calcium_homeo.jl

### A Pluto.jl notebook ###
# v0.19.26

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ bf5d2811-82db-4068-859d-0b2e6c6366e5
using PlutoUI, MinimallyDisruptiveCurves, Statistics, ModelingToolkit, Ipopt, Plots, LinearAlgebra, OrdinaryDiffEq, PlutoUI, JLD2, JuMP, ForwardDiff

# ╔═╡ e58a9316-ab83-11ec-277d-493810607706
md"""
Author: Andrea Ramirez-Hincapie air28@cam.ac.uk
# Calcium homeostasis in a STG neuron 

### Introduction
- Intracellular calcium concentration is an important signal to monitor in any neuron because it is involved in various metabolic and neuromodulatory processes. 

- For example, calcium acts as a second-messenger signalling through regulatory pathways for growth, energy efficiency, and genetic expression.

- This means that keeping calcium concentration within a tight range is essential to achieve proper function. We'll refer to this as calcium homeostasis. 


- But changes in the electrical activity imply changes in calcium concentration. Why? 

- Calcium closely correlates with a neuron's electrical activity due to voltage dependent $Ca^{2+}$ channels and buffering mechanisms that average out fluctuations in time and space. 

We'll breifly explain how the homeostatic constraints interact with naturally ocurring and ongoing electrical variations.

#### Homeostatic control of conductances

- Calcium is known to regulate the genetic expression of the proteins that conform ion channels through calcium-dependent promotion or repression of transcription factors. 

- The ion channel densities (or maximal conductances, $g$) determine the electrical activity of the neuron, which, in turn, determines the influx of calcium to the cell.

The feedback loop involving intracellular calcium concentration, $[Ca^{2+}]$, can be schematized as follows:
"""

# ╔═╡ 20283b0e-f892-4a9e-b970-e2455d69b466
PlutoUI.Resource("https://i.imgur.com/oJAfdlM.png")

# ╔═╡ 57e19144-a755-44ef-a98f-32e4a734af71
md"""
Image adapted from _O'Leary et. al. "Cell Types, Network Homeostasis, and Pathological Compensation from a Biologically Plausible Ion Channel Expression Model" Neuron 82.4 (2014)_
"""

# ╔═╡ 25702c07-af1c-4a24-94ef-9a972060b242
md"""

So intracellular calcium concentration 
- acts as a sensor, monitoring average activity over time, and 
- adjusts membrane conductances to achieve some kind of target calcium concentration or _homeostatic set-point_.

## What is done in this notebook

In this notebook we'll explore which sets of neuronal parameters (conductances) are compatible with the _homeostatic set-point_ by evolving curves where the change in average calcium is minimally disruptive. While we won't model the homeostatic controller, the minimally disruptive curves **for an appropriate cost function** can adjust the parameters to prevent deviation from calcium target.
"""

# ╔═╡ f470c746-7c63-4f95-a3cd-8803a75c1631
md"""
### 1. Build neuron model
- ``p`` is the vector of parameters, in this case the seven maximal conductances: 
fast sodium, Na ; delayed rectifier potassium, Kd ; fast transient and slow calcium, CaT and CaS; Ca-dependent potassium, KCa ; fast transient potassium, A and mixed cation, H.
- ``p_0`` are the initial parameters corresponding to a bursting behavior.
- Conductance values come from **Liu et. al. "A Model Neuron with Activity-Dependent Conductances Regulated by Multiple Calcium Sensors" J.Neuroscience (1998)**
"""

# ╔═╡ e1178ed2-a209-4759-bed8-e7402bc8681a
begin
	tspan = (0.0, 5000.0)
	p0 = [700.0, 4.0, 2.0, 50.0, 40.0, 70.0, 0.03]
end

# ╔═╡ a933890e-ca6d-4d00-9f30-cd5e56d0e531
function neuron(g_params)

    gNa, gCaS, gCaT, gKa, gKCa, gKd, gH = g_params
	
    τCa = 20.0
	Ca∞ = 0.05
	fxarea = 14.96 * 0.0628 #ca flux multiplier to get intracellular ca from currents

	defaults = Dict(Voltage => BasicVoltageDynamics(),
    Calcium => Liu.CalciumDynamics(τCa, Ca∞, fxarea), 
    Reversal{Calcium} => Liu.CaReversalDynamics())

	somatic_parameters = Dict(
    Reversal{Sodium} => 50.0,
    Reversal{Potassium} => -80.0,
    Reversal{Leak} => -50.0,
    Reversal{Proton} => -20.0,
    Voltage => -60.0,
    Calcium => 0.05,
    Reversal{Calcium} => 0.0)

