Hodgkin-Huxley benchmark
Hodgkin-Huxley is a four-dimensional ODE, which can be stiff or non-stiff depending on the parameters; here we consider a non-stiff version. We see that:
Code:
using LinearAlgebra, Statistics, Distributions
+using DiffEqDevTools, SciMLBase, OrdinaryDiffEq, Plots, SimpleUnPack
+using ProbNumDiffEq
+
+Plots.theme(
+ :dao;
+ markerstrokewidth=0.5,
+ legend=:outertopright,
+ margin=5Plots.mm,
+ xticks=10.0 .^ (-16:1:16),
+ yticks=10.0 .^ (-6:1:5),
+)
Code:
αm(V, VT) = -0.32 * (V - VT - 13) / (exp(-(V - VT - 13) / 4) - 1)
+βm(V, VT) = 0.28 * (V - VT - 40) / (exp((V - VT - 40) / 5) - 1)
+
+αn(V, VT) = -0.032 * (V - VT - 15) / (exp(-(V - VT - 15) / 5) - 1)
+βn(V, VT) = 0.5 * exp(-(V - VT - 10) / 40)
+
+αh(V, VT) = 0.128 * exp(-(V - VT - 17) / 18)
+βh(V, VT) = 4 / (1 + exp(-(V - VT - 40) / 5))
+
+const current_tspan = (5, 40)
+Inj(t) = (current_tspan[1] <= t <= current_tspan[2]) ? 500one(t) : zero(t)
+
+function f(du, u, p, t)
+ @unpack gNa, gK, ENa, EK, area, C, Eleak, VT, gleak = p
+
+ V, m, n, h = u
+
+ I_inj = Inj(t) * 1e-6
+
+ du[2] = dmdt = (αm(V, VT) * (1 - m) - βm(V, VT) * m)
+ du[3] = dndt = (αn(V, VT) * (1 - n) - βn(V, VT) * n)
+ du[4] = dhdt = (αh(V, VT) * (1 - h) - βh(V, VT) * h)
+
+ INa = gNa * m^3 * h * (V - ENa) * area
+ IK = gK * n^4 * (V - EK) * area
+ Ileak = gleak * (V - Eleak) * area
+ Cm = C * area
+ du[1] = dVdt = -(Ileak + INa + IK - I_inj) / Cm
+end
+
+p = (gNa=20.0, gK=15.0, ENa = 53, EK = -107, area = 15e-5, C = 1, Eleak = -70, VT = -60, gleak = 0.1, V0 = -70)
+
+m_inf(V, VT) = 1 / (1 + βm(V, VT) / αm(V, VT))
+n_inf(V, VT) = 1 / (1 + βn(V, VT) / αn(V, VT))
+h_inf(V, VT) = 1 / (1 + βh(V, VT) / αh(V, VT))
+u0 = [p.V0, m_inf(p.V0, p.VT), n_inf(p.V0, p.VT), h_inf(p.V0, p.VT)]
+
+prob = ODEProblem{true,SciMLBase.FullSpecialize()}(f, u0, (0.0, 50.0), p)
+
+test_sol = solve(prob, Vern7(), abstol=1/10^14, reltol=1/10^14)
+plot(test_sol,
+ legend=false,
+ layout=(4,1),
+ title=["Hodgkin-Huxley Solution" "" "" ""],
+ ylabel=["V(t)" "m(t)" "n(t)" "h(t)"],
+ xlabel=["" "" "" "t"],
+ size = (1000, 600),
+ color=[1 2 3 4],
+ xticks=:auto, yticks=:auto
+)
Solver comparison: EK0
vs. EK1
vs RosenbrockExpEK
Without smoothing
Code:
DENSE = SAVE_EVERYSTEP = false
+
+_setups = [
+ "EK0(2)" => Dict(:alg=>EK0(order=2, smooth=DENSE))
+ "EK0(3)" => Dict(:alg=>EK0(order=3, smooth=DENSE))
+ "EK1(2)" => Dict(:alg=>EK1(order=2, smooth=DENSE))
+ "EK1(3)" => Dict(:alg=>EK1(order=3, smooth=DENSE))
+ "RosenbrockExpEK1(2)" => Dict(:alg=>RosenbrockExpEK(order=2, smooth=DENSE))
+ "RosenbrockExpEK1(3)" => Dict(:alg=>RosenbrockExpEK(order=3, smooth=DENSE))
+]
+
+labels = first.(_setups)
+setups = last.(_setups)
+colors = [1 1 2 2 3 3]
+
+abstols = 1.0 ./ 10.0 .^ (6:10)
+reltols = 1.0 ./ 10.0 .^ (3:7)
+
+wp = WorkPrecisionSet(
+ prob, abstols, reltols, setups;
+ names = labels,
+ appxsol = test_sol,
+ dense = DENSE,
+ save_everystep = SAVE_EVERYSTEP,
+ maxiters = Int(1e7),
+ numruns = 5,
+ tstops = current_tspan,
+)
+
+plot(wp, title="Adaptive steps - no smoothing", color=colors)
+
+
+_ref_setups = [
+ "Tsit5" => Dict(:alg => Tsit5())
