Articles

On the potential role of lateral connectivity in retinal anticipation

We analyse the potential effects of lateral connectivity (amacrine cells and gap junctions) on motion anticipation in the retina. Our main result is that lateral connectivity can—under conditions analysed in t...

Authors: Selma Souihel and Bruno Cessac

A bio-inspired geometric model for sound reconstruction

The reconstruction mechanisms built by the human auditory system during sound reconstruction are still a matter of debate. The purpose of this study is to propose a mathematical model of sound reconstruction b...

Authors: Ugo Boscain, Dario Prandi, Ludovic Sacchelli and Giuseppina Turco

Noisy network attractor models for transitions between EEG microstates

The brain is intrinsically organized into large-scale networks that constantly re-organize on multiple timescales, even when the brain is at rest. The timing of these dynamics is crucial for sensation, percept...

Authors: Jennifer Creaser, Peter Ashwin, Claire Postlethwaite and Juliane Britz

Neural field models with transmission delays and diffusion

A neural field models the large scale behaviour of large groups of neurons. We extend previous results for these models by including a diffusion term into the neural field, which models direct, electrical conn...

Authors: Len Spek, Yuri A. Kuznetsov and Stephan A. van Gils

Stability analysis of a neural field self-organizing map

We provide theoretical conditions guaranteeing that a self-organizing map efficiently develops representations of the input space. The study relies on a neural field model of spatiotemporal activity in area 3b...

Authors: Georgios Detorakis, Antoine Chaillet and Nicolas P. Rougier

Interactions of multiple rhythms in a biophysical network of neurons

Neural oscillations, including rhythms in the beta1 band (12–20 Hz), are important in various cognitive functions. Often neural networks receive rhythmic input at frequencies different from their natural frequ...

Authors: Alexandros Gelastopoulos and Nancy J. Kopell

Spatio-chromatic information available from different neural layers via Gaussianization

How much visual information about the retinal images can be extracted from the different layers of the visual pathway? This question depends on the complexity of the visual input, the set of transforms applied to...

Authors: Jesús Malo

A new blind color watermarking based on a psychovisual model

In this paper, we address the problem of the use of a human visual system (HVS) model to improve watermark invisibility. We propose a new color watermarking algorithm based on the minimization of the perceptio...

Authors: Pascal Lefevre, David Alleysson and Philippe Carre

Synchronization and resilience in the Kuramoto white matter network model with adaptive state-dependent delays

White matter pathways form a complex network of myelinated axons that regulate signal transmission in the nervous system and play a key role in behaviour and cognition. Recent evidence reveals that white matte...

Authors: Seong Hyun Park and Jérémie Lefebvre

Attractor-state itinerancy in neural circuits with synaptic depression

Neural populations with strong excitatory recurrent connections can support bistable states in their mean firing rates. Multiple fixed points in a network of such bistable units can be used to model memory ret...

Authors: Bolun Chen and Paul Miller

Geometry of color perception. Part 2: perceived colors from real quantum states and Hering’s rebit

Authors: M. Berthier

The geometry of rest–spike bistability

Morris–Lecar model is arguably the simplest dynamical model that retains both the slow–fast geometry of excitable phase portraits and the physiological interpretation of a conductance-based model. We augment t...

Authors: Giuseppe Ilario Cirillo and Rodolphe Sepulchre

Neurally plausible mechanisms for learning selective and invariant representations

Coding for visual stimuli in the ventral stream is known to be invariant to object identity preserving nuisance transformations. Indeed, much recent theoretical and experimental work suggests that the main cha...

Authors: Fabio Anselmi, Ankit Patel and Lorenzo Rosasco

A sub-Riemannian model of the visual cortex with frequency and phase

In this paper, we present a novel model of the primary visual cortex (V1) based on orientation, frequency, and phase selective behavior of V1 simple cells. We start from the first-level mechanisms of visual pe...

Authors: E. Baspinar, A. Sarti and G. Citti

Methods to assess binocular rivalry with periodic stimuli

Binocular rivalry occurs when the two eyes are presented with incompatible stimuli and perception alternates between these two stimuli. This phenomenon has been investigated in two types of experiments: (1) Tr...

Authors: Farzaneh Darki and James Rankin

Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review

Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the em...

Authors: Christian Bick, Marc Goodfellow, Carlo R. Laing and Erik A. Martens

Spatially extended balanced networks without translationally invariant connectivity

Networks of neurons in the cerebral cortex exhibit a balance between excitation (positive input current) and inhibition (negative input current). Balanced network theory provides a parsimonious mathematical mo...

