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Notebook 5 — Incomplete parabolas

SWC ENC 2026 · ephys-one module · independent of Notebooks 4, 6, 7

The parabola of Notebook 3 runs from "all closed" to "all open". Real channels rarely oblige. If the test voltage only opens a third of them, you see only the first third of the curve; if they inactivate, the open probability never gets high. What can you still trust? The short answer — the unitary current survives almost anything, the channel count only when you can see the curve bend back — is the most practically useful lesson of the whole method.

In this notebook you will:

1. Vary the test voltage and watch how much of the parabola you get to see.
2. Fit each partial parabola and see which estimates hold up.
3. Meet an inactivating channel, and see that activation and inactivation trace the same parabola.

import numpy as np
import matplotlib.pyplot as plt
import picopatch as pp

scheme = pp.two_state()

1. The test voltage sets how far you get

In Notebook 1 the open probability rose with voltage along a smooth curve. Each test voltage therefore reaches a different maximum open probability $p_{max}$ — and since the mean current is $N i p$, a different distance along the parabola. Let's mark five test voltages on the $p(V)$ curve:

v_steps = [30, 40, 50, 65, 85]
V = np.linspace(0, 100, 200)
p_inf = np.array([scheme.open_probability(v) for v in V])
p_reached = [scheme.open_probability(v) for v in v_steps]

plt.figure(figsize=(6, 3.6))
plt.plot(V, p_inf, "k")
for v, p, c in zip(v_steps, p_reached, plt.cm.plasma(np.linspace(0, 0.85, 5))):
    plt.plot(v, p, "o", color=c, ms=9, label=f"step to {v} mV → p_max = {p:.2f}")
plt.xlabel("test voltage (mV)"); plt.ylabel("open probability reached"); plt.legend(fontsize=8); plt.show()

Now simulate the same patch — 100 channels, 1 pA each, 400 sweeps — stepped to each of these voltages, and plot variance against mean. The dashed red curve is the full true parabola; the dots are the part the experiment actually visits:

exps = {v: pp.make_experiment(N=100, i=1.0, scheme=scheme, protocol=pp.step_protocol(v_step=v, t_step_ms=15),
                              n_sweeps=400, seed=v) for v in v_steps}

fig, axes = plt.subplots(1, 5, figsize=(18, 3.6), sharex=True, sharey=True)
for ax, v in zip(axes, v_steps):
    e = exps[v]; m, var = pp.isochrone_stats(e.sweeps)
    pp.plotting.plot_variance_mean(m, var, truth=e.truth, ax=ax, s=6, color="tab:blue",
                                   title=f"step to {v} mV  (p_max = {e.truth.p_max:.2f})")
    ax.get_legend().remove(); ax.set_ylabel("variance (pA²)" if v == v_steps[0] else "")
plt.tight_layout(); plt.show()

At 30 mV the dots cover only the first 8% of the curve and look like a straight line. At 50 mV they reach the peak. Only at 65 and 85 mV do they come back down the far arm. Think about what a fit can learn from each: a straight line has a slope (that's $i$) but no curvature — and $N$ is the curvature.

Exercise 1 (~8 min). For each test voltage, fit the parabola and bootstrap the uncertainty (use pp.bootstrap_fit(sweeps, n_boot=100) for the resampling — or your own from Notebook 4). Plot the estimated $i$ and $N$, with their 95% intervals, against the $p_{max}$ reached (e.truth.p_max).

Check / unstuck. $i$ should sit near 1 pA with tight bars at every voltage; $N$ should be wildly uncertain (bars running off the plot, or infinite) below $p_{max} \approx 0.5$ and settle near 100 only at 65 and 85 mV. Stuck? pp.count_channels(e) returns the fit plus bootstrap samples (.i_boot, .N_boot).

rows = []
for v in v_steps:
    e = exps[v]
    m, var = pp.isochrone_stats(e.sweeps)
    fit = pp.fit_parabola(m, var)
    i_b, N_b = pp.bootstrap_fit(e.sweeps, n_boot=100, rng=0)
    N_b = N_b[np.isfinite(N_b)]
    N_lo, N_hi = np.percentile(N_b, [2.5, 97.5]) if len(N_b) > 1 else (np.nan, np.nan)
    i_lo, i_hi = np.percentile(i_b, [2.5, 97.5])
    rows.append((e.truth.p_max, fit.i, i_lo, i_hi, fit.N, N_lo, N_hi))
    print(f"{v} mV  p_max={e.truth.p_max:.2f}:  i = {fit.i:.2f}  N = {fit.N:.0f}")
rows = np.array(rows)

fig, axes = plt.subplots(1, 2, figsize=(11, 3.8))
axes[0].errorbar(rows[:, 0], rows[:, 1], yerr=[rows[:, 1] - rows[:, 2], rows[:, 3] - rows[:, 1]], fmt="o", capsize=4)
axes[0].axhline(1.0, color="tab:red", ls="--", label="truth"); axes[0].set_ylim(0, 2)
axes[0].set_xlabel("p_max reached"); axes[0].set_ylabel("estimated i (pA)"); axes[0].set_title("unitary current: fine everywhere"); axes[0].legend()
axes[1].errorbar(rows[:, 0], rows[:, 4], yerr=[np.clip(rows[:, 4] - rows[:, 5], 0, None), np.clip(rows[:, 6] - rows[:, 4], 0, None)],
                 fmt="o", capsize=4)
axes[1].axhline(100, color="tab:red", ls="--", label="truth"); axes[1].set_ylim(0, 400)
axes[1].set_xlabel("p_max reached"); axes[1].set_ylabel("estimated N"); axes[1].set_title("channel count: needs the far arm"); axes[1].legend()
plt.tight_layout(); plt.show()

