Here, I illustrate simple ideas to understand *multi-region dynamics*. The goal is to describe the *joint state* of a set of brain regions, and how it evolves temporally.

Imagine a system of *n* brain regions labeled each of which with an activation (or firing rate) strength that varies as a function of time denoted and so on. We can group these activities into a vector. Recall that a vector is simply an ordered set of values, such as *x*, *y*, and *z* in three dimensions. At time t_{1}, the vector specifies the *state* of the regions (that is, their activations) at time t_{1}. By plotting how this vector moves as a function of time, it is possible to visualize the temporal evolution of the system as the behavior in question unfolds[1]. We can call the succession of states at t_{1}, t_{2}, etc., visited by the system a *trajectory*. Now, suppose an animal performs two tasks, A and B, and that we collect responses across three brain regions, at multiple time points. We can then generate a trajectory for each task (Figure 1). Each trajectory provides a potentially unique *signature* for the task in question^{[2]}. We’ve created a four-dimensional representation of each task: it considers three locations in space (the regions where the signals were recorded from) and one dimension of time. Thus, each task is summarized in terms of responses across multiple brain locations and time. Of course, we can record from more than three places, that only depends on what our measuring technique allows us to do. If we record from *n* spatial locations then we’ll be dealing with an *n*+1 dimensional situation (the +1 comes from adding the dimension of time). Whereas we can’t plot that in a piece of paper, fortunately the mathematics is the same, so this poses no problems the for data analysis.

Thinking in terms of spatiotemporal trajectories brings with it multiple features. The object of interest – the trajectory – is spatially distributed and, of course, dynamic. It also encourages a process-oriented framework, instead of trying to figure out how a brain region responds to a certain stimulus. The process view also changes the typical focus on billiard-ball causation – the white ball hits the black ball, or region A excites region B. Experimentally, a central goal then becomes estimating trajectories robustly from available data. Some readers may feel that, yes, trajectories are fine, but aren’t we merely describing the system but not *explaining* it? Why is the trajectory of task A different from that of task B, for example? Without a doubt, a trajectory is not the be-all and end-all of the story. Deciphering how it comes about is ultimately the goal, which will require more elaborate models, and here computational models of brain function will be key. In other words, what kind of system, and what kind of interactions among system elements generate similar trajectories, given similar inputs and conditions?

[1] Rabinovich et al. (2008); Buonomano and Maass (2009).

[2] The proximity of trajectories depends on the dimensionality of the system in question (which is usually unknown) and the dimensionality of the space where data are being considered (say, after dimensionality reduction). Naturally, points projected onto a lower-dimensional representation might be closer than in the original higher-dimensional space.

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