    Liu_conv = Cm

    channels = [Liu.Na(gNa * Liu_conv), Liu.CaS(gCaS * Liu_conv), Liu.CaT(gCaT * Liu_conv), Liu.Ka(gKa * Liu_conv), Liu.KCa(gKCa * Liu_conv), Liu.Kdr(gKd * Liu_conv), Liu.H(gH * Liu_conv), Liu.leak(0.01)]

	b = BasicNeuron(NoGeometry(Cm), defaults, somatic_parameters, channels, :Liu_neuron)

	neur = b() #no synapses

    prob = ODEProblem(neur.sys, [], (0.0, 5000.0), [])

    sol = solve(prob, AutoTsit5(Rosenbrock23()), save_idxs=[1, 2], abstol=1e-9, reltol=1e-9)

    return prob, sol
end

# ╔═╡ b8ae9d21-1c1f-48fd-8ec2-a976c7064532
begin
    _, sol = neuron(p0)
    plot(sol)
end

# ╔═╡ e3edc18b-7771-41a7-a4cd-b255be08e3e2
md"""
### 2. Build a differentiable loss function

- Our output is a features-function ``f``. It is the low-pass filtered and averaged calcium trace over 2 seconds.

- Let's take our inital bursting neuron's average calcium as the set-point. That is, the output from our starting set of parameters is the target.
$Ca_{tgt} = mean(LPF(Ca(p_0,tspan)))$
$Ca_{tgt} = f(p_0)$

"""

# ╔═╡ 6388a94b-8d5e-41bb-a3a4-0895919f2e1b
function LowPassFilter(input, Twindow)
    f(u, p, t) = 1 / Twindow * (input(t) - u)
    u0 = mean(input(0:Twindow))
    prob = ODEProblem(f, u0, tspan)
    filt = solve(prob, Tsit5(); abstol=1e-6, reltol=1e-6)
    return filt
end

# ╔═╡ a8a68b13-2309-4afc-bdf8-02154d9a8149
function features(p)
    prob, sol = neuron(p)
    ca(t) = sol(t; idxs=2)
    filt_ca = LowPassFilter(ca, 2000)
    mean(tspan[1]:1:tspan[end]) do t
        filt_ca(t)
    end
end

# ╔═╡ 121d3961-a2b2-49e0-8c08-537b2fa2e1b6
Ca_tgt = features(p0)

# ╔═╡ 47dc83c5-31db-4d86-b274-9e0f212d0d80
md"""
Now we define the loss function 

$|f(p)-Ca_{tgt}|^2$
Remember that flowing along the valleys of this cost function traces a path in conductance space where calcium doesn't deviate too much from target. 

We use ForwardDiff to get the gradient of the loss function with respect to parameters.
"""

# ╔═╡ 92828ce2-89c9-4ca6-9fa0-fb4b70678993
function loss(p)
    loss = abs2(features(p) - Ca_tgt)
    return loss
end

# ╔═╡ 1293afc6-3d04-47d4-97da-4bb7a8907b39
function loss(p, g)
    g[:] = ForwardDiff.gradient(p) do p
        loss(p)
    end
    return loss(p)
end

# ╔═╡ 3c45b18a-2676-49e3-bcb6-224dd7824379
begin
    #check that loss is 0 at p0 and gradient is (almost) zero
    grad_template = deepcopy(p0)
    @show loss(p0, grad_template)
    @show grad_template
end

# ╔═╡ b851d3e9-ad60-41ed-b0c5-c68ff938cb64
md"""
#### Transforming the loss function
Since the values of the 7 conductances differ quite a bit in orders of magnitude, transforming the loss function with `log_abs` will help to see relative changes in conductances.
"""

# ╔═╡ 4bac91af-7699-4194-9f2b-397549dcdcdb
begin
    LA = logabs_transform(p0)
    C, θnew = transform_cost(loss, p0, LA)
    @show C(θnew, grad_template)
    @show grad_template
end

# ╔═╡ af151969-8880-4f3a-a3f0-6829dfabcf8e
md"""
### 3. Get initial curve directions

Next we generate potential directions just like in the NFκB example notebook. We select a few sparse (involving only a couple of parameters at a time) initial directions, which does require computing the hessian of our loss function once.
"""

