Interaction-based transition from passivity to excitability

In this paper we study the process of transition from passive to excitable behavior due to interaction between nonlinear dynamical systems. We show that under certain conditions a passive unit may demonstrate qualitatively new excitable dynamics. We study the properties of an excitable medium constructed on the basis of the proposed transition. The effects are demonstrated with the realistic Luo-Rudy model. Application to the cardiac dynamics and functioning is discussed. The qualitative analytic and numerical description is also given for the phenomenological FitzHugh-Nagumo system.

Figure 1

Schematic diagram of the Markov process describing dynamics of time and voltage dependent outward current $I_{\mathrm{Shkr}}$ of the Sachse fibroblast model.

Figure 2

The dependency of RT (relaxation time, blue circles, left axis) and RA (response amplitude, green squares, right axis) of the single passive fibroblast coupled with single excitable myocyte on the value of the initial voltage shift $V_p^{\mathrm{ini}}$ (i.e., shift of the initial conditions).

Figure 3

The dependencies of threshold manifolds $V_p^{\mathrm{thresh}}$ of the passive fibroblast (blue circles) and $V_e^{\mathrm{thresh}}$ of the excitable myocyte (green squares) on the diffusive coupling parameter between the two cells $D_{ep}$. Note, that starting from some critical value of $D_{ep}\approx 0.2$ the $V_p^{\mathrm{thresh}}$ starts to exist. This means that the fibroblast is now an effectively excitable system rather than passive.

Figure 4

(a) Schematic representation of the experiment setup (see text for details). (b) Space-time diagram of the chain pulse propagation experiment. Horizontal axis represents time. Vertical axis represents space. The color (brightness) encodes the membrane voltage of cells. There are 160 cells forming the chain: the first 30 and last 30 are excitable myocytes; the 100 middle units are passive fibroblasts. Each fibroblast has an extra excitable myocyte. Interaction with extras is switched on at time moment $t=2500$ ms. Then the passive medium starts to propagate pulses. Note the changes in the pulse speed and width (see text for details).

Figure 5

The dependencies of propagation speed in the chain of passive cells coupled to the excitable extras (blue circles) and vice versa, i.e., in the chain of excitable myocytes coupled with passive extras (green rectangles) on $D_{ep}$.

Figure 6

The parameter space $(D_p,D_{ep})$ and the regions of this space where the passive medium becomes excitable and starts to conduct excitation (upper right).

Figure 7

(a) The dependency of APD on DI and D2; (b) same for $\partial \left(\mathrm{APD}\right)/\partial \left(\mathrm{DI}\right)$.

Figure 8

(a) The dependency of APD on DI and $T_{\mathrm{base}}$; (b) same for $\partial \left(\mathrm{APD}\right)/\partial \left(\mathrm{DI}\right)$.

Figure 9

(a) Lattice with 50% of fibroblasts and $D_{ep}=0$ at three consecutive moments of time—wave front break up with formation of free ends. (b) Same for 55% of fibroblasts—wave breaks up and fails to propagate to the other side. (c) Same as (b) but with $D_{ep}=1$—passive elements became excitable and wave is propagated without breaking up.

Figure 10

Visualization of the small stimulation (below the threshold) in the system (9). No excitation is observed. (a) Time series of $x$ (lower) and $z$ (upper). (b) $(x,z)$ plane projection of the phase space of the system. (c) 3D phase space with the response trajectory (see text for details).

Figure 11

Visualization of the excitation process. (a) Time series of $x$ (lower) and $z$ (upper). (b) $(x,z)$ plane projection of the phase space of the system. (c) 3D phase space with the response trajectory (see text for details).

Figure 12

Dependencies of the excitation threshold $Z_0^{\mathrm{th}}$ on coupling $d$. Circle magenta is numerical; solid red with squares is first analytic approach without additional corrections; solid green with x markers is analytic dependency with $\kappa$ heuristics; and solid blue is $\kappa$ heuristics and $\varepsilon$ neighborhood taken into account.