Table of Contents Preface; 7. Subthreshold response to synaptic input; 8. Theory of the action potential; 9. The stochastic activity of neurons; The analysis of stochastic neuronal activity; References; Index. Average Review. Write a Review. Cambridge University Press. Cambridge Studies in Mathematical Biology , 8. Indeed the existence of limit cycle oscillations was an important feature of the original deterministic Wilson—Cowan network [ 27 , 58 , 59 ] which provided an early explanation for the oscillations observed in EEG recordings; synchronization in Wilson—Cowan networks has been widely investigated [ 18 ].
The contribution by Fasoli et al. The authors develop a novel formalism for evaluating the cross-correlation structure of a finite-size recurrently connected network. They incorporate three sources of variability in the initial voltages, in the additive white noise current, and in the synaptic weight distribution and five small parameters: the noise intensity, standard deviation of the initial voltage ensemble, standard deviation of the synaptic weight ensemble, amplitude of the time-varying component of the synaptic weights, and amplitude of the time-varying component of the input current.
By expanding in these small parameters, and assuming Gaussian distributions for each source of variability, the authors obtain analytic expressions for the n -fold covariance of the voltages of each cell in the network, as well as asymptotically the covariances of the firing rates. The article demonstrates how one can relate anatomical and functional connectivity analytically, and fully develops the cases of certain regular graphs, specifically networks with block-circulant and hypercube topologies.
Interestingly, they find that pairwise correlations in the network decrease when the constant DC component of the driving current is increased to large values, because saturation of the sigmoidal activation function weakens the effective connectivity when the driving current is large. In addition, Fasoli et al. Leen and Shea-Brown [ 38 ] further extend the analysis of n -point correlations induced by common noise inputs by studying the emergence of beyond-pairwise correlations in two spiking neuron models: the exponential integrate-and-fire EIF model with cells driven by partially correlated white noise currents, and a linear—nonlinear LNL spiking model, a doubly stochastic point process derived from the EIF.
The binned spike trains obtained from these models exhibit stronger higher-order spike count correlations than could be predicted under a pairwise maximum entropy Ansatz from the first- and second-order statistics alone. Beginning in the s Cowan and his students pioneered the application of equivariant bifurcation theory to pattern formation in cortical networks. Similar methods yielded results on the formation of connectivity patterns underlying phenomena such as orientation preference, ocular dominance, and spatial frequency maps in primary visual cortex area V1 [ 2 , 6 , 8 , 9 , 50 — 52 ].
Two contributions in this collection report on continued advances in understanding pattern formation in cortical activity and connectivity patterns. This last element is interpreted in terms of the irreducible factors in the Plancherel decomposition of the Hilbert space of square-integrable functions on the Euclidean plane, the sphere, and the hyperbolic plane, respectively.
Veltz et al. They investigate a model in which isotropic local coupling is perturbed by weak anisotropic lateral coupling in order to understand how long-range connections with discrete lattice symmetry would affect the stability of different patterns of spontaneous activity hallucination patterns.
Their analysis shows that the possible periodic lattices of pinwheels orientation preference singularities is a subset of the wallpaper groups of Euclidean symmetries, and that the simplest spontaneously bifurcating dynamics generated by these networks are determined by the perturbation of invariant tori. Afgoustidis A.
Orientation maps in V1 and non-Euclidean geometry. J Math Neurosci. Spontaneous pattern formation and pinning in the primary visual cortex. J Physiol Paris. When do microcircuits produce beyond-pairwise correlations? Front Comput Neurosci. Avalanches in a stochastic model of spiking neurons. PLoS Comput Biol.
Bressloff PC. Path-integral methods for analyzing the effects of fluctuations in stochastic hybrid neural networks. An amplitude equation approach to contextual effects in primary visual cortex. Neural Comput. Spontaneous pattern formation in primary visual cortex. In: Nonlinear dynamics and chaos: where do we go from here? The functional geometry of local and horizontal connections in a model of V1. Spherical model of orientation and spatial frequency tuning in a cortical hypercolumn.
