Impulses and Physiological States in Theoretical Models of Nerve Membrane

1961/07/01 by Richard FitzHugh, Richard Fitzhugh · 6,317 citations
Agricultural and Biological Sciences · Mathematics · Neuroscience · #Classical mechanics #Control theory (sociology) #Hodgkin–Huxley model #Impulse (physics) #Limit (mathematics) #Limit cycle #Mathematical analysis #Mathematics #Neural dynamics and brain function #Neuroscience #Neuroscience and Neural Engineering #Nonlinear system #Phase (matter) #Phase diagram #Phase plane #Phase space #Physics #Plant and Biological Electrophysiology Studies #Quantum mechanics #Refractory period

paper · doi:10.1016/s0006-3495(61)86902-6

published in Biophysical Journal 1(6), 445-466 (Elsevier BV)

openalex publication_date 1961/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Cited by

• Dynamical principles in neuroscience
• SPHinXsys: an open-source multi-physics and multi-resolution library based on smoothed particle hydrodynamics
• Deconstructing sex: Strategies for undoing binary thinking in neuroendocrinology and behavior
• Physics-informed learning of governing equations from scarce data
• Discovering network dynamics with neural symbolic regression
• Spatially-Enhanced Temporal Fusion Transformer: Interpretable Multi-Output Prediction for Parametric Dynamical Systems with Time-Varying Inputs
• Electrical instability in cardiac muscle: Phase singularities and rotors
• A Josephson junction-coupled neuron with double capacitive membranes
• Multipeak solutions for an elliptic system of Fitzhugh-Nagumo type
• Standing waves in the FitzHugh-Nagumo system and a problem in combinatorial geometry
• Pattern formation outside of equilibrium
• A Continuum Neuronal Model for the Instigation and Propagation of Cortical Spreading Depression
• Distribution Control of Stochastic Oscillators via Periodic Feedforward and Population-Level Feedback
• Graph Coloring Approach to Solving Sudoku with Oscillatory Neural Networks
• Positive-Allocation Companion Predictors for Nonlinear Dynamics and Their Finite-Difference Diagnostics
• A Projected Drift-Randomized Milstein Method for SDEs with Non-differentiable and Super-linear Drift Coefficients
• An agglomeration-based multigrid solver for the discontinuous Galerkin discretization of cardiac electrophysiology
• Turing patterns on non-fluctuating surfaces under mechanical stresses
• Supersymmetric pairing of Lambert W-kink nerve impulses
• Are Statistical Methods Obsolete in the Era of Deep Learning? A Study of ODE Inverse Problems
• Dynamical Effects of Electromagnetic Flux on Chialvo Neuron Map: Nodal and Network Behaviors
• Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution
• Stochastic reaction-diffusion systems with Hölder continuous multiplicative noise
• Semidefinite Approximations of Reachable Sets for Discrete-time Polynomial Systems
• Enabling Local Neural Operators to perform Equation-Free System-Level Analysis
• The NeuroML ecosystem for standardized multi-scale modeling in neuroscience
• Neuromorphic Processing: A Unifying Tutorial
• Diverse dynamical responses of a memristive bi-membrane photosensitive neuron model
• Bifurcation Analysis of the Driven FitzHugh-Nagumo Oscillator: Prediction and Experiment
• Deep Learning in Neural Networks: An Overview
• On the accuracy and computational cost of spiking neuron implementation
• Heterogeneous noise-induced extreme events and synchronization in a globally coupled network of FitzHugh-Nagumo oscillators
• Joint Bayesian Inference of Parameter and Discretization Error Uncertainties in ODE Models
• Dynamics of Periodically-kicked Oscillators
• Fast approximate Bayesian inference for stable differential equation models
• Beat Frequency Induced Transitions in Synchronization Dynamics
• On the Contraction of Excitable Systems
• Learning stochasticity: a nonparametric framework for intrinsic noise estimation
• Symmetry-based selection rules for higher-order interactions in coupled oscillators
• Final state sensitivity and fractal basin boundaries from coupled Chialvo neurons
• Reliability entails input-selective contraction and regulation in excitable networks
• On the contraction of excitable systems
