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Solving Inverse Problems with Spectral Dynamic Causal Modeling

Introduction

Neuroblox provides you with a comprehensive environment for simulations as we have explored previously, but its functionality doesn't stop there. We will now pivot and turn our attention to a different kind of problem: inferring model parameters, that is solving inverse problems, from time series. The method of choice is one of the most widely spread in imaging neuroscience, spectral Dynamic Causal Modeling (spDCM)[1,2]. In this tutorial we will introduce how to perform a spDCM analysis on simulated data. To do so we roughly reproduce the procedure in the SPM12 script DEM_demo_induced_fMRI.m in Neuroblox. This work was also presented in Hofmann et al.[2]

In this tutorial we will define a circuit of three linear neuronal mass models, all driven by an Ornstein-Uhlenbeck process. We will model fMRI data by a balloon model and BOLD signal on top. After simulation of this simple model we will use spectral Dynamic Causal Modeling to infer some of the model parameters from the simulation time series.

Workflow illustration

A brief outline of the procedure we will pursue:

  • define the graph, add blocks -> section A, B and C in the figure
  • simulate the model -> instead we could also use actual data, section D in figure
  • compute the cross spectral density
  • setup the DCM
  • estimate parameters
  • plot the results
using Neuroblox
using LinearAlgebra
using StochasticDiffEq
using DataFrames
using OrderedCollections
using CairoMakie
using ModelingToolkit

Model simulation

Define the model

We will define a model of 3 regions. This means first of all to define a graph. To this graph we will add three linear neuronal mass models which constitute the (hidden) neuronal dynamics. These constitute three nodes of the graph. Next we will also need some input that stimulates the activity, we use simple Ornstein-Uhlenbeck blocks to create stochastic inputs. One per region. We want to simulate fMRI signals thus we will need to also add a BalloonModel per region. Note that the Ornstein-Uhlenbeck block will feed into the linear neural mass which in turn will feed into the BalloonModel blox. This needs to be represented by the way we define the edges.

nr = 3             # number of regions
g = MetaDiGraph()
regions = [];   # list of neural mass blocks to then connect them to each other with an adjacency matrix `A_true`

Now add the different blocks to each region and connect the blocks within each region:

for i = 1:nr
    region = LinearNeuralMass(;name=Symbol("r$(i)₊lm"))
    push!(regions, region)          # store neural mass model for connection of regions

    # add Ornstein-Uhlenbeck block as noisy input to the current region
    input = OUBlox(;name=Symbol("r$(i)₊ou"), σ=0.1)
    add_edge!(g, input => region; :weight => 1/16)   # Note that 1/16 is taken from SPM12, this stabilizes the balloon model simulation. Alternatively the noise of the Ornstein-Uhlenbeck block or the weight of the edge connecting neuronal activity and balloon model could be reduced to guarantee numerical stability.

    # simulate fMRI signal with BalloonModel which includes the BOLD signal on top of the balloon model dynamics
    measurement = BalloonModel(;name=Symbol("r$(i)₊bm"))
    add_edge!(g, region => measurement; :weight => 1.0)
end

Next we define the between-region connectivity matrix and make sure that it is diagonally dominant to guarantee numerical stability (see Gershgorin theorem).

A_true = 0.1*randn(nr, nr)
A_true -= diagm(map(a -> sum(abs, a), eachrow(A_true)))    # ensure diagonal dominance of matrix
3×3 Matrix{Float64}:
 -0.220731  -0.123353   0.0612029
 -0.121674  -0.140556  -0.0188823
 -0.177178   0.179645  -0.660217

Instead of a random matrix use the same matrix as is defined in [3]