Scalar and pseudoscalar bifurcations motivated by pattern formation on the visual cortex. Geometric visual hallucinations, Euclidean symmetry, and the functional architecture of visual cortex.
What geometric visual hallucinations tell us about the visual cortex. Spike initiation by transmembrane current: a white-noise analysis. J Physiol. Field-theoretic approach to fluctuation effects in neural networks. Statistical mechanics of the neocortex. Prog Biophys Mol Biol. Systematic fluctuation expansion for neural network activity equations.
Transient potentials in dendritic systems of arbitrary geometry.
Biophys J. Campbell S, Wang D. Synchronization and desynchronization in a network of locally coupled Wilson—Cowan oscillators. Path integral methods for stochastic differential equations. Cowan JD. Statistical mechanics of nervous nets. In: Neural networks. Berlin: Springer; Spontaneous symmetry breaking in large scale nervous activity. Int J Quant Chem. A personal account of the development of the field theory of large-scale brain activity from onward.
In: Neural fields. Simple spin models for the development of ocular dominance columns and iso-orientation patches. Advances in neural information processing systems. San Mateo: Morgan Kaufmann; Wilson—Cowan equations for neocortical dynamics. Self-organized criticality in a network of interacting neurons. Random Variables 5. The Extension Theorem 6. Product Spaces. Sequences of Independent Variables 7.
Null Sets. Distributions and Expectations 2. Preliminaries 3. Densities 4. Convolutions 5. Symmetrization 6. Integration by Parts. Existence of Moments 7.
Further Inequalities. Convex Functions 9. Simple Conditional Distributions. Mixtures Conditional Distributions Conditional Expectations Stable Distributions in R 1 2. Examples 3. Infinitely Divisible Distributions in R 1 4. Processes with Independent Increments 5.
Hydrodynamic regime drives flow reversals in suction feeding larval fishes during early ontogeny. Credits: 4. A thermodynamics analogy. Wiener originated path-integral methods for studying stochastic processes in the s, and they have found wide application in quantum field theory and statistical mechanics. The Advanced Data Science Option is the mechanism by which students get credit on their transcripts for their focus on data science Student Info Academic Year Plan Course projections are tentative and subject to change. Continuing Education Department Approved Providers.
Ruin Problems in Compound Poisson Processes 6. Renewal Processes 7. Examples and Problems 8. Random Walks 9. The Queuing Process Persistent and Transient Random Walks General Markov Chains Martingales Applications in Analysis 1. Main Lemma and Notations 2. Bernstein Polynomials.
Absolutely Monotone Functions 3. Moment Problems 4.
Application to Exchangeable Variables 5. Generalized Taylor Formula and Semi-Groups 6. Inversion Formulas for Laplace Transforms 7. Strong Laws 9. Generalization to Martingales Convergence of Measures 2. Special Properties 3. Distributions as Operators 4. The Central Limit Theorem 5.
Infinite Convolutions 6. Selection Theorems 7. Ergodic Theorems for Markov Chains 8. Regular Variation 9.
to Theoretical Neurobiology. Volume 2. Nonlinear and Stochastic Theories It begins with an introduction to the effects of reversal potentials on response to. Buy Introduction to Theoretical Neurobiology: Volume 2, Nonlinear and Stochastic Theories (Cambridge Studies in Mathematical Biology) on Amazon. com.
Asymptotic Properties of Regularly Varying Functions Orientation 2. Convolution Semi-Groups 3. Preparatory Lemmas 4. Finite Variances 5. The Main Theorems 6. Example: Stable Semi-Groups 7. Triangular Arrays with Identical Distributions 8. Domains of Attraction 9. Variable Distributions. The Three-Series Theorem The Pseudo-Poisson Type 2. A Variant: Linear Increments 3.
Jump Processes 4. Diffusion Processes in R 1 5. The Forward Equation. Boundary Conditions 6. Diffusion in Higher Dimensions 7. Subordinated Processes 8.