2021-12-06 • Local shape of \dot{V} in 2D HH
Contents
2021-12-06 • Local shape of \(\dot{V}\) in 2D HH¶
What’s the shape of dV/dt (V) for fixed \(n\) in the Hodgkin-Huxley model simplified to 2D (as done in Izhikevich book sec 5.2 or here in the epfl book)?
[Prelude]¶
from voltage_to_wiring_sim.notebook_init import *
Preloading: numpy, numba, matplotlib.pyplot, seaborn.
Importing from submodules … ✔
Imported `np`, `mpl`, `plt`, `sns`, `pd`
Imported codebase (`voltage_to_wiring_sim`) as `v`
Imported `*` from `v.support.units`
Setup autoreload
v.print_reproducibility_info()
This cell was last run by lpxtf3 on DUIP74576 on Wed 08 Dec 2021, at 19:15 (UTC+0000).
Last git commit (Tue 07 Dec 2021, 03:59).
Uncommited changes to 4 files.
Params¶
From HH1952 / Izh book sec 2.3.1.
(Alternatively, params could be taken from Ch 2.2 of the epfl book, which has HH params fitted to a cortical pyramidal neuron, instead of params for the squid giant axon used here).
E_shift = 65 * mV;
c = 1 * (μF / cm**2)
E_K = -12 * mV - E_shift
E_Na = 120 * mV - E_shift
E_L = 10.6 * mV - E_shift
g_K = 36 * (mS / cm**2)
g_Na = 120 * (mS / cm**2)
g_L = 0.03 * (mS / cm**2)
α_m = lambda V: 0.1 * (25 - V/mV) / (np.exp((25-V/mV)/10) - 1) / ms
β_m = lambda V: 4 * np.exp(-V/mV/18) / ms
m_inf = lambda V: α_m(V) / (α_m(V) + β_m(V))
α_n = lambda V: 0.01 * (10 - V/mV) / (np.exp((10-V/mV)/10) - 1) / ms
β_n = lambda V: 0.125 * np.exp(-V/mV/80) / ms
n_inf = lambda V: α_n(V) / (α_n(V) + β_n(V))
τ_n = lambda V: 1 / (α_n(V) + β_n(V));
Simplified Hodgkin-Huxley model¶
With \(τ_m = 0\) and \(n(t) \propto h(t)\), the 4D system is reduced to just 2D.
h = lambda n: 0.89 - 1.1 * n
I_K = lambda V, n: - g_K * n**4 * (V - E_K)
I_Na = lambda V, n: - g_Na * m_inf(V)**3 * h(n) * (V - E_Na)
I_L = lambda V: - g_L * (V - E_L)
I_tot = lambda V, n, I: I + I_K(V,n) + I_Na(V,n) + I_L(V)
dV_dt = lambda V, n, I: I_tot(V,n,I) / c
dn_dt = lambda V, n: (n_inf(V) - n) / τ_n(V);
The equation for \(\dot{n}\) is only used in the simulation at the end. The other sections all analyse \(\dot{V}\) independently from \(\dot{n}\).
\(dV/dt\)¶
def horizontal_ylabel(ax, text, dx=-6, dy=-2, ha="right", x=0, y=1, bg_pad=0.4, **text_kwargs):
""" dx and dy are in points, i.e. 1/72th of an inch. """
offset = mpl.transforms.ScaledTranslation(dx/72, dy/72, ax.figure.dpi_scale_trans)
bbox=dict(facecolor=ax.get_facecolor(), edgecolor='none', boxstyle=f'square,pad={bg_pad}')
return ax.text(x, y, text, transform=ax.transAxes + offset,
ha=ha, va="top", bbox=bbox, **text_kwargs)
def make_V_dV_ax(V_min=-100*mV, V_max=5*mV):
fig, ax = plt.subplots()
ax.axhline(y=0, color='k')
ax.set_xlabel("$V$ (mV)")
horizontal_ylabel(ax, "$\dot{V}$ (mV/mS)")
V = np.linspace(V_min, V_max, num=1000)
return ax, V
ax, V = make_V_dV_ax()
ax.plot(V / mV, dV_dt(V, n=0.0, I=0) / (mV / ms));
from scipy.optimize import newton
f = partial(dV_dt, n=0, I=0)
newton(f, [-60*mV, 0*mV]) / mV
array([-54.4, 2.007])
There is a stable node at \(-\)54.4 mV ( → ⚫ ←, resting potential)
and an unstable node at 2 mV ( ← ⚪ →, spike threshold).
