We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

Over the past 20 years, neuroimaging has become a predominant technique in systems neuroscience. One might envisage that over the next 20 years the neuroimaging of distributed processing and connectivity will play a major role in disclosing the brain's functional architecture and operational principles. The inception of this journal has been foreshadowed by an ever-increasing number of publications on functional connectivity, causal modeling, connectomics, and multivariate analyses of distributed patterns of brain responses. I accepted the invitation to write this review with great pleasure and hope to celebrate and critique the achievements to date, while addressing the challenges ahead.

/pdf/functional-and-effective-connectivity-a-review-s5oik3vige.pdf

Functional and effective connectivity: a review.

Synchronous rhythms represent a core mechanism for sculpting temporal coordination of neural activity in the brain-wide network. This review focuses on oscillations in the cerebral cortex that occur during cognition, in alert behaving conditions. Over the last two decades, experimental and modeling work has made great strides in elucidating the detailed cellular and circuit basis of these rhythms, particularly gamma and theta rhythms. The underlying physiological mechanisms are diverse (ranging from resonance and pacemaker properties of single cells to multiple scenarios for population synchronization and wave propagation), but also exhibit unifying principles. A major conceptual advance was the realization that synaptic inhibition plays a fundamental role in rhythmogenesis, either in an interneuronal network or in a reciprocal excitatory-inhibitory loop. Computational functions of synchronous oscillations in cognition are still a matter of debate among systems neuroscientists, in part because the notion of regular oscillation seems to contradict the common observation that spiking discharges of individual neurons in the cortex are highly stochastic and far from being clocklike. However, recent findings have led to a framework that goes beyond the conventional theory of coupled oscillators and reconciles the apparent dichotomy between irregular single neuron activity and field potential oscillations. From this perspective, a plethora of studies will be reviewed on the involvement of long-distance neuronal coherence in cognitive functions such as multisensory integration, working memory, and selective attention. Finally, implications of abnormal neural synchronization are discussed as they relate to mental disorders like schizophrenia and autism.

/pdf/neurophysiological-and-computational-principles-of-cortical-48xnk4ixsg.pdf

Neurophysiological and Computational Principles of Cortical Rhythms in Cognition

Many observers see geometric visual hallucinations after taking hallucinogens such as LSD, cannabis, mescaline or psilocybin; on viewing bright flickering lights; on waking up or falling asleep; in "near-death" experiences; and in many other syndromes. Kluver organized the images into four groups called form constants: (I) tunnels and funnels, (II) spirals, (III) lattices, including honeycombs and triangles, and (IV) cobwebs. In most cases, the images are seen in both eyes and move with them. We interpret this to mean that they are generated in the brain. Here, we summarize a theory of their origin in visual cortex (area V1), based on the assumption that the form of the retino-cortical map and the architecture of V1 determine their geometry. (A much longer and more detailed mathematical version has been published in Philosophical Transactions of the Royal Society B, 356 [2001].)We model V1 as the continuum limit of a lattice of interconnected hypercolumns, each comprising a number of interconnected iso-orientation columns. Based on anatomical evidence, we assume that the lateral connectivity between hypercolumns exhibits symmetries, rendering it invariant under the action of the Euclidean group E(2), composed of reflections and translations in the plane, and a (novel) shift-twist action. Using this symmetry, we show that the various patterns of activity that spontaneously emerge when V1's spatially uniform resting state becomes unstable correspond to the form constants when transformed to the visual field using the retino-cortical map. The results are sensitive to the detailed specification of the lateral connectivity and suggest that the cortical mechanisms that generate geometric visual hallucinations are closely related to those used to process edges, contours, surfaces, and textures.

