FitzHugh–Nagumo Waves

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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

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

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