Articles

A Rate-Reduced Neuron Model for Complex Spiking Behavior

We present a simple rate-reduced neuron model that captures a wide range of complex, biologically plausible, and physiologically relevant spiking behavior. This includes spike-frequency adaptation, postinhibit...

Authors: Koen Dijkstra, Yuri A. Kuznetsov, Michel J. A. M. van Putten and Stephan A. van Gils

The Dynamics of Neural Fields on Bounded Domains: An Interface Approach for Dirichlet Boundary Conditions

Continuum neural field equations model the large-scale spatio-temporal dynamics of interacting neurons on a cortical surface. They have been extensively studied, both analytically and numerically, on bounded a...

Authors: Aytül Gökçe, Daniele Avitabile and Stephen Coombes

Fundamental Limits of Forced Asynchronous Spiking with Integrate and Fire Dynamics

Authors: Anirban Nandi, Heinz Schättler, Jason T. Ritt and ShiNung Ching

Symmetries Constrain Dynamics in a Family of Balanced Neural Networks

We examine a family of random firing-rate neural networks in which we enforce the neurobiological constraint of Dale’s Law—each neuron makes either excitatory or inhibitory connections onto its post-synaptic t...

Authors: Andrea K. Barreiro, J. Nathan Kutz and Eli Shlizerman

An Analysis of Waves Underlying Grid Cell Firing in the Medial Enthorinal Cortex

Layer II stellate cells in the medial enthorinal cortex (MEC) express hyperpolarisation-activated cyclic-nucleotide-gated (HCN) channels that allow for rebound spiking via an ...

Authors: Mayte Bonilla-Quintana, Kyle C. A. Wedgwood, Reuben D. O’Dea and Stephen Coombes

A Stochastic Version of the Jansen and Rit Neural Mass Model: Analysis and Numerics

Neural mass models provide a useful framework for modelling mesoscopic neural dynamics and in this article we consider the Jansen and Rit neural mass model (JR-NMM). We formulate a stochastic version of it whi...

Authors: Markus Ableidinger, Evelyn Buckwar and Harald Hinterleitner

Fast–Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultra-slow Transitions of Classes

Bursting is a phenomenon found in a variety of physical and biological systems. For example, in neuroscience, bursting is believed to play a key role in the way information is transferred in the nervous system...

Authors: Maria Luisa Saggio, Andreas Spiegler, Christophe Bernard and Viktor K. Jirsa

Regularization of Ill-Posed Point Neuron Models

Point neuron models with a Heaviside firing rate function can be ill-posed. That is, the initial-condition-to-solution map might become discontinuous in finite time. If a Lipschitz continuous but steep firing rat...

Authors: Bjørn Fredrik Nielsen

Finite-Size Effects on Traveling Wave Solutions to Neural Field Equations

Neural field equations are used to describe the spatio-temporal evolution of the activity in a network of synaptically coupled populations of neurons in the continuum limit. Their heuristic derivation involves...

Authors: Eva Lang and Wilhelm Stannat

How Adaptation Makes Low Firing Rates Robust

Low frequency firing is modeled by Type 1 neurons with a SNIC, but, because of the vertical slope of the square-root-like f–I curve, low f only occurs over a narrow range of I. When an adaptive current is added, ...

Authors: Arthur S. Sherman and Joon Ha

Timescales and Mechanisms of Sigh-Like Bursting and Spiking in Models of Rhythmic Respiratory Neurons

Neural networks generate a variety of rhythmic activity patterns, often involving different timescales. One example arises in the respiratory network in the pre-Bötzinger complex of the mammalian brainstem, wh...

Authors: Yangyang Wang and Jonathan E. Rubin

Emergent Dynamical Properties of the BCM Learning Rule

The Bienenstock–Cooper–Munro (BCM) learning rule provides a simple setup for synaptic modification that combines a Hebbian product rule with a homeostatic mechanism that keeps the weights bounded. The homeosta...

Authors: Lawrence C. Udeigwe, Paul W. Munro and G. Bard Ermentrout

Stable Control of Firing Rate Mean and Variance by Dual Homeostatic Mechanisms

Homeostatic processes that provide negative feedback to regulate neuronal firing rates are essential for normal brain function. Indeed, multiple parameters of individual neurons, including the scale of afferen...

