Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

From Knowledge, Knowability and the Search for Objective Randomness to a New Vision of Complexity

Herein we consider various concepts of entropy as measures of the complexity of phenomena and in so doing encounter a fundamental problem in physics that affects how we understand the nature of reality. In essence the difficulty has to do with our understanding of randomness, irreversibility and unpredictability using physical theory, and these in turn undermine our certainty regarding what we can and what we cannot know about complex phenomena in general. The sources of complexity examined herein appear to be channels for the amplification of naturally occurring randomness in the physical world. Our analysis suggests that when the conditions for the renormalization group apply, this spontaneous randomness, which is not a reflection of our limited knowledge, but a genuine property of nature, does not realize the conventional thermodynamic state, and a new condition, intermediate between the dynamic and the thermodynamic state, emerges. We argue that with this vision of complexity, life, which with ordinary statistical mechanics seems to be foreign to physics, becomes a natural consequence of dynamical processes.Comment: Phylosophica

Interpretation of percolation in terms of infinity computations

In this paper, a number of traditional models related to the percolation theory has been considered by means of new computational methodology that does not use Cantor's ideas and describes infinite and infinitesimal numbers in accordance with the principle `The part is less than the whole'. It gives a possibility to work with finite, infinite, and infinitesimal quantities numerically by using a new kind of a computer - the Infinity Computer - introduced recently in by Ya.D. Sergeyev in a number of patents. The new approach does not contradict Cantor. In contrast, it can be viewed as an evolution of his deep ideas regarding the existence of different infinite numbers in a more applied way. Site percolation and gradient percolation have been studied by applying the new computational tools. It has been established that in an infinite system the phase transition point is not really a point as with respect of traditional approach. In light of new arithmetic it appears as a critical interval, rather than a critical point. Depending on "microscope" we use this interval could be regarded as finite, infinite and infinitesimal short interval. Using new approach we observed that in vicinity of percolation threshold we have many different infinite clusters instead of one infinite cluster that appears in traditional consideration.Comment: 22 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:1203.4140, arXiv:1203.316

Excitable media in open and closed chaotic flows

We investigate the response of an excitable medium to a localized perturbation in the presence of a two-dimensional smooth chaotic flow. Two distinct types of flows are numerically considered: open and closed. For both of them three distinct regimes are found, depending on the relative strengths of the stirring and the rate of the excitable reaction. In order to clarify and understand the role of the many competing mechanisms present, simplified models of the process are introduced. They are one-dimensional baker-map models for the flow and a one-dimensional approximation for the transverse profile of the filaments.Comment: 14 pages, 16 figure

The Continuous time random walk, still trendy, fifty-year history, state of art and outlook

In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond

The Continuous time random walk, still trendy, fifty-year history, state of art and outlook

In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond

Characterizing the diffusional behavior and trafficking pathways of Kv2.1 using single particle tracking in live cells

2013 Spring.Includes bibliographical references.Studying the diffusion pattern of membrane components yields valuable information regarding membrane structure, organization, and dynamics. Single particle tracking serves as an excellent tool to probe these events. We are investigating of the dynamics of the voltage gated potassium channel, Kv2.1. Kv2.1 uniquely localizes to stable, micro-domains on the cell surface where it plays a non-conducting role. The work reported here examines the diffusion pattern of Kv2.1 and determines alternate functional roles of surface clusters by investigating recycling pathways using single particle tracking in live cells. The movement of Kv2.1 on the cell surface is found to be best modeled by the combination of a stationary and non-stationary process, namely a continuous time random walk in a fractal geometry. Kv2.1 surface structures are shown to be specialized platforms involved in trafficking of Kv channels to and from the cell surface in hippocampal neurons and transfected HEK cells. Both Kv2.1 and Kv1.4, a non-clustering membrane protein, are inserted and retrieved from the plasma membrane at the perimeter of Kv2.1 clusters. From the distribution of cluster sizes, using a Fokker-Planck formalism, we find there is no evidence of a feedback mechanism controlling Kv2.1 domain size on the cell surface. Interestingly, the sizes of Kv2.1 clusters are rather governed by fluctuations in the endocytic and exocytic machinery. Lastly, we pinpoint the mechanism responsible for inducing Kv2.1 non-ergodic dynamics: the capture of Kv2.1 into growing clathrin-coated pits via transient binding to pit proteins