# ╔═╡ d14c3869-5526-493d-9d62-ebd78d4af822
# @time H = ForwardDiff.hessian(C, θnew)

## we've precomputed H to save you some time
H =[ 37.7619    74.3368    77.8368   -52.6974   20.4354   -39.7287   13.5836
  74.3368   146.626    153.509   -103.983    40.2349   -78.3274   26.7624
  77.8368   153.509    160.705   -108.848    42.1341   -82.0099   28.009
 -52.6974  -103.983   -108.848     73.7499  -28.52      55.5434  -18.9764
  20.4354    40.2349    42.1341   -28.52     11.0615   -21.5011    7.35147
 -39.7287   -78.3274   -82.0099    55.5434  -21.5011    41.8504  -14.3015
  13.5836    26.7624    28.009    -18.9764    7.35147  -14.3015    4.89876]

# ╔═╡ 3155c379-a950-4697-b77d-02c981d6da2a
function sparse_init_dir(hessian; orthogonal_to=nothing, λ=1.0, start=randn(size(hessian)[1]), trim_level=1e-5)
    n = size(hessian)[1]
    model = Model(Ipopt.Optimizer)
    JuMP.set_silent(model)
    JuMP.@variable(model, x[1:n])
    JuMP.@variable(model, z[1:n])
    for i = 1:n
        set_start_value(x[i], start[i])
    end

    @objective(model, Min, x' * hessian * x + λ * sum(z))
    @constraint(model, x' * x == 1.0)
    @constraint(model, z .>= x)
    @constraint(model, z .>= -x)
    if !(orthogonal_to === nothing)
        for el in orthogonal_to
            @constraint(model, x' * el == 0)
        end
    end

    JuMP.optimize!(model)
    out = value.(x)
    out[abs.(out).<trim_level] .= 0
    val = out' * hessian * out
    return out, val
end

# ╔═╡ 4ef8bd5c-2cb0-44e3-aa87-4545c6babb00
function potential_dirs(H; λ=1.0)
    potential_dirs = [sparse_init_dir(H; λ)[1] for i = 1:100]
    pd_projs = [el' * H * el for el in potential_dirs] # ie how good is each guess
    inds = sortperm(pd_projs) # sort in order
    potential_dirs = potential_dirs[inds]

    duplicates = [] # remove potential_directions that are approximately equal (norm(a-b) < 1e-5) to previous directions
    for (i, el) in enumerate(potential_dirs)
        for (j, comp) in enumerate(potential_dirs[i+1:end])
            if ((norm(el - comp) < 1e-5) || (norm(el + comp) < 1e-5))
                push!(duplicates, i + j)
            end
        end
    end
    potential_dirs[unique(duplicates)]
    potential_dirs = potential_dirs[setdiff(1:length(potential_dirs), duplicates)]
end

# ╔═╡ 3b5d0c6b-02c4-455c-8378-ed7a79a91a01
dirs = potential_dirs(H)

# ╔═╡ 212ebe25-92e1-4870-b43f-da50aa13c933
num_dirs = size(dirs, 1)

# ╔═╡ 9b89335a-167d-40c6-9704-0eaa980b3c14
md"""
### 4. Evolve curves

- Experimental results show that 1/10 up to 7-fold variations in conductances can produce consistent electrical activity (a phenomenon known as degeneracy). 
- We can add these biologically-inspired bounds on our parameters with the callback `ParameterBounds` 

"""

# ╔═╡ d70bdeab-7956-45d9-9343-c7cf32d59569
function curve(δp0, half_dist, C, θ)
    init_dir = δp0
    span = (-1.0, half_dist)
    momentum = 1.0
    curve_prob = MDCProblem(C, θ, init_dir, momentum, span)
    cb = [Verbose([CurveDistance(0.0:0.1:span[2])]),
        ParameterBounds([i for i = 1:length(θ)], log(0.1) .+ θ, log(7) .+ θ)]
    @time mdc = evolve(curve_prob, Tsit5; momentum_tol=NaN, mdc_callback=cb,
        dense=true, adaptive=false, dt=0.05)
    return mdc
end

# ╔═╡ 7280f393-474e-401f-aa91-4cb2091c5f93
dist = 3.0 # the curve span in positive direction. You can make it shorter if you want

# ╔═╡ e5a239b0-4f2f-4165-bdea-c684a8aac64a
md"""
#### Choose a direction

Since it can take a couple of minutes to generate curves, we've commented the cell that calls `curve` so that it doesn't run again everytime you tweak the code above. 
You could also disable/enable that cell.

