Critical avalanches and subsampling in map-based neural networks coupled with noisy synapses

Devil's staircase inside shrimp-shaped regions reveals periodicity of plateau spikes and bursts

Slow-fast dynamics are intrinsically related to complex phenomena and are responsible for many of the homeostatic dynamics that keep biological systems healthy functioning. We study a discrete-time membrane potential model that can generate a diverse set of spiking behavior depending on the choice of slow-fast time scales, from fast spiking to bursting, or plateau action potentials-also known as cardiac spikes since they are characteristic in heart myocytes. The plateau of cardiac spikes can lose stability, generating early or delayed afterdepolarizations (EADs and DADs, respectively), both of which are related to cardiac arrhythmia. We show the periodicity changes along the transition from the healthy action potentials to these impaired oscillations. We show that while EADs are mainly periodic attractors, DADs usually come with chaos. EADs are found inside shrimp-shaped regions of the parameter space. However, in our system, multiple periodic attractors live within a shrimp-shaped region, giving it an internal structure made of infinite transitions between periodicities forming a complete devil's staircase. Understanding the periodicity of plateau attractors in slow-fast systems could be useful in unveiling the characteristics of heart myocyte behaviors that are linked to cardiac arrhythmias.

Maximum-entropy-based metrics for quantifying critical dynamics in spiking neuron data

An important working hypothesis to investigate brain activity is whether it operates in a critical regime. Recently, maximum-entropy phenomenological models have emerged as an alternative way of identifying critical behavior in neuronal data sets. In the present paper, we investigate the signatures of criticality from a firing rate-based maximum-entropy approach on data sets generated by computational models, and we compare them to experimental results. We found that the maximum entropy approach consistently identifies critical behavior around the phase transition in models and rules out criticality in models without phase transition. The maximum-entropy-model results are compatible with results for cortical data from urethane-anesthetized rats data, providing further support for criticality in the brain.

Optimal input reverberation and homeostatic self-organization toward the edge of synchronization

Transient or partial synchronization can be used to do computations, although a fully synchronized network is sometimes related to the onset of epileptic seizures. Here, we propose a homeostatic mechanism that is capable of maintaining a neuronal network at the edge of a synchronization transition, thereby avoiding the harmful consequences of a fully synchronized network. We model neurons by maps since they are dynamically richer than integrate-and-fire models and more computationally efficient than conductance-based approaches. We first describe the synchronization phase transition of a dense network of neurons with different tonic spiking frequencies coupled by gap junctions. We show that at the transition critical point, inputs optimally reverberate through the network activity through transient synchronization. Then, we introduce a local homeostatic dynamic in the synaptic coupling and show that it produces a robust self-organization toward the edge of this phase transition. We discuss the potential biological consequences of this self-organization process, such as its relation to the Brain Criticality hypothesis, its input processing capacity, and how its malfunction could lead to pathological synchronization and the onset of seizure-like activity.

Similar local neuronal dynamics may lead to different collective behavior

This report is concerned with the relevance of the microscopic rules that implement individual neuronal activation, in determining the collective dynamics, under variations of the network topology. To fix ideas we study the dynamics of two cellular automaton models, commonly used, rather in-distinctively, as the building blocks of large-scale neuronal networks. One model, due to Greenberg and Hastings (GH), can be described by evolution equations mimicking an integrate-and-fire process, while the other model, due to Kinouchi and Copelli (KC), represents an abstract branching process, where a single active neuron activates a given number of postsynaptic neurons according to a prescribed "activity" branching ratio. Despite the apparent similarity between the local neuronal dynamics of the two models, it is shown that they exhibit very different collective dynamics as a function of the network topology. The GH model shows qualitatively different dynamical regimes as the network topology is varied, including transients to a ground (inactive) state, continuous and discontinuous dynamical phase transitions. In contrast, the KC model only exhibits a continuous phase transition, independently of the network topology. These results highlight the importance of paying attention to the microscopic rules chosen to model the interneuronal interactions in large-scale numerical simulations, in particular when the network topology is far from a mean-field description. One such case is the extensive work being done in the context of the Human Connectome, where a wide variety of types of models are being used to understand the brain collective dynamics.

