoes a shift towards sub-critical dynamics due to successful consolidation of the CFM. Based on these findings, we propose that dynamical features associated with criticality may be universally necessary for storing new memories.

Cortical Circuit Dynamics Are Homeostatically Tuned to Criticality In Vivo

Homeostatic mechanisms stabilize neuronal activity in vivo, but whether this process gives rise to balanced network dynamics is unknown. Here, we continuously monitored the statistics of network spiking in visual cortical circuits in freely behaving rats for 9 days. Under control conditions in light and dark, networks were robustly organized around criticality, a regime that maximizes information capacity and transmission. When input was perturbed by visual deprivation, network criticality was severely disrupted and subsequently restored to criticality over 48 h. Unexpectedly, the recovery of excitatory dynamics preceded homeostatic plasticity of firing rates by >30 h. We utilized model investigations to manipulate firing rate homeostasis in a cell-type-specific manner at the onset of visual deprivation. Our results suggest that criticality in excitatory networks is established by inhibitory plasticity and architecture. These data establish that criticality is consistent with a homeostatic set point for visual cortical dynamics and suggest a key role for homeostatic regulation of inhibition.

Topological phase transitions in functional brain networks

Functional brain networks are often constructed by quantifying correlations between time series of activity of brain regions. Their topological structure includes nodes, edges, triangles, and even higher-dimensional objects. Topological data analysis (TDA) is the emerging framework to process data sets under this perspective. In parallel, topology has proven essential for understanding fundamental questions in physics. Here we report the discovery of topological phase transitions in functional brain networks by merging concepts from TDA, topology, geometry, physics, and network theory. We show that topological phase transitions occur when the Euler entropy has a singularity, which remarkably coincides with the emergence of multidimensional topological holes in the brain network. The geometric nature of the transitions can be interpreted, under certain hypotheses, as an extension of percolation to high-dimensional objects. Due to the universal character of phase transitions and noise robustness of TDA, our findings open perspectives toward establishing reliable topological and geometrical markers for group and possibly individual differences in functional brain network organization.

Hierarchical Connectome Modes and Critical State Jointly Maximize Human Brain Functional Diversity

The brain requires diverse segregated and integrated processing to perform normal functions in terms of anatomical structure and self-organized dynamics with critical features, but the fundamental relationships between the complex structural connectome, critical state, and functional diversity remain unknown. Herein, we extend the eigenmode analysis to investigate the joint contribution of hierarchical modular structural organization and critical state to brain functional diversity. We show that the structural modes inherent to the hierarchical modular structural connectome allow a nested functional segregation and integration across multiple spatiotemporal scales. The real brain hierarchical modular organization provides large structural capacity for diverse functional interactions, which are generated by sequentially activating and recruiting the hierarchical connectome modes, and the critical state can best explore the capacity to maximize the functional diversity. Our results reveal structural and dynamical mechanisms that jointly support a balanced segregated and integrated brain processing with diverse functional interactions, and they also shed light on dysfunctional segregation and integration in neurodegenerative diseases and neuropsychiatric disorders.

Critical excitation-inhibition balance in dense neural networks

The "edge of chaos" phase transition in artificial neural networks is of renewed interest in light of recent evidence for criticality in brain dynamics. Statistical mechanics traditionally studied this transition with connectivity k as the control parameter and an exactly balanced excitation-inhibition ratio. While critical connectivity has been found to be low in these model systems, typically around k=2, which is unrealistic for natural neural systems, a recent study utilizing the excitation-inhibition ratio as the control parameter found a new, nearly degree independent, critical point when connectivity is large. However, the new phase transition is accompanied by an unnaturally high level of activity in the network. Here we study random neural networks with the additional properties of (i) a high clustering coefficient and (ii) neurons that are solely either excitatory or inhibitory, a prominent property of natural neurons. As a result, we observe an additional critical point for networks with large connectivity, regardless of degree distribution, which exhibits low activity levels that compare well with neuronal brain networks.