+ "Vern7" => Dict(:alg => Vern7())
+ "RadauIIA5" => Dict(:alg => RadauIIA5())
+]
+ref_labels = first.(_ref_setups)
+ref_setups = last.(_ref_setups)
+ref_wp_final = WorkPrecisionSet(
+ prob, abstols ./ 1000, reltols ./ 1000, ref_setups;
+ names = ref_labels,
+ appxsol = test_sol,
+ dense = false,
+ save_everystep = false,
+ maxiters = Int(1e7),
+ tstops = current_tspan,
+)
+ref_wp_dense = WorkPrecisionSet(
+ prob, abstols ./ 1000, reltols ./ 1000, ref_setups;
+ names = ref_labels,
+ appxsol = test_sol,
+ dense = true,
+ save_everystep = true,
+ maxiters = Int(1e7),
+ tstops = current_tspan,
+)
+
+plot!(ref_wp_final, x=:final, color=:gray, alpha=0.7, linestyle=:dash)
With smoothing
Code:
DENSE = SAVE_EVERYSTEP = true
+
+_setups = [
+ "EK0(2)" => Dict(:alg=>EK0(order=2, smooth=DENSE))
+ "EK0(3)" => Dict(:alg=>EK0(order=3, smooth=DENSE))
+ "EK1(2)" => Dict(:alg=>EK1(order=2, smooth=DENSE))
+ "EK1(3)" => Dict(:alg=>EK1(order=3, smooth=DENSE))
+ "RosenbrockExpEK1(2)" => Dict(:alg=>RosenbrockExpEK(order=2, smooth=DENSE))
+ "RosenbrockExpEK1(3)" => Dict(:alg=>RosenbrockExpEK(order=3, smooth=DENSE))
+]
+
+labels = first.(_setups)
+setups = last.(_setups)
+colors = [1 1 2 2 3 3]
+
+abstols = 1.0 ./ 10.0 .^ (6:10)
+reltols = 1.0 ./ 10.0 .^ (3:7)
+
+wp = WorkPrecisionSet(
+ prob, abstols, reltols, setups;
+ names = labels,
+ appxsol = test_sol,
+ dense = DENSE,
+ save_everystep = SAVE_EVERYSTEP,
+ maxiters = Int(1e7),
+ numruns = 5,
+ tstops = current_tspan,
+)
+
+plot(wp, title="Adaptive steps - with smoothing", color=colors)
+plot!(ref_wp_dense, x=:final, color=:gray, alpha=0.7, linestyle=:dash)
Interoplation errors (L2):
Code:
plot(wp, x=:L2, title="Adaptive steps - with smoothing", color=colors)
+plot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)
Calibration
Code:
plot(wp; x=:final, y=:chi2_final, yguide="Chi-squared (final)", color=colors)
+
+# Should be distributed according to a Chi-squared distribution:
+function plot_chisq_interval!(df, q=0.01)
+ dist = Chisq(df)
+ low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)
+ hline!([low, high], linestyle=:dash, color=:black, label="",
+ fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)
+ hline!([mid], linestyle=:solid, color=:black, label="")
+end
+plot_chisq_interval!(4)
Fixed-step solver comparison
Without smothing:
Code:
DENSE = SAVE_EVERYSTEP = false
+
+dts = 10.0 .^ range(-2, -3, length=10)[begin:end-1]
+abstols = reltols = repeat([missing], length(dts))
+
+DM = FixedDiffusion()
+_setups = [
+ "EK0(3)" => Dict(:alg=>EK0(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)
+ "EK1(3)" => Dict(:alg=>EK1(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)
+ "RosenbrockExpEK1(3)" => Dict(:alg=>RosenbrockExpEK(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)
+]
+
+labels = first.(_setups)
+setups = last.(_setups)
+colors = [1 2 3]
+
+wp = WorkPrecisionSet(
+ prob, abstols, reltols, setups;
+ adaptive = false,
+ names = labels,
+ appxsol = test_sol,
+ dense = DENSE,
+ save_everystep = SAVE_EVERYSTEP,
+ maxiters = Int(1e7),
+ numruns = 5,
+)
+