Run a single curve by selecting a particular direction.
"""

# ╔═╡ d7233d7e-2cd1-4119-813a-c8814c947f11
@bind i html"""<label for="dir_num">Initial direction i = </label> <input id="dir_num" value="1" type="number" min="1" max="17"/> """

# ╔═╡ c0292c71-7410-4ede-93f1-174a75fcd25d
#begin
#	 mdc = curve(dirs[i], dist, C, θnew)
#	 save_object("mdc_$i.jld2", mdc)
#end

# ╔═╡ 4dce4235-9971-43c9-ae89-54258d0c65dd
md"""
#### You can save a curve as an abstract object using JLD2 and easily recover it later! 
If after closing this notebook you want to opening a new session and load a curve, there's no need to generate it all over again, simply do :
"""

# ╔═╡ c4b40826-8fc4-4a49-841c-08cae0a05d57
mdc = load_object("calcium_homeo/mdc_1.jld2");

# ╔═╡ 4be2a848-5b18-49de-9e7f-38b661df159d

md"""
### 5. Animating the curves

The code below makes an animation by sampling points along the curve and simulating the neuron with those parameters. 

There are 3 plots to a gif.
Shown in the bottom are changes to conductances, ordered in the histogram by biggest changes, and the trajectory of the parameters in log scale. At the top, the voltage and average calcium of the neuron. We've zoomed in a bit on the x axis to see just three seconds of neuronal activity.

"""

# ╔═╡ 130376a2-b74d-445a-acd9-4a3328091c08
function sol_at_p(p)
        sol = neuron(p)[2]
        return sol.t, sol[1, :]
end

# ╔═╡ 0675f53c-ecd7-4934-961f-841ddfc8c228
function neur_traj(mdc, s)
        dist = round(s, digits=3)
        plot(sol_at_p(exp.(mdc(s)[:states])), label="V(t)", xlims=(2000,5000), ylims=(-75, 50), xlabel="Time (ms)", ylabel="mV", legend=:outertopright, title="distance $dist", fg_legend=:transparent, background_color_legend=nothing)
        plot!(t -> features(exp.(mdc(s)[:states])), xlabel="Time (ms)", ylabel="mV", color=:red, label="<Ca>")
    end

# ╔═╡ 5f5891ee-c2d5-4527-9da2-a90a7c9ea312
begin
@userplot varying_hist
    @recipe function f(vh::varying_hist)
        mdc, s = vh.args
        n = 7
        idxs = biggest_movers(mdc, n) #sort them
        tfirst = mdc.sol.t[1]
        orig_p = mdc(tfirst)[:states][idxs] #relative changes since parameters were log-transformed to run the curve
        now_p = mdc(s)[:states][idxs]

        pnames = ["gNa" "gCaS" "gCaT" "gKa" "gKCa" "gKd" "gH"]
        xnames = pnames[idxs]
        legend := false
        ylims := [-1.6, 1.7]
        seriestype := bar
        tickfontsize := 9
        xrotation := 45
        xticks := 1:n, xnames

        @series begin
            seriesalpha := 0.6
            seriescolor := [:blue, :pink, :green, :orange, :cyan, :purple, :yellow][idxs]
            1:n, now_p .- orig_p
        end
    end
end

# ╔═╡ 1fde1b55-cf75-4d26-9305-7b864d5f65bb
begin
 @userplot static_mdc
    @recipe function f(sm::static_mdc)
        mdc, s = sm.args
        now_p = mdc(s)[:states]
        cutoff = max(findfirst(distances(mdc) .> s), 1)
        legend := :outertopright
        fg_legend := :transparent
        background_color_legend := nothing
        legendfontsize --> 8
        label := ["gNa" "gCaS" "gCaT" "gKa" "gKCa" "gKd" "gH"]
        linecolor := [:blue :pink :green :orange :cyan :purple :yellow]
        xguide := "distance"
        xlims := [mdc.sol.t[1], mdc.sol.t[end]]
        yguide := "Δ parameters (log)"

        @series begin
            linewidth := 2.5
            distances(mdc)[1:cutoff], Δ(mdc)[:, 1:cutoff]'
        end

        @series begin
            label := nothing
            linestyle := :dash
            linewidth := 1.5
            distances(mdc), Δ(mdc)'
        end

        @series begin
            label := nothing
            seriestype := scatter
            markersize := 7
            seriescolor := [:blue, :pink, :green, :orange, :cyan, :purple, :yellow]
            repeat([s], 7), Δ(mdc, s)
        end
    end
end