This \(\dot{V}\) function has the same shape as in the AdEx (aka aEIF) model: linear + exponential:
\[
C \dot{V} = k \left( -(V - V_r) + Δ_T e^{(V - V_{rh}) / Δ_T} \right) - w + I
\]
(This is eq 6.3 here, where we equated the membrane conductance \(1/R\) with Izhikevich’s \(k\)).
It is not the same shape as in the Izhikevich model. There, \(\dot{V}\) is quadratic:
\[
C \dot{V} = k (V - V_r) (V - V_t) - w + I
\]
(\(w\) is Izhikevich’s \(u\), is the 2D HH model’s \(n\)).
Izhikevich’s \(dV/dt\)¶
C = 1320 * pF # 100 pF
k = 0.7 * (nS/mV)
V_r = -54.4 * mV # -65 mV
V_t = 2 * mV # -40 mV
dV_dt_Izh = lambda V: (k * (V - V_r) * (V - V_t)) / C
ax, V = make_V_dV_ax(V_max=20*mV)
# ax, V = make_V_dV_ax(V_min=-54.5*mV, V_max=-54.3*mV)
ax.plot(V / mV, dV_dt_Izh(V) / (mV / ms), label="Izhikevich")
ax.plot(V / mV, dV_dt(V,0,0) / (mV / ms), label="HH 2D")
ax.set_xlim(-100, 20)
ax.set_ylim(-1.7, 4.9)
label_lines(ax.lines, xvals=[-88, -18], outline_width=7);
The parabola’s zeros (\(V_r\) and \(V_t\)) were chosen to match those of the 2D Hodgkin-Huxley model.
The capacitance \(C\) was chosen so that the slopes of the two models approximately match at the resting potential.
Input current \(I\)¶
ax, V = make_V_dV_ax(V_max = 10*mV)
for I in np.array([0, -10, 10, 20]) * nA/mm**2:
ax.plot(V / mV, dV_dt(V, n=0.0, I=I) / (mV / ms), label=f"$I = {I/(nA/mm**2):g}$ nA/mm²")
ax.set_xlim(-83, 10)
ax.set_ylim(-2.8, 3.1)
label_lines(ax.lines, xvals=[-38]*4, outline_width=7);
Negative input current → lower stable node aka resting potential.
Somewhere between +10 and +20 nA/mm², the stable node and the unstable node (‘saddle’) join and disappear (‘annihilate’); this is the saddle–node AKA fold bifurcation. There are no more fixed points, the neuron spikes continuously. (Thanks to a voltage return/reset not visible here).
Slow current gating variable \(n\)¶
ax, V = make_V_dV_ax(V_max = 10*mV)
for n in [0, 0.1, 0.2, 0.3]:
ax.plot(V / mV, dV_dt(V, n, I=0) / (mV / ms), label=f"$n = {n}$")
ax.set_xlim(-100, 10)
labels = label_lines(ax.lines); # labels after lims, so rotations are correct (issue 36)
labels[0].set_position((-50, 0))
labels[1].set_position((-32, -1.1))
labels[2].set_position((-23, -4))
\(n ∈ (0, 1)\) (it is a gating variable).
