Breathing pulses in the damped-soliton model for nerves

Unlike the Hodgkin-Huxley picture in which the nerve impulse results from ion exchanges across the cell membrane through ion-gate channels, in the so-called soliton model the impulse is seen as an electromechanical process related to thermodynamical phenomena accompanying the generation of the action potential. In this work, account is taken of the effects of damping on the nerve impulse propagation, within the framework of the soliton model. Applying the reductive perturbation expansion on the resulting KdV-Burgers equation, a damped nonlinear Schrödinger equation is derived and shown to admit breathing-type solitary wave solutions. Under specific constraints, these breathing pulse solitons become self-trapped structures in which the damping is balanced by nonlinearity such that the pulse amplitude remains unchanged even in the presence of damping.

Figure 1

(a) Gain spectrum for different values of viscosity. (b) Three-dimensional representation of the gain. Parameters chosen are $\xi =40$ and $a=0.0002$, $A_0=0.2$, and $t={10}^{-4}$.

Figure 2

Time variations of the amplitude $A_1$ (a), width $l_e$ (b), and carrier speed $u_e$ of the envelope soliton (c) for $a=0.2$ and $k=1$. Parameters are taken from the works of Heimburg and Jackson on unicellular DPPC vesicles [12].

Figure 3

Evolution of the breathing soliton ($\mathrm{\Delta }{\rho }^A$) along the nerve $\left(x\right)$, at different times. Parameters were chosen as $\varepsilon =0.02$, $l_e=0.8$, $a=0.0001$, $k=0.008$, $u_e=100$, $u_c=10$, $\nu =0.05$, $h=2.22$.

Figure 4

Evolution of the breathing soliton ($\mathrm{\Delta }{\rho }^A$) along the nerve $\left(x\right)$, for different values of the perturbation parameter $a$ and for $t=0.1$, $l_e=0.8$, $\varepsilon =0.02$, $k=0.008$, $u_e=100$, $u_c=10$, $\nu =0.05$, $h=2.22$. The profile (a) represents a typical breathing mode soliton while (b) and (c) are typical longitudinal and transverse profiles, respectively.

Figure 5

Evolution of the nerve impulse nerve with changes in viscosity at different time intervals. The nerve impulse is completely damped due to an increase in the viscosity. Parameters were chosen as $\varepsilon =0.02$, $l_e=0.8$, $a=0.0001$, $k=0.008$, $u_e=100$, $u_c=10$, $\nu =0.05$, $h=2.22$.

Figure 6

(a) Amplitude of the soliton envelope at different times $t$. (b) Three-dimensional propagation of the nerve impulse for $\varepsilon =0.02$, $l_e=0.8$, $a=0.0001$, $k=0.008$, $u_e=100$, $u_c=10$, $\nu =0.05$, $h=2.22$. The signal propagates with constant profile without dissipation of energy.