/pdf/what-geometric-visual-hallucinations-tell-us-about-the-sk8cffonp5.pdf

What geometric visual hallucinations tell us about the visual cortex

A well-defined stochastic theory for neural activity, which permits the calculation of arbitrary statistical moments and equations governing them, is a potentially valuable tool for theoretical neuroscience. We produce such a theory by analyzing the dynamics of neural activity using field theoretic methods for nonequilibrium statistical processes. Assuming that neural network activity is Markovian, we construct the effective spike model, which describes both neural fluctuations and response. This analysis leads to a systematic expansion of corrections to mean field theory, which for the effective spike model is a simple version of the Wilson-Cowan equation. We argue that neural activity governed by this model exhibits a dynamical phase transition which is in the universality class of directed percolation. More general models (which may incorporate refractoriness) can exhibit other universality classes, such as dynamic isotropic percolation. Because of the extremely high connectivity in typical networks, it is expected that higher-order terms in the systematic expansion are small for experimentally accessible measurements, and thus, consistent with measurements in neocortical slice preparations, we expect mean field exponents for the transition. We provide a quantitative criterion for the relative magnitude of each term in the systematic expansion, analogous to the Ginsburg criterion. Experimental identification of dynamic universality classes in vivo is an outstanding and important question for neuroscience.

Field-theoretic approach to fluctuation effects in neural networks.

Neuronal avalanches are a form of spontaneous activity widely observed in cortical slices and other types of nervous tissue, both in vivo and in vitro. They are characterized by irregular, isolated population bursts when many neurons fire together, where the number of spikes per burst obeys a power law distribution. We simulate, using the Gillespie algorithm, a model of neuronal avalanches based on stochastic single neurons. The network consists of excitatory and inhibitory neurons, first with all-to-all connectivity and later with random sparse connectivity. Analyzing our model using the system size expansion, we show that the model obeys the standard Wilson-Cowan equations for large network sizes ( neurons). When excitation and inhibition are closely balanced, networks of thousands of neurons exhibit irregular synchronous activity, including the characteristic power law distribution of avalanche size. We show that these avalanches are due to the balanced network having weakly stable functionally feedforward dynamics, which amplifies some small fluctuations into the large population bursts. Balanced networks are thought to underlie a variety of observed network behaviours and have useful computational properties, such as responding quickly to changes in input. Thus, the appearance of avalanches in such functionally feedforward networks indicates that avalanches may be a simple consequence of a widely present network structure, when neuron dynamics are noisy. An important implication is that a network need not be “critical” for the production of avalanches, so experimentally observed power laws in burst size may be a signature of noisy functionally feedforward structure rather than of, for example, self-organized criticality.

/pdf/avalanches-in-a-stochastic-model-of-spiking-neurons-481j2p7i18.pdf

Avalanches in a stochastic model of spiking neurons.

Hunter, John D., John G. Milton, Peter J. Thomas, and Jack D. Cowan. Resonance effect for neural spike time reliability. J. Neurophysiol. 80: 1427–1438, 1998. The spike timing reliability of Aplysi...

/pdf/resonance-effect-for-neural-spike-time-reliability-5acf9fbw1t.pdf

Resonance Effect for Neural Spike Time Reliability

Population rate or activity equations are the foundation of a common approach to modeling for neural networks. These equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity. The shortcoming of these equations is that they take into account only the average firing rate, while leaving out higher-order statistics like correlations between firing. A stochastic theory of neural networks that includes statistics at all orders was recently formulated. We describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms. Each level of the approximation yields closed equations; they depend only on the mean and specific correlations of interest, without an ad hoc criterion for doing so. We show in an example of an all-to-all connected network how our system of generalized activity equations captures phenomena missed by the mean field rate equations alone.

/pdf/systematic-fluctuation-expansion-for-neural-network-activity-42je67aymt.pdf

Jack D. Cowan

Papers

What geometric visual hallucinations tell us about the visual cortex

Field-theoretic approach to fluctuation effects in neural networks.

Avalanches in a stochastic model of spiking neurons.

Resonance Effect for Neural Spike Time Reliability

Systematic fluctuation expansion for neural network activity equations