Spiking at the edge: Excitability at interfaces in reaction–diffusion systems
Edited by Jacques Prost, Institut Curie, Paris, France; received May 12, 2023; accepted October 25, 2023 by Editorial Board Member Paul Chaikin
Significance
Spiking is a general phenomenon that is crucial in the firing of neurons, beating of hearts, and spread of diseases. In homogeneous media, spiking arises from a local competition between amplifying and suppressing forces. But most real-world systems are far from homogeneous. Here, we demonstrate that inhomogeneities such as interfaces and boundaries (that spatially segregate these two forces) can promote spiking, even if the system does not spike when these forces are evenly mixed. We mathematically derive a spiking phase diagram in terms of interfacial diffusion and amplification strength. Our findings apply to chemical reactions, predator–prey dynamics, and recent electrophysiology experiments, in which localized action potentials were observed at the interface of distinct, nonspiking bioelectric tissues.
Abstract
Excitable media, ranging from bioelectric tissues and chemical oscillators to forest fires and competing populations, are nonlinear, spatially extended systems capable of spiking. Most investigations of excitable media consider situations where the amplifying and suppressing forces necessary for spiking coexist at every point in space. In this case, spikes arise due to local bistabilities, which require a fine-tuned ratio between local amplification and suppression strengths. But, in nature and engineered systems, these forces can be segregated in space, forming structures like interfaces and boundaries. Here, we show how boundaries can generate and protect spiking when the reacting components can spread out: Even arbitrarily weak diffusion can cause spiking at the edge between two non-excitable media. This edge spiking arises due to a global bistability, which can occur even if amplification and suppression strengths do not allow spiking when mixed. We analytically derive a spiking phase diagram that depends on two parameters: i) the ratio between the system size and the characteristic diffusive length-scale and ii) the ratio between the amplification and suppression strengths. Our analysis explains recent experimental observations of action potentials at the interface between two non-excitable bioelectric tissues. Beyond electrophysiology, we highlight how edge spiking emerges in predator–prey dynamics and in oscillating chemical reactions. Our findings provide a theoretical blueprint for a class of interfacial excitations in reaction–diffusion systems, with potential implications for spatially controlled chemical reactions, nonlinear waveguides and neuromorphic computation, as well as spiking instabilities, such as cardiac arrhythmias, that naturally occur in heterogeneous biological media.
Data, Materials, and Software Availability
All data and code supporting this article are available in Zenodo (DOI: https://www.doi.org/10.5281/zenodo.10426295) (62).
Acknowledgments
We would like to thank William Bialek, Ned Wingreen, Joshua W. Shaevitz, David J. Schwab, James P. Sethna, Itai Cohen, Eric R. Dufresne, Erwin Frey, Fridtjof Brauns, Pankaj Mehta, Mary Silber, Brent Dorion, Massimo Vergassola, Suraj Shankar, Noah P. Mitchell, Danny S. Seara, Michel Fruchart, and Tali Khain for useful discussions. C.S. acknowledges support from the Bloomenthal Fellowship and the NSF Graduate Research Fellowship under Grant No. 1746045. V.V. acknowledges support from the Simons Foundation, the Army Research Office under grants W911NF-22-2-0109 and W911NF-23-1-0212, and the University of Chicago Materials Research Science and Engineering Center, which is funded by the NSF under Award No. DMR-2011854. This research was also supported from the National Science Foundation under grant DMR-2118415 and through the Center for Living Systems (grant no. 2317138). A.E.C. acknowledges support from a Vannevar Bush Faculty Fellowship grant N00014-18-1-2859, and NSF Quantum Sensing for Biophysics and Bioengineering (QuBBE) Quantum Leap Challenge Institute (QLCI) grant OMA-2121044. H.O. acknowledges support from an European Molecular Biology Organization (EMBO) Fellowship ALTF 543-2020.
Author contributions
C.S., H.O., A.E.C., and V.V. designed research; C.S., H.O., A.E.C., and V.V. performed research; C.S. and H.O. analyzed data; and C.S., H.O., A.E.C., and V.V. wrote the paper.
Competing interests
The authors declare no competing interest.
Supporting Information
Appendix 01 (PDF)
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Movie S1.
Experimental observation of spiking at a bioelectric interface: Two tissues of human embryonic kidney (HEK293) cells were genetically modified to express either sodium (NaV1.5) or potassium (Kir2.1) channels. Neither tissue alone is able to spike. However, upon stimulation by a laser, an action potential is observed to propagate along their interface, as revealed by a voltage sensitive red dye. Adapted from Ref. (8).
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Movie S2.
2D nonlinear waveguides built from excitable interfaces: A network of excitable interfaces forms a two-dimensional nonlinear waveguide capable of performing computations. Simulations of the binary half adder in Fig. 5c-d are shown. See SI §5 for details of the numerics.
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Copyright © 2024 the Author(s). Published by PNAS. This article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).
Data, Materials, and Software Availability
All data and code supporting this article are available in Zenodo (DOI: https://www.doi.org/10.5281/zenodo.10426295) (62).
Submission history
Received: May 12, 2023
Accepted: October 25, 2023
Published online: January 12, 2024
Published in issue: January 16, 2024
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Acknowledgments
We would like to thank William Bialek, Ned Wingreen, Joshua W. Shaevitz, David J. Schwab, James P. Sethna, Itai Cohen, Eric R. Dufresne, Erwin Frey, Fridtjof Brauns, Pankaj Mehta, Mary Silber, Brent Dorion, Massimo Vergassola, Suraj Shankar, Noah P. Mitchell, Danny S. Seara, Michel Fruchart, and Tali Khain for useful discussions. C.S. acknowledges support from the Bloomenthal Fellowship and the NSF Graduate Research Fellowship under Grant No. 1746045. V.V. acknowledges support from the Simons Foundation, the Army Research Office under grants W911NF-22-2-0109 and W911NF-23-1-0212, and the University of Chicago Materials Research Science and Engineering Center, which is funded by the NSF under Award No. DMR-2011854. This research was also supported from the National Science Foundation under grant DMR-2118415 and through the Center for Living Systems (grant no. 2317138). A.E.C. acknowledges support from a Vannevar Bush Faculty Fellowship grant N00014-18-1-2859, and NSF Quantum Sensing for Biophysics and Bioengineering (QuBBE) Quantum Leap Challenge Institute (QLCI) grant OMA-2121044. H.O. acknowledges support from an European Molecular Biology Organization (EMBO) Fellowship ALTF 543-2020.
Author contributions
C.S., H.O., A.E.C., and V.V. designed research; C.S., H.O., A.E.C., and V.V. performed research; C.S. and H.O. analyzed data; and C.S., H.O., A.E.C., and V.V. wrote the paper.
Competing interests
The authors declare no competing interest.
Notes
This article is a PNAS Direct Submission. J.P. is a guest editor invited by the Editorial Board.
*
†
For simplicity, in this example, we are using the same hopping rate within the forest as between the forest and desert. This distinction becomes irrelevant in the continuum limit (large and large N) because this subextensive heterogeneity is absorbed into the Dirichlet boundary condition at an edge or into the continuity requirements across an interface.
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Spiking at the edge: Excitability at interfaces in reaction–diffusion systems, Proc. Natl. Acad. Sci. U.S.A.
121 (3) e2307996120,
https://doi.org/10.1073/pnas.2307996120
(2024).
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