Criticality in neural networks : a study of the interplay between experimental tools and theoretical models

In the human brain trillions of neurons transmit information “firing” electrical pulses called action potentials or spikes. Neurons are connected to each other and form highly complex networks in which a single neuron may be connected to thousands of other neurons. The activity of single neurons and, more recently, the activity of groups of neurons have been monitored extensively using intracellular and extracellular recordings. One of the most striking observations arisen from such recordings is the fact that neuronal activity seems to be characterized by “avalanches” whose size and lifetime distributions obey a power law, which is typical of self-organized critical systems. Such critical behavior has been confirmed also by theoretical models, but the way avalanches are defined and detected in the experimental analysis is very different from the way they are defined and detected in theoretical simulations. In this work, after a brief review of the concept of Self-Organized Criticality, we describe the experiment that led to the observation of neuronal avalanches. Then, we describe the Millman model, a neuronal network model that reproduces the critical behavior observed in real networks. Finally, we investigate the differences between theoretical and experimental avalanches. In particular, we analyze the data from numerical simulations with the methods used to detect avalanches in real networks. We show that, if the methods of analysis change, the critical behavior is no longer observed. Introduction: Self-Organized Criticality The concept of Self-Organized Criticality was introduced for the first time in 1987 by Bak, Tang and Wiesenfeld [1]. They were trying to explain why the 1/f noise, i.e. a signal whose power spectral density goes like the inverse of the frequency1, was so ubiquitous in nature, having been observed in systems as different as the flow of the river Nile, highway traffic and the luminosity of quasars. To illustrate the concept we can consider the example of the sandpile. Consider a flat table where grains of sand are added slowly at random positions. At first each grain will stay in the same position where it lands; if we continue to add sand, the pile will grow and become steeper and steeper, grains will begin to slide and some of them will hit other grains, causing them to slide too: small avalanches are generated. While in the beginning these slides and avalanches have a local nature and grains at distant parts of the pile do not affect each other, if we continue to add sand we will eventually reach a state where the slope does not increase anymore. In this state there is a balance between the amount of sand added and the amount of sand that leaves the pile falling from the edges: the state is stationary. Also, for the first time there might be avalanches spanning the whole pile: the dynamics stops being local and becomes global. This state is called Self-Organized Critical State. Bak, Tang and Wiesenfeld made a simple model for the sandpile [2]. They found that in the stationary state the distributions of the avalanche sizes and times obey power laws: the system has no characteristic length scale or time scale. The power law distribution of lifetimes is equivalent to a power law 1/f frequency spectrum, that is what Bak, Tang and Wiesenfeld were trying to explain in the first place. The absence of a characteristic scale is typical of critical phenomena. In the Ising Model, for instance, there are two distinct phases; the disordered phase, with zero magnetization, occurs when the temperature is higher than the critical temperature while the ordered phase, with non zero magnetizaSpectra going like 1/f instead of 1/f are also commonly referred to as 1/f noise.