Isostable reduction with applications to time-dependent partial differential equations

Isostables and isostable reduction, analogous to isochrons and phase reduction for oscillatory systems, are useful in the study of nonlinear equations which asymptotically approach a stationary solution. In this work, we present a general method for isostable reduction of partial differential equations, with the potential power to reduce the dimensionality of a nonlinear system from infinity to 1. We illustrate the utility of this reduction by applying it to two different models with biological relevance. In the first example, isostable reduction of the Fokker-Planck equation provides the necessary framework to design a simple control strategy to desynchronize a population of pathologically synchronized oscillatory neurons, as might be relevant to Parkinson's disease. Another example analyzes a nonlinear reaction-diffusion equation with relevance to action potential propagation in a cardiac system.

Figure 1

A qualitative illustration of isostables for a PDE which approaches a stationary solution. Surfaces of constant isostable are represented by colored lines ${\psi }^1, {\psi }^2$, and ${\psi }^3$. Black dots represent snapshots of the PDE solution in time. Initial conditions along the same isostable surface approach the fixed point with the same asymptotic convergence as defined by (3). Initial conditions along the same isostable cross successive isostables at the same time. Here, $\gamma$ is some trajectory for which we wish to characterize the change in isostables in response to spatiotemporal perturbations.

Figure 2

Illustration of isostables in the Fokker-Planck equation. Panel (a) shows an initial distribution in black and several associated Fourier modes as dashed lines. After some time passes, panel (b) shows that all other modes except for the first have decayed substantially. Panel (c) shows the same initial distribution with the first mode shown in red for reference, to which we add one of three perturbations given in panel (d). After $4T$ has elapsed, the distribution resulting from the blue, green, and red perturbation is shown as a solid line in panels (e), (f), and (g), respectively, with the unperturbed distribution after $4T$ has elapsed shown as a dashed line. The perturbed and unperturbed distributions are indistinguishable in panel (g).

Figure 3

Isostable reduction applied to the Fokker-Planck equation. Panel (a) shows $Z\left(\theta \right)$, used in the desynchronizing control strategy. Panel (b) shows the iIRC, $\mathcal{I}(\theta ,\psi )/\beta$ where $\beta \equiv 2/A_1^{*}{B}^2\pi$. Panels (c)–(h) show the result of applying the control strategy to the PDE (25). The control input (in $\mu \mathrm{A}/\mu \mathrm{F}$) in D drives the first mode of the distribution from 0.5 to 0.05. Panel (c) gives the amplitude of this mode as a function of time for the equations with control applied as well as the system without control, shown as a solid and dashed-dotted line, respectively. Panel (e) gives the location of the maximum of the distribution, ${\theta }_{\max }$, as a function of time. Panels (f)–(h) show a comparison of the controlled distribution to the uncontrolled distribution at $t=7.4,41.8$, and $75.8$ ms. The horizontal dashed line shows the ideal, uniform distribution.

Figure 4

The top and bottom panels shows the state of the full equations (25) and the reduced equations (26) with control (27). Larger isostable values correspond to more desynchronized states.

Figure 5

The top panel shows the approximated control strategy (29) applied to a population of Hodgkin-Huxley neurons, with the middle and bottom panels showing the applied control and the Kuramoto order parameter, $R$, for the network, respectively.

Figure 6

At $\psi =0$ a perturbation to $u$ creates a traveling wave in the excitable system. The wave is absorbed by the right boundary. The variable $v$ approaches the stationary solution slowly relative to $u$.

Figure 7

Top-left and top-right panels show the iIRC at $\psi =200$ as solid lines for perturbations to $u$ and $v$, respectively. Dashed lines show $u\left(r\right)$ for reference. Bottom left and right panels compare theoretical predictions (black lines) to results from direct numerical simulation (red crosses) for perturbations to $u\left(r\right)$ and $v\left(r\right)$, respectively. Here, the variable $\zeta$ represents the signed distance from where the perturbation is centered relative to the peak of $u\left(r\right)$.

Figure 8

Panel (a) shows the numerically calculated isostable difference coupling from (40). The red dashed line gives $f\left(\phi \right)$ linearized about $\phi =0$, as used in (41). Panel (b) shows snapshots of a traveling wave at two points in time in the 2D system (37). The isostable difference as a function of $x$ (the shorter dimension) is plotted in panel (c) for the snapshots from panel (c). Black lines from panels (d) and (e) show the numerically calculated maximum difference in isostables as a function of time. The dashed red line in panel (e) is calculated from (43) using numerically determined isostable data at $t=700$ as an initial condition.