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Hodgkin-Huxley benchmark

Summary

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 smoothing:

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 1.12.4
Commit 01a2eadb047 (2026-01-06 16:56 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD Ryzen Threadripper PRO 7985WX 64-Cores
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver4)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 128 virtual cores)
Environment:
  LD_LIBRARY_PATH = /.singularity.d/libs
Package information:
using Pkg
Pkg.status()
Status `/home/nrbosch/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`
  [f3b72e0c] DiffEqDevTools v2.49.0
  [31c24e10] Distributions v0.25.123
  [7073ff75] IJulia v1.34.3
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  [961ee093] ModelingToolkit v11.10.0
  [54ca160b] ODEInterface v0.5.0
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  [1dea7af3] OrdinaryDiffEq v6.108.0
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  [91a5bcdd] Plots v1.41.5
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Status `/home/nrbosch/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`
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  [988b38a3] ReadOnlyArrays v0.2.0
  [795d4caa] ReadOnlyDicts v1.0.1
  [3cdcf5f2] RecipesBase v1.3.4
  [01d81517] RecipesPipeline v0.6.12
  [731186ca] RecursiveArrayTools v3.48.0
  [189a3867] Reexport v1.2.2
  [05181044] RelocatableFolders v1.0.1
  [ae029012] Requires v1.3.1
  [ae5879a3] ResettableStacks v1.2.0
  [79098fc4] Rmath v0.9.0
  [47965b36] RootedTrees v2.25.0
  [7e49a35a] RuntimeGeneratedFunctions v0.5.17
  [9dfe8606] SCCNonlinearSolve v1.10.0
  [94e857df] SIMDTypes v0.1.0
  [476501e8] SLEEFPirates v0.6.43
  [0bca4576] SciMLBase v2.138.1
  [19f34311] SciMLJacobianOperators v0.1.12
  [a6db7da4] SciMLLogging v1.9.0
  [c0aeaf25] SciMLOperators v1.15.1
  [431bcebd] SciMLPublic v1.0.1
  [53ae85a6] SciMLStructures v1.10.0
  [505e40e9] SciPyDiffEq v0.2.2
  [6c6a2e73] Scratch v1.3.0
  [91c51154] SentinelArrays v1.4.9
  [efcf1570] Setfield v1.1.2
  [1277b4bf] ShiftedArrays v2.0.0
  [992d4aef] Showoff v1.0.3
  [777ac1f9] SimpleBufferStream v1.2.0
  [727e6d20] SimpleNonlinearSolve v2.10.0
  [699a6c99] SimpleTraits v0.9.5
  [ce78b400] SimpleUnPack v1.1.0
  [a2af1166] SortingAlgorithms v1.2.2
  [0a514795] SparseMatrixColorings v0.4.23
  [276daf66] SpecialFunctions v2.7.1
  [860ef19b] StableRNGs v1.0.4
  [64909d44] StateSelection v1.3.0
  [aedffcd0] Static v1.3.1
  [0d7ed370] StaticArrayInterface v1.9.0
  [90137ffa] StaticArrays v1.9.16
  [1e83bf80] StaticArraysCore v1.4.4
  [10745b16] Statistics v1.11.1
  [82ae8749] StatsAPI v1.8.0
  [2913bbd2] StatsBase v0.34.10
  [4c63d2b9] StatsFuns v1.5.2
  [3eaba693] StatsModels v0.7.9
  [7792a7ef] StrideArraysCore v0.5.8
  [69024149] StringEncodings v0.3.7
  [892a3eda] StringManipulation v0.4.2
  [09ab397b] StructArrays v0.7.2
  [c3572dad] Sundials v5.1.0
  [3384d301] SymbolicCompilerPasses v0.1.2
  [2efcf032] SymbolicIndexingInterface v0.3.46
  [19f23fe9] SymbolicLimits v1.1.0
  [d1185830] SymbolicUtils v4.18.2
  [0c5d862f] Symbolics v7.15.1
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.12.1
  [ed4db957] TaskLocalValues v0.1.3
  [92b13dbe] TaylorIntegration v0.18.2
  [6aa5eb33] TaylorSeries v0.20.10
  [62fd8b95] TensorCore v0.1.1
  [8ea1fca8] TermInterface v2.0.0
  [8290d209] ThreadingUtilities v0.5.5
  [a759f4b9] TimerOutputs v0.5.29
  [c751599d] ToeplitzMatrices v0.8.5
  [3bb67fe8] TranscodingStreams v0.11.3
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.6.1
