Similarities between action potentials and acoustic pulses in a van der Waals fluid

Abstract

An action potential is typically described as a purely electrical change that propagates along the membrane of excitable cells. However, recent experiments have demonstrated that non-linear acoustic pulses that propagate along lipid interfaces and traverse the melting transition, share many similar properties with action potentials. Despite the striking experimental similarities, a comprehensive theoretical study of acoustic pulses in lipid systems is still lacking. Here we demonstrate that an idealized description of an interface near phase transition captures many properties of acoustic pulses in lipid monolayers, as well as action potentials in living cells. The possibility that action potentials may better be described as acoustic pulses in soft interfaces near phase transition is illustrated by the following similar properties: correspondence of time and velocity scales, qualitative pulse shape, sigmoidal response to stimulation amplitude (an 'all-or-none' behavior), appearance in multiple observables (particularly, an adiabatic change of temperature), excitation by many types of stimulations, as well as annihilation upon collision. An implication of this work is that crucial functional information of the cell may be overlooked by focusing only on electrical measurements.

Figure 1

Schematic phase diagram in the w–p plane of the van der Waals model. Four different isotherms, along with the Spinodal (S) and coexistence (C) curves are plotted. The critical point is denoted by a filled circle. LE: liquid-expanded phase, MLE: metastable liquid-expanded phase, COE: coexistence phase, MLC: metastable liquid-condensed phase, and LC: liquid-condensed phase.

Figure 2

(a) Density pulse as a function of time, as measured at distance x/L = 1 from the excitation point, with an initial density of ${\tilde{\rho }}_0$ = 0.67 (solid line) and 0.33 (dotted-dashed line). (b) A projection of phase space into the w–p plane (w = ρ⁻¹). Several isotherms are plotted for reference (shades of grey lines) as well as the coexistence and spinodal curves (dark blue and blue lines, respectively). The trajectory of the two pulses shown in (a) is plotted in green and purple respectively. (c) Density field of the entire fluid as a function of space (x-axis) and time (y-axis) with initial density ${\tilde{\rho }}_0$ = 0.67. Dashed yellow line marks the solid line solution depicted in (a). Parameters of the model were $({\tilde{c}}_v,\tilde{k},\tilde{C})$ = (600, 100, 1), additional initial conditions were $({\tilde{v}}_0,{\tilde{\theta }}_0)$ = (0, 0.93), and excitation parameters were $({\tilde{x}}_0,{\tilde{t}}_0,{\tilde{p}}_{exc},\lambda )$ = (0, 0.1, 300, 0.088). Numerical calculation was conducted with 4096 grid points, x-domain [−3π/2, 3π/2] and dt = 5 ⋅ 10⁻⁴.

Figure 3

Excitation by a local injection of (a) velocity (middle of Eq. (1)), (b) pressure (upper Eq. (4)), (c) temperature (lower Eq. (4)), and (d) energy (lower Eq. (1)). Excitation parameters were $(\tilde{A},t_0)$ = (300, 0.1), (300, 0.2), (3000, 0.1) and (2 · 10⁴, 0.2), respectively. A is the normalized amplitude of excitation (p_exc/p_c, p_exc/p_c, θ_exc/θ_c and E_exc/U², respectively). Density as a function of time was plotted at x/L = 0.8, 1.0, 0.8 and 1.4, respectively.

Figure 4

(a) Pulse shape at four amplitudes of the excitation, as reflected by the change of density. (b) Non-linear response of the amplitude of the density pulse (ρ_max) to stimulation amplitude (p_exc). (c) Pressure and (d) temperature aspects that co-appear with the density pulse. The first two pressure curves are indistinguishable, as the state did not reach the liquid-condensed phase. Initial density was ${\tilde{\rho }}_0$ = 0.67, and pulse was measured at distance x/L = 1.2 from the excitation point. Other parameters appear in the caption of Fig. 2.

Figure 5

Collision between two pulses as appeared in the (a) density, (b) pressure and (c) temperature fields. x- and y-axis represent space and time, respectively. A local description of the (d) density, (e) pressure and (f) temperature is plotted as a function of time at the collision point (x/L = 0, solid line) and at some distance away from it (x/L = 0.4, dashed-dotted line). The location of these solutions in space is marked in (c) by solid and dashed-dotted line, respectively. Fluid initial density was ${\tilde{\rho }}_0$ = 0.77. Excitation was conducted at ${\tilde{x}}_0$ = ±0.4 π. Numerical calculation was performed with 8192 grid points, and the x-domain was [−0.4π, 0.4π]. Other parameters are similar to those given in caption of Fig. 2.