Traveling pulses in type-I excitable media

Letter

Traveling pulses in type-I excitable media

Andreu Arinyo-i-Prats, Pablo Moreno-Spiegelberg, Manuel A. Matias, and Damià Gomila

Phys. Rev. E 104, L052203 – Published 22 November 2021

Abstract

We consider a general model exhibiting type-I excitability mediated by a homoclinic and a saddle node on the invariant circle bifurcations. We show how the distinct properties of type-I with respect to type-II excitability confer unique features to traveling pulses in excitable media. They inherit the characteristic divergence of type-I excitable trajectories at threshold exhibiting analogous scalings in the spatial thickness of the pulses. Our results pave the way to identify basic underlying mechanisms behind type-I excitable pulses based solely on the characteristics of the pulse.

Received 4 January 2021
Accepted 28 October 2021

DOI:https://doi.org/10.1103/PhysRevE.104.L052203

Physics Subject Headings (PhySH)

Bifurcations Chemical waves Ecological pattern formation Emergent biological functions from complex interactions Neuronal dynamics Pattern formation Population dynamics

Coherent structures Reaction diffusion systems Solitons

Physics of Living SystemsNonlinear DynamicsFluid Dynamics

Authors & Affiliations

Andreu Arinyo-i-Prats ^1,2, Pablo Moreno-Spiegelberg ¹, Manuel A. Matias¹, and Damià Gomila ^1,*

¹IFISC (CSIC-UIB), Instituto de Física Interdisciplinar y Sistemas Complejos, E-07122 Palma de Mallorca, Spain
²Institute of Computer Science, Czech Academy of Sciences, 182 07 Praha 8, Czech Republic

^*damia@ifisc.uib-csic.es

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Issue

Vol. 104, Iss. 5 — November 2021

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Images

Figure 1
Examples of traveling pulses in excitable media. (a) Waves in a Belousov-Zhabotinsky chemical reaction. (b) Rings of vegetation growing radially in intertidal salt marshes in Nanhui shoal, Shanghai. (c) Spiral formed by a seagrass traveling pulse in the coast of Mallorca in the Mediterranean Sea, Spain. (d) Geographic tongue caused by fungiform papillae, adapted from Ref. [10]. (e), (f) Two kinds of Rhizocarpon lichen in rocks, Mallorca. (a) and (d) are modeled as type-II excitable media, while (b) and (c) fit within the type-I framework. The excitable type of (e) and (f) has not yet been identified.
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Figure 2
Phase diagram of the system (1) including the bifurcation lines of homogeneous and traveling pulse solutions. The diagram is organized by a SNSL codimension-2 point (red dot) from which a SNIC (blue dot-dashed line) and homoclinic of a stable cycle (red line) bifurcation lines emerge for the temporal dynamics. The diagram is completed by the heteroclinic bifurcation line of traveling pulses (dashed black line). For the temporal system, excitable excursions occur in the area above the SNIC and homoclinic lines. For the spatiotemporal system, excitable traveling pulses can form in this green shaded area. The insets show a schematic representation of the phase space for the temporal system in each corresponding area. $P_{1}, P_{2}$ , and $P_{3}$ indicate the fixed, saddle, and unstable fixed points, respectively, for each region, and the limit cycle and its remnants are shown as a red line.
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Figure 3
Comparison of the temporal dynamics of an excitable excursion and the spatial profile of a 1D traveling pulse sustained by this excitable dynamics. (a) Temporal (spatially homogeneous) excitable trajectory of $u$ (red solid line) and $v$ (blue dashed line) starting from an initial condition just above the saddle point $P_{2}$ . (b) Stable 1D traveling pulse as a function of the spatial coordinate $ξ$ in the moving reference frame. (c),(d) The temporal excitable excursion and the traveling pulse in the $(u, v)$ phase space, respectively. $P_{1}, P_{2}$ , and $P_{3}$ indicate stable, saddle, and unstable fixed points. Here, $μ_{1} = 0.3$ and $μ_{2} = 1.0$ .
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Figure 4
(a) Divergence of the duration of the excitable excursion in the temporal system approaching the homoclinic bifurcation, and (b) divergence of the plateau in the pulses approaching the heteroclinic bifurcation. Here, $μ_{2} = 0.4$ and $μ_{1} = μ_{1 c} - Δ μ_{1}$ , where the homoclinic bifurcation occurs at $μ_{1 c} = 0.075 603 955 87$ for the temporal system, and the heteroclinic bifurcation occurs at $μ_{1 c} = 0.081 078 760 02$ for the spatial system, and $Δ μ_{1} = 10^{- 3}$ (red solid line) and $Δ μ_{1} = 10^{- 12}$ (black dashed line). In all panels, $P_{1}, P_{2}$ signal the stable and saddle fixed points, respectively. (c), (d) A zoom of the most relevant region in the phase space $(u, v)$ for the temporal and spatial dynamics.
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Figure 5
Same as in Fig. 4 for the SNIC bifurcation. $u - u_{P_{1}}$ is plotted in (a) and (b), where $u_{P_{1}}$ is the value of $u$ at the stable fixed point $P_{1}$ . Here, $μ_{2} = 2.0$ and $μ_{1 c} = 1.0887$ with $Δ μ_{1} = - 10^{- 4}$ and $Δ μ_{1} = - 10^{- 1}$ for the black dashed and red solid curves, respectively. (c), (d) The phase space $(u, v)$ for the temporal and spatial dynamics, respectively.
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Figure 6
(a) Scaling of the pulse width, $γ$ , approaching the heteroclinic bifurcation. $γ$ is defined as the distance from where the pulse separates $10^{- 2}$ from the stable homogeneous solution until it comes back to the same distance (black points). The expected scaling is $γ = \frac{1}{λ_{1}} ln (| μ_{1} - μ_{1 c} |)$ (red line in the plot), where $λ_{1}$ is the closest eigenvalue to zero of the saddle point in the spatial dynamics. (b) Scaling of the rate $λ$ at which the tail of the traveling pulse exponentially approaches the saddle, as a function of the parameter distance to the SNIC bifurcation. The expected scaling is a power law with exponent $1 / 2$ : $λ \propto \sqrt{| μ_{1} - μ_{1 c} |}$ (red line).
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