Authors: Jonathan Cannon and Paul Miller

A Theoretical Study on the Role of Astrocytic Activity in Neuronal Hyperexcitability by a Novel Neuron-Glia Mass Model

Recent experimental evidence on the clustering of glutamate and GABA transporters on astrocytic processes surrounding synaptic terminals pose the question of the functional relevance of the astrocytes in the r...

Authors: Aurélie Garnier, Alexandre Vidal and Habib Benali

Analytic Modeling of Neural Tissue: I. A Spherical Bidomain

Presented here is a model of neural tissue in a conductive medium stimulated by externally injected currents. The tissue is described as a conductively isotropic bidomain, i.e. comprised of intra and extracell...

Authors: Benjamin L. Schwartz, Munish Chauhan and Rosalind J. Sadleir

Responses of Leaky Integrate-and-Fire Neurons to a Plurality of Stimuli in Their Receptive Fields

A fundamental question concerning the way the visual world is represented in our brain is how a cortical cell responds when its classical receptive field contains a plurality of stimuli. Two opposing models ha...

Authors: Kang Li, Claus Bundesen and Susanne Ditlevsen

Ill-Posed Point Neuron Models

We show that point-neuron models with a Heaviside firing rate function can be ill posed. More specifically, the initial-condition-to-solution map might become discontinuous in finite time. Consequently, if finite...

Authors: Bjørn Fredrik Nielsen and John Wyller

Entrainment Ranges for Chains of Forced Neural and Phase Oscillators

Sensory input to the lamprey central pattern generator (CPG) for locomotion is known to have a significant role in modulating lamprey swimming. Lamprey CPGs are known to have the ability to entrain to a bendin...

Authors: Nicole Massarelli, Geoffrey Clapp, Kathleen Hoffman and Tim Kiemel

Wave Generation in Unidirectional Chains of Idealized Neural Oscillators

We investigate the dynamics of unidirectional semi-infinite chains of type-I oscillators that are periodically forced at their root node, as an archetype of wave generation in neural networks. In previous stud...

Authors: Bastien Fernandez and Stanislav M. Mintchev

Stochastic Network Models in Neuroscience: A Festschrift for Jack Cowan. Introduction to the Special Issue

Jack Cowan’s remarkable career has spanned, and molded, the development of neuroscience as a quantitative and mathematical discipline combining deep theoretical contributions, rigorous mathematical work and gr...

Authors: Paul C. Bressloff, Bard Ermentrout, Olivier Faugeras and Peter J. Thomas

Neural Field Models with Threshold Noise

The original neural field model of Wilson and Cowan is often interpreted as the averaged behaviour of a network of switch like neural elements with a distribution of switch thresholds, giving rise to the class...

Authors: Rüdiger Thul, Stephen Coombes and Carlo R. Laing

Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to s...

Authors: Peter Ashwin, Stephen Coombes and Rachel Nicks

Wilson–Cowan Equations for Neocortical Dynamics

In 1972–1973 Wilson and Cowan introduced a mathematical model of the population dynamics of synaptically coupled excitatory and inhibitory neurons in the neocortex. The model dealt only with the mean numbers o...

Authors: Jack D. Cowan, Jeremy Neuman and Wim van Drongelen

A Mechanistic Neural Field Theory of How Anesthesia Suppresses Consciousness: Synaptic Drive Dynamics, Bifurcations, Attractors, and Partial State Equipartitioning

With the advances in biochemistry, molecular biology, and neurochemistry there has been impressive progress in understanding the molecular properties of anesthetic agents. However, there has been little focus ...

Authors: Saing Paul Hou, Wassim M. Haddad, Nader Meskin and James M. Bailey

Clarification and Complement to “Mean-Field Description and Propagation of Chaos in Networks of Hodgkin–Huxley and FitzHugh–Nagumo Neurons”

In this note, we clarify the well-posedness of the limit equations to the mean-field N-neuron models proposed in (Baladron et al. in J. Math. Neurosci. 2:10, 2012) and we prove the associated propagation of chaos...

Authors: Mireille Bossy, Olivier Faugeras and Denis Talay

A Simple Mechanism for Beyond-Pairwise Correlations in Integrate-and-Fire Neurons

The collective dynamics of neural populations are often characterized in terms of correlations in the spike activity of different neurons. We have developed an understanding of the circuit mechanisms that lead to...