+plot(wp, title="Fixed steps - no smoothing", color=colors)
With smoothing:
Code:
DENSE = SAVE_EVERYSTEP = true
+
+dts = 10.0 .^ range(-2, -3, length=10)[begin:end-1]
+abstols = reltols = repeat([missing], length(dts))
+
+DM = FixedDiffusion()
+_setups = [
+ "EK0(3)" => Dict(:alg=>EK0(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)
+ "EK1(3)" => Dict(:alg=>EK1(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)
+ "RosenbrockExpEK1(3)" => Dict(:alg=>RosenbrockExpEK(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)
+]
+
+labels = first.(_setups)
+setups = last.(_setups)
+colors = [1 2 3]
+
+wp = WorkPrecisionSet(
+ prob, abstols, reltols, setups;
+ adaptive = false,
+ names = labels,
+ appxsol = test_sol,
+ dense = DENSE,
+ save_everystep = SAVE_EVERYSTEP,
+ maxiters = Int(1e7),
+ numruns = 5,
+)
+
+plot(wp, title="Fixed steps - with smoothing", color=colors)
Appendix
Computer information:
using InteractiveUtils
+InteractiveUtils.versioninfo()
Julia Version 1.10.0
+Commit 3120989f39b (2023-12-25 18:01 UTC)
+Build Info:
+ Official https://julialang.org/ release
+Platform Info:
+ OS: Linux (x86_64-linux-gnu)
+ CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz
+ WORD_SIZE: 64
+ LIBM: libopenlibm
+ LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)
+ Threads: 17 on 12 virtual cores
+Environment:
+ JULIA_NUM_THREADS = auto
Package information:
using Pkg
+Pkg.status()
Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`
+ [f3b72e0c] DiffEqDevTools v2.44.1
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+Info Packages marked with ⌃ have new versions available and may be upgradable.
Full manifest:
Pkg.status(mode=Pkg.PKGMODE_MANIFEST)
Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`
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+ [aae0fff6] LSODA_jll v0.1.2+0
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+ [935fb764] Xorg_libXcursor_jll v1.2.0+4
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+ [8ba89e20] Distributed
+ [f43a241f] Downloads v1.6.0
+ [7b1f6079] FileWatching
+ [9fa8497b] Future
+ [b77e0a4c] InteractiveUtils
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+ [b27032c2] LibCURL v0.6.4
+ [76f85450] LibGit2
+ [8f399da3] Libdl
+ [37e2e46d] LinearAlgebra
+ [56ddb016] Logging
+ [d6f4376e] Markdown
+ [a63ad114] Mmap
+ [ca575930] NetworkOptions v1.2.0
+ [44cfe95a] Pkg v1.10.0
+ [de0858da] Printf
+ [9abbd945] Profile
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+ [9a3f8284] Random
+ [ea8e919c] SHA v0.7.0
+ [9e88b42a] Serialization
+ [1a1011a3] SharedArrays
+ [6462fe0b] Sockets
+ [2f01184e] SparseArrays v1.10.0
+ [10745b16] Statistics v1.10.0
+ [4607b0f0] SuiteSparse
+ [fa267f1f] TOML v1.0.3
+ [a4e569a6] Tar v1.10.0
+ [8dfed614] Test
+ [cf7118a7] UUIDs
+ [4ec0a83e] Unicode
+ [e66e0078] CompilerSupportLibraries_jll v1.0.5+1
+ [deac9b47] LibCURL_jll v8.4.0+0
+ [e37daf67] LibGit2_jll v1.6.4+0
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+ [8e850b90] libblastrampoline_jll v5.8.0+1
+ [8e850ede] nghttp2_jll v1.52.0+1
+ [3f19e933] p7zip_jll v17.4.0+2
+Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`