Articles

Special Issue from the 2017 International Conference on Mathematical Neuroscience

The ongoing acquisition of large and multifaceted data sets in neuroscience requires new mathematical tools for quantitatively grounding these experimental findings. Since 2015, the International Conference on...

Authors: Zachary P. Kilpatrick, Julijana Gjorgjieva and Robert Rosenbaum

M-Current Expands the Range of Gamma Frequency Inputs to Which a Neuronal Target Entrains

Theta (4–8 Hz) and gamma (30–80 Hz) rhythms in the brain are commonly associated with memory and learning (Kahana in J Neurosci 26:1669–1672, 2006; Quilichini et al. in J Neurosci 30:11128–11142, 2010). The preci...

Authors: Yujia Zhou, Theodore Vo, Horacio G. Rotstein, Michelle M. McCarthy and Nancy Kopell

Stochastic Hybrid Systems in Cellular Neuroscience

We review recent work on the theory and applications of stochastic hybrid systems in cellular neuroscience. A stochastic hybrid system or piecewise deterministic Markov process involves the coupling between a ...

Authors: Paul C. Bressloff and James N. Maclaurin

Data Assimilation Methods for Neuronal State and Parameter Estimation

This tutorial illustrates the use of data assimilation algorithms to estimate unobserved variables and unknown parameters of conductance-based neuronal models. Modern data assimilation (DA) techniques are wide...

Authors: Matthew J. Moye and Casey O. Diekman

Bifurcations of Limit Cycles in a Reduced Model of the Xenopus Tadpole Central Pattern Generator

We present the study of a minimal microcircuit controlling locomotion in two-day-old Xenopus tadpoles. During swimming, neurons in the spinal central pattern generator (CPG) generate anti-phase oscillations betwe...

Authors: Andrea Ferrario, Robert Merrison-Hort, Stephen R. Soffe, Wen-Chang Li and Roman Borisyuk

What Is Required for Neuronal Calcium Waves? A Numerical Parameter Study

Neuronal calcium signals propagating by simple diffusion and reaction with mobile and stationary buffers are limited to cellular microdomains. The distance intracellular calcium signals can travel may be signi...

Authors: Markus Breit and Gillian Queisser

Investigating the Correlation–Firing Rate Relationship in Heterogeneous Recurrent Networks

The structure of spiking activity in cortical networks has important implications for how the brain ultimately codes sensory signals. However, our understanding of how network and intrinsic cellular mechanisms...

Authors: Andrea K. Barreiro and Cheng Ly

Effect of Neuromodulation of Short-term Plasticity on Information Processing in Hippocampal Interneuron Synapses

Neurons in a micro-circuit connected by chemical synapses can have their connectivity affected by the prior activity of the cells. The number of synapses available for releasing neurotransmitter can be decreas...

Authors: Elham Bayat Mokhtari, J. Josh Lawrence and Emily F. Stone

Analysis of an Attractor Neural Network’s Response to Conflicting External Inputs

The theory of attractor neural networks has been influential in our understanding of the neural processes underlying spatial, declarative, and episodic memory. Many theoretical studies focus on the inherent pr...

Authors: Kathryn Hedrick and Kechen Zhang

Saddle Slow Manifolds and Canard Orbits in \(\mathbb{R}^{4}\) and Application to the Full Hodgkin–Huxley Model

Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron models typically involve observable differences in time scales. The Hodgkin–Huxley model i...

Authors: Cris R. Hasan, Bernd Krauskopf and Hinke M. Osinga

The Dynamics of Networks of Identical Theta Neurons

We consider finite and infinite all-to-all coupled networks of identical theta neurons. Two types of synaptic interactions are investigated: instantaneous and delayed (via first-order synaptic processing). Ext...

Authors: Carlo R. Laing

Kernel Reconstruction for Delayed Neural Field Equations

Understanding the neural field activity for realistic living systems is a challenging task in contemporary neuroscience. Neural fields have been studied and developed theoretically and numerically with conside...

Authors: Jehan Alswaihli, Roland Potthast, Ingo Bojak, Douglas Saddy and Axel Hutt

Sparse Functional Identification of Complex Cells from Spike Times and the Decoding of Visual Stimuli

We investigate the sparse functional identification of complex cells and the decoding of spatio-temporal visual stimuli encoded by an ensemble of complex cells. The reconstruction algorithm is formulated as a ...

Authors: Aurel A. Lazar, Nikul H. Ukani and Yiyin Zhou

Robust Exponential Memory in Hopfield Networks

The Hopfield recurrent neural network is a classical auto-associative model of memory, in which collections of symmetrically coupled McCulloch–Pitts binary neurons interact to perform emergent computation. Alt...

Authors: Christopher J. Hillar and Ngoc M. Tran

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