Authors: David A. Leen and Eric Shea-Brown

Extensive Four-Dimensional Chaos in a Mesoscopic Model of the Electroencephalogram

In a previous work (Dafilis et al. in Chaos 23(2):023111, 2013), evidence was presented for four-dimensional chaos in Liley’s mesoscopic model of the electroencephalogram. The study was limited to one paramete...

Authors: Mathew P. Dafilis, Federico Frascoli, Peter J. Cadusch and David T. J. Liley

Neural Excitability and Singular Bifurcations

We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define e...

Authors: Peter De Maesschalck and Martin Wechselberger

Conditions for Multi-functionality in a Rhythm Generating Network Inspired by Turtle Scratching

Rhythmic behaviors such as breathing, walking, and scratching are vital to many species. Such behaviors can emerge from groups of neurons, called central pattern generators, in the absence of rhythmic inputs. ...

Authors: Abigail C. Snyder and Jonathan E. Rubin

The Minimal k-Core Problem for Modeling k-Assemblies

The concept of cell assembly was introduced by Hebb and formalized mathematically by Palm in the framework of graph theory. In the study of associative memory, a cell assembly is a group of neurons that are st...

Authors: Cynthia I. Wood and Illya V. Hicks

Stochastic Synchronization in Purkinje Cells with Feedforward Inhibition Could Be Studied with Equivalent Phase-Response Curves

Simple-spike synchrony between Purkinje cells projecting to a common neuron in the deep cerebellar nucleus is emerging as an important factor in the encoding of output information from cerebellar cortex. A phe...

Authors: Sergio Verduzco-Flores

Orientation Maps in V1 and Non-Euclidean Geometry

In the primary visual cortex, the processing of information uses the distribution of orientations in the visual input: neurons react to some orientations in the stimulus more than to others. In many species, o...

Authors: Alexandre Afgoustidis

On the Effects on Cortical Spontaneous Activity of the Symmetries of the Network of Pinwheels in Visual Area V1

This paper challenges and extends earlier seminal work. We consider the problem of describing mathematically the spontaneous activity of V1 by combining several important experimental observations including (1...

Authors: Romain Veltz, Pascal Chossat and Olivier Faugeras

Monochromaticity of Orientation Maps in V1 Implies Minimum Variance for Hypercolumn Size

In the primary visual cortex of many mammals, the processing of sensory information involves recognizing stimuli orientations. The repartition of preferred orientations of neurons in some areas is remarkable: ...

Authors: Alexandre Afgoustidis

Noise-Induced Precursors of State Transitions in the Stochastic Wilson–Cowan Model

The Wilson–Cowan neural field equations describe the dynamical behavior of a 1-D continuum of excitatory and inhibitory cortical neural aggregates, using a pair of coupled integro-differential equations. Here ...

Authors: Ehsan Negahbani, D. Alistair Steyn-Ross, Moira L. Steyn-Ross, Marcus T. Wilson and Jamie W. Sleigh

Modeling Focal Epileptic Activity in the Wilson–Cowan Model with Depolarization Block

Measurements of neuronal signals during human seizure activity and evoked epileptic activity in experimental models suggest that, in these pathological states, the individual nerve cells experience an activity...

Authors: Hil G. E. Meijer, Tahra L. Eissa, Bert Kiewiet, Jeremy F. Neuman, Catherine A. Schevon, Ronald G. Emerson, Robert R. Goodman, Guy M. McKhann Jr., Charles J. Marcuccilli, Andrew K. Tryba, Jack D. Cowan, Stephan A. van Gils and Wim van Drongelen

Path Integral Methods for Stochastic Differential Equations

Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult. Here, we give a self-contained pedagogical review of perturbative field theoretic...

Authors: Carson C. Chow and Michael A. Buice

A Formalism for Evaluating Analytically the Cross-Correlation Structure of a Firing-Rate Network Model

We introduce a new formalism for evaluating analytically the cross-correlation structure of a finite-size firing-rate network with recurrent connections. The analysis performs a first-order perturbative expans...