A_true = [[-0.5 -2 0]; [0.4 -0.5 -0.3]; [0 0.2 -0.5]]
for idx in CartesianIndices(A_true)
    add_edge!(g, regions[idx[1]] => regions[idx[2]]; :weight => A_true[idx[1], idx[2]])
end

finally we compose the simulation model

@named simmodel = system_from_graph(g, split=false)

\[ \begin{align} \frac{\mathrm{d} \mathtt{r1.ou.x}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{r1.ou.\mu} - \mathtt{r1.ou.x}\left( t \right)}{\mathtt{r1.ou.\tau}} \\ \frac{\mathrm{d} \mathtt{r1.lm.x}\left( t \right)}{\mathrm{d}t} &= \mathtt{w\_r1.lm\_r1.lm} \mathtt{r1.lm.x}\left( t \right) + \mathtt{w\_r1.ou\_r1.lm} \mathtt{r1.ou.x}\left( t \right) + \mathtt{w\_r2.lm\_r1.lm} \mathtt{r2.lm.x}\left( t \right) + \mathtt{w\_r3.lm\_r1.lm} \mathtt{r3.lm.x}\left( t \right) \\ \frac{\mathrm{d} \mathtt{r1.bm.s}\left( t \right)}{\mathrm{d}t} &= - 0.32 \left( -1 + e^{\mathtt{r1.bm.lnu}\left( t \right)} \right) + \mathtt{w\_r1.lm\_r1.bm} \mathtt{r1.lm.x}\left( t \right) - 0.64 \mathtt{r1.bm.s}\left( t \right) e^{\mathtt{r1.bm.ln\kappa}} \\ \frac{\mathrm{d} \mathtt{r1.bm.lnu}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{r1.bm.s}\left( t \right)}{e^{\mathtt{r1.bm.lnu}\left( t \right)}} \\ \frac{\mathrm{d} \mathtt{r1.bm.ln\nu}\left( t \right)}{\mathrm{d}t} &= \frac{e^{\mathtt{r1.bm.lnu}\left( t \right)} - \left( e^{\mathtt{r1.bm.ln\nu}\left( t \right)} \right)^{3.125}}{2 e^{\mathtt{r1.bm.ln\tau}} e^{\mathtt{r1.bm.ln\nu}\left( t \right)}} \\ \frac{\mathrm{d} \mathtt{r1.bm.lnq}\left( t \right)}{\mathrm{d}t} &= \frac{\frac{2.5 e^{\mathtt{r1.bm.lnu}\left( t \right)} \left( 1 - 0.6^{\frac{1}{e^{\mathtt{r1.bm.lnu}\left( t \right)}}} \right)}{e^{\mathtt{r1.bm.lnq}\left( t \right)}} - \left( e^{\mathtt{r1.bm.ln\nu}\left( t \right)} \right)^{2.125}}{2 e^{\mathtt{r1.bm.ln\tau}}} \\ \frac{\mathrm{d} \mathtt{r2.ou.x}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{r2.ou.\mu} - \mathtt{r2.ou.x}\left( t \right)}{\mathtt{r2.ou.\tau}} \\ \frac{\mathrm{d} \mathtt{r2.lm.x}\left( t \right)}{\mathrm{d}t} &= \mathtt{w\_r1.lm\_r2.lm} \mathtt{r1.lm.x}\left( t \right) + \mathtt{w\_r2.lm\_r2.lm} \mathtt{r2.lm.x}\left( t \right) + \mathtt{w\_r2.ou\_r2.lm} \mathtt{r2.ou.x}\left( t \right) + \mathtt{w\_r3.lm\_r2.lm} \mathtt{r3.lm.x}\left( t \right) \\ \frac{\mathrm{d} \mathtt{r2.bm.s}\left( t \right)}{\mathrm{d}t} &= - 0.32 \left( -1 + e^{\mathtt{r2.bm.lnu}\left( t \right)} \right) + \mathtt{w\_r2.lm\_r2.bm} \mathtt{r2.lm.x}\left( t \right) - 0.64 \mathtt{r2.bm.s}\left( t \right) e^{\mathtt{r2.bm.ln\kappa}} \\ \frac{\mathrm{d} \mathtt{r2.bm.lnu}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{r2.bm.s}\left( t \right)}{e^{\mathtt{r2.bm.lnu}\left( t \right)}} \\ \frac{\mathrm{d} \mathtt{r2.bm.ln\nu}\left( t \right)}{\mathrm{d}t} &= \frac{e^{\mathtt{r2.bm.lnu}\left( t \right)} - \left( e^{\mathtt{r2.bm.ln\nu}\left( t \right)} \right)^{3.125}}{2 e^{\mathtt{r2.bm.ln\nu}\left( t \right)} e^{\mathtt{r2.bm.ln\tau}}} \\ \frac{\mathrm{d} \mathtt{r2.bm.lnq}\left( t \right)}{\mathrm{d}t} &= \frac{\frac{2.5 \left( 1 - 0.6^{\frac{1}{e^{\mathtt{r2.bm.lnu}\left( t \right)}}} \right) e^{\mathtt{r2.bm.lnu}\left( t \right)}}{e^{\mathtt{r2.bm.lnq}\left( t \right)}} - \left( e^{\mathtt{r2.bm.ln\nu}\left( t \right)} \right)^{2.125}}{2 e^{\mathtt{r2.bm.ln\tau}}} \\ \frac{\mathrm{d} \mathtt{r3.ou.x}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{r3.ou.