Spiking at the edge: Excitability at interfaces in reaction-diffusion systems.

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.

Dynamic clamp constructed phase diagram for the Hodgkin and Huxley model of excitability.

Excitability-a threshold-governed transient in transmembrane voltage-is a fundamental physiological process that controls the function of the heart, endocrine, muscles, and neuronal tissues. The 1950s Hodgkin and Huxley explicit formulation provides a mathematical framework for understanding excitability, as the consequence of the properties of voltage-gated sodium and potassium channels. The Hodgkin-Huxley model is more sensitive to parametric variations of protein densities and kinetics than biological systems whose excitability is apparently more robust. It is generally assumed that the model's sensitivity reflects missing functional relations between its parameters or other components present in biological systems. Here we experimentally assembled excitable membranes using the dynamic clamp and voltage-gated potassium ionic channels (Kv1.3) expressed in Xenopus oocytes. We take advantage of a theoretically derived phase diagram, where the phenomenon of excitability is reduced to two dimensions defined as combinations of the Hodgkin-Huxley model parameters, to examine functional relations in the parameter space. Moreover, we demonstrate activity dependence and hysteretic dynamics over the phase diagram due to the impacts of complex slow inactivation kinetics. The results suggest that maintenance of excitability amid parametric variation is a low-dimensional, physiologically tenable control process. In the context of model construction, the results point to a potentially significant gap between high-dimensional models that capture the full measure of complexity displayed by ion channel function and the lower dimensionality that captures physiological function.

Electric-Induced Reversal of Morphogenesis in Hydra.

Morphogenesis involves the dynamic interplay of biochemical, mechanical, and electrical processes. Here, we ask to what extent can the course of morphogenesis be modulated and controlled by an external electric field? We show that at a critical amplitude, an external electric field can halt morphogenesis in Hydra regeneration. Moreover, above this critical amplitude, the electric field can lead to reversal dynamics: a fully developed Hydra folds back into its incipient spheroid morphology. The potential to renew morphogenesis is reexposed when the field is reduced back to amplitudes below criticality. These dynamics are accompanied by modulations of the Wnt3 activity, a central component of the head organizer in Hydra. The controlled backward-forward cycle of morphogenesis can be repeated several times, showing that the reversal trajectory maintains the integrity of the tissue and its regeneration capability. Each cycle of morphogenesis leads to a newly emerged body plan in the redeveloped folded tissue, which is not necessarily similar to the one before the reversal process. Reversal of morphogenesis is shown to be triggered by enhanced electrical excitations in the Hydra tissue, leading to intensified calcium activity. Folding back of the body-plan morphology together with the decay of a central biosignaling system, indicate that electrical processes are tightly integrated with biochemical and mechanical-structural processes in morphogenesis and play an instructive role to a level that can direct developmental trajectories. Reversal of morphogenesis by external fields calls for extending its framework beyond program-like, forward-driven, hierarchical processes based on reaction diffusion and positional information.

Cellular function given parametric variation in the Hodgkin and Huxley model of excitability.

How is reliable physiological function maintained in cells despite considerable variability in the values of key parameters of multiple interacting processes that govern that function? Here, we use the classic Hodgkin-Huxley formulation of the squid giant axon action potential to propose a possible approach to this problem. Although the full Hodgkin-Huxley model is very sensitive to fluctuations that independently occur in its many parameters, the outcome is in fact determined by simple combinations of these parameters along two physiological dimensions: structural and kinetic (denoted S and K, respectively). Structural parameters describe the properties of the cell, including its capacitance and the densities of its ion channels. Kinetic parameters are those that describe the opening and closing of the voltage-dependent conductances. The impacts of parametric fluctuations on the dynamics of the system-seemingly complex in the high-dimensional representation of the Hodgkin-Huxley model-are tractable when examined within the S-K plane. We demonstrate that slow inactivation, a ubiquitous activity-dependent feature of ionic channels, is a powerful local homeostatic control mechanism that stabilizes excitability amid changes in structural and kinetic parameters.