Authors: Diego Fasoli, Olivier Faugeras and Stefano Panzeri

A Mathematical Model of a Midbrain Dopamine Neuron Identifies Two Slow Variables Likely Responsible for Bursts Evoked by SK Channel Antagonists and Terminated by Depolarization Block

Midbrain dopamine neurons exhibit a novel type of bursting that we call “inverted square wave bursting” when exposed to Ca²⁺-activated small conductance (SK) K⁺ channel blockers in vitro. This type of bursting ha...

Authors: Na Yu and Carmen C. Canavier

Path-Integral Methods for Analyzing the Effects of Fluctuations in Stochastic Hybrid Neural Networks

We consider applications of path-integral methods to the analysis of a stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is...

Authors: Paul C. Bressloff

Uncertainty Propagation in Nerve Impulses Through the Action Potential Mechanism

We investigate the propagation of probabilistic uncertainty through the action potential mechanism in nerve cells. Using the Hodgkin–Huxley (H-H) model and Stochastic Collocation on Sparse Grids, we obtain an ...

Authors: Aldemar Torres Valderrama, Jeroen Witteveen, Maria Navarro and Joke Blom

Coarse-Grained Clustering Dynamics of Heterogeneously Coupled Neurons

The formation of oscillating phase clusters in a network of identical Hodgkin–Huxley neurons is studied, along with their dynamic behavior. The neurons are synaptically coupled in an all-to-all manner, yet the...

Authors: Sung Joon Moon, Katherine A Cook, Karthikeyan Rajendran, Ioannis G Kevrekidis, Jaime Cisternas and Carlo R Laing

Shifting Spike Times or Adding and Deleting Spikes—How Different Types of Noise Shape Signal Transmission in Neural Populations

We study a population of spiking neurons which are subject to independent noise processes and a strong common time-dependent input. We show that the response of output spikes to independent noise shapes inform...

Authors: Sergej O Voronenko, Wilhelm Stannat and Benjamin Lindner

Numerical Bifurcation Theory for High-Dimensional Neural Models

Numerical bifurcation theory involves finding and then following certain types of solutions of differential equations as parameters are varied, and determining whether they undergo any bifurcations (qualitativ...

Authors: Carlo R Laing

Adaptation and Fatigue Model for Neuron Networks and Large Time Asymptotics in a Nonlinear Fragmentation Equation

Motivated by a model for neural networks with adaptation and fatigue, we study a conservative fragmentation equation that describes the density probability of neurons with an elapsed time s after its last dischar...

Authors: Khashayar Pakdaman, Benoît Perthame and Delphine Salort

Frequency Preference Response to Oscillatory Inputs in Two-dimensional Neural Models: A Geometric Approach to Subthreshold Amplitude and Phase Resonance

We investigate the dynamic mechanisms of generation of subthreshold and phase resonance in two-dimensional linear and linearized biophysical (conductance-based) models, and we extend our analysis to account fo...

Authors: Horacio G Rotstein

Network Symmetry and Binocular Rivalry Experiments

Hugh Wilson has proposed a class of models that treat higher-level decision making as a competition between patterns coded as levels of a set of attributes in an appropriately defined network (Cortical Mechani...

Authors: Casey O Diekman and Martin Golubitsky

Effects of Synaptic Plasticity on Phase and Period Locking in a Network of Two Oscillatory Neurons

We study the effects of synaptic plasticity on the determination of firing period and relative phases in a network of two oscillatory neurons coupled with reciprocal inhibition. We combine the phase response c...

Authors: Zeynep Akcay, Amitabha Bose and Farzan Nadim

Approximate, not Perfect Synchrony Maximizes the Downstream Effectiveness of Excitatory Neuronal Ensembles

The most basic functional role commonly ascribed to synchrony in the brain is that of amplifying excitatory neuronal signals. The reasoning is straightforward: When positive charge is injected into a leaky tar...

Authors: Christoph Börgers, Jie Li and Nancy Kopell

Identification of Criticality in Neuronal Avalanches: II. A Theoretical and Empirical Investigation of the Driven Case

The observation of apparent power laws in neuronal systems has led to the suggestion that the brain is at, or close to, a critical state and may be a self-organised critical system. Within the framework of sel...

Authors: Caroline Hartley, Timothy J Taylor, Istvan Z Kiss, Simon F Farmer and Luc Berthouze