\mu} - \mathtt{r3.ou.x}\left( t \right)}{\mathtt{r3.ou.\tau}} \\ \frac{\mathrm{d} \mathtt{r3.lm.x}\left( t \right)}{\mathrm{d}t} &= \mathtt{w\_r1.lm\_r3.lm} \mathtt{r1.lm.x}\left( t \right) + \mathtt{w\_r2.lm\_r3.lm} \mathtt{r2.lm.x}\left( t \right) + \mathtt{w\_r3.lm\_r3.lm} \mathtt{r3.lm.x}\left( t \right) + \mathtt{w\_r3.ou\_r3.lm} \mathtt{r3.ou.x}\left( t \right) \\ \frac{\mathrm{d} \mathtt{r3.bm.s}\left( t \right)}{\mathrm{d}t} &= - 0.32 \left( -1 + e^{\mathtt{r3.bm.lnu}\left( t \right)} \right) + \mathtt{w\_r3.lm\_r3.bm} \mathtt{r3.lm.x}\left( t \right) - 0.64 e^{\mathtt{r3.bm.ln\kappa}} \mathtt{r3.bm.s}\left( t \right) \\ \frac{\mathrm{d} \mathtt{r3.bm.lnu}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{r3.bm.s}\left( t \right)}{e^{\mathtt{r3.bm.lnu}\left( t \right)}} \\ \frac{\mathrm{d} \mathtt{r3.bm.ln\nu}\left( t \right)}{\mathrm{d}t} &= \frac{e^{\mathtt{r3.bm.lnu}\left( t \right)} - \left( e^{\mathtt{r3.bm.ln\nu}\left( t \right)} \right)^{3.125}}{2 e^{\mathtt{r3.bm.ln\nu}\left( t \right)} e^{\mathtt{r3.bm.ln\tau}}} \\ \frac{\mathrm{d} \mathtt{r3.bm.lnq}\left( t \right)}{\mathrm{d}t} &= \frac{\frac{2.5 \left( 1 - 0.6^{\frac{1}{e^{\mathtt{r3.bm.lnu}\left( t \right)}}} \right) e^{\mathtt{r3.bm.lnu}\left( t \right)}}{e^{\mathtt{r3.bm.lnq}\left( t \right)}} - \left( e^{\mathtt{r3.bm.ln\nu}\left( t \right)} \right)^{2.125}}{2 e^{\mathtt{r3.bm.ln\tau}}} \\ 0 &= - \mathtt{r1.bm.bold}\left( t \right) + 4 \left( 3.7726 + \frac{ - 0.4 e^{\mathtt{r1.bm.ln\epsilon}} e^{\mathtt{r1.bm.lnq}\left( t \right)}}{e^{\mathtt{r1.bm.ln\nu}\left( t \right)}} - 0.6 e^{\mathtt{r1.bm.ln\epsilon}} - 2.7726 e^{\mathtt{r1.bm.lnq}\left( t \right)} + \left( -1 + e^{\mathtt{r1.bm.ln\epsilon}} \right) e^{\mathtt{r1.bm.ln\nu}\left( t \right)} \right) \\ 0 &= - \mathtt{r2.bm.bold}\left( t \right) + 4 \left( 3.7726 - 0.6 e^{\mathtt{r2.bm.ln\epsilon}} - 2.7726 e^{\mathtt{r2.bm.lnq}\left( t \right)} + \frac{ - 0.4 e^{\mathtt{r2.bm.ln\epsilon}} e^{\mathtt{r2.bm.lnq}\left( t \right)}}{e^{\mathtt{r2.bm.ln\nu}\left( t \right)}} + \left( -1 + e^{\mathtt{r2.bm.ln\epsilon}} \right) e^{\mathtt{r2.bm.ln\nu}\left( t \right)} \right) \\ 0 &= - \mathtt{r3.bm.bold}\left( t \right) + 4 \left( 3.7726 - 0.6 e^{\mathtt{r3.bm.ln\epsilon}} - 2.7726 e^{\mathtt{r3.bm.lnq}\left( t \right)} + \frac{ - 0.4 e^{\mathtt{r3.bm.ln\epsilon}} e^{\mathtt{r3.bm.lnq}\left( t \right)}}{e^{\mathtt{r3.bm.ln\nu}\left( t \right)}} + e^{\mathtt{r3.bm.ln\nu}\left( t \right)} \left( -1 + e^{\mathtt{r3.bm.ln\epsilon}} \right) \right) \end{align} \]