# ╔═╡ 101cf791-e10e-4f7e-a879-2dcfc150ada4
function animate_it(mdc, name)
        anim = @animate for s = mdc.sol.t[1]:0.1:mdc.sol.t[end]
            l = @layout [
                a{1.0w}
                b{0.25w} c{0.75w}
            ]
            p1 = neur_traj(mdc, s)
            p2 = plot(varying_hist(mdc, s))
            p3 = plot(static_mdc(mdc, s))
            plot(p1, p2, p3, layout=l)
        end
        gif(anim, "$name.gif", fps=1.2)
end

# ╔═╡ 1fb2caa4-bfc9-41ab-a183-34a5b1fca899
#animate_it(mdc, "curve1")

# ╔═╡ cdc273c4-faee-41b0-bd88-404dbbfd6f35
md"""
$(PlutoUI.LocalResource("calcium_homeo/curve1.gif"))
"""

# ╔═╡ 2b7d38bb-2264-43c0-89a3-cf3bc0757603
md"""
### 6. Scientific analysis

We analyse the first curve. Its initial direction involves only T-type and slow calcium conductances. 

- Slow calcium and T-type calcium, as well as calcium-activated potassium channels, are a source of ultraslow negative feedback to the membrane potential and therefore crucial for bursting. 

- They mostly cooperate in shaping adaptation properties of the neuron.

- However, CaS and CaT tune different properties: 

	- Slow calcium channel density modulates intra-burst frequency

	- T-type calcium channels also play a role in fast excitability properties : they can generate spike upstroke (in the absence of sodium this is interesting!)

#### Static curve and cost

Around the starting point of the curve, see the interval (-0.3, 0.5), only CaS and CaT change. 

- They change **linearly**, indicating that correlations between the expression of ion channels are behind the observed degeneracy. In other words, it's not the absolute values of conductances that determine a cell's activity but the relative ratios of the conductances that it expresses.

- This agrees with many experimental and computational results ([Schulz et al., 2007](https://www.pnas.org/doi/full/10.1073/pnas.0705827104); [Zhao and Golowasch, 2012](https://www.jneurosci.org/content/32/39/13380); [O’Leary et al., 2013](https://www.pnas.org/doi/abs/10.1073/pnas.1309966110); [O’Leary et al., 2014](https://www.cell.com/neuron/pdf/S0896-6273(14)00292-X.pdf))


- They change in **opposite directions** because they both control burstiness. Increasing them together would result in higher burstiness. 
- In particular, increasing CaT would lead to a depolarization block. (The effect of increasing both calcium densities is predicted analytically [here](https://www.eneuro.org/content/2/1/ENEURO.0031-14.2015)).
- But then voltage-gated calcium channels which open upon depolarizarion would allow more calcium to flow inside the cell.

"""

# ╔═╡ e5dc02e9-005f-4736-8983-1db70c204c93
function threeplts(mdc)
    ps = ["gNa", "gCaS", "gCaT", "gKa", "gKCa", "gKd", "gH"]
    cc = [C(el) for el in eachcol(trajectory(mdc))]
    plt1 = plot(mdc; pnames=ps, what=:trajectory, title="log change in parameters", legendfontsize=6, fg_legend=:transparent, background_color_legend=nothing)
    plt2 = plot(distances(mdc), cc, ylabel="cost", xlabel="distance", title="cost over curve", legend=false, ylims=(0, 2))
    plt3 = plot(mdc; pnames=ps, what=:final_changes)
    plt = plot(plt1, plt2, plt3, layout=(3, 1), legendfontsize=6, fg_legend=:transparent)
end

# ╔═╡ 20226001-560d-4131-bf8c-896f1a840534
threeplts(mdc)

# ╔═╡ 0b3c43cd-1099-47e9-aede-00a128f4904f
md"""
When the initial direction is no longer a good one, more conductances start to move. 

- See the interval (0.5, 1.5) As T-type calcium and sodium increase, all potassium conductances decrease. This balances the excitability levels of the neuron


- In the last portion of this curve, changes in conductances roughly preserve behavior and therefore average calcium too.


