Fig 1.
The pool consists of two responsive voxels and the two color disks represent the trial-by-trial response distributions evoked by two different stimuli. Panel A illustrates the original state of the population responses. Decoding performance can be improved via either a bigger separation of the mean population response (panel B) or changes in the covariance structure (panel C). Representational structures in panels B and C indicate improved population codes but have distinct underlying mechanisms. Panel D illustrates that certain covariance changes can worsen decoding.
Fig 2.
Neuron- and voxel-encoding models.
The neuron-encoding model (panel A) proposes a neuronal population with orientation-selective tuning curves. Each neuron has Poisson-like response variance and the noise correlation between two neurons can be specified with different structures and strength (see Materials and Methods). The voxel-encoding model proposes a similar neuronal population and the response of a single voxel is the linear combination of the responses of multiple neurons. The noise correlation between two voxels can be specified using similar methods (see Materials and Methods). Note that voxelwise NCs can come from the response variability at both neuronal and voxel levels (see Fig 6). Using the neuron- and the voxel-encoding models, we can generate many trials of neuronal and voxel population responses and perform conventional MVPA on the simulated data. The goal is to examine multivariate decoding performance as a function of the NC structure and strength between either neurons or voxels.
Table 1.
List of symbols.
Fig 3.
Example noise correlation matrices simulated in a neuronal (panels A-C) and a voxel population (D, E). In the neuronal population (180 neurons), the angular-based TCNC matrix, the curve-based TCNC matrix, and the SFNC matrix are illustrated from left to right. Neurons are sorted according to their preferred orientation from 1 to 180°. In the voxel population (180 voxels), the curve-based TCNC matrix and the SFNC matrix are illustrated. Note that we do not sort the voxels according to their tuning preferences. The NC coefficients (cneuron or cvxs) are set to 1 in matrices from A-E. Panels F-H illustrate the cTCNC matrices with NC coefficient (cneuron) values 0, 0.5 and 1, respectively. Note that panels B and H are identical.
Fig 4.
TCNCs impair population codes in a neuronal population.
The multivariate classification accuracy (panels A-C) and maximum likelihood estimation efficiency (panels D-F) are depicted as a function of the magnitude of the aTCNC (panels A, D), TCNC (panels B, E) and the SFNC (panels C, F). Both classification accuracy and estimation efficiency decline as the strength of aTCNC and cTCNC increases. Conversely, increasing the strength of SFNC improves decoding accuracy.
Fig 5.
Decoding accuracy as U-shaped functions of cTCNCs in a voxel population.
The multivariate classification accuracy (panels A, B) and estimation efficiency (panels C, D) are depicted as a function of the magnitude of cTCNCs (panels A, C) and SFNCs (panels B, D). Decoding accuracy exhibits U-shaped functions as cTCNCs increase. Similar to a neuronal population, SFNCs always improve decoding accuracy.
Fig 6.
The impacts of neuronal and voxelwise cTCNCs on stimulus classification (A) and estimation (B). In both tasks, decoding performance exhibits U-shaped functions of the strength of voxelwise cTCNCs (i.e., cvxs). Neuronal cTCNCs (i.e., cneuron) have small impacts on classification accuracy, because the voxel-level noise primarily limits information. Neuronal cTCNCs have a more prominent detrimental effect in the estimation task. These results are consistent with the results when two levels of cTCNCs are manipulated independently.
Fig 7.
Amount of information in neuronal and voxel populations with diverse forms and strength of NCs.
The upper and the bottom rows depict the amount of information as a function of increasing strength of NCs and the increasing number of units in the population, respectively. Panels A-C and F-H illustrate the amount of information in a neuronal population and correspond to Fig 4. Panels D-E and I-J illustrate the amount of information in a voxel population and correspond to Fig 5. Note that here we only illustrate the information in the stimulus-estimation task. We consider three types of NC—aTCNC (panels A, F), cTCNC (panels B, G), and SFNCs (panels C, H) in the neuronal population, as already shown in Fig 4. Similar treatments are performed for the voxel population, as shown in Fig 5. The calculation of information largely mirrors the decoding results shown in Fig 4 and Fig 5. Critically, the amount of information in the voxel population exhibits U-shaped functions of increasing strength of cTCNCs (panel D) and cTCNCs do not limit information as the number of voxels increases (panel I). These results clearly differ from the effects of aTCNCs (panels A, F) and cTCNCs (panels B, G) in the neuronal population.
Fig 8.
Interaction between cTCNC and tuning heterogeneity on population codes.
500 voxels were simulated (see Materials and Methods). A larger value of chomo indicates more homogeneous voxel tuning curves. Note that the simulated voxel tuning curves are identical to neuronal tuning curves when chomo = 1. Panel A illustrates some sample tuning curves of the simulated voxels. Due to the uncertain neuron-to-voxel connections (i.e., linear weighting matrix W), the endowed voxel tuning curves also exhibit irregular forms. Panels B and C illustrate the raw and normalized amount of information as a function of cTCNC under different tuning heterogeneity levels. The raw information is normalized to the condition when cvxs = 0 (panel C). As voxel tuning homogeneity increases, the shape of the functions changes from U-shaped to monotonically decreasing. Panel D illustrates sample voxel tuning curves with different heterogeneity levels.
Fig 9.
The detrimental effects of TCNCs and the beneficial effects of tuning heterogeneity.
Panel A illustrates the scenario of homogeneous tuning curves of two units. Panels B and C depict the cases of classifying stimuli a and b and stimuli c and d, respectively. In both panels B and C, the noise correlation is detrimental. The dots or squares on the x- and y-axes indicate the mean responses of the two units towards the two stimuli. Panels D-F are similar to Panels A-C but illustrate the scenario of heterogeneous tuning curves. The noise correlation between the two units is detrimental to the classification of stimuli a and b, but beneficial to the classification of stimuli c and d. Panels D-F show how tuning heterogeneity can mitigate the detrimental effect of TCNC.
Fig 10.
Simulation of voxel population responses based2 on realistic fMRI data.
A. noise correlation as an exponential function of signal correlation. The magenta curve is estimated from van Bergen and Jehee [22] and we scale it to increase the magnitude of noise correlations (i.e., the blue curve). B,C. simulated effects of noise correlations in the regimes of the magenta and the blue curves, respectively. Noise correlations reduce the amount of information because the overall magnitude is relatively week (i.e., the magenta curve). We again observe U-shaped functions if the overall magnitude of noise correlations are large (i.e., the blue curve).
Fig 11.
A three-voxel simulation illustrating the disproportional benefits of good covariance in multivariate decoding.
Panel A illustrates the trial-by-trial responses of voxels X and Y towards two stimuli. The covariance structure of X and Y enhances classification accuracy. Similarly, panel B illustrates that the covariance structure of voxels Y and Z impairs classification. Voxels X and Z have no systematic NC (panel C). Panel D depicts the classification accuracy based on population responses of X and Y, Y and Z, X and Z, and all three units. We also include a situation where we set all NCs among three units to 0 and keep other settings the same (i.e., X, Y&Z without NC). We add this condition because in most empirical scenarios we are interested in comparing a population code with and without NCs. The beneficial and detrimental effects of the covariance structures in panels A and B do not cancel each other if all three voxels are combined.