Fig 1.
Changes in individual RGC feature selectivities are consistent with a switch between rod and cone dominated circuitry.
A. Color-dependent STA for a sample ganglion cell (Experiment #1). Lighter shades measured in the light condition. Red, green and blue colors correspond to the monitor gun. B. Projections of the ganglion cell color profiles onto the red rod (zi,r) on the y-axis plotted versus the projection onto the green cone (zi,g) for N = 111 ganglion cells (see Supplement); dark blue dots for the dark, light blue dots for light condition. Absolute values are used to provide invariance to the sign of the color vector (ON- vs. OFF-type cells). The rod-cone overlap (similarity of the two photopigments) is plotted as red squares. C. The distribution of the rod-cone spectral separations (see Supplement), plotted for the dark (dark blue) and light (light blue) conditions. D. Responses of a sample ganglion cell to repeated presentations of the same stimulus segment, in the two conditions. E. Auto- and cross-correlations of spike trains for the cell in panel D, calculated excluding spikes from the same trial; dark-dark auto-correlation (dark blue), light-light auto-correlation (light blue), dark-light cross-correlation (black). These are normalized by the probability of a spike, so that this measure tends to 1 at long time shifts. F. Distribution over ganglion cells of the reproducibility (see main text) of spikes across light adapted conditions, with an arrow indicating the sample cell in D,E.
Fig 2.
Statistically significant changes in population structure across stimulus conditions.
A. Firing rate in the light condition plotted against firing rate in the dark for all N = 128 cells. Error bars are given by the standard error of the mean (SE) and are smaller than the plotted points. B. Pairwise correlation in the light plotted against pairwise correlation in the dark, for all 128 ⋅ 127/2 pairs of cells. Error Bars are not shown (but see panel D). Inset is the probability density function (PDF), on a log scale, of correlation coefficients. C. Measured P(K) in the two light conditions. K is the number of active cells in a state. Error bars are given by the SE. D. The PDF of z-scores of changes in correlation coefficients. The change in correlation coefficient is normalized by the error (). These error bars are standard deviations over bootstrap resamples of the data, estimated per cell pair. Data compares the light and dark adapted conditions (thick black line), the control compares a random half of the dark dataset to the other half (gray), and a numerical gaussian is plotted in red for comparison.
Fig 3.
Phase transitions robustly present in both stimulus conditions.
A. For each value of N, we selected 10 groups of cells (out of the N = 128, M1, light recording). For each group we inferred the k-pairwise maximum entropy model as in [19], and estimated the specific heat. Shaded areas are the SE. B,C. Peak temperature, Tmax, and peak specific heat, C(Tmax), plotted as a function of inverse system size, 1/N. D. Specific heat plotted versus temperature for the full network (N = 128) and for smaller subnetworks (N = 20) for both the dark and light conditions. Error bars as in A. E. Comparison of the specific heat of the full network to that of an independent network estimated from shuffled data, for both the dark and light conditions.
Fig 4.
Robustness of phase transition to alternate adaptive mechanisms.
A. Firing rate in the “leaves” natural movie (M1) plotted against firing rate in the “water” natural movie (M2) condition, for all N = 140 cells. Error bars are the SE. B. The distribution of changes in correlation coefficient across the two stimulus conditions, for all 140 ⋅ 139/2 pairs of cells. C. Distributions of the spike count, P(K), across the two stimulus conditions. D. Specific heats for full data and shuffled data (calculated as in Fig 3), in the two natural movie conditions.
Fig 5.
Dependence on correlation structure of naturalistic stimuli.
A. Distributions of correlation coefficients in the naturalistic and artificial stimulus conditions (Experiment #1, light, N = 111 ganglion cells). B. Specific heats in the naturalistic and artificial stimulus conditions. Here the independent curve was calculated analytically based on the firing rates in the checkerboard condition.
Fig 6.
Model LN neurons were fit to the measured receptive fields as described in the text, N = 111 ganglion cells. A. Distributions of correlation coefficients over pairs of cells estimated in the training data (responding to checkerboard), and the simulated network of LN neurons. B. The specific heats in the checkerboard and simulated LN network. The independent curve is the same analytic estimate as in Fig 5.
Fig 7.
Targeted manipulations of the correlation matrix.
A. The specific heat plotted versus temperature for several values of the connectivity, L (see text). B. The specific heat plotted versus temperature for several values of the global covariance α (see text). C. The specific heat at the operating point of the network (T = 1) plotted as a function of α. For subnetworks of different sizes, we inferred the pairwise maximum entropy models for a given α. Error bars are standard error of the mean over ten different choices of cells per subnetwork size. Red dashed line indicates our estimate of the discontinuity (which occurs between the values of α = 0.225 and α = 0.25). D. Same calculation as C, but for correlations and firing rates measured in the dark.
Fig 8.
Emergence of structure in the energy landscape at low covariance strength.
All states in the light dataset with five or more spike counts were sorted by spike count and grouped into 100 equally populated groups. Averages were calculated over the states within each group (for clarity here indexed by p); the color of the line indicates the average spike count within a group (1613 states per group). A. Average similarity in direction of sampling between full model and model at correlation strength α. This is estimated as the average over data states, p, within a given group of states: . B. Similarities of magnitudes, estimated as 〈||X(Rp, α)|| / ||X(Rp, α = 1)||〉p. Dotted black lines on both panels indicate α* and unity.
Fig 9.
Visualizing the structure of the models through the dwell times in sampling.
A. Distributions of dwell times for three sample states, estimated over 104 separate instantiations of MC sampling from the full model. The persistence indices for states 1,2 and 3 were 0.092, 0.54, and 0.92, respectively. B. For the same states as in (A), distributions of dwell times estimated on shuffled (independent) data. C. Across the N = 3187 states with K = 12 spiking cells recorded in the data (M1, light) we measured the average dwell time (over 103 MC runs) in the full (red) and independent (blue) models. These are plotted vs. the PI given by the full model. Note the logarithmic scale on the y-axis. D. The persistence indices for the same group of states are estimated using the maximum entropy model fitting the natural movie in the light (x-axis) and the dark (y-axis) adapted conditions.
Fig 10.
The phase space of the retinal population code.
Sketches of the phase space of the nearest neighbor Ising ferromagnet (A), the Sherrington-Kirkpatrick model (B, following Fig.1 in Ref. [67]), and our work (C). Black lines indicate boundaries between phases which correspond to first-order phase transitions, while gray dots and gray lines correspond to transitions of higher-order (second- or third-order, depending on the model). Colored arrows indicate phase transitions of different orders. A. The line of first-order phase transitions is centered on an applied field h = 0, the critical point here is second-order. B. All transitions here are second-order, except in the spin glass limit, where they are third-order (the spin glass limit, μJ ≪ σJ, is denoted by the dotted line). As explained later on in the text, the second-order transitions here are marked by a discontinuity, not divergence, in the specific heat. C. In our work, a third-order phase transition as a function of correlation (at α = α*, T = 0) is the origin of a line of first-order phase transitions as a function of temperature. The location of the real neural population code is denoted by a star.
Fig 11.
A sharp increase in firing rates accompanies the phase transition.
A. Firing rates averaged over cells plotted as a function of the temperature, T, during the same annealing procedure used to estimate the specific heat (B, reproduced from Fig 3 for convenience). Model parameters were inferred for the k-pairwise maximum entropy model, in the light condition. Error bars (shaded regions) are the standard error of the mean.