Subsampled Directed-Percolation Models Explain Scaling Relations Experimentally Observed in the Brain

Recent experimental results on spike avalanches measured in the urethane-anesthetized rat cortex have revealed scaling relations that indicate a phase transition at a specific level of cortical firing rate variability. The scaling relations point to critical exponents whose values differ from those of a branching process, which has been the canonical model employed to understand brain criticality. This suggested that a different model, with a different phase transition, might be required to explain the data. Here we show that this is not necessarily the case. By employing two different models belonging to the same universality class as the branching process (mean-field directed percolation) and treating the simulation data exactly like experimental data, we reproduce most of the experimental results. We find that subsampling the model and adjusting the time bin used to define avalanches (as done with experimental data) are sufficient ingredients to change the apparent exponents of the critical point. Moreover, experimental data is only reproduced within a very narrow range in parameter space around the phase transition.

Between Perfectly Critical and Fully Irregular: A Reverberating Model Captures and Predicts Cortical Spike Propagation

Knowledge about the collective dynamics of cortical spiking is very informative about the underlying coding principles. However, even most basic properties are not known with certainty, because their assessment is hampered by spatial subsampling, i.e., the limitation that only a tiny fraction of all neurons can be recorded simultaneously with millisecond precision. Building on a novel, subsampling-invariant estimator, we fit and carefully validate a minimal model for cortical spike propagation. The model interpolates between two prominent states: asynchronous and critical. We find neither of them in cortical spike recordings across various species, but instead identify a narrow "reverberating" regime. This approach enables us to predict yet unknown properties from very short recordings and for every circuit individually, including responses to minimal perturbations, intrinsic network timescales, and the strength of external input compared to recurrent activation "thereby informing about the underlying coding principles for each circuit, area, state and task.

25 years of criticality in neuroscience - established results, open controversies, novel concepts

Twenty-five years ago, Dunkelmann and Radons (1994) showed that neural networks can self-organize to a critical state. In models, the critical state offers a number of computational advantages. Thus this hypothesis, and in particular the experimental work by Beggs and Plenz (2003), has triggered an avalanche of research, with thousands of studies referring to it. Nonetheless, experimental results are still contradictory. How is it possible, that a hypothesis has attracted active research for decades, but nonetheless remains controversial? We discuss the experimental and conceptual controversy, and then present a parsimonious solution that (i) unifies the contradictory experimental results, (ii) avoids disadvantages of a critical state, and (iii) enables rapid, adaptive tuning of network properties to task requirements.

Measuring neuronal avalanches in disordered systems with absorbing states

Power-law-shaped avalanche-size distributions are widely used to probe for critical behavior in many different systems, particularly in neural networks. The definition of avalanche is ambiguous. Usually, theoretical avalanches are defined as the activity between a stimulus and the relaxation to an inactive absorbing state. On the other hand, experimental neuronal avalanches are defined by the activity between consecutive silent states. We claim that the latter definition may be extended to some theoretical models to characterize their power-law avalanches and critical behavior. We study a system in which the separation of driving and relaxation time scales emerges from its structure. We apply both definitions of avalanche to our model. Both yield power-law-distributed avalanches that scale with system size in the critical point as expected. Nevertheless, we find restricted power-law-distributed avalanches outside of the critical region within the experimental procedure, which is not expected by the standard theoretical definition. We remark that these results are dependent on the model details.

Phase diagrams and dynamics of a computationally efficient map-based neuron model

We introduce a new map-based neuron model derived from the dynamical perceptron family that has the best compromise between computational efficiency, analytical tractability, reduced parameter space and many dynamical behaviors. We calculate bifurcation and phase diagrams analytically and computationally that underpins a rich repertoire of autonomous and excitable dynamical behaviors. We report the existence of a new regime of cardiac spikes corresponding to nonchaotic aperiodic behavior. We compare the features of our model to standard neuron models currently available in the literature.

