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Register a new userScaling of causal neural avalanches in a neutral model
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Neural avalanches are collective firings of neurons that exhibit emergent scale-free behavior. Understanding the nature and distribution of these avalanches is an important element in understanding how the brain functions. We study a model of neural avalanches for which the dynamics are governed by neutral theory. The neural avalanches are defined using causal connections between the firing neurons. We analyze the scaling of causal neural avalanches as the critical point is approached from the absorbing phase. By using cluster analysis tools from percolation theory, we characterize the critical properties of the neural avalanches. We identify the tuning parameters consistent with experiments. The scaling hypothesis provides a unified explanation of the power laws which characterize the critical point. The critical exponents characterizing the avalanche distributions and divergence of the response functions are consistent with the predictions of the scaling hypothesis. We use a universal scaling function for the avalanche profile to find that the firing rates for avalanches of different durations show data collapse after appropriate rescaling. We also find data collapse for the avalanche distribution functions, which is stronger evidence of criticality than just the existence of power laws. Critical slowing-down and power law relaxation of avalanches is observed as the system is tuned to its critical point. We discuss how our results motivate future empirical studies of criticality in the brain.
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Based on a theoretical model for opinion spreading on a network, through avalanches, the effect of external field is now considered, by using methods from non-equilibrium statistical mechanics. The original part contains the implementation that the avalanche is only triggered when a local variable (a so called awareness) reaches and goes above a threshold. The dynamical rules are constrained to be as simple as possible, in order to sort out the basic features, though more elaborated variants are proposed. Several results are obtained for a Erdos-Renyi network and interpreted through simple analytical laws, scale free or logistic map-like, i.e., (i) the sizes, durations, and number of avalanches, including the respective distributions, (ii) the number of times the external field is applied to one possible node before all nodes are found to be above the threshold, (iii) the number of nodes still below the threshold and the number of hot nodes (close to threshold) at each time step.
The accumulation of potassium in the narrow space outside nerve cells is a classical subject of biophysics that has received much attention recently. It may be involved in potassium accumulation textcolor{black}{including} spreading depression, perhaps migraine and some kinds of epilepsy, even (speculatively) learning. Quantitative analysis is likely to help evaluate the role of potassium clearance from the extracellular space after a train of action potentials. Clearance involves three structures that extend down the length of the nerve: glia, extracellular space, and axon and so need to be described as systems distributed in space in the tradition used for electrical potential in the `cable equations of nerve since the work of Hodgkin in 1937. A three-compartment model is proposed here for the optic nerve and is used to study the accumulation of potassium and its clearance. The model allows the convection, diffusion, and electrical migration of water and ions. We depend on the data of Orkand et al to ensure the relevance of our model and align its parameters with the anatomy and properties of membranes, channels, and transporters: our model fits their experimental data quite well. The aligned model shows that glia has an important role in buffering potassium, as expected. The model shows that potassium is cleared mostly by convective flow through the syncytia of glia driven by osmotic pressure differences. A simplified model might be possible, but it must involve flow down the length of the optic nerve. It is easy for compartment models to neglect this flow. Our model can be used for structures quite different from the optic nerve that might have different distributions of channels and transporters in its three compartments. It can be generalized to include a fourth (distributed) compartment representing blood vessels to deal with the glymphatic flow into the circulatory system.
The existence of power-law distributions is only a first requirement in the validation of the critical behavior of a system. Long-range spatio-temporal correlations are fundamental for the spontaneous neuronal activity to be the expression of a system acting close to a critical point. This chapter focuses on temporal correlations and avalanche dynamics in the spontaneous activity of cortex slice cultures and in the resting fMRI BOLD signal. Long-range correlations are investigated by means of the scaling of power spectra and of Detrended Fluctuations Analysis. The existence of 1/f decay in the power spectrum, as well as of power-law scaling in the root mean square fluctuations function for the appropriate balance of excitation and inhibition suggests that long-range temporal correlations are distinctive of healthy brains. The corresponding temporal organization of neuronal avalanches can be dissected by analyzing the distribution of inter-event times between successive events. In rat cortex slice cultures this distribution exhibits a non-monotonic behavior, not usually found in other natural processes. Numerical simulations provide evidences that this behavior is a consequence of the alternation between states of high and low activity, leading to a dynamic balance between excitation and inhibition that tunes the system at criticality. In this scenario, inter-times show a peculiar relation with avalanche sizes, resulting in a hierarchical structure of avalanche sequences. Large avalanches correspond to low-frequency oscillations, and trigger cascades of smaller avalanches that are part of higher frequency rhythms. The self-regulated balance of excitation and inhibition observed in cultures is confirmed at larger scales, i.e. on fMRI data from resting brain activity, and appears to be closely related to critical features of avalanche activity.
In physics of living systems, a search for relationships of a few macroscopic variables that emerge from many microscopic elements is a central issue. We evolved gene regulatory networks so that the expression of target genes (partial system) is insensitive to environmental changes. Then, we found the expression levels of the remaining genes autonomously increase as a plastic response. Negative proportionality was observed between the average changes in target and remnant genes, reflecting reciprocity between the macroscopic robustness of homeostatic genes and plasticity of regulator genes. This reciprocity follows the lever principle, which was satisfied throughout the evolutionary course, imposing an evolutionary constraint.
An avalanche or cascade occurs when one event causes one or more subsequent events, which in turn may cause further events in a chain reaction. Avalanching dynamics are studied in many disciplines, with a recent focus on average avalanche shapes, i.e., the temporal profiles that characterize the growth and decay of avalanches of fixed duration. At the critical point of the dynamics the average avalanche shapes for different durations can be rescaled so that they collapse onto a single universal curve. We apply Markov branching process theory to derive a simple equation governing the average avalanche shape for cascade dynamics on networks. Analysis of the equation at criticality demonstrates that nonsymmetric average avalanche shapes (as observed in some experiments) occur for certain combinations of dynamics and network topology; specifically, on networks with heavy-tailed degree distributions. We give examples using numerical simulations of models for information spreading, neural dynamics, and behaviour adoption and we propose simple experimental tests to quantify whether cascading systems are in the critical state.
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