Neural systems include interactions that occur across many scales. Two divergent methods for characterizing such interactions have drawn on the physical analysis of critical phenomena and the mathematical study of information. Inferring criticality in neural systems has traditionally rested on fitting power laws to the property distributions of “neural avalanches” (contiguous bursts of activity), but the fractal nature of avalanche shapes has recently emerged as another signature of criticality. On the other hand, neural complexity, an information theoretic measure, has been used to capture the interplay between the functional localization of brain regions and their integration for higher cognitive functions. Unfortunately, treatments of all three methods—power-law fitting, avalanche shape collapse, and neural complexity—have suffered from shortcomings. Empirical data often contain biases that introduce deviations from true power law in the tail and head of the distribution, but deviations in the tail have often been unconsidered; avalanche shape collapse has required manual parameter tuning; and the estimation of neural complexity has relied on small data sets or statistical assumptions for the sake of computational efficiency. In this paper we present technical advancements in the analysis of criticality and complexity in neural systems. We use maximum-likelihood estimation to automatically fit power laws with left and right cutoffs, present the first automated shape collapse algorithm, and describe new techniques to account for large numbers of neural variables and small data sets in the calculation of neural complexity. In order to facilitate future research in criticality and complexity, we have made the software utilized in this analysis freely available online in the MATLAB NCC (Neural Complexity and Criticality) Toolbox.

The analysis of neural systems leverages tools from many different fields. Drawing on techniques from the study of critical phenomena in statistical mechanics, several studies have reported signatures of criticality in neural systems, including power-law distributions, shape collapses, and optimized quantities under tuning. Independently, neural complexity—an information theoretic measure—has been introduced in an effort to quantify the strength of correlations across multiple scales in a neural system. This measure represents an important tool in complex systems research because it allows for the quantification of the complexity of a neural system. In this analysis, we studied the relationships between neural complexity and criticality in neural culture data. We analyzed neural avalanches in 435 recordings from dissociated hippocampal cultures produced from rats, as well as neural avalanches from a cortical branching model. We utilized recently developed maximum likelihood estimation power-law fitting methods that account for doubly truncated power-laws, an automated shape collapse algorithm, and neural complexity and branching ratio calculation methods that account for sub-sampling, all of which are implemented in the freely available Neural Complexity and Criticality MATLAB toolbox. We found evidence that neural systems operate at or near a critical point and that neural complexity is optimized in these neural systems at or near the critical point. Surprisingly, we found evidence that complexity in neural systems is dependent upon avalanche profiles and neuron firing rate, but not precise spiking relationships between neurons. In order to facilitate future research, we made all of the culture data utilized in this analysis freely available online.

A basic paradigm underlying the Hookean mechanics of amorphous, isotropic solids is that small deformations are proportional to the magnitude of external forces. However, slender bodies may undergo large deformations even under minute forces, leading to nonlinear responses rooted in purely geometric effects. Here we identify stiffening and softening as two prominent motifs of such a material-independent, geometrically-nonlinear response of thin sheets, and we show how both effects emanate from nontrivial yet generic features of the stress and displacement fields. Our insights are borne out of studying the indentation of a polymer film on a liquid bath, using experiments, simulations, and theory. We find that stiffening is due to growing anisotropy of the stress field whereas softening is due to changes in shape. We discuss similarities with the mechanics of fiber networks, suggesting that our results are relevant to a wide range of two-and three-dimensional materials.A major challenge in the mechanics of materials and structures is bridging the gap between a system's local material response and its global stiffness. The connection from microscopic to macroscopic scales is often complicated by subtle geometric effects [1]. One conceptually simple example is given by an elastic rod, which may drastically change its shape in response to loading; such elastica problems captured the attention of Galileo, the Bernoullis, and Euler, and variations on it continue to fascinate and push our understanding of slender bodies today [2][3][4]. A twodimensional sheet may also carry tensile loads in one direction while buckling in a perpendicular direction, leading to large anisotropies that control the transmission of forces through the material [5,6]. Understanding such geometrically-nonlinear behaviors is important to a wide range of applications involving thin sheets, from stretchable electronics [7] to large-scale inflatable structures [8]. There is also a diverse array of structures that are essentially composed of many connected rods or sheets, from buckling-or origami-based mechanical metamaterials [9] to soft-robotic systems [10] to disordered networks of elastic fibers including biopolymer gels and the cytoskeleton [11,12]. While much work has been carried out to understand specific systems, much less is known about generic mechanisms underlying their global mechanical response.Here we identify two general types of geometrically-nonlinear behaviors by focusing on the response of a floating thin film to indentation [ Fig. 1a,b], using experiments and simulations spanning four decades in indentation depth and theory that addresses small and large deflections. In the first behavior, the onset of anisotropy in the stress field leads to more effective transmission of forces away from a point of loading, causing the observed force to stiffen (i.e., F/δ increases where F is the force due to the imposed displacement δ) [13]. In the second behavior, the geometry of the deformations causes the transmitted force to saturate at lar...