Table 1 Forward simulation

From: Geometry of neural dynamics along the cortical attractor landscape reflects changes in attention

\(X\) represents the fMRI activity time series matrix of size (200 parcels \(\times\) time).

\(U\) represents the input time series matrix of size (100 stimulus embedding PCs \(\times\) time).

Fit a dynamical systems model to estimate parameters (W, D, α, β) specific to the fMRI run.

For each time step, with \({x}_{t}\) as the initial condition and \({u}_{t}\) as the input at that corresponding moment, run a forward simulation (5000 iterations).

 for t from 1 to the number of time steps:

  for 5000 iterations:

   if 1st iteration:

    \(\,{x}_{t+1}={x}_{t}+W{\psi }_{\alpha }\left({x}_{t}\right)-D\odot {x}_{t}+\beta {u}_{t}\)

   else:

    \({\,x}_{t+1}={x}_{t}+W{\psi }_{\alpha }\left({x}_{t}\right)-D\odot {x}_{t}\)

   \(\begin{array}{c}\begin{array}{c}{x}_{t}={x}_{t+1}\end{array}\end{array}\)

  1. For every time step of the neural data, we simulated the model continuously to predict the likely path of neural dynamics, assuming it strictly follows the equation of the model. For each time step, \(\beta {u}_{t}\) was simulated only one time step in the future because stimulus-driven activity for future simulations cannot be predicted. In contrast, the intrinsic drift could be predicted through simulations, by using the predicted \({x}_{t+1}\) as the next \({x}_{t}\) and iterating this 5000 times. If the forward simulation reached a state where no further change occurred within the last ten steps of the iterations (\({{||}\dot{x}{||}}_{\infty }\)< 1e-6), we considered it to have converged to a fixed point. Across all time steps, fixed points separated by less than 0.1 in Euclidean distance were grouped as the same attractor. The attractors identified this way represent states that the neural activity tend toward, assuming no further perturbation beyond what’s driven by the input \({u}_{t}\).
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