Run the simulation and plot the results

setup simulation of the model, time in seconds

tspan = (0.0, 612.0)
prob = SDEProblem(simmodel, [], tspan)
dt = 2.0   # two seconds as measurement interval for fMRI
sol = solve(prob, ImplicitRKMil(), saveat=dt);

plot bold signal time series

idx_m = get_idx_tagged_vars(simmodel, "measurement")    # get index of bold signal
f = Figure()
ax = Axis(f[1, 1],
    title = "fMRI time series",
    xlabel = "Time [s]",
    ylabel = "BOLD",
)
lines!(ax, sol, idxs=idx_m)
f
Example block output

We note that the initial spike is not meaningful and a result of the equilibration of the stochastic process thus we remove it.

dfsol = DataFrame(sol[ceil(Int, 101/dt):end]);

Estimate and plot the cross-spectral densities

data = Matrix(dfsol[:, idx_m]);

We compute the cross-spectral density by fitting a linear model of order p and then compute the csd analytically from the parameters of the multivariate autoregressive model

p = 8
mar = mar_ml(data, p)   # maximum likelihood estimation of the MAR coefficients and noise covariance matrix
ns = size(data, 1)
freq = range(min(128, ns*dt)^-1, max(8, 2*dt)^-1, 32)
csd = mar2csd(mar, freq, dt^-1);

Now plot the cross-spectrum:

fig = Figure(size=(1200, 800))
grid = fig[1, 1] = GridLayout()
for i = 1:nr
    for j = 1:nr
        ax = Axis(grid[i, j])
        lines!(ax, freq, real.(csd[:, i, j]))
    end
end
fig
Example block output

Model Inference

We will now assemble a new model that is used for fitting the previous simulations. This procedure is similar to before with the difference that we will define global parameters and use tags such as [tunable=false/true] to define which parameters we will want to estimate. Note that parameters are tunable by default.

g = MetaDiGraph()
regions = [];   # list of neural mass blocks to then connect them to each other with an adjacency matrix `A`

The following parameters are shared accross regions, which is why we define them here.

