Recent developments in the quantitative analysis of complex networks, based largely on graph theory, have been rapidly translated to studies of brain network organization. The brain's structural and functional systems have features of complex networks--such as small-world topology, highly connected hubs and modularity--both at the whole-brain scale of human neuroimaging and at a cellular scale in non-human animals. In this article, we review studies investigating complex brain networks in diverse experimental modalities (including structural and functional MRI, diffusion tensor imaging, magnetoencephalography and electroencephalography in humans) and provide an accessible introduction to the basic principles of graph theory. We also highlight some of the technical challenges and key questions to be addressed by future developments in this rapidly moving field.

Complex brain networks: graph theoretical analysis of structural and functional systems

http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

I have developed "tennis elbow" from lugging this book around the past four weeks, but it is worth the pain, the effort, and the aspirin. It is also worth the (relatively speaking) bargain price. Including appendixes, this book contains 894 pages of text. The entire panorama of the neural sciences is surveyed and examined, and it is comprehensive in its scope, from genomes to social behaviors. The editors explicitly state that the book is designed as "an introductory text for students of biology, behavior, and medicine," but it is hard to imagine any audience, interested in any fragment of neuroscience at any level of sophistication, that would not enjoy this book. The editors have done a masterful job of weaving together the biologic, the behavioral, and the clinical sciences into a single tapestry in which everyone from the molecular biologist to the practicing psychiatrist can find and appreciate his or

Principles of Neural Science

Prelude. Cycle 1. Introduction. Cycle 2. Structure defines function. Cycle 3. Diversity of cortical functions is provided by inhibition. Cycle 4. Windows on the brain. Cycle 5. A system of rhythms: from simple to complex dynamics. Cycle 6. Synchronization by oscillation. Cycle 7. The brain's default state: self-organized oscillations in rest and sleep. Cycle 8. Perturbation of the default patterns by experience. Cycle 9. The gamma buzz: gluing by oscillations in the waking brain. Cycle 10. Perceptions and actions are brain state-dependent. Cycle 11. Oscillations in the "other cortex:" navigation in real and memory space. Cycle 12. Coupling of systems by oscillations. Cycle 13. The tough problem. References.

Rhythms of the brain

This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology "Dynamical Systems in Neuroscience" presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties The book introduces dynamical systems starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems Each chapter proceeds from the simple to the complex, and provides sample problems at the end The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum - or taught by math or physics department in a way that is suitable for students of biology This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience

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Dynamical Systems in Neuroscience

Channel noise in neurons

http://www.bu.edu/hasselmo/KrausRobinsonWhiteEichenbaumHasselmo2013.pdf

Hippocampal “Time Cells”: Time versus Path Integration

We study some mechanisms responsible for synchronous oscillations and loss of synchrony at physiologically relevant frequencies (10-200 Hz) in a network of heterogeneous inhibitory neurons. We focus on the factors that determine the level of synchrony and frequency of the network response, as well as the effects of mild heterogeneity on network dynamics. With mild heterogeneity, synchrony is never perfect and is relatively fragile. In addition, the effects of inhibition are more complex in mildly heterogeneous networks than in homogeneous ones. In the former, synchrony is broken in two distinct ways, depending on the ratio of the synaptic decay time to the period of repetitive action potentials (tau s/T), where T can be determined either from the network or from a single, self-inhibiting neuron. With tau s/T > 2, corresponding to large applied current, small synaptic strength or large synaptic decay time, the effects of inhibition are largely tonic and heterogeneous neurons spike relatively independently. With tau s/T < 1, synchrony breaks when faster cells begin to suppress their less excitable neighbors; cells that fire remain nearly synchronous. We show numerically that the behavior of mildly heterogeneous networks can be related to the behavior of single, self-inhibiting cells, which can be studied analytically.

/pdf/synchronization-and-oscillatory-dynamics-in-heterogeneous-37r026aued.pdf

Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons.

Spontaneous action potentials due to channel fluctuations

Most sensory stimuli are actively sampled, yet the role of sampling behavior in shaping sensory codes is poorly understood. Mammals sample odors by sniffing, a complex behavior that controls odorant access to receptor neurons. Whether sniffing shapes the neural code for odors remains unclear. We addressed this question by imaging receptor input to the olfactory bulb of awake rats performing odor discriminations that elicited different sniffing behaviors. High-frequency sniffing of an odorant attenuated inputs encoding that odorant, whereas lower sniff frequencies caused little attenuation. Odorants encountered later in a sniff bout were encoded as the combination of that odorant and the background odorant during low-frequency sniffing, but were encoded as the difference between the two odorants during high-frequency sniffing. Thus, sniffing controls an adaptive filter for detecting changes in the odor landscape. These data suggest an unexpected functional role for sniffing and show that sensory codes can be transformed by sampling behavior alone.

John A. White

Papers

Channel noise in neurons

Hippocampal “Time Cells”: Time versus Path Integration

Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons.

Spontaneous action potentials due to channel fluctuations

Sniffing controls an adaptive filter of sensory input to the olfactory bulb