Articles

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

Cross-Correlations and Joint Gaussianity in Multivariate Level Crossing Models

A variety of phenomena in physical and biological sciences can be mathematically understood by considering the statistical properties of level crossings of random Gaussian processes. Notably, a growing number ...

Authors: Elena Di Bernardino, José León and Tatjana Tchumatchenko

Editorial for the Special Issue on Uncertainty in the Brain

Authors: Olivier Faugeras and Michèle Thieullen

Measuring Edge Importance: A Quantitative Analysis of the Stochastic Shielding Approximation for Random Processes on Graphs

Mathematical models of cellular physiological mechanisms often involve random walks on graphs representing transitions within networks of functional states. Schmandt and Galán recently introduced a novel stochast...

Authors: Deena R Schmidt and Peter J Thomas

How Uncertainty Bounds the Shape Index of Simple Cells

We propose a theoretical motivation to quantify actual physiological features, such as the shape index distributions measured by Jones and Palmer in cats and by Ringach in macaque monkeys. We will adopt the un...

Authors: D Barbieri, G Citti and A Sarti

Fokker–Planck and Fortet Equation-Based Parameter Estimation for a Leaky Integrate-and-Fire Model with Sinusoidal and Stochastic Forcing

Analysis of sinusoidal noisy leaky integrate-and-fire models and comparison with experimental data are important to understand the neural code and neural synchronization and rhythms. In this paper, we propose ...

Authors: Alexandre Iolov, Susanne Ditlevsen and André Longtin

Goodness-of-Fit Tests and Nonparametric Adaptive Estimation for Spike Train Analysis

When dealing with classical spike train analysis, the practitioner often performs goodness-of-fit tests to test whether the observed process is a Poisson process, for instance, or if it obeys another type of p...

Authors: Patricia Reynaud-Bouret, Vincent Rivoirard, Franck Grammont and Christine Tuleau-Malot

A Density Model for a Population of Theta Neurons

Population density models that are used to describe the evolution of neural populations in a phase space are closely related to the single neuron model that describes the individual trajectories of the neurons...

Authors: Grégory Dumont, Jacques Henry and Carmen Oana Tarniceriu

Large Deviations for Nonlocal Stochastic Neural Fields

We study the effect of additive noise on integro-differential neural field equations. In particular, we analyze an Amari-type model driven by a Q-Wiener process, and focus on noise-induced transitions and escape....

Authors: Christian Kuehn and Martin G Riedler

Stabilization of Memory States by Stochastic Facilitating Synapses

Bistability within a small neural circuit can arise through an appropriate strength ofexcitatory recurrent feedback. The stability of a state of neural activity, measured bythe mean dwelling time before a nois...

Authors: Paul Miller

Analysis of a Scenario for Chaotic Quantal Slowing Down of Inspiration

On exposure to opiates, preparations from rat brain stems have been observed to continue to produce regular expiratory signals, but to fail to produce some inspiratory signals. The numbers of expirations betwe...

Authors: C Baesens and RS MacKay

Mechanisms of Intermittent State Transitions in a Coupled Heterogeneous Oscillator Model of Epilepsy

We investigate the dynamic mechanisms underlying intermittent state transitions in a recently proposed neural mass model of epilepsy. A low dimensional model is constructed, which preserves two key features of...

Authors: Marc Goodfellow and Paul Glendinning

Analytical Insights on Theta-Gamma Coupled Neural Oscillators

In this paper, we study the dynamics of a quadratic integrate-and-fire neuron, spiking in the gamma (30–100 Hz) range, coupled to a delta/theta frequency (1–8 Hz) neural oscillator. Using analytical and semian...

Authors: Lorenzo Fontolan, Maciej Krupa, Alexandre Hyafil and Boris Gutkin

Gap Junctions, Dendrites and Resonances: A Recipe for Tuning Network Dynamics

Gap junctions, also referred to as electrical synapses, are expressed along the entire central nervous system and are important in mediating various brain rhythms in both normal and pathological states. These ...

Authors: Yulia Timofeeva, Stephen Coombes and Davide Michieletto

An Interactive Channel Model of the Basal Ganglia: Bifurcation Analysis Under Healthy and Parkinsonian Conditions

Oscillations in the basal ganglia are an active area of research and have been shown to relate to the hypokinetic motor symptoms of Parkinson’s disease. We study oscillations in a multi-channel mean field mode...

Authors: Robert Merrison-Hort, Nada Yousif, Felix Njap, Ulrich G Hofmann, Oleksandr Burylko and Roman Borisyuk

Phase-Amplitude Response Functions for Transient-State Stimuli

The phase response curve (PRC) is a powerful tool to study the effect of a perturbation on the phase of an oscillator, assuming that all the dynamics can be explained by the phase variable. However, factors li...

Authors: Oriol Castejón, Antoni Guillamon and Gemma Huguet

Excitable Neurons, Firing Threshold Manifolds and Canards

We investigate firing threshold manifolds in a mathematical model of an excitable neuron. The model analyzed investigates the phenomenon of post-inhibitory rebound spiking due to propofol anesthesia and is ada...

Authors: John Mitry, Michelle McCarthy, Nancy Kopell and Martin Wechselberger

A Computational Study of Spike Time Reliability in Two Types of Threshold Dynamics

Spike time reliability (STR) refers to the phenomenon in which repetitive applications of a frozen copy of one stochastic signal to a neuron trigger spikes with reliable timing while a constant signal fails to...

Authors: Na Yu, Yue-Xian Li and Rachel Kuske

Editorial for Special Issue on Neurodynamics

“Neurodynamics” is an interdisciplinary area of mathematics where dynamical systems theory (deterministic and stochastic) is the primary tool for elucidating the fundamental mechanisms responsible for the beha...

Authors: Stephen Coombes and Yulia Timofeeva