• Milking a spherical cow: Toy models in neuroscience
• La fente et la représentation successorale
• Global Attractors for Hindmarsh-Rose Equations in Neurodynamics
• Dynamics, synchronization and traveling wave patterns of flux coupled network of Chay neurons
• Can Single Neurons Solve MNIST? The Computational Power of Biological Dendritic Trees
• Improved energy-adaptive coupling for synchronization of neurons with nonlinear and memristive membranes
• Fitting of dynamic recurrent neural network models to sensory stimulus-response data
• Modeling Biological Multifunctionality with Echo State Networks
• Control functionals for Monte Carlo integration
• Self-induced stochastic resonance: A physics-informed machine learning approach
• Koopman Mode Decomposition of Thermodynamic Dissipation in Nonlinear Langevin Dynamics
• Reduction of conductance-based neuron models
• Optimal interaction functions realizing higher-order Kuramoto dynamics with arbitrary limit-cycle oscillators
• An Upper Bound on the Number of Discrete States Possible for the Human\n Brain
• Models of cardiac tissue electrophysiology: Progress, challenges and open questions
• Reducing local minima in fitness landscapes of parameter estimation by using piecewise evaluation and state estimation
• Asymptotics of conduction velocity restitution in models of electrical excitation in the heart
• Machine Learning Classification Informed by a Functional Biophysical System
• Zero-Shot Forecasting of Network Dynamics through Weight Flow Matching
• Stability of Traveling Fronts of the FitzHugh-Nagumo Equations on Cylindrical Surfaces
• Optimal coupling functions for fast and global synchronization of weakly coupled limit-cycle oscillators
• Dynamic threshold curves and response precision in forced excitable systems
• Predicting individual learning trajectories in zebrafish via the free-energy principle
• Sharp Interface Dynamics in a Minimal Non-Reciprocal Cahn-Hilliard System
• Quantization of nonlinear non-Hamiltonian systems
• Self-tuned criticality: Controlling a neuron near its bifurcation point via temporal correlations
• Synaptic pruning facilitates online Bayesian model selection
• Eponyms in Science: how long can they get?
• A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology
• Observation of synchronization between two quantum van der Pol oscillators in trapped ions
• Slowing and Stopping of Chemical Waves in a Narrowing Canal
• Quantifying Mental States in Work Environment: Mathematical Perspectives
• Interpretable neural network system identification method for two families of second-order systems based on characteristic curves
• Turing patterns on adaptive networks
• Data-Driven Discovery of Emergent Dynamics in Reaction-Diffusion Systems from Sparse and Noisy Observations
• Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation
• Computational modelling of Parkinson's disease: A multiscale approach with deep brain stimulation and stochastic noise
• Reinforced Data-Driven Estimation for Spectral Properties of Koopman Semigroup in Stochastic Dynamical Systems
• Manifold lifting: scaling MCMC to the vanishing noise regime
• A Brain-Inspired Gating Mechanism Unlocks Robust Computation in Spiking Neural Networks
• Where To Go and How To Go: a Theoretical Study of Different Leader Roles in Networked Systems
• Transient localized wave patterns and their application to migraine
• Effect Of Site Selective Ion Channel Blocking on Action Potential
• MAGI-X: Manifold-Constrained Gaussian Process Inference for Unknown System Dynamics
• A SIMPLE CHAOTIC NEURON MODEL: STOCHASTIC BEHAVIOR OF NEURAL NETWORKS
• Steady states of FitzHugh-Nagumo-type systems with sign-changing coefficients
• Strong Lyapunov functions for rough systems
• Noise-induced chimera states in a neural network
• Hyperparameter Optimization in the Estimation of PDE and Delay-PDE models from data
• Output Feedback Stabilization of Semilinear Parabolic PDEs using Backstepping
• An organizing center in a planar model of neuronal excitability
• Regular spiking in asymmetrically delay-coupled FitzHugh-Nagumo systems
