A Linear Analysis of Coupled Wilson-Cowan Neuronal Populations

Neves, L. L.; Monteiro, Luiz Henrique Alves

doi:10.1155/2016/8939218

Cited by 8 publications

(5 citation statements)

References 31 publications

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“…Besides non-linearity of differential equations makes arduous mathematical challenges. Thus, mathematical analysis is replaced by numerical methods (Latham et al, 2000; Maruyama et al, 2014; Neves and Monteiro, 2016).…”

Section: Methodsmentioning

confidence: 99%

On the Influence of Structural Connectivity on the Correlation Patterns and Network Synchronization

Nazemi

Jamali

2019

Front. Comput. Neurosci.

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Since brain structural connectivity is the foundation of its functionality, in order to understand brain abilities, studying the relation between structural and functional connectivity is essential. Several approaches have been applied to measure the role of the structural connectivity in the emergent correlation/synchronization patterns. In this study, we investigates the cross-correlation and synchronization sensitivity to coupling strength between neural regions for different topological networks. We model the neural populations by a neural mass model that express an oscillatory dynamic. The results highlight that coupling between neural ensembles leads to various cross-correlation patterns and local synchrony even on an ordered network. Moreover, as the network departs from an ordered organization to a small-world architecture, correlation patterns, and synchronization dynamics change. Interestingly, at a certain range of the synaptic strength, by fixing the structural conditions, different organized patterns are seen at the different input signals. This variety switches to a bifurcation region by increasing the synaptic strength. We show that topological variations is a major factor of synchronization behavior and lead to alterations in correlated local clusters. We found the coupling strength (between cortical areas) to be especially important at conversions of correlation and synchronization states. Since correlation patterns generate functional connections and transitions of functional connectivity have been related to cognitive operations, these diverse correlation patterns may be considered as different dynamical states corresponding to various cognitive tasks.

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Section: Methodsmentioning

confidence: 99%

On the Influence of Structural Connectivity on the Correlation Patterns and Network Synchronization

Nazemi

Jamali

2019

Front. Comput. Neurosci.

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“…This permanently changed the course of development of models in this field, and directly led to the modern formulations of neural field theory. The dynamical system analyses of resulting fixed points and limit cycles that led to these conclusions and consequent developments are summarised by Zetterberg et al [224], Ermentrout and Terman [60], and Neves and Monteiro [152]. A brief summary of the applications and extensions of the Wilson and Cowan model can be found in the reviews by Kilpatrick [106] and Chow and Karimipanah [38].…”

Section: Origins Of Neural Field Theorymentioning

confidence: 99%

Neural Field Models: A mathematical overview and unifying framework

Cook¹,

Peterson²,

Woldman³

et al. 2022

Mathematical Neuroscience and Applications

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Mathematical modelling of the macroscopic electrical activity of the brain is highly non-trivial and requires a detailed understanding of not only the associated mathematical techniques, but also the underlying physiology and anatomy. Neural field theory is a population-level approach to modelling the non-linear dynamics of large populations of neurons, while maintaining a degree of mathematical tractability. This class of models provides a solid theoretical perspective on fundamental processes of neural tissue such as state transitions between different brain activities as observed during epilepsy or sleep. Various anatomical, physiological, and mathematical assumptions are essential for deriving a minimal set of equations that strike a balance between biophysical realism and mathematical tractability. However, these assumptions are not always made explicit throughout the literature. Even though neural field models (NFMs) first appeared in the literature in the early 1970's, the relationships between them have not been systematically addressed. This may partially be explained by the fact that the inter-dependencies between these models are often implicit and non-trivial. Herein we provide a review of key stages of the history and development of neural field theory and contemporary uses of this branch of mathematical neuroscience. First, the principles of the theory are summarised throughout a discussion of the pioneering models by Wilson and Cowan, Amari and Nunez. Upon thorough review of these models, we then present a unified mathematical framework in which all neural field models can be derived by applying different assumptions. We then use this framework to i) derive contemporary models by Robinson, Jansen and Rit, Wendling, Liley, and Steyn-Ross, and ii) make explicit the many significant inherited assumptions that exist in the current literature.

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“…In this context, a linear function can be seen as the limit of such a function when increasing its stiffness, and linear activations have actually been considered as well (see e.g. [28]).…”

Section: Properties Of the Equationsmentioning

confidence: 99%

A Connection Between Image Processing and Artificial Neural Networks Layers Through a Geometric Model of Visual Perception

Batard¹,

Maldonado

Steidl

et al. 2019

Lecture Notes in Computer Science

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In this paper, we establish a connection between image processing, visual perception, and deep learning by introducing a mathematical model inspired by visual perception from which neural network layers and image processing models for color correction can be derived. Our model is inspired by the geometry of visual perception and couples a geometric model for the organization of some neurons in the visual cortex with a geometric model of color perception. More precisely, the model is a combination of a Wilson-Cowan equation describing the activity of neurons responding to edges and textures in the area V1 of the visual cortex and a Retinex model of color vision. For some particular activation functions, this yields a color correction model which processes simultaneously edges/textures, encoded into a Riemannian metric, and the color contrast, encoded into a nonlocal covariant derivative. Then, we show that the proposed model can be assimilated to a residual layer provided that the activation function is nonlinear and to a convolutional layer for a linear activation function. Finally, we show the accuracy of the model for deep learning by testing it on the MNIST dataset for digit classification.

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A Linear Analysis of Coupled Wilson-Cowan Neuronal Populations

Cited by 8 publications

References 31 publications

On the Influence of Structural Connectivity on the Correlation Patterns and Network Synchronization

On the Influence of Structural Connectivity on the Correlation Patterns and Network Synchronization

Neural Field Models: A mathematical overview and unifying framework

A Connection Between Image Processing and Artificial Neural Networks Layers Through a Geometric Model of Visual Perception

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