2021-12-08 • Cortical params for Hodgkin-Huxley
Contents
2021-12-08 • Cortical params for Hodgkin-Huxley¶
This is as /notebooks/2021-12-06__local_HH_dV_shape, but with params for HH model not for squid giant axon but a coritcal RS neuron. Most saliently, the firing threshold is way lower, closer to reality and to Izhikevich cortical RS.
[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 20:00 (UTC+0000).
Last git commit (Tue 07 Dec 2021, 03:59).
Uncommited changes to 5 files.
Params¶
Params from Ch 2.2 of the epfl book.
c = 1 * (μF / cm**2)
E_K = -77 * mV
E_Na = 55 * mV
E_L = -65 * mV
g_K = 35 * (mS / cm**2)
g_Na = 40 * (mS / cm**2)
g_L = 0.3 * (mS / cm**2)
α_m = lambda V: 0.182 * (V/mV + 35) / (1 - np.exp((-35-V/mV)/9)) / ms
β_m = lambda V: -0.124 * (V/mV + 35) / (1 - np.exp((V/mV+35)/9)) / ms
m_inf = lambda V: α_m(V) / (α_m(V) + β_m(V))
α_n = lambda V: 0.02 * (V/mV - 25) / (1 - np.exp((25-V/mV)/9)) / ms
β_n = lambda V: -0.002 * (V/mV - 25) / (1 - np.exp((V/mV-25)/9)) / 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=-77*mV, V_max=-52*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
f = partial(dV_dt, n=0.25, I=0);
ax, V = make_V_dV_ax()
ax.plot(V / mV, f(V) / (mV / ms));
Izhikevich’s \(dV/dt\)¶
from scipy.optimize import newton
C = 22.5 * pF # 100 pF
k = 0.7 * (nS/mV)
V_r, V_t = newton(f, [-70*mV, -55*mV])
dV_dt_Izh = lambda V: (k * (V - V_r) * (V - V_t)) / C
ax, V = make_V_dV_ax()
# ax, V = make_V_dV_ax(V_r-0.1*mV, V_r+0.1*mV)
ax.plot(V / mV, dV_dt_Izh(V) / (mV / ms), label="Izhikevich")
ax.plot(V / mV, f(V) / (mV / ms), label="HH 2D")
# ax.set_xlim(-100, 20)
# ax.set_ylim(-1.7, 4.9)
ax.set_xlim()
# label_lines(ax.lines, xvals=[-88, -18], outline_width=7);
ax.legend();
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.
Slow current gating variable \(n\)¶
ax, V = make_V_dV_ax()
for n in [0, 0.2, 0.3, 0.4]:
ax.plot(V / mV, dV_dt(V, n, I=0) / (mV / ms), label=f"$n = {n}$")
labels = label_lines(ax.lines); # labels after lims, so rotations are correct (issue 36)
n = n_inf(V_r)
0.00023997233576810773
Neuron spikes without input current, that ain’t right.
I_Na(V_r, n=0.3) * (nA/mm**2)
5.505099801252533e-07
I_K(V_r, n=0.3) * (nA/mm**2)
-1.775847400062642e-05
I_L(V_r) * (nA/mm**2)
1.7207964020501135e-05
Simulation¶
Time
duration = 2 * second
duration = 500 * 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) * 400 * 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) * 0 * nA/mm**2;
ODE integration
V = np.ones(N) * V_r
n = np.ones(N) * 0.3
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(bottom=0)
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)
axs[3].plot(t, I / (nA/mm**2))
ylabel_inside(axs[3], "Input current $I$ (nA/mm²)")
axs[3].set_xlabel("Time (ms)");
..¶
Eh. I’d have to use full HH equations, not 2D approx.
Params not from epfl book guess, but eg from https://brian2.readthedocs.io/en/stable/examples/COBAHH.html . (that’s.. hippocampal pyramidal? (Traub ref).
I just want to know how ‘wrong’ Izhikevich is compared to reality.
Let’s reproduce dV curve from Badel 2008, fig 2 B&C.
(fig 2 is for cortical neuron, fig 1 for something else).
See new notebook.
[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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