Griffiths phase and long-range correlations in a biologically motivated visual cortex model

Activity in the brain propagates as waves of firing neurons, namely avalanches. These waves' size and duration distributions have been experimentally shown to display a stable power-law profile, long-range correlations and 1/f (b) power spectrum in vivo and in vitro. We study an avalanching biologically motivated model of mammals visual cortex and find an extended critical-like region - a Griffiths phase - characterized by divergent susceptibility and zero order parameter. This phase lies close to the expected experimental value of the excitatory postsynaptic potential in the cortex suggesting that critical be-havior may be found in the visual system. Avalanches are not perfectly power-law distributed, but it is possible to collapse the distributions and define a cutoff avalanche size that diverges as the network size is increased inside the critical region. The avalanches present long-range correlations and 1/f (b) power spectrum, matching experiments. The phase transition is analytically determined by a mean-field approximation.

Spike avalanches in vivo suggest a driven, slightly subcritical brain state

In self-organized critical (SOC) systems avalanche size distributions follow power-laws. Power-laws have also been observed for neural activity, and so it has been proposed that SOC underlies brain organization as well. Surprisingly, for spiking activity in vivo, evidence for SOC is still lacking. Therefore, we analyzed highly parallel spike recordings from awake rats and monkeys, anesthetized cats, and also local field potentials from humans. We compared these to spiking activity from two established critical models: the Bak-Tang-Wiesenfeld model, and a stochastic branching model. We found fundamental differences between the neural and the model activity. These differences could be overcome for both models through a combination of three modifications: (1) subsampling, (2) increasing the input to the model (this way eliminating the separation of time scales, which is fundamental to SOC and its avalanche definition), and (3) making the model slightly sub-critical. The match between the neural activity and the modified models held not only for the classical avalanche size distributions and estimated branching parameters, but also for two novel measures (mean avalanche size, and frequency of single spikes), and for the dependence of all these measures on the temporal bin size. Our results suggest that neural activity in vivo shows a mélange of avalanches, and not temporally separated ones, and that their global activity propagation can be approximated by the principle that one spike on average triggers a little less than one spike in the next step. This implies that neural activity does not reflect a SOC state but a slightly sub-critical regime without a separation of time scales. Potential advantages of this regime may be faster information processing, and a safety margin from super-criticality, which has been linked to epilepsy.

Undersampled critical branching processes on small-world and random networks fail to reproduce the statistics of spike avalanches

The power-law size distributions obtained experimentally for neuronal avalanches are an important evidence of criticality in the brain. This evidence is supported by the fact that a critical branching process exhibits the same exponent . Models at criticality have been employed to mimic avalanche propagation and explain the statistics observed experimentally. However, a crucial aspect of neuronal recordings has been almost completely neglected in the models: undersampling. While in a typical multielectrode array hundreds of neurons are recorded, in the same area of neuronal tissue tens of thousands of neurons can be found. Here we investigate the consequences of undersampling in models with three different topologies (two-dimensional, small-world and random network) and three different dynamical regimes (subcritical, critical and supercritical). We found that undersampling modifies avalanche size distributions, extinguishing the power laws observed in critical systems. Distributions from subcritical systems are also modified, but the shape of the undersampled distributions is more similar to that of a fully sampled system. Undersampled supercritical systems can recover the general characteristics of the fully sampled version, provided that enough neurons are measured. Undersampling in two-dimensional and small-world networks leads to similar effects, while the random network is insensitive to sampling density due to the lack of a well-defined neighborhood. We conjecture that neuronal avalanches recorded from local field potentials avoid undersampling effects due to the nature of this signal, but the same does not hold for spike avalanches. We conclude that undersampled branching-process-like models in these topologies fail to reproduce the statistics of spike avalanches.