- Note that the cost function stays well below 1 most of the time, wich means that the deviation from target calcium is << 10% (since $Ca_{tgt}$ is 10 $\mu M$).
"""

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git-tree-sha1 = "f548a9e9c490030e545f72074a41edfd0e5bcdd7"
uuid = "a759f4b9-e2f1-59dc-863e-4aeb61b1ea8f"
version = "0.5.23"

[[deps.Tokenize]]
git-tree-sha1 = "90538bf898832b6ebd900fa40f223e695970e3a5"
uuid = "0796e94c-ce3b-5d07-9a54-7f471281c624"
version = "0.5.25"

[[deps.TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "9a6ae7ed916312b41236fcef7e0af564ef934769"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.13"

[[deps.Transducers]]
deps = ["Adapt", "ArgCheck", "BangBang", "Baselet", "CompositionsBase", "DefineSingletons", "Distributed", "InitialValues", "Logging", "Markdown", "MicroCollections", "Requires", "Setfield", "SplittablesBase", "Tables"]
git-tree-sha1 = "25358a5f2384c490e98abd565ed321ffae2cbb37"
uuid = "28d57a85-8fef-5791-bfe6-a80928e7c999"
version = "0.4.76"

[[deps.TreeViews]]
deps = ["Test"]
git-tree-sha1 = "8d0d7a3fe2f30d6a7f833a5f19f7c7a5b396eae6"
uuid = "a2a6695c-b41b-5b7d-aed9-dbfdeacea5d7"
version = "0.3.0"

[[deps.TriangularSolve]]
deps = ["CloseOpenIntervals", "IfElse", "LayoutPointers", "LinearAlgebra", "LoopVectorization", "Polyester", "Static", "VectorizationBase"]
git-tree-sha1 = "31eedbc0b6d07c08a700e26d31298ac27ef330eb"
uuid = "d5829a12-d9aa-46ab-831f-fb7c9ab06edf"
version = "0.1.19"

[[deps.Tricks]]
git-tree-sha1 = "aadb748be58b492045b4f56166b5188aa63ce549"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.7"

[[deps.TruncatedStacktraces]]
deps = ["InteractiveUtils", "MacroTools", "Preferences"]
git-tree-sha1 = "7bc1632a4eafbe9bd94cf1a784a9a4eb5e040a91"
uuid = "781d530d-4396-4725-bb49-402e4bee1e77"
version = "1.3.0"

[[deps.URIs]]
git-tree-sha1 = "074f993b0ca030848b897beff716d93aca60f06a"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.4.2"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
version = "1.0.2"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unitful]]
deps = ["ConstructionBase", "Dates", "LinearAlgebra", "Random"]
git-tree-sha1 = "ba4aa36b2d5c98d6ed1f149da916b3ba46527b2b"
uuid = "1986cc42-f94f-5a68-af5c-568840ba703d"
version = "1.14.0"

    [deps.Unitful.extensions]
    InverseFunctionsUnitfulExt = "InverseFunctions"

    [deps.Unitful.weakdeps]
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"

[[deps.UnitfulLatexify]]
deps = ["LaTeXStrings", "Latexify", "Unitful"]
git-tree-sha1 = "e2d817cc500e960fdbafcf988ac8436ba3208bfd"
uuid = "45397f5d-5981-4c77-b2b3-fc36d6e9b728"
version = "1.6.3"

[[deps.Unityper]]
deps = ["ConstructionBase"]
git-tree-sha1 = "d5f4ec8c22db63bd3ccb239f640e895cfde145aa"
uuid = "a7c27f48-0311-42f6-a7f8-2c11e75eb415"
version = "0.1.2"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.VectorizationBase]]
deps = ["ArrayInterface", "CPUSummary", "HostCPUFeatures", "IfElse", "LayoutPointers", "Libdl", "LinearAlgebra", "SIMDTypes", "Static", "StaticArrayInterface"]
git-tree-sha1 = "b182207d4af54ac64cbc71797765068fdeff475d"
uuid = "3d5dd08c-fd9d-11e8-17fa-ed2836048c2f"
version = "0.21.64"

[[deps.VertexSafeGraphs]]
deps = ["Graphs"]
git-tree-sha1 = "8351f8d73d7e880bfc042a8b6922684ebeafb35c"
uuid = "19fa3120-7c27-5ec5-8db8-b0b0aa330d6f"
version = "0.2.0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "ed8d92d9774b077c53e1da50fd81a36af3744c1c"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.21.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "93c41695bc1c08c46c5899f4fe06d6ead504bb73"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.10.3+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+0"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.5+0"

[[deps.ZygoteRules]]
deps = ["ChainRulesCore", "MacroTools"]
git-tree-sha1 = "977aed5d006b840e2e40c0b48984f7463109046d"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.3"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.7.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

# ╔═╡ Cell order:
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