@parameters lnκ=0.0 [tunable=false] lnϵ=0.0 [tunable=false] lnτ=0.0 [tunable=false]   # lnκ: decay parameter for hemodynamics; lnϵ: ratio of intra- to extra-vascular components, lnτ: transit time scale
@parameters C=1/16 [tunable=false]   # note that C=1/16 is taken from SPM12 and stabilizes the balloon model simulation. See also comment above.

for i = 1:nr
    region = LinearNeuralMass(;name=Symbol("r$(i)₊lm"))
    push!(regions, region)
    input = ExternalInput(;name=Symbol("r$(i)₊ei"))
    add_edge!(g, input => region; :weight => C)

    # we assume fMRI signal and model them with a BalloonModel
    measurement = BalloonModel(;name=Symbol("r$(i)₊bm"), lnτ=lnτ, lnκ=lnκ, lnϵ=lnϵ)
    add_edge!(g, region => measurement; :weight => 1.0)
end

A_prior = 0.01*randn(nr, nr)
A_prior -= diagm(diag(A_prior))    # remove the diagonal
3×3 Matrix{Float64}:
 0.0          -0.00639533  -0.0176601
 0.013165      0.0          0.0103215
 0.000412903  -0.0207301    0.0

Since we want to optimize these weights we turn them into symbolic parameters: Add the symbolic weights to the edges and connect regions.

A = []
for (i, a) in enumerate(vec(A_prior))
    symb = Symbol("A$(i)")
    push!(A, only(@parameters $symb = a))
end

With the function untune!` we can list indices of parameters whose tunable flag should be set to false. For instance the first element in the second row:

untune!(A, [])
for (i, idx) in enumerate(CartesianIndices(A_prior))
    if idx[1] == idx[2]
        add_edge!(g, regions[idx[1]] => regions[idx[2]]; :weight => -exp(A[i])/2)  # -exp(A[i])/2: treatement of diagonal elements in SPM12 to make diagonal dominance (see Gershgorin Theorem) more likely but it is not guaranteed
    else
        add_edge!(g, regions[idx[2]] => regions[idx[1]]; :weight => A[i])
    end
end

@named fitmodel = system_from_graph(g, split=false)