Note the weak influence of lower values of \(n\), but the strongly growing influence for higher values, from \(n > ± 0.1\) onwards.
dn = 0.001
n = np.linspace(0, 1, num=round(1/dn), endpoint=False)
fig, ax = plt.subplots()
Vdot = dV_dt(0 * mV, n, I=0) / (mV/ms)
ax.plot(n, Vdot)
# ax.set_xscale("log")
ax.set_yscale("symlog", linthresh=1e-6) # this is just -(log(-x))
ax.set_xlabel("$n$")
horizontal_ylabel(ax, "$\dot{V}$ at $V = 0$ mV"+"\n"+f"(mV/ms)", dx=-22)
ax.xaxis.set_major_formatter(mpl.ticker.FormatStrFormatter("%g"));
ax.yaxis.set_major_formatter(mpl.ticker.FormatStrFormatter("$%g$"));
fig, ax = plt.subplots()
l = ax.plot(n[1:], np.diff(Vdot) / Vdot[1:])
ax.set_xlabel("$n$")
# ax.set_xscale("log")
horizontal_ylabel(ax, "$Δ\dot{V}/Δn$ / $\dot{V}$"+"\n"+"at $V = 0$ mV"+"\n"+f"($Δn = {dn:g}$)", dx=-36)
ax.set_ylim(bottom=0)
ax.yaxis.set_major_formatter(mpl.ticker.PercentFormatter(xmax=1, decimals=1))
ax.xaxis.set_major_formatter(mpl.ticker.FormatStrFormatter("%g"));
Simulation¶
Time
duration = 2 * second
# duration = 200 * ms
Δt = 0.01 * ms
N = round(duration / Δt)
t = np.linspace(0, duration / ms, N, endpoint=False);
Input
from scipy.signal.windows import gaussian
np.random.seed(1)
I = np.random.randn(N) * 1400 * nA/mm**2
σ = 6 * ms
M = round(2 * 4 * σ / Δt)
I = np.convolve(I, gaussian(M, std=σ / Δt) / M, "same"); # local weighted average, to smooth
# I = np.ones(N) * 60 * nA/mm**2;
ODE integration
V = np.ones(N) * -54.4 * mV
n = n_inf(V)
for i in range(N - 1):
V[i+1] = V[i] + dV_dt(V[i], n[i], I[i]) * Δt
n[i+1] = n[i] + dn_dt(V[i], n[i]) * Δt
nrows = 4
fig, axs = plt.subplots(nrows, **v.figsize(width=800, aspect=1), sharex=True)
ylabel_inside = partial(horizontal_ylabel, ha="left", dx=6, dy=0, bg_pad=0.2)
axs[0].plot(t, V / mV)
ylabel_inside(axs[0], "Membrane voltage $V$ (mV)")
axs[1].plot(t, n)
axs[1].set_ylim(-0.01, 1)
ylabel_inside(axs[1], "Slow current gating variable $n$")
axs[2].plot(t, I_K(V, n) / (nA/mm**2), label="$I_{K}$")
axs[2].plot(t, I_Na(V, n) / (nA/mm**2), label="$I_{Na}$")
axs[2].plot(t, I_L(V) / (nA/mm**2), label="$I_{L}$")
# axs[2].set_ylim(-32, 32)
ylabel_inside(axs[2], "Membrane currents (nA/mm²)")
label_lines(axs[2].lines, align=False, shrink_factor=0.3)
# label_lines(axs[2].lines, xvals=[620, 60, 140], align=False)
axs[3].plot(t, I / (nA/mm**2))
ylabel_inside(axs[3], "Input current $I$ (nA/mm²)")
axs[3].set_xlabel("Time (ms)");
[Reproducibility]¶
v.print_reproducibility_info(verbose=True)
This cell was last run by lpxtf3 on DUIP74576
on Tue 07 Dec 2021, at 02:33 (UTC+0000).
Last git commit (Mon 06 Dec 2021, 20:02).
Uncommited changes to:
M notebooks/2021-12-06__local_HH_dV_shape.ipynb
?? notebooks/2021-11-18__constant_input_spikes.ipynb
?? notebooks/2021-11-18__prototype_new_sim.ipynb
Platform:
Windows-10
CPython 3.9.6 (C:\miniforge3\python.exe)
Intel(R) Xeon(R) W-2123 CPU @ 3.60GHz
Dependencies of voltage_to_wiring_sim and their installed versions:
numpy 1.21.1
matplotlib 3.4.2
numba 0.53.1
joblib 1.0.1
seaborn 0.11.1
scipy 1.7.0
preload 2.2
nptyping 1.4.2
Full conda list:
# packages in environment at C:\miniforge3:
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