Authors: Christopher Ebsch and Robert Rosenbaum

Geometry of color perception. Part 1: structures and metrics of a homogeneous color space

This is the first half of a two-part paper dealing with the geometry of color perception. Here we analyze in detail the seminal 1974 work by H.L. Resnikoff, who showed that there are only two possible geometri...

Authors: Edoardo Provenzi

Correction to: Linking demyelination to compound action potential dispersion with a spike-diffuse-spike approach

Following publication of the original article (Naud and Longtin in J Math Neurosci 9:3, 2019), the authors noticed a mistake in the first paragraph within “Altered propagation”.

Authors: Richard Naud and André Longtin

Mesoscopic population equations for spiking neural networks with synaptic short-term plasticity

Coarse-graining microscopic models of biological neural networks to obtain mesoscopic models of neural activities is an essential step towards multi-scale models of the brain. Here, we extend a recent theory f...

Authors: Valentin Schmutz, Wulfram Gerstner and Tilo Schwalger

Phase-dependence of response curves to deep brain stimulation and their relationship: from essential tremor patient data to a Wilson–Cowan model

Essential tremor manifests predominantly as a tremor of the upper limbs. One therapy option is high-frequency deep brain stimulation, which continuously delivers electrical stimulation to the ventral intermedi...

Authors: Benoit Duchet, Gihan Weerasinghe, Hayriye Cagnan, Peter Brown, Christian Bick and Rafal Bogacz

Sparse identification of contrast gain control in the fruit fly photoreceptor and amacrine cell layer

The fruit fly’s natural visual environment is often characterized by light intensities ranging across several orders of magnitude and by rapidly varying contrast across space and time. Fruit fly photoreceptors...

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

The hyperbolic model for edge and texture detection in the primary visual cortex

The modeling of neural fields in the visual cortex involves geometrical structures which describe in mathematical formalism the functional architecture of this cortical area. The case of contour detection and ...

Authors: Pascal Chossat

Exact solutions to cable equations in branching neurons with tapering dendrites

Neurons are biological cells with uniquely complex dendritic morphologies that are not present in other cell types. Electrical signals in a neuron with branching dendrites can be studied by cable theory which ...

Authors: Lu Yihe and Yulia Timofeeva

Greedy low-rank algorithm for spatial connectome regression

Recovering brain connectivity from tract tracing data is an important computational problem in the neurosciences. Mesoscopic connectome reconstruction was previously formulated as a structured matrix regressio...

Authors: Patrick Kürschner, Sergey Dolgov, Kameron Decker Harris and Peter Benner

Correction to: Linking demyelination to compound action potential dispersion with a spike-diffuse-spike approach

Following publication of the original article [1], the authors noticed a mistake in the first paragraph within “Altered propagation”.

Authors: Richard Naud and André Longtin

The uncoupling limit of identical Hopf bifurcations with an application to perceptual bistability

We study the dynamics arising when two identical oscillators are coupled near a Hopf bifurcation where we assume a parameter ϵ uncouples the system at

Authors: Alberto Pérez-Cervera, Peter Ashwin, Gemma Huguet, Tere M. Seara and James Rankin

Data-driven inference for stationary jump-diffusion processes with application to membrane voltage fluctuations in pyramidal neurons

The emergent activity of biological systems can often be represented as low-dimensional, Langevin-type stochastic differential equations. In certain systems, however, large and abrupt events occur and violate ...

Authors: Alexandre Melanson and André Longtin

Drift–diffusion models for multiple-alternative forced-choice decision making

The canonical computational model for the cognitive process underlying two-alternative forced-choice decision making is the so-called drift–diffusion model (DDM). In this model, a decision variable keeps track...

Authors: Alex Roxin

A modified Hodgkin–Huxley model to show the effect of motor cortex stimulation on the trigeminal neuralgia network

Trigeminal neuralgia (TN) is a severe neuropathic pain, which has an electric shock-like characteristic. There are some common treatments for this pain such as medicine, microvascular decompression or radio fr...

Authors: Mohammadreza Khodashenas, Golnaz Baghdadi and Farzad Towhidkhah

Linking demyelination to compound action potential dispersion with a spike-diffuse-spike approach

To establish and exploit novel biomarkers of demyelinating diseases requires a mechanistic understanding of axonal propagation. Here, we present a novel computational framework called the stochastic spike-diff...

Authors: Richard Naud and André Longtin

Efficient calculation of heterogeneous non-equilibrium statistics in coupled firing-rate models

Understanding nervous system function requires careful study of transient (non-equilibrium) neural response to rapidly changing, noisy input from the outside world. Such neural response results from dynamic in...

Authors: Cheng Ly, Woodrow L. Shew and Andrea K. Barreiro

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