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [41fe7b60] Unzip v0.2.0
  [3d5dd08c] VectorizationBase v0.21.72
  [81def892] VersionParsing v1.3.0
  [d30d5f5c] WeakCacheSets v0.1.0
  [44d3d7a6] Weave v0.10.12
  [1b915085] WinReg v1.0.0
  [ddb6d928] YAML v0.4.16
  [c2297ded] ZMQ v1.5.1
  [0518478a] deSolveDiffEq v1.0.0
  [6e34b625] Bzip2_jll v1.0.9+0
  [83423d85] Cairo_jll v1.18.5+1
  [ee1fde0b] Dbus_jll v1.16.2+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
  [2e619515] Expat_jll v2.7.3+0
  [b22a6f82] FFMPEG_jll v8.0.1+0
  [f5851436] FFTW_jll v3.3.11+0
  [a3f928ae] Fontconfig_jll v2.17.1+0
  [d7e528f0] FreeType2_jll v2.13.4+0
  [559328eb] FriBidi_jll v1.0.17+0
  [0656b61e] GLFW_jll v3.4.1+0
  [d2c73de3] GR_jll v0.73.22+0
  [b0724c58] GettextRuntime_jll v0.22.4+0
  [61579ee1] Ghostscript_jll v9.55.1+0
  [7746bdde] Glib_jll v2.86.3+0
  [3b182d85] Graphite2_jll v1.3.15+0
  [2e76f6c2] HarfBuzz_jll v8.5.1+0
  [1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
  [aacddb02] JpegTurbo_jll v3.1.4+0
  [c1c5ebd0] LAME_jll v3.100.3+0
  [88015f11] LERC_jll v4.0.1+0
  [1d63c593] LLVMOpenMP_jll v18.1.8+0
  [aae0fff6] LSODA_jll v0.1.2+0
  [dd4b983a] LZO_jll v2.10.3+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
  [7e76a0d4] Libglvnd_jll v1.7.1+1
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.41.3+0
  [89763e89] Libtiff_jll v4.7.2+0
  [38a345b3] Libuuid_jll v2.41.3+0
  [856f044c] MKL_jll v2025.2.0+0
  [c8ffd9c3] MbedTLS_jll v2.28.1010+0
  [c771fb93] ODEInterface_jll v0.0.2+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [656ef2d0] OpenBLAS32_jll v0.3.30+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.6.1+0
  [36c8627f] Pango_jll v1.57.0+0
⌅ [30392449] Pixman_jll v0.44.2+0
  [c0090381] Qt6Base_jll v6.8.2+2
  [629bc702] Qt6Declarative_jll v6.8.2+1
  [ce943373] Qt6ShaderTools_jll v6.8.2+1
  [e99dba38] Qt6Wayland_jll v6.8.2+2
  [f50d1b31] Rmath_jll v0.5.1+0
  [ca45d3f4] SuiteSparse32_jll v7.12.1+0
  [fb77eaff] Sundials_jll v7.5.0+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.2+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.13+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.8+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.2+0
  [a51aa0fd] Xorg_libXi_jll v1.8.3+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.2.0+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.44.0+0
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [8f1865be] ZeroMQ_jll v4.3.6+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.14+0
  [214eeab7] fzf_jll v0.61.1+0
  [a4ae2306] libaom_jll v3.13.1+0
  [0ac62f75] libass_jll v0.17.4+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
  [b53b4c65] libpng_jll v1.6.55+0
  [a9144af2] libsodium_jll v1.0.21+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.0.0+1
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.7.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.3.0
  [44cfe95a] Pkg v1.12.1
  [de0858da] Printf v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.12.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.3.0+1
  [deac9b47] LibCURL_jll v8.15.0+0
  [e37daf67] LibGit2_jll v1.9.0+0
  [29816b5a] LibSSH2_jll v1.11.3+1
  [14a3606d] MozillaCACerts_jll v2025.11.4
  [4536629a] OpenBLAS_jll v0.3.29+0
  [05823500] OpenLibm_jll v0.8.7+0
  [458c3c95] OpenSSL_jll v3.5.4+0
  [efcefdf7] PCRE2_jll v10.44.0+1
  [bea87d4a] SuiteSparse_jll v7.8.3+2
  [83775a58] Zlib_jll v1.3.1+2
  [8e850b90] libblastrampoline_jll v5.15.0+0
  [8e850ede] nghttp2_jll v1.64.0+1
  [3f19e933] p7zip_jll v17.7.0+0
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`