\[ \begin{align} \frac{\mathrm{d} \mathtt{r1.lm.x}\left( t \right)}{\mathrm{d}t} &= \mathtt{r1.lm.jcn}\left( t \right) \\ \frac{\mathrm{d} \mathtt{r1.bm.s}\left( t \right)}{\mathrm{d}t} &= \mathtt{r1.bm.jcn}\left( t \right) - 0.32 \left( -1 + e^{\mathtt{r1.bm.lnu}\left( t \right)} \right) - 0.64 e^{\mathtt{ln\kappa}} \mathtt{r1.bm.s}\left( t \right) \\ \frac{\mathrm{d} \mathtt{r1.bm.lnu}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{r1.bm.s}\left( t \right)}{e^{\mathtt{r1.bm.lnu}\left( t \right)}} \\ \frac{\mathrm{d} \mathtt{r1.bm.ln\nu}\left( t \right)}{\mathrm{d}t} &= \frac{e^{\mathtt{r1.bm.lnu}\left( t \right)} - \left( e^{\mathtt{r1.bm.ln\nu}\left( t \right)} \right)^{3.125}}{2 e^{\mathtt{ln\tau}} e^{\mathtt{r1.bm.ln\nu}\left( t \right)}} \\ \frac{\mathrm{d} \mathtt{r1.bm.lnq}\left( t \right)}{\mathrm{d}t} &= \frac{\frac{2.5 e^{\mathtt{r1.bm.lnu}\left( t \right)} \left( 1 - 0.6^{\frac{1}{e^{\mathtt{r1.bm.lnu}\left( t \right)}}} \right)}{e^{\mathtt{r1.bm.lnq}\left( t \right)}} - \left( e^{\mathtt{r1.bm.ln\nu}\left( t \right)} \right)^{2.125}}{2 e^{\mathtt{ln\tau}}} \\ \frac{\mathrm{d} \mathtt{r2.lm.x}\left( t \right)}{\mathrm{d}t} &= \mathtt{r2.lm.jcn}\left( t \right) \\ \frac{\mathrm{d} \mathtt{r2.bm.s}\left( t \right)}{\mathrm{d}t} &= - 0.32 \left( -1 + e^{\mathtt{r2.bm.lnu}\left( t \right)} \right) + \mathtt{r2.bm.jcn}\left( t \right) - 0.64 e^{\mathtt{ln\kappa}} \mathtt{r2.bm.s}\left( t \right) \\ \frac{\mathrm{d} \mathtt{r2.bm.lnu}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{r2.bm.s}\left( t \right)}{e^{\mathtt{r2.bm.lnu}\left( t \right)}} \\ \frac{\mathrm{d} \mathtt{r2.bm.ln\nu}\left( t \right)}{\mathrm{d}t} &= \frac{e^{\mathtt{r2.bm.lnu}\left( t \right)} - \left( e^{\mathtt{r2.bm.ln\nu}\left( t \right)} \right)^{3.125}}{2 e^{\mathtt{ln\tau}} e^{\mathtt{r2.bm.ln\nu}\left( t \right)}} \\ \frac{\mathrm{d} \mathtt{r2.bm.lnq}\left( t \right)}{\mathrm{d}t} &= \frac{\frac{2.5 \left( 1 - 0.6^{\frac{1}{e^{\mathtt{r2.bm.lnu}\left( t \right)}}} \right) e^{\mathtt{r2.bm.lnu}\left( t \right)}}{e^{\mathtt{r2.bm.lnq}\left( t \right)}} - \left( e^{\mathtt{r2.bm.ln\nu}\left( t \right)} \right)^{2.125}}{2 e^{\mathtt{ln\tau}}} \\ \frac{\mathrm{d} \mathtt{r3.lm.x}\left( t \right)}{\mathrm{d}t} &= \mathtt{r3.lm.jcn}\left( t \right) \\ \frac{\mathrm{d} \mathtt{r3.bm.s}\left( t \right)}{\mathrm{d}t} &= - 0.32 \left( -1 + e^{\mathtt{r3.bm.lnu}\left( t \right)} \right) + \mathtt{r3.bm.jcn}\left( t \right) - 0.64 e^{\mathtt{ln\kappa}} \mathtt{r3.bm.s}\left( t \right) \\ \frac{\mathrm{d} \mathtt{r3.bm.lnu}\left( t \right)}{\mathrm{d}t} &= \frac{\mathtt{r3.bm.s}\left( t \right)}{e^{\mathtt{r3.bm.lnu}\left( t \right)}} \\ \frac{\mathrm{d} \mathtt{r3.bm.ln\nu}\left( t \right)}{\mathrm{d}t} &= \frac{e^{\mathtt{r3.bm.lnu}\left( t \right)} - \left( e^{\mathtt{r3.bm.ln\nu}\left( t \right)} \right)^{3.125}}{2 e^{\mathtt{ln\tau}} e^{\mathtt{r3.bm.ln\nu}\left( t \right)}} \\ \frac{\mathrm{d} \mathtt{r3.bm.lnq}\left( t \right)}{\mathrm{d}t} &= \frac{\frac{2.5 \left( 1 - 0.6^{\frac{1}{e^{\mathtt{r3.bm.lnu}\left( t \right)}}} \right) e^{\mathtt{r3.bm.lnu}\left( t \right)}}{e^{\mathtt{r3.bm.lnq}\left( t \right)}} - \left( e^{\mathtt{r3.bm.ln\nu}\left( t \right)} \right)^{2.125}}{2 e^{\mathtt{ln\tau}}} \\ 0 &= 1 - \mathtt{r1.ei.u}\left( t \right) \\ 0 &= - \mathtt{r1.bm.bold}\left( t \right) + 4 \left( 3.7726 + \frac{ - 0.4 e^{\mathtt{ln\epsilon}} e^{\mathtt{r1.bm.lnq}\left( t \right)}}{e^{\mathtt{r1.bm.ln\nu}\left( t \right)}} - 0.6 e^{\mathtt{ln\epsilon}} - 2.7726 e^{\mathtt{r1.bm.lnq}\left( t \right)} + \left( -1 + e^{\mathtt{ln\epsilon}} \right) e^{\mathtt{r1.bm.ln\nu}\left( t \right)} \right) \\ 0 &= 1 - \mathtt{r2.ei.u}\left( t \right) \\ 0 &= - \mathtt{r2.bm.bold}\left( t \right) + 4 \left( 3.7726 + \frac{ - 0.4 e^{\mathtt{r2.bm.lnq}\left( t \right)} e^{\mathtt{ln\epsilon}}}{e^{\mathtt{r2.bm.ln\nu}\left( t \right)}} - 2.7726 e^{\mathtt{r2.bm.lnq}\left( t \right)} - 0.6 e^{\mathtt{ln\epsilon}} + e^{\mathtt{r2.bm.ln\nu}\left( t \right)} \left( -1 + e^{\mathtt{ln\epsilon}} \right) \right) \\ 0 &= 1 - \mathtt{r3.ei.u}\left( t \right) \\ 0 &= - \mathtt{r3.bm.bold}\left( t \right) + 4 \left( 3.7726 - 2.7726 e^{\mathtt{r3.bm.lnq}\left( t \right)} - 0.6 e^{\mathtt{ln\epsilon}} + \frac{ - 0.4 e^{\mathtt{r3.bm.lnq}\left( t \right)} e^{\mathtt{ln\epsilon}}}{e^{\mathtt{r3.bm.ln\nu}\left( t \right)}} + e^{\mathtt{r3.bm.ln\nu}\left( t \right)} \left( -1 + e^{\mathtt{ln\epsilon}} \right) \right) \end{align} \]

