Donders is dead: cortical traveling waves and the limits of mental chronometry in cognitive neuroscience

Abstract

An assumption nearly all researchers in cognitive neuroscience tacitly adhere to is that of space-time separability. Historically, it forms the basis of Donders' difference method, and to date, it underwrites all difference imaging and trial-averaging of cortical activity, including the customary techniques for analyzing fMRI and EEG/MEG data. We describe the assumption and how it licenses common methods in cognitive neuroscience; in particular, we show how it plays out in signal differencing and averaging, and how it misleads us into seeing the brain as a set of static activity sources. In fact, rather than being static, the domains of cortical activity change from moment to moment: Recent research has suggested the importance of traveling waves of activation in the cortex. Traveling waves have been described at a range of different spatial scales in the cortex; they explain a large proportion of the variance in phase measurements of EEG, MEG and ECoG, and are important for understanding cortical function. Critically, traveling waves are not space-time separable. Their prominence suggests that the correct frame of reference for analyzing cortical activity is the dynamical trajectory of the system, rather than the time and space coordinates of measurements. We illustrate what the failure of space-time separability implies for cortical activation, and what consequences this should have for cognitive neuroscience.

Keywords: Cortex; Difference method; Traveling waves.

Fig. 1

Examples of space–time separability and inseparability in neuroscience. a Standard fMRI imaging techniques decompose the BOLD response into a hemodynamic response function and a voxel map of amplitudes. Here we show a map of cortical amplitudes to which a threshold has also been applied, for ease of visualization. Convolving the response function with the unthresholded amplitude map reproduces the original signal, under the assumption that the two are space–time separable. b Many orientation-selective receptive fields in the primary visual cortex are space–time separable. This means that the time course of the response differs by only amplitude and sign, and not in shape, depending on the exact spatial location and direction of the visual stimulation. However, many direction-selective cells have receptive fields that are space–time inseparable. This means that the time course of the response will change shape depending on the exact spatial location and direction of the stimulation. c MEG traveling wave shown as the cosine of the phase within the time–space plot. The phase of the wave is a function of both space and time, showing the characteristic diagonal symmetry (left). However, the axis symmetry changes from trial to trial, washing out most of the signal when trial averages are made (middle). A transformation of coordinates is required in order for traveling waves to be additive (right)

Fig. 2

Clustering trials by phase values reveals a wide range of dynamics in sub-averages. These data are taken from a single subject. The phase values over one temporal cycle (108 ms duration for one cycle at 9.2 Hz) are grouped together over all the measurement sites. K-means cluster is used to group these time by space matrices into patterns of phase that are similar over trials. The traveling wave model is then fit to mean pattern indicated by each cluster. a The trial-averaged data from all the trials, showing a static pattern of activity (vertical stripes) that changes sign over the time cycle. b The sub-averaged data from each of the six clusters found by k-means. There is a wide variety of behaviors apparent, from traveling waves (B1) to standing waves (B6). The top row shows the raw MEG signal, in units of Tesla. The number of trials in each average is given by n. The second row shows the cosine of the trial-averaged phase. The third row shows the model wave estimated from the trial-averaged phase. v is the normalized velocity, ξ is the spatial frequency and r is the fit of the wave model to data. These example data are centered on 150 ms post-button press. The sensor ordering is shown by the letters on the left of each graph. ‘A’ is the most anterior site; ‘P’ is the most posterior, ‘L’ left, ‘R’ right and ‘S’ superior

Fig. 3

Schematic illustration of components of the EEG signal. Space is represented on the horizontal axis, signal amplitude on the vertical axis. The signal at different times is represented by multiple curves. a Time-locked and space-locked components are enhanced when signal is averaged over trials. The trial average at sample time t = 1 is shown in darkest gray, with a maximum amplitude over x = 0. Over consecutive samples (lighter gray), the trial averages retain the same spatial envelope, and the amplitude varies. b Individual trials from which (a) is computed at time t = 1. All these curves are averaged together to form one curve in (a). The amplitude of the signal declines from the spatial location of the maximum, but less steeply than appears in (a). The curves have more jitter near the edges of the array, while they are more phase-locked in the center. Most of the amplitude information in (a) is due to the degree of phase jitter across trials, rather than changes in amplitude, per se. c An individual trial from the set shown in (b), over times t = 1 to 4. The sequence across consecutive time samples is indicated by lightening of the gray curve. While the amplitude of the signal declines slowly from the center, the most prominent component of the signal is the motion of the wave from left to right. The degree of jitter in (b) can be understood as due to different velocities of wave propagation in different trials. The maximum amplitude in (a) reflects the coordinate in space and time at which the peak in the waves most consistently intersect each other. The trial averages in (a) are therefore entirely consistent with the time course of the traveling wave, shown in this example, when averaged over the many trials shown in (b). However, (a) and (c) are quite different spatiotemporal patterns. In fact, the amplitude of (a) is a measure of the degree of constructive/destructive interference of the kind of signal shown in (c)