Articles

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

Distributed Nonlocal Feedback Delays May Destabilize Fronts in Neural Fields, Distributed Transmission Delays Do Not

The spread of activity in neural populations is a well-known phenomenon. To understand the propagation speed and the stability of stationary fronts in neural populations, the present work considers a neural fi...

Authors: Axel Hutt and Linghai Zhang

Inhomogeneous Sparseness Leads to Dynamic Instability During Sequence Memory Recall in a Recurrent Neural Network Model

Theoretical models of associative memory generally assume most of their parameters to be homogeneous across the network. Conversely, biological neural networks exhibit high variability of structural as well as...

Authors: Daniel Medina and Christian Leibold

Transient Localized Wave Patterns and Their Application to Migraine

Transient dynamics is pervasive in the human brain and poses challenging problems both in mathematical tractability and clinical observability. We investigate statistical properties of transient cortical wave ...

Authors: Markus A Dahlem and Thomas M Isele

Derived Patterns in Binocular Rivalry Networks

Binocular rivalry is the alternation in visual perception that can occur when the two eyes are presented with different images. Wilson proposed a class of neuronal network models that generalize rivalry to mul...

Authors: Casey O Diekman, Martin Golubitsky and Yunjiao Wang

Identification of Criticality in Neuronal Avalanches: I. A Theoretical Investigation of the Non-driven Case

In this paper, we study a simple model of a purely excitatory neural network that, by construction, operates at a critical point. This model allows us to consider various markers of criticality and illustrate ...

Authors: Timothy J Taylor, Caroline Hartley, Péter L Simon, Istvan Z Kiss and Luc Berthouze

A Network Model of the Periodic Synchronization Process in the Dynamics of Calcium Concentration in GnRH Neurons

Mathematical neuroendocrinology is a branch of mathematical neurosciences that is specifically interested in endocrine neurons, which have the uncommon ability of secreting neurohormones into the blood. One of...

Authors: Maciej Krupa, Alexandre Vidal and Frédérique Clément

A Simple Algorithm for Averaging Spike Trains

Although spike trains are the principal channel of communication between neurons, a single stimulus will elicit different spike trains from trial to trial. This variability, in both spike timings and spike num...

Authors: Hannah Julienne and Conor Houghton

Phase-Amplitude Descriptions of Neural Oscillator Models

Phase oscillators are a common starting point for the reduced description of many single neuron models that exhibit a strongly attracting limit cycle. The framework for analysing such models in response to wea...

Authors: Kyle CA Wedgwood, Kevin K Lin, Ruediger Thul and Stephen Coombes

Laws of Large Numbers and Langevin Approximations for Stochastic Neural Field Equations

In this study, we consider limit theorems for microscopic stochastic models of neural fields. We show that the Wilson–Cowan equation can be obtained as the limit in uniform convergence on compacts in probabili...

Authors: Martin G Riedler and Evelyn Buckwar

Predicted Selective Increase of Cortical Magnification Due to Cortical Folding

The cortical magnification matrix M is introduced founded on a notion similar to that of the scalar cortical magnification factor M. Unlike M, this matrix is suitable to describe anisotropy in cortical magnificat...

Authors: A Markus Dahlem and Jan Tusch

Multiscale analysis of slow-fast neuronal learning models with noise

This paper deals with the application of temporal averaging methods to recurrent networks of noisy neurons undergoing a slow and unsupervised modification of their connectivity matrix called learning. Three ti...

Authors: Mathieu Galtier and Gilles Wainrib

Stability of the splay state in networks of pulse-coupled neurons

We analytically investigate the stability of splay states in the networks of N globally pulse-coupled phase-like models of neurons. We develop a perturbative technique which allows determining the Floquet exponen...

Authors: Simona Olmi, Antonio Politi and Alessandro Torcini

Computational Convergence of the Path Integral for Real Dendritic Morphologies

Neurons are characterised by a morphological structure unique amongst biological cells, the core of which is the dendritic tree. The vast number of dendritic geometries, combined with heterogeneous properties ...

Authors: Quentin Caudron, Simon R Donnelly, Samuel PC Brand and Yulia Timofeeva

Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons

We derive the mean-field equations arising as the limit of a network of interacting spiking neurons, as the number of neurons goes to infinity. The neurons belong to a fixed number of populations and are repre...

Authors: Javier Baladron, Diego Fasoli, Olivier Faugeras and Jonathan Touboul

Interface dynamics in planar neural field models

Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where ...

Authors: Stephen Coombes, Helmut Schmidt and Ingo Bojak

Analysis of stability and bifurcations of fixed points and periodic solutions of a lumped model of neocortex with two delays

A lumped model of neural activity in neocortex is studied to identify regions of multi-stability of both steady states and periodic solutions. Presence of both steady states and periodic solutions is considere...

Authors: Sid Visser, Hil GE Meijer, Michel JAM van Putten and Stephan A van Gils

Dynamical systems analysis of spike-adding mechanisms in transient bursts

Transient bursting behaviour of excitable cells, such as neurons, is a common feature observed experimentally, but theoretically, it is not well understood. We analyse a five-dimensional simplified model of af...

Authors: Jakub Nowacki, Hinke M Osinga and Krasimira Tsaneva-Atanasova

From invasion to extinction in heterogeneous neural fields

In this paper, we analyze the invasion and extinction of activity in heterogeneous neural fields. We first consider the effects of spatial heterogeneities on the propagation of an invasive activity front. In c...

Authors: Paul C Bressloff