Setup spectral DCM

max_iter = 128;            # maximum number of iterations
# attribute initial conditions to states
sts, _ = get_dynamic_states(fitmodel);

the following step is needed if the model's Jacobian would give degenerate eigenvalues if expanded around 0 (which is the default expansion)

perturbedfp = Dict(sts .=> abs.(0.001*rand(length(sts))))     # slight noise to avoid issues with Automatic Differentiation. TODO: find different solution, this is hacky.
Dict{SymbolicUtils.BasicSymbolic{Real}, Float64} with 15 entries:
  r3₊lm₊x(t)   => 0.000650114
  r3₊bm₊lnq(t) => 0.000922559
  r2₊bm₊lnq(t) => 9.61488e-5
  r1₊bm₊lnν(t) => 0.000500157
  r2₊bm₊s(t)   => 0.000671477
  r3₊bm₊lnu(t) => 0.000911254
  r2₊bm₊lnν(t) => 0.000462573
  r1₊bm₊s(t)   => 0.000657441
  r3₊bm₊lnν(t) => 0.000670559
  r1₊lm₊x(t)   => 0.000948512
  r1₊bm₊lnu(t) => 0.000298179
  r3₊bm₊s(t)   => 0.000753007
  r2₊lm₊x(t)   => 0.000422999
  r2₊bm₊lnu(t) => 0.000896388
  r1₊bm₊lnq(t) => 0.000952494

We can use the default prior function to use standardized prior values as given in SPM12.

pmean, pcovariance, indices = defaultprior(fitmodel, nr)

priors = (μθ_pr = pmean,
          Σθ_pr = pcovariance
         );

Setup hyper parameter prior as well:

hyperpriors = Dict(:Πλ_pr => 128.0*ones(1, 1),   # prior metaparameter precision, needs to be a matrix
                   :μλ_pr => [8.0]               # prior metaparameter mean, needs to be a vector
                  );

To compute the cross spectral densities we need to provide the sampling interval of the time series, the frequency axis and the order of the multivariate autoregressive model:

csdsetup = (mar_order = p, freq = freq, dt = dt);