• Harmonic fractal transformation for modeling complex neuronal effects: from bursting and noise shaping to waveform sensitivity and noise-induced subthreshold spiking
• Spectral Methods for Partial Differential Equations in Irregular Domains: The Spectral Smoothed Boundary Method
• Emergent Properties in a V1-Inspired Network of Hodgkin-Huxley Neurons
• Control of sub-excitable waves in neural networks by nonlocal coupling
• Ubiquity of Uncertainty in Neuron Systems
• Exponential Attractor for Hindmarsh-Rose Equations in Neurodynamics
• Time-optimal synchronisation to self-sustained oscillations under bounded control
• Efficient control of transient wave forms to prevent spreading depolarizations
• Synchronization and temporal nonreciprocity of optical microresonators via spontaneous symmetry breaking
• Smale-Williams Solenoids in a System of Coupled Bonhoeffer-van der Pol Oscillators
• Feedback Synchronization of FHN Cellular Neural Networks
• Clique Topology Reveals Intrinsic Geometric Structure in Neural Correlations: An Overview
• Time series analysis of coupled slow-fast neuron models: From Hurst exponent to Granger causality
• Genetic Algorithmic Parameter Optimisation of a Recurrent Spiking Neural Network Model
• Stochastic Adversarial Koopman Model for Dynamical Systems
• Spatial coherence resonance in excitable media
• On-line detection of qualitative dynamical changes in nonlinear systems: the resting-oscillation case
• Variational Bayes method for ordinary differential equation models
• Stability and Bifurcation Analysis of Coupled Fitzhugh-Nagumo Oscillators
• On Optimality in Auditory Information Processing
• Regularity of pullback attractors for non-autonomous stochastic FitzHugh-Nagumo systems with additive noises on unbounded domains
• Learning from Imperfect Data: Robust Inference of Dynamic Systems using Simulation-based Generative Model
• From biophysical to integrate-and-fire modelling
• Automatic Dynamic Parallelotope Bundles for Reachability Analysis of Nonlinear Systems
• Excitable mechanics embodied in a walking cilium
• Fast Wavelet-Based Visual Classification
• A hybrid model for large neural network description
• Pattern dynamics analysis of a time-space discrete FitzHugh-Nagumo (FHN) model based on coupled map lattices
• A preconditioner based on sine transform for two-dimensional Riesz space factional diffusion equations in convex domains
• Fast and accurate high-order method for high dimensional space-fractional reaction-diffusion equation with general boundary conditions
• Universal mechanism of low-frequency brain rhythm formation through nonlinear coupling of high-frequency spiking-like activity
• Robust Instability Radius for Multi-agent Dynamical Systems with Cyclic Structure
• Manifold-Constrained Gaussian Processes for Inference of Mixed-effects Ordinary Differential Equations with Application to Pharmacokinetics
• Oscillations and Random Perturbations of a FitzHugh-Nagumo System
• Dynamic link switching induces stable synchronized states in sparse networks
• Shaping dynamical neural computations using spatiotemporal constraints
• Synchronization of Dirac-Bianconi driven oscillators
• Dynamics in a Discrete Nagumo Equation: Spatial Topological Chaos
• rd-spiral: An open-source Python library for learning 2D reaction-diffusion dynamics through pseudo-spectral method
• Random Attractor for Stochastic Hindmarsh-Rose Equations with Multiplicative Noise
• Pattern Formation in Robotic Mechanical Metamaterial
• Simulation of parametrized cardiac electrophysiology in three dimensions using physics-informed neural networks
• A Numerical Verification Method for a System of FitzHugh-Nagumo Type
• Dynamics and Synchronization of Boundary Coupled FitzHugh-Nagumo Neural Networks
• Stable Synchronous Propagation in Feedforward Networks for Biped Locomotion
• Synchronous Propagation of Periodic Signals in Feedforward Networks of Standard Model Neurons
• Equivalence of stationary dynamical solutions in a directed chain and a Delay Differential Equation of neuroscientific relevance
• Hysteresis bifurcation and application to delayed Fitzhugh-Nagumo neural systems
• Periodicity induced by noise and interaction in the kinetic mean-field FitzHugh-Nagumo model
