Bursting Regimes in a Reaction-Diffusion System with Action Potential-Dependent Equilibrium

Abstract

The equilibrium Nernst potential plays a critical role in neural cell dynamics. A common approximation used in studying electrical dynamics of excitable cells is that the ionic concentrations inside and outside the cell membranes act as charge reservoirs and remain effectively constant during excitation events. Research into brain electrical activity suggests that relaxing this assumption may provide a better understanding of normal and pathophysiological functioning of the brain. In this paper we explore time-dependent ionic concentrations by allowing the ion-specific Nernst potentials to vary with developing transmembrane potential. As a specific implementation, we incorporate the potential-dependent Nernst shift into a one-dimensional Morris-Lecar reaction-diffusion model. Our main findings result from a region in parameter space where self-sustaining oscillations occur without external forcing. Studying the system close to the bifurcation boundary, we explore the vulnerability of the system with respect to external stimulations which disrupt these oscillations and send the system to a stable equilibrium. We also present results for an extended, one-dimensional cable of excitable tissue tuned to this parameter regime and stimulated, giving rise to complex spatiotemporal pattern formation. Potential applications to the emergence of neuronal bursting in similar two-variable systems and to pathophysiological seizure-like activity are discussed.

Dynamics of signal propagation and collision in axons

“…Conversely, it has been suggested that some biological axons may show properties of crossing action potentials [16], and that Hodgkin-Huxley type models may be inadequate for representing some axons. This view has recently been challenged in a study [47] demonstrating that in the particular case of small neurons, modeled with a reduced Hodgkin-Huxley system (Morris-Lecar equations [48,49]), leading-order effects elicit soliton-like behaviors. In specific, by incorporating a variable Nernst potential into the conventional Morris-Lecar reaction-diffusion model, emergent complex spatiotemporal dynamics of axon excitability, including soliton-like behaviors, were demonstrated [47].…”

“…This view has recently been challenged in a study [47] demonstrating that in the particular case of small neurons, modeled with a reduced Hodgkin-Huxley system (Morris-Lecar equations [48,49]), leading-order effects elicit soliton-like behaviors. In specific, by incorporating a variable Nernst potential into the conventional Morris-Lecar reaction-diffusion model, emergent complex spatiotemporal dynamics of axon excitability, including soliton-like behaviors, were demonstrated [47]. Thus, soliton-like regimes may be common in excitable membranes, and indeed there is experimental evidence in the form of action potential reflection at axon branch points [50].…”

“…Furthermore, an accumulation of extracellular potassium and the subsequent change in the potassium Nernst potential may be involved in changing axon properties [61][62][63][64], making the study of modified Hodgkin-Huxley models such as in Ref. [47] necessary for determining axon propagation dynamics. This is particularly true given the presence of various metabotropic receptors in the axonal membrane, rendering axon properties conditional to the neuromodulatory state at hand.…”

Gap junctions set the speed and nucleation rate of stage I retinal waves

“…The cells are coupled by the Ohmic currents through GJs which corresponds to the discretized version of a diffusive coupling (see e.g. [28] for a recent example of complex pattern generation with such a coupling); for recent studies of wave propagation using the alternative spatially extended coupling by an integral kernel, see e.g. [29, 30].…”

Induced mitochondrial membrane potential for modeling solitonic conduction of electrotonic signals

“…In this paper, we extend the above approaches by deriving a cable model that considers the effects of changes to ionic concentration gradients through a conduction process, which leads to changes in equilibrium potentials when ions are in solution and ionic flow is inhomogeneous [ 28 ]. This is the first study in which electrical conduction of polarization-induced capacitive current in a homogenous core-conductor reflects upon ionic concentration gradients without explicitly modeling electrodiffusion of ions (since cable theory ignores the effects of changes in ionic concentrations that lead to changes in Nernst potentials when molecular ions are in bulk solutions).…”