Fitzhugh–Nagumo Waves

L. Edelstein-Keshet (Author)

Y. Xiao (Contributor)

L. Edelstein-Keshet: Mathematical Models in Cell Biology

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Waves produced by an excitable Fitzhugh–Nagumo PDE.

Introduction

We will here take a closer look at waves produced by an excitable Fitzhugh–Nagumo PDE

in a periodic 1D domain,
in a periodic 2D domain.

Description

Consider an excitable system such as the Fitzhugh-Nagumo model that can produce traveling waves and pulses:

$$\begin{align} \frac{\partial u}{\partial t} &= D\Delta u + \gamma u(1-u)(u-a) - v + I_0(x,y) \\ \frac{\partial v}{\partial t} &= \epsilon (u-bv) \\ \end{align}$$

Results

Simulations of the model in 1D with periodic BoundaryConditions, showing a traveling pulse (top) and in 2D, showing a set of spiral waves (bottom figures).

The Fitzhugh–Nagumo model in a periodic 1D domain produces a tavelling pulse. Left: kymograph of $u$ over time. Right: the shape of $u$ and $v$ for the pulse. Produced with [`FNwaves_main.xml`](#model). — The Fitzhugh–Nagumo model in a periodic 1D domain produces a tavelling pulse. Left: kymograph of $u$ over time. Right: the shape of $u$ and $v$ for the pulse. Produced with `FNwaves_main.xml`.

Simulation video of FNwaves_main.xml ( | )

Simulation video of FN_PDE_in_2D.xml ( | ) with Fitzhugh–Nagumo equations solved in 2D. There is current injected at the ‘origin’ to kickstart the waves. The initial profiles of $u$ (‘voltage’ or excitable variable) and $v$ (‘refractory’ variable) are asymmetric to provoke spirals.

The Fitzhugh–Nagumo PDEs can sustain spiral waves when stimulated in a 2D domain. Left: $u$. Right: $v$ at $t = 100$. Produced with [`FNspiralwaves2D.xml`](#downloads). — The Fitzhugh–Nagumo PDEs can sustain spiral waves when stimulated in a 2D domain. Left: $u$. Right: $v$ at $t = 100$. Produced with `FNspiralwaves2D.xml`.

Simulation video of model FNspiralwaves2D.xml ( | )

Model

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Download: FNwaves_main.xml

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<MorpheusModel version="4">
    <Description>
        <Title>FNwaves1D</Title>
        <Details>Full title:		Fitzhugh–Nagumo Waves
Authors:		L. Edelstein-Keshet
Contributors:	Y. Xiao
Date:		23.06.2022
Software:		Morpheus (open-source). Download from https://morpheus.gitlab.io
Model ID:		https://identifiers.org/morpheus/M2014
File type:		Main model
Reference:		L. Edelstein-Keshet: Mathematical Models in Cell Biology
Comment:		Waves produced by an excitable Fitzhugh–Nagumo PDE in a periodic 1D domain. u is the "voltage", or excitable variable, v is the "refractory" variable. The initial conditions are a peak in u and a shallow ramp in v. The parameter values are as suggested by James Keener.</Details>
    </Description>
    <Global>
        <Constant symbol="L" value="dx*size.x"/>
        <Function symbol="x">
            <Expression>dx*space.x</Expression>
        </Function>
        <Field symbol="u" name="u" value="exp(-(x-0.2)^2/0.1)">
            <Diffusion rate="0.01"/>
        </Field>
        <Field symbol="v" name="v" value="0.1*x/L"/>
        <System solver="Dormand-Prince [adaptive, O(5)]">
            <DiffEqn symbol-ref="u">
                <Expression>H*(-u*(u-1)*(u-a)-v)</Expression>
            </DiffEqn>
            <DiffEqn symbol-ref="v">
                <Expression>eps*(u-b*v)</Expression>
            </DiffEqn>
            <Constant symbol="eps" value="0.021"/>
            <Constant symbol="b" value="2"/>
            <Constant symbol="a" value="0.1"/>
            <Constant symbol="H" value="20"/>
        </System>
    </Global>
    <Space>
        <Lattice class="linear">
            <Size symbol="size" value="200, 0, 0"/>
            <BoundaryConditions>
                <Condition type="periodic" boundary="x"/>
            </BoundaryConditions>
            <NodeLength symbol="dx" value="0.015"/>
            <Neighborhood>
                <Order>1</Order>
            </Neighborhood>
        </Lattice>
        <SpaceSymbol symbol="space"/>
    </Space>
    <Time>
        <StartTime value="0"/>
        <StopTime value="100"/>
        <TimeSymbol symbol="t" name="time"/>
    </Time>
    <Analysis>
        <Logger time-step="1">
            <Input>
                <Symbol symbol-ref="u"/>
                <Symbol symbol-ref="v"/>
                <Symbol symbol-ref="L"/>
            </Input>
            <Output>
                <TextOutput/>
            </Output>
            <Plots>
                <Plot time-step="5" title="Spatial profiles">
                    <Style decorate="true" line-width="3.0" style="lines"/>
                    <Terminal terminal="png"/>
                    <X-axis>
                        <Symbol symbol-ref="x"/>
                    </X-axis>
                    <Y-axis>
                        <Symbol symbol-ref="u"/>
                        <Symbol symbol-ref="v"/>
                    </Y-axis>
                    <Range>
                        <Time mode="current"/>
                    </Range>
                </Plot>
                <Plot time-step="-1" title="Time-space plot">
                    <Style style="points"/>
                    <Terminal terminal="png"/>
                    <X-axis>
                        <Symbol symbol-ref="x"/>
                    </X-axis>
                    <Y-axis>
                        <Symbol symbol-ref="t"/>
                    </Y-axis>
                    <Color-bar>
                        <Symbol symbol-ref="u"/>
                    </Color-bar>
                </Plot>
            </Plots>
        </Logger>
        <ModelGraph format="dot" reduced="false" include-tags="#untagged"/>
    </Analysis>
</MorpheusModel>

Downloads

Files associated with this model:

1D 2D Actin Actin Waves Asymmetry BoundaryConditions Contributed Examples FitzHugh–Nagumo Fitzhugh–Nagumo Wave Kymograph Partial Differential Equation PDE Periodic Periodic Boundary Conditions Periodic Domain Pulse Spiral Spiral Wave Traveling Pulse Traveling Wave Voltage Wave

Last updated on Tue, 25 Apr 2023