• Invariant algebraic surfaces of the FitzHugh-Nagumo system
• Mechanistic Model to Replace Hodgkin-Huxley Equations
• Constructive approach to limiting periodic orbits with exponential and power law dynamics
• On the bifurcation curve for an elliptic system of FitzHugh–Nagumo type
• A regularization method for the parameter estimation problem in ordinary differential equations via discrete optimal control theory
• Pullback Attractors for the Non-autonomous FitzHugh-Nagumo System on Unbounded Domains
• Parameter Estimation with Dense and Convolutional Neural Networks Applied to the FitzHugh-Nagumo ODE
• Self-organization to multicriticality
• Signal anticipation and delay in excitable media: group delay of the FitzHugh-Nagumo model
• 2D Path Solutions from a Single Layer Excitable CNN Model
• Functional modeling of neural–glial interaction
• Modeling the modulation of neuronal bursting: a singularity theory approach
• A condensing approach to multiple shooting neural ordinary differential equation
• Fast automated adjoints for spectral PDE solvers
• A model of neuronal bursting using three coupled first order differential equations
• Information theoretic analysis of computational models as a tool to understand the neural basis of behaviors
• Dynamic Parameter Estimation of Brain Mechanisms
• Modeling cardiac pacemakers with relaxation oscillators
• Bridging Physics-based and Data-driven modeling for Learning Dynamical Systems
• On modelling of physical effects accompanying the propagation of action potentials in nerve fibres
• The constructive role of diversity on the global response of coupled neuron systems
• A stochastic phase model with reflective boundary and induced beating for the cardiac muscle cells
• A Generalized Strong-Inversion CMOS Circuitry for Neuromorphic Applications
• Geometric analysis on the unidirectionality of the pulmonary veins for atrial reentry
• Learning High-dimensional Ionic Model Dynamics Using Fourier Neural Operators
• Saddle-node canard cycles in planar piecewise linear differential systems
• LQResNet: A Deep Neural Network Architecture for Learning Dynamic Processes
• Precision medicine in human heart modeling
• Dynamic homeostasis in relaxation and bursting oscillations
• Slow manifold and parameter estimation for a nonlocal fast-slow stochastic evolutionary system
• Hypoelliptic diffusions: discretization, filtering and inference from complete and partial observations
• Learning cardiac activation and repolarization times with operator learning
• Optimal control of the bidomain system (IV): Corrected proofs of the stability and regularity theorems
• Propagation and Its Failure in Coupled Systems of Discrete Excitable Cells
• Bifurcation and Stability Analysis of Bistable Neuromodules
• Dissipative solitons
• Spiking input-output relation for general biophysical neuron models
• Visual Grouping by Neural Oscillators
• A robust method to identify chimera states
• Spatial coherence resonance on diffusive and small-world networks of Hodgkin–Huxley neurons
• A normal form for excitable media
• Links, Knots, and Knotted Labyrinths in Bistable Systems
• Dissecting the Phase Response of a Model Bursting Neuron
• Predict-prevent control method for perturbed excitable systems
• Asymptotic analysis and analytical solutions of a model of cardiac excitation
• Dynamical Mechanism of Anticipating Synchronization in Excitable Systems
• Waves in Excitable Neural Fields
• Triggering synchronized oscillations through arbitrarily weak diversity in close-to-threshold excitable media
• Machine Discovery of Partial Differential Equations from Spatiotemporal Data
• Dynamical mean-field theory of noisy spiking neuron ensembles: Application to the Hodgkin-Huxley model
• Delayed feedback control of synchronization in weakly coupled oscillator networks
• Solutions with interior and boundary peaks for the Neumann problem of an elliptic system of FitzHugh-Nagumo type
• Chaotic Oscillator Networks for Classification Tasks
• Implementation of PDE models of cardiac dynamics on GPUs using OpenCL
• Structural stability and renormalization group for propagating fronts
• Intrusion prevention systems: How do they prevent intrusion?
