Sensory choices as logistic classification

Abstract

Logistic classification is a simple way to make choices based on a set of factors: give each factor a weight, sum the results, and use the sum to set the log odds of a random draw. This operation is known to describe human and animal choices based on value (economic decisions). There is increasing evidence that it also describes choices based on sensory inputs (perceptual decisions), presented across sensory modalities (multisensory integration) and combined with non-sensory factors such as prior probability, expected value, overall motivation, and recent actions. Logistic classification can also capture the effects of brain manipulations such as local inactivations. The brain may implement it by thresholding stochastic inputs (as in signal detection theory) acquired over time (as in the drift diffusion model). It is the optimal strategy under certain conditions, and the brain appears to use it as a heuristic in a wider set of conditions.

Figure 1.. Logistic classification describes sensory choices between two alternatives.

a Diagram of logistic classification, in the case of 3 factors $x_1,x_2$, and $x_3$. The weighted factors are added to a constant bias b and passed through a logistic function. The result is used to bias a coin, which is then flipped. b. The logistic sigmoid, plotting probability of choice A vs. decision variable (negative if in favor of choice B, positive if in favor of choice A). c. The logistic sigmoid becomes a line when plotting the odds (in logarithmic scale) vs. the decision variable. d A monkey performing the random dots task, indicating whether most dots move Left or Right. e. Performance of an example monkey, showing probability of Right choices as a function of coherence (negative for leftward motion, positive for rightward motion). Data were taken from Ref. using a data grabber (grabit.m). Curve shows fit of the logistic classification model. f Same data and fit, in log odds.

Figure 2.. Logistic classification describes choices based on stimuli, priors, and values.

a. A random dots task where a human is told that rightward motion is more likely than leftward motion (blue) or that they are equally likely (gray). b. Data from an observer showing that probability of choosing Right depends not only on stimulus coherence (abscissa) but also on prior probability (blue vs. gray). Data grabbed from Ref.. Curves show fit of the logistic classification model with same weight w but two different values for the bias b. c. The same data and curves plotted as odds of choosing Right vs. Left, in logarithmic scale. d-f. Same, for a mouse indicating whether a visual stimulus appears on the Left or the Right, presented in alternating blocks where Left was more likely (red) or less likely (blue) than Right. Data grabbed from Ref.. g. A position discrimination task where the observer indicates whether the top stimulus is to the Right of the bottom one, in conditions where correct Left responses were rewarded less (red) or more (blue) than correct Right responses. h. Data from an observer showing that probability of choosing Right depends not only on stimulus offset (abscissa) but also on reward condition (blue vs. red). Data grabbed from Ref.. Curves show fit of the logistic classification model with same weight $W$ but two different values for the bias $B$. The red and blue ticks in the central axis indicate the intercepts predicted by the optimal strategy. i. The same data and curves plotted as odds of choosing Right vs. Left, in logarithmic scale. j-l. Same, for a monkey performing a random dots task where stimuli are presented in alternating blocks where correct Left choices were rewarded less (red) or more (blue) than correct Right choices. Data grabbed from Ref.. m-o. Same, for a mouse indicating whether a visual stimulus appears on the Left or the Right, presented in alternating blocks where correct Left choices were rewarded less (red) or more (blue) than correct Right choices. Data grabbed from Ref..

Figure 3.. Logistic classification describes audiovisual choices.

a. The audiovisual task: images appear on the Left or right, and sounds are emitted at one of 3–5 positions. b. Probability that a mouse chooses Right based on the contrast of the visual stimulus (abscissa, negative for stimuli on the Left) and on the position of the auditory stimulus (colors as in a). c. Same, for a mouse with higher auditory sensitivity. d. Results in a task with 5 auditory positions. e-g. The data in b-d, replotted as odds of Right vs. Left choices, in logarithmic scale. The quantities on the abscissa (horizontal plane in g) are compressed (raised to an exponent $q<1$). Data from Ref.. Therefore, the probability of choosing $R$ is a logistic function $p=\sigma \left(z\right)$ with three terms: one that depends on vision, one that depends on audition, and a constant.

