A power law of cortical adaptation

Abstract

How do neural populations adapt to the time-varying statistics of sensory input? To investigate, we measured the activity of neurons in primary visual cortex adapted to different environments, each associated with a distinct probability distribution over a stimulus set. Within each environment, a stimulus sequence was generated by independently sampling form its distribution. We find that two properties of adaptation capture how the population responses to a given stimulus, viewed as vectors, are linked across environments. First, the ratio between the response magnitudes is a power law of the ratio between the stimulus probabilities. Second, the response directions are largely invariant. These rules can be used to predict how cortical populations adapt to novel, sensory environments. Finally, we show how the power law enables the cortex to preferentially signal unexpected stimuli and to adjust the metabolic cost of its sensory representation to the entropy of the environment.

Fig 1.

Experimental protocol. a, Sessions included the presentation of three environments, $A,B$, and $C$ each associated with a different distribution over the stimulus set. b, A session consisted of six blocks, each containing a unique permutation of all three environments. Each environment was presented for 5 min. Within an environment, stimuli were drawn from the corresponding distribution and flashed at a rate of 3 per second. The presentation protocol was meant to mimic the changes of the retinal image during saccadic eye movements^,,,. A blank screen was shown for 1 min between environments. From one session to the next, the order of the permutations was randomized. c, Each session began with a coarse retinotopic mapping, where we determined the average locations of receptive fields within different sectors of the field of view, numbered 1 to 9. The bottom panel shows the center of the receptive fields for each sector mapped on the computer monitor. The background image represents the aggregate receptive field of the entire field of view. The red circle denotes its center. The dashed circle represents the circular window used during visual stimulation.

Fig 2.

Characterizing adaptation in neural populations. Each row shows results from a different experiment. Axes are labeled in the top row and, unless otherwise noted, they have the same scale in all other rows. (i) The distribution of orientations associated with each environment. (ii) Mean responses of cells to an orientation. Each column represents a tuning curve. Cells have been arranged according to their preferred orientation. (iii) Logarithmic plot of the ratios between probabilities versus the ratio between magnitudes across the 3 possible pairs of environments. Colors indicate the corresponding pairs. Solid line represents the best fitting line (without intercept). Fit statistics appear at the inset. (iv) Distribution of cosine distance scatter. The mean value appears at the inset. (v) Calculation of the equivalent angular distance. The resulting value is noted at the inset. (vi) Using the power law to predict magnitudes of population responses in a new environment. Best fitting line (without intercept) is shown as a solid line. Fit statistics appear at the inset. (vii) Testing for population homeostasis. (viii) Correlation between the $l_2$ and $l_1$ norms across stimuli and environments. Solid line represents best linear fit (without intercept). Fit statistics appear at the inset.

Fig 3.

Results using peaked distributions. Each panel shows the result of one experiment. The top rows are organized as in Fig 2. Bottom rows: (ix) Distributions obtained after smoothing the actual probabilities in (i) with the optimal von Mises kernel with concentration ${\kappa }_{opt}$. (x) The adjusted R-squared as a function of the smoothing parameter $\kappa$. The curve has an inverted Ushape with the maximum goodness of fit attained at an intermediate value. (xi) Restoration of the power law under the assumption the cortex is using a smoothed estimate of the actual probabilities. (xii) Predictions using the power law relationship derived from (xi).

Fig 4.

Testing the power law using naturalistic, orientation distributions. a, Orientation distributions in natural image patches photographed by the authors on the UCLA campus. b-d, Results using naturalistic environments. The panels are formatted exactly as Fig 3.

Fig 5.

Testing the power law using movie clips. a, Stills from movie clips obtained from National Geographic documentaries available online. Each clip was present for the 333 ms. b-d, Each row shows the results obtained in separate experiments. Each panel has the same layout. i, Movie clips were assigned ID from 1 to 18 in a random order. Environments were defined using the same type of distributions used in the experiments described in Fig 4. li, Direction scatter, expressed in terms of the cosine distance, is shown by the orange bars. This is the same calculation shown in Figs 2–4 (iv) for sinusoidal grating data. The blue bars show a histogram of cosine distances between the mean population responses evoked by pairs of movie clips. The formatting of the remaining panels is the same as in prior Figures.

Fig 6.

Dynamics of adaptation and relative entropy. a-b, Modulation of response magnitude by a stimulus with orientation $\mathrm{\Delta }\theta$ away, shown $T$ seconds earlier in the sequence. Adaptation is fast – stimuli presented beyond 2 sec into the past have no influence on the population response. b, Same data as in a, for $\mathrm{\Delta }\theta =0$ and $\mathrm{\Delta }\theta =90$ with solid lines showing exponential fits to the data. We refer to ${\tau }_d$ as the depletion time constant and ${\tau }_r$ as the recovery time constant. The terms are used for convenience and are not meant to imply we know the mechanism behind adaptation is synaptic depression. c, Modulatory effect of an immediately preceding stimulus jointly as a function of relative shifts in orientation and spatial phase. The data for $\mathrm{\Delta }\theta =0$ shows that adaptation is sensitive to spatial phase. d, As expected from the theory, relative entropy of $p_X$ from $p_Y$ is correlated with the expected value (in environment $X$) with $\log \left(r_X\left(s_i\right)/r_Y\left(s_i\right))\right.$. These data are pooled from 16 different experiments.

Fig 7.

Consistency of direction invariance and tuning curve shifts. a, Responses of a homogenous population in a uniform environment. b, Modulation function evoked by an adaptor at 90deg. Each row in a is multiplied by its corresponding gain to yield the responses of the population under adaptation in c. d, Examples of a few tuning curves (columns of c) under adaptation. Solid curves show two tuning curves near the adaptor. The flanks of the tuning curves closer to the adaptor fall more rapidly than those facing away, shifting their preferred orientations. d, Shifts in the preferred orientation of tuning curves under adaptation relative to the uniform environment.