• Wave trains, self-oscillations and synchronization in discrete media
• Dynamical mean-field theory of spiking neuron ensembles: Response to a single spike with independent noises
• Waiting on the Fringe: cell autonomy and signaling delays in segmentation clocks
• What are you synching about? Emerging complexity of Notch signaling in the segmentation clock
• Causality and Scientific Inquiry: Lessons from Space Physics and Medical Sciences
• Neuronal electricality founded in murburn-thermodynamic principles: 2. Comparisons, evidenced explanations, and predictions
• Neuronal electricality founded in murburn-thermodynamic principles: 1. Background and basic theoretical formulation
• Inferring coupling strength and natural frequency distribution in coupled Stuart-Landau oscillators using linear response
• Axonal excitability revisited
• Playing the Devil's advocate: is the Hodgkin–Huxley model useful?
• Voltage oscillations in the barnacle giant muscle fiber
• An Active Pulse Transmission Line Simulating Nerve Axon
• Local stability of the nerve impulse
• Varieties of spiral wave behavior: An experimentalist’s approach to the theory of excitable media
• Multi-Tongue Frequency Fractal Dynamics in Hodgkin-Huxley Neurons Induced by Temporal Interference Stimulation
• Superexcitability in Chiral Oscillators with Spatial Degrees of Freedom
• Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models
• Analysis of inverse stochastic resonance: Effects of neural excitability and timescale separation
• Influence of external forcings on abrupt millennial-scale climate changes: a statistical modelling study
• Quantifying the irregularity of a time series
• Noise-Induced Coherence in Neural Networks
• Bifurcation analysis of a normal form for excitable media: Are stable dynamical alternans on a ring possible?
• Stable concurrent synchronization in dynamic system networks
• Modal synchronization of coupled bistable van der Pol oscillators
• Solitary shock waves and adiabatic phase transition in lipid interfaces and nerves
• Observability and Controllability of Nonlinear Networks: The Role of Symmetry
• Oscillation patterns in tori of modified FHN neurons
• An Analytically Solvable Asymptotic Model of Atrial Excitability
• The complex parameter space of a two-mode oscillator model
• Asymptotic Phase and Amplitude for Classical and Semiclassical Stochastic Oscillators via Koopman Operator Theory
• Scroll waves in isotropic excitable media: Linear instabilities, bifurcations, and restabilized states
• Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh–Nagumo model
• Reduced order optimal control of the convective FitzHugh–Nagumo equations
• Global Dynamics of Nonautonomous Hindmarsh-Rose Equations
• Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions
• Rings on strings in excitable media
• Neuromodulation of Neuromorphic Circuits
• Jarzynski equality in van der Pol and Rayleigh oscillators
• Spectra and waiting-time densities in firing resonant and nonresonant neurons
• Applications of optimal nonlinear control to a whole-brain network of FitzHugh-Nagumo oscillators
• Penetration of Action Potentials During Collision in the Median and Lateral Giant Axons of Invertebrates
• Lyapunov functions for a non-linear model of the X-ray bursting of the microquasar GRS 1915+105
• Interplay of the mechanisms of synchronization by common noise and global coupling for a general class of limit-cycle oscillators
• Cellular function given parametric variation in the Hodgkin and Huxley model of excitability
• Optimal Rate of Direct Estimators in Systems of Ordinary Differential Equations Linear in Functions of the Parameters
• Optimal Self-Induced Stochastic Resonance in Multiplex Neural Networks: Electrical vs. Chemical Synapses
• Experimental observation of a transition from amplitude to oscillation death in coupled oscillators
• EDEN: A High-Performance, General-Purpose, NeuroML-Based Neural Simulator
• Observability and synchronization of neuron models