Figure 4.. Logistic classification describes audiovisual and vestibulo-visual choices with stimuli differing in feature and amplitude.

a Average choices of a human performing a rate discrimination task based on auditory sequences. The ordinate plots the probability that the observer judges the rate of the sequence to be higher than a reference (10.5 Hz) when stimuli have low amplitude (Hard) or high amplitude (Easy). Curves in this panel and subsequent ones are fits of the logistic classification model. b. Same, for visual stimuli. c. Same, for stimuli obtained by summing an easy auditory stimulus to a hard visual stimulus. As illustrated in the inset, the rate of the auditory stimulus was lower (blue), equal (green), or higher (red) than the rate of the visual stimulus. The abscissa plots the average rate of the visual and auditory stimuli. d. Same, for stimuli obtained by summing a hard auditory stimulus to an easy visual stimulus. e-h: The same data as a-d, plotted in terms of odds, in logarithmic scale. Data are from Human 1 in Ref.. i Average choices of a monkey performing a heading discrimination task based on unisensory vestibular stimulation. The ordinate plots the probability that the observer judges the heading to be Right. Curves are predictions of the logistic classification model. j. Same, for unisensory visual stimuli varying in coherence from 12% (hardest) to 96% (easiest). k. Same, for multisensory stimuli obtained by summing the vestibular stimulus to a hard visual stimulus, with the former having a heading left of (blue), same as (green), or right of (red) the latter. The abscissa plots the average heading of the vestibular and visual stimuli. l. Same, for stimuli obtained by summing the vestibular stimulus to an easy visual stimulus. m-p: The same data as a-d, plotted in terms of odds, in logarithmic scale. Open symbols indicate values where the probability is close to 0 or 1, where the odds would require a high precision, and are thus likely to lie outside the range of the ordinate. Data were grabbed from Ref..

Figure 5.. Logistic classification describes effects of history.

a. Model of logistic classification where choices are determined not only by a stimulus but also by prior wins or losses (Eq. 12). Adapted from Ref.. The subsequent panels show results of simulations with $W=0.8,S=2,F=-1,B=0$. See Fig. 2c of Ref. for similar data. b. Simulated data showing the probability of $R$ choices as a function of contrast, compared to what would be observed based on the sensory factor alone (dashed). c. Same, plotted as the logarithm of the odds. d-e. Same as b-c, contingent on previous trial being a success or a failure (red vs. blue) on the $L$ or $R$ (open vs. closed). The dashed curve shows the result that would be observed based on the sensory factor alone. f. Distribution of values of the decision variable $z$, as predicted by the stimuli alone (gray) or by the stimuli plus the history terms (green). g. Binned values of the decision variables for the two models, on top of a logistic curve. h. Same, plotted as the logarithm of the odds.

Figure 6.. Logistic classification describes effects of brain manipulations.

a.Effects of inactivating right visual cortex in mice that perform an audiovisual localization task. Showing the probability of Right choices as a function of visual contrast (abscissa) in the presence of sounds from the Left (blue) and Right (red), in control conditions (dashed) and during inactivation (data and continuous curves). The curves are predictions of the model (Eq. 14) allowing only one parameter to change with inactivation: the weight applied to visual stimuli on the left, $F_{1L}$. b. Same data, plotted as log odds. c,d. Effects of inactivating the right prefrontal cortex. The model captured them via changes not only in the contralateral visual weight but also in the contralateral auditory weight and in the overall bias. Replotted from Ref. . e. Effects of prefrontal inactivation in a rat performing rate discrimination based on auditory (green), visual (blue) and audiovisual (red) cues, showing the probability of choosing “Right” (for high rates) as a function of rate (expressed relative to the reference). f. Same data, in log odds. Data from Ref. .

Figure 7.. Three implementations of logistic classification.

a. The implementation described in most of this paper, based on deterministic utilities $z_L,z_R$ informing a stochastic decision. b. An alternative implementation where the two utilities are stochastic across trials, and are then subtracted and thresholded. This is analogous to signal detection theory^–. c. An implementation where the decision variable $z$ controls the drift of a random walk, and the choice is made when the walk hits one of two boundaries. This is the widely used drift diffusion model^,–.

Figure 8.. Logistic classification with more than two choices.

a. Multinomial logistic classification among 3 choices, showing probability of choices 1,2, and 3 (left to right) as a function of $z1$ (abscissa) and $z2$ (ordinate), when $z3=0$. b. The 3-choice version of the position detection task, where mice choose “Left” or “Right” to indicate stimulus position, and “NoGo” if the stimulus is absent. c-e. Probabilities of the three choices for a mouse in this task (Mouse I from Fig. 3 of Ref.), as a function of stimulus contrast. f-h. The odds of the three choices relative to the third choice.