A42 Stability of a Nonlinear Steady State

What we have learned about stability of steady states for linear systems can be transferred partially to nonlinear ODEs. To be specific, let us consider a biological membrane with a gated ion channel. To do this we combine a model of ion gating with an expression that governs the membrane potential (see Chapter 1 and Chapter 2). For simplicity, we will consider only one conductance. If n represents the gating variable and V the voltage, then the two are coupled by the differential equations

where VTev is the reversal potential. We assume that n^ has the following voltage dependence:

with V0 5 and S positive constants. Equations (A.49) and (A.50) are both nonlinear due to the factor n(V — V-ev) in (A.49) and the voltage dependence of n^ in (A.50).

To analyze the stability of the steady states of these equations we first must find the steady states by setting the right-hand sides of the equations equal to zero. This gives gnss(Vss — Vrev) = lapp (A.52)

which can be written as a single nonlinear equation to solve for Vss:

This equation cannot be solved in closed form, and a much simpler way to locate the steady state is graphically in the (V, n) phase plane using the nullclines. Setting the left-hand sides of (A.49) and (A.50) separately equal to zero and solving for n as a function of V gives n(V) = ,JapPr . (V-nullcline), (A.55)

n(V)= (V )=i + exp(—¿ + Vo,)/S) (n-nullcline)- (A.56)

The V- and n-nullclines are plotted in Figure A.5A, along with representative trajectories. Due to the nonlinearities in (A.49) and (A.50) the nullclines are curved rather than straight lines. This curvature influences the shape of the trajectories, which must cross the nullcline perpendicular to the axis of the variable. Close to the steady state, however, both nullclines become approximately straight lines, as is seen in Figure A.5B, which is the same phase plane as in Figure A.5A, but zoomed in around the steady state.

If we restrict the initial conditions for trajectories to be close to the steady state, then the nonlinear equations are well approximated by a 2 X 2 linear system. This can be seen in detail if we define as new variables x = V — Vss and x2 = n — nss, the deviations of the voltage and gating variable from their steady-state values. Since the steady-state values are constants, it follows that d^/dt = dV/dt and dx2/dt = dn/dt, so that we can use (A.49) and (A.50) to obtain differential equations for x1 and x2. In particular, if the initial conditions are close to the steady state, then we can substitute V = Vss + x1 and n = nss + x2 into the right-hand sides of (A.49) and (A.50) and then use a Taylor series expansion in the small deviations x1 and x2. Explicitly:

Figure A.5 Phase plane plots for (A.49)-(A.51) (dashed line), and the n-nullcline (broken dashed lir that the nullclines are approximately straight lines

showing typical trajectories (full lines), the V-nullcline e). (B) iszoomed-in around the steady state, illustrating near the steady state.

= [gnBB(VBB - Vrev) + /app]/C + (gnBBXi + g(VBB - )x2)/C

+ gXiX2/C, (A.57) dx2/dt = — (nBB + x2 — n^(VBB + x1)) /t

In the second equality in both (A.57) and (A.58) the terms in square brackets vanish because of the steady—state conditions in (A.52) and (A.52); the second terms are linear in X1 and X2; and the third terms are quadratic or of higher order in X1 and X2. Thus keeping the lowest—order terms gives the linear equations dXx/dt = (gnBB/C>1 + (g(VBB - Vrev)/C>2, (A.59)

Once the elements of the matrix of this 2 X 2 linear equation have been evaluated, the behavior of the solution in a neighborhood of the steady state can be evaluated. This type of linear analysis, which gives information only about trajectories nearby the steady state, is called linear stability analysis.

The trajectories in Figure A.5B make it clear that the steady state is asymptotically stable, and according to the catalogue of possibilities in Figure A.4, the steady state is a stable spiral. It is also possible to find the steady states numerically and,

Figure A.6 Bifurcations of new fixed points: (A) saddle-node bifurcation, (B) transcritcial bifurcation, (C) supercritical pitchfork bifurcation, (D) subcritical pitchfork bifurcation. Stable fixed points are solid, and unstable are dashed.

in addition, determine the stability of the steady state by numerical evaluation of the eigenvalues. Combining the analytical tools developed in this chapter with the numerical tools available in various software packages, we are ready to explore the dynamics of a variety of cellular and neural dynamical systems in the remaining chapters.

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