• SYNCHRONIZATION OF COUPLED NEURAL OSCILLATORS WITH HETEROGENEOUS DELAYS
• Three-dimensional nonlinear dynamics of prestressed active filaments: Flapping, swirling, and flipping
• Stability properties of a two-dimensional system involving one Caputo derivative and applications to the investigation of a fractional-order Morris–Lecar neuronal model
• Control of spiral-wave dynamics using feedback signals from line detectors
• Network resilience of FitzHugh-Nagumo neurons in the presence of nonequilibrium dynamics
• Synchronization-desynchronization transitions in complex networks: An interplay of distributed time delay and inhibitory nodes
• Modelling the heart as a communication system
• Canards, folded nodes and mixed-mode oscillations in piecewise-linear slow-fast systems
• Causal Scale of Rotors in a Cardiac System
• Period–amplitude co‐variation in biomolecular oscillators
• The effect of distributed time-delays on the synchronization of neuronal networks
• Generative System Dynamics in Recurrent Neural Networks
• Role of modularity in self-organization dynamics in biological networks
• Nucleation of reaction-diffusion waves on curved surfaces
• From empirical brain networks towards modeling music perception -- a perspective
• Variable Metric Splitting Methods for Neuromorphic Circuits Simulation
• PINNverse: Accurate parameter estimation in differential equations from noisy data with constrained physics-informed neural networks
• Action potential [wikipedia]
• Nonlinear interaction between shunting and adaptation controls a switch between integration and coincidence detection in pyramidal neurons. [europepmc]
• Multi-scale computational modelling in biology and physiology. [europepmc]
• Robust, tunable biological oscillations from interlinked positive and negative feedback loops. [europepmc]
• From mitochondrial ion channels to arrhythmias in the heart: computational techniques to bridge the spatio-temporal scales. [europepmc]
• Cortical network dynamics with time delays reveals functional connectivity in the resting brain. [europepmc]
• The ups and downs of p53: understanding protein dynamics in single cells. [europepmc]
• Dendritic vulnerability in neurodegenerative disease: insights from analyses of cortical pyramidal neurons in transgenic mouse models. [europepmc]
• Biased excitable networks: how cells direct motion in response to gradients. [europepmc]
• Dynamic primitives of motor behavior. [europepmc]
• The Virtual Brain: a simulator of primate brain network dynamics. [europepmc]
• An excitable signal integrator couples to an idling cytoskeletal oscillator to drive cell migration. [europepmc]
• Nonlinear and Stochastic Dynamics in the Heart. [europepmc]
• The Living Heart Project: A robust and integrative simulator for human heart function. [europepmc]
• Three-dimensional cardiac computational modelling: methods, features and applications. [europepmc]
• Robust and efficient parameter estimation in dynamic models of biological systems. [europepmc]
• Wilson-Cowan Equations for Neocortical Dynamics. [europepmc]
• Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience. [europepmc]
• Multiphysics and multiscale modelling, data-model fusion and integration of organ physiology in the clinic: ventricular cardiac mechanics. [europepmc]
• Excitable Dynamics and Yap-Dependent Mechanical Cues Drive the Segmentation Clock. [europepmc]
• An excitable Rho GTPase signaling network generates dynamic subcellular contraction patterns. [europepmc]
• The modulation of neural gain facilitates a transition between functional segregation and integration in the brain. [europepmc]
• Biological plausibility and stochasticity in scalable VO 2 active memristor neurons. [europepmc]
• Electrically induced bacterial membrane-potential dynamics correspond to cellular proliferation capacity. [europepmc]
• Brainstem Modulation of Large-Scale Intrinsic Cortical Activity Correlations. [europepmc]
• Precision medicine in human heart modeling : Perspectives, challenges, and opportunities. [europepmc]