_, s_bold = get_eqidx_tagged_vars(fitmodel, "measurement");    # get bold signal variables

Prepare the DCM:

(state, setup) = setup_sDCM(dfsol[:, String.(Symbol.(s_bold))], fitmodel, perturbedfp, csdsetup, priors, hyperpriors, indices, pmean, "fMRI");

# HACK: on machines with very small amounts of RAM, Julia can run out of stack space while compiling the code called in this loop
# this should be rewritten to abuse the compiler less, but for now, an easy solution is just to run it with more allocated stack space.
with_stack(f, n) = fetch(schedule(Task(f, n)));

We are ready to run the optimization procedure! :)

with_stack(5_000_000) do  # 5MB of stack space
    for iter in 1:max_iter
        state.iter = iter
        run_sDCM_iteration!(state, setup)
        print("iteration: ", iter, " - F:", state.F[end] - state.F[2], " - dF predicted:", state.dF[end], "\n")
        if iter >= 4
            criterion = state.dF[end-3:end] .< setup.tolerance
            if all(criterion)
                print("convergence\n")
                break
            end
        end
    end
end
iteration: 1 - F:0.0 - dF predicted:537.966642288462
iteration: 2 - F:390.8593747753753 - dF predicted:892.8426748346687
iteration: 3 - F:798.8442268181116 - dF predicted:221.168867155898
iteration: 4 - F:1370.9053560021039 - dF predicted:839.2878878142993
iteration: 5 - F:1885.6717314715543 - dF predicted:361.13884586831904
iteration: 6 - F:2259.2901333180857 - dF predicted:188.9524827833402
iteration: 7 - F:2653.538755815906 - dF predicted:513.5601520223227
iteration: 8 - F:2825.1096377240565 - dF predicted:52.694185096165256
iteration: 9 - F:2858.633462353034 - dF predicted:17.656027190858346
iteration: 10 - F:2921.0685202249488 - dF predicted:12.013022015817718
iteration: 11 - F:2929.1611675092495 - dF predicted:8.52420479429981
iteration: 12 - F:2934.3550795728634 - dF predicted:6.285594595931094
iteration: 13 - F:2937.8691929853085 - dF predicted:4.1167117117217575
iteration: 14 - F:2939.98297838148 - dF predicted:3.4229517508880503
iteration: 15 - F:2940.258348809658 - dF predicted:6.178864491522212
iteration: 16 - F:2940.258348809658 - dF predicted:4.218412029715452
iteration: 17 - F:2941.9501861806507 - dF predicted:0.024755471809055095
iteration: 18 - F:2941.977837728756 - dF predicted:0.01939451844290315
iteration: 19 - F:2941.9930165481937 - dF predicted:0.021146248714467457
iteration: 20 - F:2942.012948917589 - dF predicted:0.02401685902392889
convergence

Plot Results

Plot the free energy evolution over optimization iterations:

freeenergy(state)
Example block output

Plot the estimated posterior of the effective connectivity and compare that to the true parameter values. Bar hight are the posterior mean and error bars are the standard deviation of the posterior.

ecbarplot(state, setup, A_true)
Example block output

References

[1] Novelli, Leonardo, Karl Friston, and Adeel Razi. “Spectral Dynamic Causal Modeling: A Didactic Introduction and Its Relationship with Functional Connectivity.” Network Neuroscience 8, no. 1 (April 1, 2024): 178–202.
[2] Hofmann, David, Anthony G. Chesebro, Chris Rackauckas, Lilianne R. Mujica-Parodi, Karl J. Friston, Alan Edelman, and Helmut H. Strey. “Leveraging Julia’s Automated Differentiation and Symbolic Computation to Increase Spectral DCM Flexibility and Speed.” bioRxiv: The Preprint Server for Biology, 2023.
[3] Friston, Karl J., Joshua Kahan, Bharat Biswal, and Adeel Razi. “A DCM for Resting State fMRI.” NeuroImage 94 (July 2014): 396–407.

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