30 mV  p_max=0.08:  i = 1.01  N = 79
40 mV  p_max=0.31:  i = 1.02  N = 109
50 mV  p_max=0.68:  i = 0.95  N = 105
65 mV  p_max=0.96:  i = 1.05  N = 95

85 mV  p_max=1.00:  i = 0.98  N = 102

This is the paper's Fig. 9 in one plot. The practical rules:

• The unitary current is robust. Even a short stretch of the parabola fixes the initial slope. If all you can see is a straight line, report $i$ and stop.
• The channel count needs curvature. Until the dots clearly bend over — in practice $p_{max} \gtrsim 0.5$ — any $N$ you fit is a guess with enormous error bars. Never quote $N$ without them.
• $p_{max}$ inherits $N$'s problems, since $p_{max} = \langle I\rangle_{max}/(iN)$.

▸ Go deeper: why the curvature is so poorly determined (optional)

Write the parabola as $\sigma^2 = i\,m - m^2/N$. For small $m$ the second term is smaller than the first by a factor $m/(iN) = p$. If the data only reach $p = 0.1$, the quadratic term is at most 10% of the variance — and the variance itself is measured with ~10% noise from a few hundred sweeps. The fit is being asked to measure a 10% effect to 10% precision: it can't. Statistically, the two regressors $m$ and $m^2$ are nearly collinear over a short range, so the design matrix is ill-conditioned and $1/N$ comes with a huge standard error, while $i$ (the well-determined direction) stays sharp.

2. Channels that inactivate

Many channels — sodium channels, A-type potassium channels — open on depolarisation and then shut themselves off while the depolarisation lasts: they inactivate, entering a non-conducting state from which they can't reopen until the membrane is repolarised. The scheme gains a third state:

inact = pp.inactivating()
fig, ax = plt.subplots(figsize=(8, 2.6))
pp.plotting.plot_scheme(ax, inact, V=65, title="C ⇌ O ⇌ I  (rates at the test voltage)")
plt.show()

Here is what one such channel does on a step, and what a thousand of them do together. Each channel opens quickly, flickers for a while, and then goes quiet for good; the macroscopic current rises fast and then decays to a low steady level.

protocol = pp.step_protocol(v_step=65, t_step_ms=15)
states = pp.simulate_single(inact, protocol, n_sweeps=200, rng=1)
open_1 = states == 1

fig, axes = plt.subplots(3, 1, figsize=(10, 7), sharex=True, gridspec_kw=dict(height_ratios=[1, 3, 2]))
pp.plotting.plot_voltage(axes[0], protocol.t_ms, protocol.V)
pp.plotting.plot_open_raster(axes[1], open_1, protocol.t_ms, n_show=40, title="40 sweeps of ONE inactivating channel")
e_in = pp.make_experiment(N=1000, i=1.0, scheme=inact, protocol=protocol, n_sweeps=300, seed=2)
for k in range(15):
    axes[2].plot(e_in.t_ms, e_in.sweeps[k], color="k", lw=0.5, alpha=0.6)
axes[2].set_ylabel("current (pA)"); axes[2].set_xlabel("time (ms)"); axes[2].set_title("1000 channels: 15 sweeps")
plt.tight_layout(); plt.show()

Now the key observation. Time never appeared in $\sigma^2 = i\langle I\rangle - \langle I\rangle^2/N$. The parabola doesn't care whether the current is on its way up or on its way down — only how many channels are open. So the fast rising phase and the slow decay should land on the same curve, the rising phase sparsely (few samples, since it's quick) and the decay densely:

m, var = pp.isochrone_stats(e_in.sweeps)
pp.plotting.plot_variance_mean(m, var, truth=e_in.truth, t_ms=e_in.t_ms,
                               title="inactivating channels: up and down the same parabola")
plt.show()

Exercise 2 (~5 min). Fit the parabola to the inactivating patch and compare with the truth (e_in.truth). Then look at the open probability time course, $p(t) = \langle I\rangle / (iN)$, and read off the peak $p$.

Check / unstuck. $i$ should be right; $N$ will be off — the peak open probability here is only ~0.25, so the dots never reach the far arm (section 1 again, in disguise). Stuck? pp.fit_parabola(m, var).

fit = pp.fit_parabola(m, var)
print(f"fit:   {fit}")
print(f"truth: i = {e_in.truth.i} pA, N = {e_in.truth.N}, peak p = {e_in.truth.p_max:.2f}")

plt.figure(figsize=(9, 3.2))
plt.plot(e_in.t_ms, m / (fit.i * fit.N), "k", label="p(t) from the fit")
plt.plot(e_in.t_ms, e_in.truth.p_open, "r--", label="truth")
plt.xlabel("time (ms)"); plt.ylabel("open probability"); plt.legend(); plt.show()

fit:   NoiseFit(i=0.969 pA, N=1204, p_max=0.22, baseline=0 pA^2)
truth: i = 1.0 pA, N = 1000, peak p = 0.25

The paper found the same (their Fig. 10: $N$ = 842 for a true 1000). Inactivation is not a flaw in the method — it's just a channel that never lets you see enough of the parabola. In practice one would push $p$ higher (a stronger step, or a drug that removes inactivation) if the channel count really mattered.

Where we are. A partial parabola still gives the unitary current, reliably. The number of channels — and therefore $p_{max}$ — needs the curve to bend back, which means an open probability well past one half. Inactivating channels are the commonest way to be stuck on the first arm.

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