Fig 1.
The stimulation protocol and population responses in L2/3 of mouse V1.
(A) Natural images were presented for 500 ms, interleaved by 1474 ms periods of blank gray screens. (B) The responses (inferred spike rate) of 103 measured neurons to the first 600 of the 1800 images presented to the animal as a part of the training set (note that due to copyright restrictions the images presented during experiments were in this figure replaced with different equally pre-processed images which are under Creative Commons CC0 license). (C) Examples of spatial activity patterns to single natural images.
Fig 2.
The model consists of a limited number of difference-of-Gaussian kernels, parameterized by the width and weight of the central and surrounding Gaussians, and the x and y coordinates of their center. This LGN layer is followed by two `cortical`layers of simple integrators with logistic-loss type transfer functions. The two layers are inter-connected by all-to-all connections and the first layer has all-to-all connections from the LGN units. Each unit in the two ‘cortical’ layers is parameterized by the set of incoming weights and the threshold of its logistic-loss transfer function. The log-loss function approximates a linear function with the slope of 1 as (x − t)→∞.
Fig 3.
The distribution of linear RFs in the cortical space obtained from the first recorded region.
Each linear RF is centered on the corresponding neuron’s cell body in the cortical region where recordings took place (displayed in lower right corner). The color of the frame indicates the NLI of the given neuron (blue-green scale bar). A wide variety of linear RFs were observed, and no specific ordering was identified. Each RF was individually normalized to remove differences due to average firing rates. The color scale of the RF maps is shown on the scale bar next to the neuron near the bottom right corner.
Fig 4.
Responses predicted from the HSM are highly correlated with recorded neuronal responses.
The measured mean activity (full circles) and the predicted responses (empty circles) for a neuron with highest correlation (R = 0.90, P<0.001)(A) and a neuron with median correlation (R = 0.53, P<0.001)(B). (C) The cumulative distribution functions of correlation coefficients across the population of 260 neurons in the three measured regions. (D) Normalized noise power [28] and the performance of the model are negatively correlated (R = -0.63, -0.55 and -0.67 for the three regions, P<0.001 for all regions). Higher normalized noise power implies lower and more variable prediction performance. The small dots indicate neurons for which the model performance (correlation between predicted and recorded responses) was not statistically significant (based on bootstrapped 95% confidence intervals).
Fig 5.
RFs estimated by HSM explain neuronal responses significantly better than existing methods.
(A) The average correlation coefficients of the rLN, BWT and HSMs, and of an HSM in which each neuron has been fitted individually HSM(SN). (B) The average fraction of explained variance (FEV) for the three compared models. (C) Scatter plot of the FEV of individual neurons by the rLN and HSMs. (D) Scatter plot of the FEV of individual neurons under the BWT and HSMs. Data from individual regions are marked by the colored lines and/or dots while the averages across regions are indicated as bars.
Fig 6.
Effect of meta-parameter choice on model performance.
(A) Relationship between the performance of the HSM and the number of LGN inputs for the training set (solid line) and the validation set (dashed line). (B) Relationship between the performance of the HSM and the number of units in the hidden layer expressed as fraction of measured neurons for the training set (solid line) and the validation set (dashed line). Note that the performance on the training set is consistently poorer than on the validation set because the validation set is an average over multiple trials while the training set is single-trial data (see Materials and Methods).
Fig 7.
Composition of RFs estimated by the HSM.
(A) The kernels of LGN HSM units fitted to the 3 imaged regions in V1. (B) The linear kernels of the hidden units of the HSM fitted to the 3 imaged regions in V1. These have been calculated as the sum of the difference-of-Gaussians kernel of the LGN units weighted by the fitted connections from the LGN to hidden units. (C) The weight matrices between the hidden and output units for HSMs fitted to the 3 imaged regions. For each output neuron the weights were individually normalized. (D) The histograms of the weight matrices shown in C with calculated kurtosis of the respective distributions. Overall, the RFs of the intermediate units in the fitted HSM differ somewhat from standard descriptions of V1 simple cell RFs estimated by rLN (or similar) methods. However, it is possible that the RFs of the hidden units are combined in the HSM output layer to form RFs that—when linearized—match those obtained via the rLN method. To verify this, we performed a rLN analysis using the training set of images and the corresponding responses of the fitted HSM, thus obtaining a linear estimate of the HSM-derived RFs. Fig 8 shows the RFs obtained via the rLN method directly from the data (A columns) and using the corresponding HSM predictions (B columns) for neurons in the first imaged region. Neurons for which a linear RF could be estimated showed a close match between rLN and HSM estimations. This indicates that the fits of the HSM are compatible with the previous rLN results. At the same time, the HSM significantly outperforms response predictions of the rLN model for most neurons, indicating that non-linearities in the HSM, which cannot be captured by the inherently linear rLN model, provide significant performance improvements.
Fig 8.
Linear components of RFs estimated with HSM are consistent with rLN-derived RFs.
For neurons where RFs can be discerned from the noise, the rLN model produces very similar RFs for both experimental responses (A columns) and HSM responses (B columns), indicating that the HSM is compatible with the RFs estimated by rLN. The numbers above the rLN filters obtained directly from data (A columns) show the performance (R) of the rLN model for the corresponding neuron. The two numbers above the rLN filters obtained using the prediction of the HSM (B columns) indicate the performance of the rLN model derived from the HSM prediction (before slash) and the performance of the full HSM for the corresponding neuron (after slash).
Fig 9.
Distribution of NLI index and HSM prediction power in cortical space.
(A-B): The scatter plots of the NLI index and HSM prediction power in cortical space from an example cortical region. The points correspond to the positions of cell bodies in cortical coordinates and the gray levels correspond to the two measures (first region shown, scale bar 50μm). (C-D) No correlation between cortical distance and NLI and HSM prediction power. The three recorded cortical regions correspond to the black, blue and orange colors. For both measures and all regions, all correlation coefficients were small (R<0.05) and their signs were not consistent across regions.